DEVELOPMENT OF NEW MEASURING AND MODELLING TECHNIQUES FOR RFICS AND THEIR NONLINEAR BEHAVIOUR

Transcription

1 Vrije Universiteit Brussel Faculteit Toegepaste Wetenschappen Departement ELEC Pleinlaan 2, B-1050 Brussels, Belgium DEVELOPMENT OF NEW MEASURING AND MODELLING TECHNIQUES FOR RFICS AND THEIR NONLINEAR BEHAVIOUR Wendy Van Moer Promotor: Prof. Dr. Ir. Yves Rolain Co-Promotor: Prof. Dr. Ir. Alain Barel Proefschrift ingediend tot het behalen van de academische graad van doctor in de toegepaste wetenschappen Juni 2001

2

3 Vrije Universiteit Brussel Faculteit Toegepaste Wetenschappen Departement ELEC Pleinlaan 2, B-1050 Brussels, Belgium DEVELOPMENT OF NEW MEASURING AND MODELLING TECHNIQUES FOR RFICS AND THEIR NONLINEAR BEHAVIOUR Wendy Van Moer Voorzitter: Prof. G. Maggetto (Vrije Universiteit Brussel) Vice-voorzitter: Prof. J. Vereecken (Vrije Universiteit Brussel) Promotor: Prof. Y. Rolain (Vrije Universiteit Brussel) Co-Promotor: Prof. A. Barel (Vrije Universiteit Brussel) Jury: Prof. R. Pollard ( University of Leeds, UK) Prof. D. Van Hoenacker (Université Catholique de Louvain) Prof. J. Schoukens (Vrije Universiteit Brussel)

4 Voor Alain, ma en pa...

5 Table of Contents Table of Contents i Preface vii List of Symbols xiii PART I A New Network Analyser Concept 1 CHAPTER 1 Nonlinear Vectorial Network Analyser Introduction Linear versus nonlinear systems Linear time invariant systems Nonlinear systems Existing nonlinear measurement techniques A nonlinear measurement system based on a broadband sampling oscilloscope A nonlinear measurement system based on a Vectorial Network Analyser A nonlinear measurement system based on a network analyser and sampling oscilloscope A nonlinear measurement system based on a Microwave Transition Analyser An alternative for the Nonlinear Vectorial Network Analyser based on a Vector Network Analyser Nonlinear vectorial network analysers Concept Harmonic sampling principle Full three-port Nonlinear Vectorial Network Analyser Conclusion 29 CHAPTER 2 Virtual Instrumentation A dreamed measurement instrument Once I had a dream Introduction of abstraction in a complex instrument framework Designing a complex instrument framework through abstraction User Interaction Layer Network Layer Abstract Module Layer Instrumentation Layer Transmission Layer Hardware Layer Instructions for dummies 46 i

6 Table of Contents 2.5 The Nonlinear Vectorial Network Analyser as implemented for the measurements Databases: the way to fully automated measurements Conclusion Appendices 54 Appendix 2.A : Implemented modules 54 Appendix 2.B : User commands 55 CHAPTER 3 Calibration of a Nonlinear Vectorial Network Analyser The need for calibration and calibration standards Classical S-parameter calibration How to obtain a better calibration accuracy? Stochastical based S-parameter calibration Building the calibration equations Taking measurement noise into account Minimizing the cost function Taking the nonrepeatability into account Accuracy bounds The algorithm implemented Properties of the estimator Why using a stochastic calibration? Comparing real-world performance of stochastic and deterministic S-parameter calibrations Determining the statistical properties Comparing the deterministic and stochastic calibration Experimental verification Conclusion Nonlinear calibration Power calibration Phase calibration Noise analysis of the nonlinear calibration A database driven calibration Conclusion Appendices 126 Appendix 3.A : Measured scattering matrix 126 Appendix 3.B : Mathematical computations 128 Appendix 3.C : Matrix algebra 136 Appendix 3.D : Taylor series 137 Appendix 3.E : Properties of the reconnection noise 137 Appendix 3.F : The noise n K on the calibration factor K 145 ii

7 Table of Contents PART II Measurement Techniques for RF Components and Systems 153 CHAPTER 4 An Automatic Harmonic Selection Scheme Introduction Selection criterion Stochastic model Automatic detection algorithm Simulations A practical example based on NVNA measurements Combined selection criterion Conclusion Appendices 168 Appendix 4.A : Student t- and t 2 - distribution 168 Appendix 4.B : The Fisher s F- distribution 169 Appendix 4.C : Sampling distribution of the t 2 - distribution 170 CHAPTER 5 Continuous Wave and Modulated Measurements Measurement cook book Measurement guidelines Nonlinear continuous wave measurements of an amplifier Device Under Test Measurement setup Experiment design Downconversion requirements Calibration Measurement results Nonlinear AM-modulated measurements of an amplifier Device Under Test Measurement setup Experiment design Downconversion requirements Calibration Measurement results Conclusion 212 iii

8 Table of Contents PART III Modelling of RF Components and Systems 213 CHAPTER 6 Measurement Based Nonlinear Modelling in Continuous Wave Regime Finding the right tool for the job Polynomial Volterra models Rational Volterra models Implicit Volterra models Proposed estimator Polynomial VIOMAP Rational VIOMAP Implicit VIOMAP Introducing the weight Modelling in the presence of switching uncertainties Proposed estimator Interpretation Simulations Conclusion Logarithmic estimator Comparison of the attainable model flexibility Model of an amplifier under continuous wave excitation Measurement setup Soft compression Hard compression Conclusion Appendices 256 Appendix 6.A : Nonrepeatability of attenuator switchings 256 Appendix 6.B : Consistent estimates 256 CHAPTER 7 Nonlinear Modelling In Modulated Regime based on CW Models Recycling continuous wave models Static approach CW polynomial VIOMAP Model based modulation simulation A practical example: modelling the modulation and spectral regrowth components of an amplifier Conclusion Quasi-static approach Conclusion Appendices 277 Appendix 7.A : Binomium of Newton 277 iv

9 Table of Contents Conclusions and Ideas for Further Research 279 References 283 Publications 289 v

10 Table of Contents vi

11 PREFACE During the last decades, linear system theory has ruled engineering applications. Almost all engineers are raised in the linear framework and a lot of techniques are available to measure and to model linear devices. Everything that is not linear or cannot be linearised is treated as a perturbation that has to be eliminated. However, the constant pressure of low-power and high bandwidth applications pushes an increasing number of devices beyond the edges of their linear range into a nonlinear operation region. Think for example of an exigent customer who wants an optimal working, cheap-priced mobile phone with a large autonomy. Most of these demands can be granted by increasing the efficiency of the transistors. The problem, however, is that increasing the efficiency of a transistor means pushing its operating point closer to the compression region. Since compression is synonymous for nonlinearity, acquiring insight in the nonlinear behaviour of a component becomes essential. By better integrating oneself in the nonlinear world, it becomes hence easier to model the nonlinear behaviour of RFICs. Eager to explore the mysterious ways of a nonlinearity, the first prototype of the Nonlinear Vectorial Network Analyser (NVNA) was developed in 1993 by a co-operation of the HP- NMDG group and the department ELEC of the Free University of Brussels. The purpose was to build an absolute wavemeter that allows to capture the whole wave spectrum in a single take. It was the first instrument able to measure the absolute magnitude of the waves as well as the absolute phase relations between the measured harmonics. vii

12 This Nonlinear Vectorial Network Analyser is the starting point of this Ph. D. thesis. The goal is to develop simple black-box models of RFICs which describe the linear as well as the nonlinear behaviour of these integrated circuits. The developed models are based on Nonlinear Vectorial Network Analyser measurements. Before starting the modelling job, it is important to explore the nonlinearity through the measured data. What can these measurements tell us about the behaviour of the nonlinearity and which type of measurements forces the nonlinearity to show its real face? However, during the measurements with the Nonlinear Vectorial Network Analyser, some problems were encountered. The NVNA was not really a user-friendly device. Hence, an abstract model based instrumentation framework was developed which allows to obtain complete instrumentation flexibility. Furthermore, during a nonlinear measurement campaign, the acquired amount of data is huge! The user must be very punctual in organizing and saving the measured data. By integrating a transparent database into the complex instrument framework, life becomes suddenly much easier! Post processing of the data is eased and the user is able to play-back the measurement completely or partly without the need for the hardware infrastructure and without a change to the measurement code. This thesis is split up in three parts: tackling the adaptations of the NVNA, the measurements and the modelling. Part I: A new network analyser concept Part II: Measurement techniques for RF components and systems Part III: Modelling of RF components and systems Part I In order to understand each other, it is important to speak the same language. Therefore, Chapter 1 defines the concepts of a linear and a nonlinear system. The class of nonlinear systems that will be studied throughout the text is specified. It considers here the NICE systems, which converge in mean squares sense to a Volterra model of eventually infinite viii

13 order. In other words which can be described by a Volterra series. Furthermore, the concept behind the Nonlinear Vectorial Network Analyser is fully explained. Chapter 2 introduces a new abstract instrumentation software framework, which allows to increase the flexibility of the NVNA and to adapt the measurement instrument to the user s needs. This software framework is based on a layered structure and a multiple-abstraction level system. Accurate measurements not only require a good measurement instrument but also an accurate calibration procedure, which removes the systematic errors introduced by the imperfections of the measurement instrument. In Chapter 3, the differences between the calibration of a Vectorial Network Analyser and a Nonlinear Vectorial Network Analyser are highlighted. Calibrating the NVNA requires a power and phase calibration on top of a linear S-parameter calibration. All these calibration steps are fully worked out. The classical analytical linear S- parameter calibration procedure has, however, some drawbacks. It depends on exactly known calibration standards and is not able to take measurement and reconnection noise into account. Therefore, a stochastic based calibration procedure is presented and its performance is compared with the classical calibration method. The stochastic calibration is based on an estimation procedure which takes the measurement uncertainty on the S-parameters, the uncertainty on the exact value of the calibration standards as well as the reconnection uncertainty of these standards into account. Part II The measured spectra of the incident and the reflected waves of a nonlinear system can contain a lot of signal carrying harmonics and intermodulation products which need to be calibrated. It is important to select for calibration all and only those spectral lines which carry signal information. Selecting too much components will make the calibration time unnecessarily long. Omitting significant spectral components will result in incomplete measurements. Chapter 4 presents a selection criterion which allows to select automatically the significant components in the measured spectra. This stochastic selection criterion is based on the Student t 2 - distribution and allows to save and to calibrate all and only significant spectral ix

14 components, without the need for wild guessing the number of signal containing harmonics in the measured data. Using this algorithm is a major step to fully automated measurements. Possessing a nonlinear measurement system is one thing, knowing what to do with it is another. Performing good nonlinear measurements is crucial to understand a nonlinear system. Chapter 5 shows which measurement results will learn us most about the behaviour of a nonlinear system. An example of an amplifier measured in continuous wave regime and modulated regime is given. Part III Even though currently available white-box models are accurate at the component level, they mostly fail to yield quantitative descriptions of the operation of a circuit as a whole. Due to a good insight in the behaviour of a nonlinear system obtained through the Nonlinear Vectorial Network Analyser measurements, developing black-box models becomes a lot easier. In Chapter 6 simple black-box models based on the Volterra theory are introduced which allow to describe the linear as well as the nonlinear behaviour of RFICs under continuous wave excitation. Different models are applied for systems under soft compression and hard compression. An example of an amplifier in compression is shown. In Chapter 7, these continuous wave models are used to predict the response of a system to modulated signals. Experimental verification of the nonlinear model prediction is performed on an amplifier under modulated excitation. This thesis is closed with some general conclusions and ideas for further research. Contributions of this work The realisation of a practical and useful measurement instrument for high frequency, nonlinear devices is a puzzle that requires several pieces, namely flexible instrumentation software, accurate calibration procedures, measurement utilities to assist the user in the configuration of the instrument and modelling tools to describe the data. Contributions have been made in each of these fields: x

15 1. Instrumentation software (Chapter 2) Development and implementation of an abstract instrumentation software framework for the Nonlinear Vectorial Network Analyser. Integration of a database system into the instrumentation software. 2. Calibration (Chapter 3) Adapting and extending a stochastical based S-parameter calibration for a multi-port Nonlinear Vectorial Network Analyser. Comparison between a stochastic and analytical S-parameter calibration. Noise analysis of the nonlinear calibration. Development and implementation of a database driven calibration. 3. Measurement utilities Development and implementation of an automatic harmonic selection method (Chapter 4). Development of measurement guidelines for continuous wave and modulated measurements (Chapter 5). A method to gain insight in a nonlinear system by searching the energy transport channels (Chapter 5). 4. Modelling Development of and comparison between a polynomial, rational and implicit Volterra model for soft and hard nonlinearities (Chapter 6). Development and implementation of an estimation technique to model in the presence of switching uncertainties (Chapter 6). Development and implementation of nonlinear models for systems under modulated excitation based on continuous wave models (Chapter 7). xi

16 Words of thank I wish to thank everybody who has contributed to this thesis and especially to Yves Rolain, his advice and help were of great value for me! Wendy Van Moer xii

17 List of Symbols OPERATORS AND NOTATIONAL CONVENTIONS the Kronecker matrix product #( ) number of diacritical ˆ estimated value diacritical measured value diacritical sample mean value subscript 0 true noise free value subscript R real part subscript I imaginary part superscript 1 inverse of a matrix (or vector) xiii

18 superscript T transposition of a matrix (or vector) superscript T transposition of the inverse of a matrix superscript H Hermitian transposition of a matrix (or vector): complex conjugate and transposition of the matrix (or vector) superscript H Hermitian transposition of the inverse of a matrix superscript * complex conjugated superscript + Moore-Penrose pseudo-inverse Im( ) imaginary part of Re( ) real part of A outline upper case denotes a set, for example, C is the set of complex numbers, while N is the set of natural numbers. X [] i () s ith entry of the vector function X() s X () i ( s) parameter X at the ith iteration A [ i, j] () s ij, th entry of the matrix function As () Ai () dependency of A on the angular frequency ω i xiv

19 X [ k] () s k th realization of a scalar, vector or matrix random process Xs () x = ( Re() x ) 2 + ( Im() x ) 2 magnitude of a (complex) number x A 2 X 2 = max σ i ( A) two norm of an n m ( n m) matrix A 1 i m = X H X two norm of the column vector X vec{ A} a column vector formed by stacking the columns of the matrix A on top of each other argminfx ( ) x a.s. lim the minimizing argument of f( x) almost sure limit, limit with probability one plim limit in probability E {} mathematical expectation Prob () probability b X = X E { X} bias of the estimate X Cov( XY, ) = E {( X E { X} )( Y E { Y} ) H } cross covariance matrix of the column vectors X and Y 2 var() x = E { x E { x} } variance of x C X = Cov( X) covariance matrix of the column vector X xv

20 R 2 1 s X = R 1 ( X k] X) ( X k] X) sample covariance matrix of R k = 1 the column vector X realizations of I m m m identity matrix 0 m m m matrix of zeros 0 m n m n matrix of zeros X 1 R = -- sample mean of realizations (experiments) R X [ k] R k = 1 of a scalar, vector or matrix random process X µ x = E { x} mean value of x σ x 2 = var() x variance of the x 2 σ xy = covar( xy, ) covariance of x and y xvi

21 SYMBOLS a = T a [ 1] a[ np ] b = T b [ 1] b[ np ] column vector of the true incident waves (dimension ( n p 1 )) column vector of the true reflected waves (dimension ( n p 1 )) D k ( S k0, t v0 ) S T T 0 1 n k0 I np T p 0 I np E matrix of constraint equations f frequency F number of frequency domain data samples Fi Fisher information matrix f s sampling frequency H M ( f 1,, f M ) VIOMAP kernel of frequencies f 1,, f M j j 2 = 1 J( θ) = εθ ( ) θ gradient of residuals εθ ( ) w.r.t. the parameters θ Ki () complex calibration factor T NL[1,1] () i frequency ω i. at xvii

22 L k ( S k0, t v0 ) W k D k ( S k0, t v0 )S T k0 S k D k ( S k0, t v0 ) I np n iteration counter N number of independent measurements N ( µ, σ) normal distribution with mean µ σ 2 and variance n X () i disturbing measurement noise on quantity X n eq number of equations in a system of equations n p number of measurement ports of an instrument n std number of calibration boxes used in a calibration procedure n v number of error correction terms to estimate n θ dimension of the parameter vector θ P power of a wave r known parameter values Si () scattering matrix of a DUT at angular frequency ω i t continuous time variable xviii

23 t v column vector of the error coefficients tv = vec( T) T NL () i full error correction matrix at angular frequency ω i Ti () error correction matrix with T NL[1,1] () i = 1 at angular frequency ω i T s sampling period U( ω), Y( ω) Fourier transform of ut () and yt () respectively (column vectors of dimension n u and n y ) ut (), yt () input and output time signals respectively (column vectors of dimension n u and n y ) v error correction terms t [ 2: ( 2np ) 2 ]v V( θ) cost function depending on the parameters θ V N ( θ) cost function based on N repeated measurements W k W H k W k = Cov 1 ( vec{ S k} ) T α = α [ 1] α[ np ] column vector of the measured incident waves (dimension ( n p 1 )) α( i, p) column vector of the measured incident waves at frequency ω i when port p is excited xix

24 T β = β [ 1] β[ np ] column vector of the measured reflected waves (dimension ( n p 1 )) β( ip, ) column vector of the measured reflected waves at frequency ω i when port p is excited εθ ( ) column vector of the model residuals Γ true reflection coefficient λ wave lenght θ column vector of the model parameters θ s S -parameters to be estimated Θ parameter vector φ( A) phase of A ρ level of significance ω angular frequency xx

25 ABBREVIATIONS AC Alternating Current ADC Analog-to-Digital Convertor AM Amplitude Modulation API Application Programming Interface AWG Arbitrary Waveform Generator CLK CLocK signal CW Continuous Wave DC Direct Current DFT Discrete Fourier Transform dof degrees of freedom DUT Device Under Test FFT Fast Fourier Transform FM Frequency Modulation FRF Frequency Response Function GPIB General Purpose Interface Bus xxi

26 iid independent identically distributed IF Intermediate Frequency IQML Iterative Quadratic Maximum Likelihood ISM Industrial, Scientific and Medical frequency bands LRM Line-Reflect-Match calibration LS Least Squares LTI Linear Time Invariant ML Maximum Likelihood MMIC Monolithic Microwave Integrated Circuit MTA Microwave Transition Analyser NAUC Network Analyser Under Calibration NICE Nonlinear system that transforms a periodic input signal into a periodic output signal with the same period NL NonLinear NVNA Nonlinear Vectorial Network Analyser xxii

27 pdf probability density function PM Phase Modulation PNP Plug and Play standard RF Radio Frequency RFIC Radio Frequency Integrated Circuit RMS Root-Mean-Square SNR Signal-to-Noise Ratio SOLT Short-Open-Load-Thru calibration SRD Step-Recovery-Diode SVD Singular Value Decomposition TLS Total Least Squares TRIG TRIGger signal TRL Thru-Reflect-Line calibration VISA Virtual Instrument Software Architecture VNA Vectorial Network Analyser xxiii

28 VXI Versa module eurocard bus extension for Instrumentation w.p.1 with probability one WLS Weighted Least Squares WNLS Weigthed Nonlinear Least Squares WTLS Weighted Total Least Squares xxiv

29 PART I A NEW NETWORK ANALYSER CONCEPT 1

30 2

31 CHAPTER 1 NONLINEAR VECTORIAL NETWORK ANALYSER Abstract: For a long time, systems were considered to be linear. Possible nonlinear effects were seen as perturbations and so stepmotherly treated. Why? Because measuring is knowing and nonlinear effects could not be fully measured and thus not be fully understood. This chapter is an attempt to make nonlinear measurements understandable and reachable for everyone. The key concepts that make the difference between linear S -parameter measurements and nonlinear measurements are introduced step-by-step. Existing nonlinear measurement techniques are highlighted and the concept of the Nonlinear Vectorial Network Analyser is explained in detail. 3

32 Nonlinear Vectorial Network Analyser 1.1 Introduction When a complex problem is encountered, good engineering practice tells to look for similar, simple problems that are already solved and to leverage the gathered knowledge in the new field. Describing even a restricted subclass of nonlinear systems certainly meets the complexity requirements. On the other hand, the theory of linear time invariant (LTI) systems is a very powerful and well understood framework for analysis, design and measurement of such a LTI system, as the numerous successful applications of this theory indicate. One of the paradoxes within this framework is that, mathematically speaking, there exists no system that fulfils the requirements of the LTI systems, namely that the impulse response does not depend at all on the applied input power. It is clear that exciting with a power tending to infinity will break down all systems. In practice, the LTI system is hence used as an approximation to the device operation characteristic in a device dependent power range. As a consequence, the LTI system theory succeeds in modelling even hard nonlinear systems such as a transistor, when the small signal operation is imposed. Hence, the nonlinearity is linearised and nonlinear phenomena are considered to be perturbations that have to be eliminated. This approach has fulfilled the needs for modelling of nonlinear components during many years. Lately, devices are pushed more and more over the boundary of small signal operation and there the linear model even fails to qualitatively describe the system. Even for mildly nonlinear devices, the linear intuition often fails because of the very rich behaviour of the nonlinearity. For the nonlinear design aspects, things are even worse as the designer is often left alone with simulation results and a few parameters such as the Third Order Intercept point (TOI) [23] or the 1dB compression point. Clearly, the few measured figures fall short to describe the general device behaviour. The analog figures simply deny phase effects and reduce a full harmonic chart to a single number. On the other hand, available models often contain a large number of parameters that have to be set very accurately. The simulated output of these models under practical conditions often suffers from poor parameter settings. The main shortcoming is that no insight is obtained in the power transfer from the input to the output from one frequency to another. 4

33 Linear versus nonlinear systems Neither the analog figures nor the brute force modelling allow the designer to obtain a better insight in the general behaviour of the nonlinear device or system. Determining if the model behaviour matches the specified figures of merit often even proves impossible due to instrumentation problems. Similarities and differences between the linear and nonlinear framework are explained to give a better insight in the pitfalls and challenges of nonlinear measurements. The basic principles of new measurement techniques are proposed that make the nonlinear world accessible to the practising microwave engineer and prepare him/her for the new measurement methods to come. 1.2 Linear versus nonlinear systems In the following two paragraphs the major differences between a linear and a nonlinear system are highlighted. It will be clear that the measurements needed to characterise both classes of systems are different Linear time invariant systems What does a linear time invariant (LTI) system looks like, how does it behave and how can one determine its behaviour out of measurements? A. Defining a linear system Definition 1.1 A system is linear if it obeys the superposition and homogenity principle. In other words, if a linear combination of the input signals results in the same linear combination of the output signals: N N If u i () t y i () t then k i u i () t k i y i () t i = 1 i = 1 5

34 Nonlinear Vectorial Network Analyser with u i () t and y i () t respectively the i th input and output signal of the system as a function of time t. k i is a constant. Since the Fast Fourier Transform (FFT) [10] is a linear operator, this means that the response spectrum of a linear system Y( ω) only contains those frequency components that were present in the input spectrum U( ω). The frequency output at ω i is only determined by the input signal at ω i. No extra harmonics are created as can be seen from Figure 1-1. U( ω) LTI DUT Y( ω) ω 1 ω ω 1 ω FIGURE 1-1. Input and output spectrum of a LTI Device Under Test (DUT) B. Linear distributed systems: the wave based description When using high-frequency excitation signals ( f» 100 MHz), the wavelength λ of the signals becomes much smaller than the dimensions of the system itself. As a result, delays appear inbetween the application of voltage and current at different points in space. It is therefore no longer possible to characterise a device by means of position-independent voltages and currents. Hence, the concept of power waves must be introduced which describes the transportation of energy through the system. Consider for example a LTI two-port Device Under Test (DUT): a [ 1] ( k) a [ 2] ( k) LTI DUT b [ 1] ( k) b [ 2] ( k) FIGURE 1-2. Definition of incident and reflected waves of a DUT 6

35 Linear versus nonlinear systems with a [ 1] ( k) and a [ 2] ( k) the incident waves at respectively port 1 and 2. b [ 1] ( k) and b [ 2] ( k) are the reflected waves at respectively port 1 and 2. The waves are a function of k which can describe the dependence on time or frequency. As a result, the behaviour of a LTI DUT can be fully characterised by its S -parameters [2], which describe the relation between the incident and reflected waves. For a two-port DUT this results in: b [ 1] ( k) b [ 2] ( k) = S [ 11, ] ( k) S [ 12, ] ( k) S [ 21, ] ( k) S [ 22, ] ( k) a [ 1] ( k) a [ 2] ( k) (1-1) in the time or frequency domain k. Only relative measurements are needed since by (1-1) the S -parameters are defined by the following ratios of waves: S [ 11, ] ( k) S [ 12, ] ( k) S [ 21, ] ( k) S [ 22, ] ( k) = = = = b [ 1] ( k) a [ 1] ( k) b [ 1] ( k) a [ 2] ( k) b [ 2] ( k) a [ 1] ( k) b [ 2] ( k) a [ 2] ( k) a [ 2] ( k) = 0 a [ 1] ( k) = 0 a [ 2] ( k) = 0 a [ 1] ( k) = 0 (1-2) (1-3) (1-4) (1-5) From equations (1-2) to (1-5) it is clear that the S -parameters are independent of the input power of the DUT. During the measurements only 1 degree of freedom is available, namely the frequency. The superposition principle for linear systems allows to reconstruct the response of 7

36 Nonlinear Vectorial Network Analyser the system to an arbitrary signal using the expansion of this signal as the sum of individual sine waves at a set of frequencies k. C. Measuring linear wave-based systems An instrument that is able to measure the single sine response of the device and is able to repeat this measurement at an arbitrary set of frequencies can totally characterise a linear system. The ideal measurement instrument for linear systems is thus a relative single frequency wave meter, such as a Vectorial Network Analyser [17] (Figure 1-3). a LTI DUT ω 1 ω φ ω 1 A b ω FIGURE 1-3. Relative S -parameter measurements D. From measurements to design: understanding the operation of a linear system To leverage the knowledge of the measurements into the design of a system, several techniques are used: Non parametric approach: out of the measured frequency response function of a system, properties such as the bandwidth, the gain or attenuation can be obtained. Parametric approach: the position of the poles and zeros gives a full description of the system. It allows to verify easily the design and the stability of the system Nonlinear systems A. Defining a nonlinear system In general all systems which do not behave as a linear system are called nonlinear systems. However, in what follows only a small subclass is studied: the NICE systems. It considers 8

37 Linear versus nonlinear systems here systems which converge in mean square sense to a Volterra series. These systems convert a periodic input to a periodic output with the same period. The hard nonlinearities, chaotic systems or systems with bifurcations such as those which create subharmonics are not considered. When a single tone is applied to the input of a NICE system, the output spectrum will not only contain a spectral component at the same frequency, but harmonics will appear (Figure 1-4). U( ω) NICE DUT Y( ω) ω 1 ω ω 1 2ω 1 ω FIGURE 1-4. Input and output spectrum of a NICE system Furthermore, the linear relationship between the input and output fundamental spectral component may be lost. Consider for example a two-port NICE system which allows a small deviation from linearity through a third order term: 3 b [ 2] () t = S [ 21, ] ()*a t [ 1] () t + S NL ()*a t [ 1] () t (1-6) where a [ 1] () t is the incident wave at port 1 and b [ 2] () t the reflected wave at port 2. The operator * denotes the convolution. When a [ 1] () t = Acos( ω 1 t), the response of the system becomes: 3A 3 b [ 2] () t = AS [ 21, ] () t S () t 4 NL * cos( ω1 t ) + A S ()* t cos( 3ω 4 NL 1 t) (1-7) It is clear from equation (1-7) that this NICE system influences the measured spectrum in two ways. First, it creates new harmonic components in the spectrum. In this case, a contribution at 3ω 1 appears. Next, it offsets the term at the fundamental frequency by a power dependent 9

38 Nonlinear Vectorial Network Analyser value: 3A 3 4. The second contribution is the most annoying one. Its presence is not understood by a linear measurement device, but is rather accounted for as a linear contribution. Instead of being a constant, the relative quantity b [ 2] () t a [ 1] () t now depends on the incident power: b [ 2] () t 3A = S a [ 1] () t [ 21, ] () t S () t + 4 NL (1-8) So that the S -parameters are now power dependent. Confusing these power dependent quantities with the linear S -parameters for a linear system can - and will - result in important errors if the power level impinging the devices is changed. As a result, during the measurements a 2 dimensional space must be scanned: the input power and frequency must be varied in an independent way to cover the whole region of interest and so obtaining a persistent excitation. B. Measuring a NICE system Since measurements performed by a classical Vectorial Network Analyser are power independent, new measurement techniques are required to fully characterise NICE systems. Instead of relative measurements, separate knowledge of the incident and reflected waves is needed. Note also that the whole wave spectrum is required to be known simultaneously. Only then the phase coherence between the different spectral components can be obtained automatically. Even under a pure single tone excitation, the output wave of the DUT contains energy at different frequencies. The spectral lines have to be measured both in amplitude and phase to characterise the system (Figure 1-5). The shape of the wave signals in the time domain is influenced by both the amplitude and phase relation between the spectral lines. Changing this relation creates a distortion between the actual system input and output and the measurement and thus invalidates the measurement of the nonlinearity. The importance of the phase distortion is illustrated in the following example. Two signals having exactly the same amplitude spectrum as shown in Figure 1-6 will 10

39 Linear versus nonlinear systems a NICE DUT b ω 1 ω ω1 3ω φ A A φ 1 ω FIGURE 1-5. Absolute network measurements be used as an excitation signal to the NICE system used before. Based on the linear intuition it can be assumed that this information, besides being very easy to get using a spectrum analyser, is also sufficient to get a good grip on the behaviour of the nonlinearity. Both test signals are however selected to have a very different phase spectrum [V] [V] Freq [xf [MHz] 0 Hz] Freq [xf [MHz] 0 Hz] [degree] IIIIIIIIIIIIIIIII Freq [MHz] [xf 0 Hz] [degree] 180 I I I I 90 I I I I 0 I I I I I -90 I I I I Freq [xf [MHz] 0 Hz] FIGURE 1-6. Amplitude and phase spectrum of the excitation signals This results in a signal shape in the time domain which is very different for both signals, even if their amplitude harmonic content is exactly equal: 11

40 Nonlinear Vectorial Network Analyser signal Time Samples Time signal Time Time Samples FIGURE 1-7. Time domain waveform of the excitation signals The nonlinear distortion that is created by the example system of equation (1-7) is shown in Figure 1-8 for both excitation signals. Figure 1-8 also shows a zoom of the amplitude characteristic around the zero level for both signals. NL signal NL signal Time Time Samples Time Time Samples FIGURE 1-8. Contribution of the nonlinear part of the system to both excitation signals Note that in this figure, the contribution of the nonlinearity for signal 1 is much more important than for signal 2, even if the exciting amplitude spectrum is equal. If the phase relation between the harmonic components of the signal impinging the DUT is not measured accurately, it will hence prove impossible to predict the difference in behaviour between both responses and this makes the whole characterisation of the nonlinearity at least very questionable. One possibility for the multi-frequency measurement of magnitude and phase relations is to acquire the spectral lines at the different frequencies separately. Phase coherence must then be maintained between the fundamental components and the harmonics of the input and output waves. Even if this type of measurement is possible with a Vectorial Network Analyser that can be run on an overtone, it will be slow as it requires one sweep for each harmonic and it will 12

41 Linear versus nonlinear systems put high demands on the medium term phase synchronisation of the device. The preferred alternative is to measure all the harmonics together using an harmonic mixer in conjunction with a wide band digital intermediate frequency vector voltmeter. Since all the spectral lines are present in the output of the harmonic mixer, the complete spectrum can then be measured in a single measurement. All the spectral lines hence share the same time reference. Phase coherence is thus obtained automatically in this case. The ideal instrument for a nonlinear system is thus an absolute wave meter that allows to capture the whole wave spectrum in a single take [39]. C. From measurement to design: understanding the operation of a NICE system To leverage the knowledge of the measurements into the design of a NICE system, several techniques can be used: Non parametric approach: a lot of insight in a NICE system is gained by knowing the major energy transport channels. How can the output energy of a system be explained by contributions of the input energy? These energy flows can be easily obtained out of the measurements of a NICE system. Furthermore, as will be seen in Chapter 5, the Volterra theory [32] gives us a great hint to determine which input energy combinations will result in the desired output component. Once these energy flows are known, a non parametric model of a NICE system is quite easily found and model validation is possible. Parametric approach: out of a SPICE schematic of the NICE system, Volterra kernels, which fully describe the behaviour of the system, can be obtained through abstract numerical calculations [50]. To extract these models, a number of simplifications are needed, which are hard to verify. However, by using the nonlinear measurements, it becomes possible to verify if the obtained Volterra kernels describe the system behaviour. Hence, one can determine which components of the NICE system influence the nonlinearities. Furthermore, by creating a link between these system models and the nonlinear measurements, it would become 13

42 Nonlinear Vectorial Network Analyser possible to control the design of NICE systems. It could then for example become possible to answer questions such as: Which components of the NICE system should be adapted to obtain the desired behaviour? Note that this technique requires further research. 1.3 Existing nonlinear measurement techniques In this paragraph, earlier prototypes of a nonlinear measurement instrument are highlighted. Advantages and disadvantages of the existing techniques are stated A nonlinear measurement system based on a broadband sampling oscilloscope In 1988, a first prototype of a nonlinear measurement system was published by Markku Sipila, Kari Lehtinen and Veikko Porra [37]. The goal was to build a system which was able to perform accurate measurements of high frequency periodic time domain voltage and current waveforms for nonlinear microwave devices. The measurements are performed in the time domain using a broadband sampling oscilloscope and Fourier transformed into the frequency domain. After error correction, the results are transformed back into the time domain. The employed vector error correction algorithm allows to take into account and correct the errors caused by losses, mismatches and imperfect directivity in the measurement system. The measurement setup consists of a generator and a two channel broadband sampling oscilloscope as can be seen from Figure 1-9. TRG RF GEN Sampling Oscilloscope b [ 1] () t b [ 2] () t Bias Tee CH1 DUT CH2 Bias Tee FIGURE 1-9. Simplified measurement setup 14

43 Existing nonlinear measurement techniques The generator delivers a sinusoidal input wave to the DUT and a trigger signal to the sampling oscilloscope. The reflected input wave b [ 1] () t and the transmitted output wave b [ 2] () t are simultaneously measured by the sampling oscilloscope. The incident wave of the DUT can be easily determined by replacing the DUT with a calibration standard, performing a measurement and some simple calculations. Note that it considers here a serious approximation, since this way of measuring the incident wave is only valid if no harmonics are created by the generator. The calibration is performed for the fundamental and each harmonic separately. The generator and the sampling heads are assumed to be perfectly matched for all harmonics. However, these mismatches can be taken into account by measuring and modelling the sampling heads. In order for the system to make accurate measurements, a number of assumptions needs to be made: 1. The load presented by the DUT does not affect the generator output. This implies the use of an attenuator at the generator output to improve the match and isolation of the generator. The generator harmonic output is hence negligible. 2. The gain of the sampling head is independent of frequency and its phase response is linear. Although the authors say that this is a safe assumption if the highest harmonic frequency under consideration is well below the upper frequency limit of the sampling head, this is only true if the measurement system is a dominant first order system or if only a small part of the instrument bandwidth (10 %) is used. 3. The sampling heads are operated in the linear region. 4. The S -parameters of the coupling networks have to be known. This is achieved by characterizing these networks with a vectorial network analyser and compensation of the measurements using the measured S -matrix. An advantage of this measurement setup is its simplicity. However, there are some limitations: 1. The measurement setup only allows harmonic distortion type measurements. It is not possible to provide excitation at both ports simultaneously during a measurement. 15

44 Nonlinear Vectorial Network Analyser 2. Sampling oscilloscopes are rather slow measurement instruments. Furthermore, since it is necessary to replace the DUT by a calibration standard in order to measure the incident wave, the measurement time will rise. Microwave sampling oscilloscopes often suffer from trigger jitter problems and have a high noise level. 3. The measurement instrument is considered not to introduce any phase distortions A nonlinear measurement system based on a Vectorial Network Analyser In 1989, one of the early foundations of today s Nonlinear Vectorial Network Analyser (NVNA) was made by Urs Lott [21]. A new method for simultaneously measuring the magnitude and phase of the harmonics generated by a microwave two-port was designed. The output harmonics of a nonlinear DUT were measured in the frequency domain with a setup including a vectorial network analyser. For the first time a power and phase calibration were performed. The power calibration was obtained by a power meter. For the phase calibration at the harmonic frequencies, a golden diode or millimetre-wave Schottky diode was used as a reference device. Figure 1-10 represents a simplified measurement setup. Power Meter n ω 0 b [ 2] P1 Network Analyser n ω 0 P2 RF GEN ω 0 Bias Tee DUT Bias Tee FIGURE Simplified measurement setup The measurement system is built around a phase-locked signal generator with internal multiplication and a vectorial network analyser. The RF source generates a fundamental signal at ω 0 (3.5 to 6.5 GHz) which is fed to the DUT. An harmonic nω 0 of this fundamental frequency is fed to the network analyser as a reference signal. The network analyser operates at 16

45 Existing nonlinear measurement techniques the frequency of the harmonic to be measured and is configured for a forward transmission measurement from port P 1 to port P 2. The coupler is used inverse and adds the reference signal from port P 1 of the network analyser to the output signal b [ 2] coming from the DUT. The directivity of this coupler is essential to prevent the reference signal from reaching the output of the DUT. In a first step, the standard forward response of the network analyser is calibrated by replacing the DUT by two matched loads. After this calibration, a forward transmission measurement of the signal coming from port P 1 of the network analyser is performed with the DUT still replaced by the two matched loads. This results in the reference signal vector r n ( ) for the n -th harmonic in Figure s n d n r n FIGURE The signal vectors at port P 2 of the measurement system The nonlinear device produces the component d n at the frequency n ω 0. The network analyser measurement with the DUT in place gives the sum vector s n. Therefore, one can calculate the vector d n from d n = s n r n = s n Note that this procedure has to be done for each harmonic component that one wants to measure. This is only the first step of the measurement procedure because the raw measurement vectors d n are still relative values (referred to the reference vectors r n ). In order to get the absolute amplitudes and phases of the spectral components d n, the vectors r n have to be denormalised by a calibration procedure. For the amplitude calibration, the DUT is disconnected and the output power of the RF source is measured by a power meter for each required frequency. This results in the denormalisation of the length of r n. The phase calibration is based on a reference element which is called a golden diode. The reference element consists of a diode connected in parallel to a 50 Ω microstrip line and works as a limiter, clipping the top of the input sine waves and thereby generating a well-defined output spectrum of harmonics. The phase relation 17

46 Nonlinear Vectorial Network Analyser between the harmonics can be exactly known through the equivalent circuit of the diode and SPICE simulations. By replacing the DUT with the reference element, measuring the harmonics and comparing them to the known phase relation between the harmonics, it becomes then possible to denormalise the phase of the reference signal vector r n. Some advantages of this nonlinear measurement setup are: 1. Using a vectorial network analyser results in a faster data acquisition with more dynamic range compared to the use of a sampling oscilloscope. This is due to the fact that a network analyser has a narrow measurement bandwidth. All the noise that is out the measurement bandwidth will be filtered out. Furthermore, the network analyser does not suffer from the trigger jitter problems that often exist in sampling oscilloscopes. 2. For the first time a full amplitude and phase calibration is proposed. Some disadvantages are: 1. The use of the setup is rather limited. Only the output wave of the DUT can be measured. The fundamental frequency is limited between 3.5 GHz and 6.5 GHz, due to the use of the multiplying principle inside the signal generator. 2. Since the exact phase relations between the harmonics of the reference diode are obtained by simulation, the uncertainty bounds on the phase is rather large due to the component characteristic uncertainty. 3. The setup relies on a coherent phase relationship (in time) between the fundamental appearing at the auxiliary output of the generator and the harmonic appearing at the generator RF output A nonlinear measurement system based on a network analyser and sampling oscilloscope In 1990, a significant extension of the nonlinear measurement system based on a sampling oscilloscope [37] was made by Gunter Kompa and Friedbert Van Raay [19]. A large-signal automatic stepped continuous wave (CW) waveform measurement system for nonlinear device characterisation was presented which combines the high accuracy of a vectorial network 18

47 Existing nonlinear measurement techniques analyser with the waveform measurement capabilities of a sampling oscilloscope. The broadband sampling oscilloscope is used to measure the harmonics present in the input and output waves of the DUT coherently. The network analyser is used to measure accurately the behaviour of the fundamental and is used for calibration purposes. The calibrated bandwidth of the system is 20 GHz. Figure 1-12 shows a simplified measurement setup. RF GEN Sampling Oscilloscope TRG CH1 CH2 a [ 1] () t REF RF Input P1 Test Set TEST a [ 2] () t or b [ 2] () t P2 TEST REF Network Analyser DUT FIGURE Simplified measurement setup The RF signal injected into the RF input of the test set is guided towards port 1 ( P 1 ) in order to excite the DUT. The incident wave at port 1 is detected and appears at the REF output of the test set. The TEST output signal is the detected transmitted b [ 2] () t or reflected a [ 2] () t wave, depending on the chosen mode of operation of the test set (reflection or transmission measurement). All the measured waves may contain harmonics. Two switches determine whether the test set REF and TEST signals are detected by a broadband sampling oscilloscope, triggered by the fundamental, or are detected by a network analyser. A calibration procedure was developed where the network analyser is used to fully characterise the test set. First, a classical relative Short-Open-Load-Thru (SOLT) calibration is done ([17] + 19

48 Nonlinear Vectorial Network Analyser Chapter 3). In a next step, three extra measurements are performed, which are sufficient to fully characterise the test set and to perform an accurate amplitude calibration. Some advantages of the measurement setup: 1. This setup is able to measure the fundamental and the harmonics of the incident and reflected/transmitted waves. Because of the calibration procedure, mismatches are accurately characterised and compensated for. 2. Since the excitation signal is measured together with the reflected or transmitted wave, the oscilloscope trigger drift is no longer a problem. Disadvantages of the measurement setup: 1. Only excitations at port 1 are possible. 2. The phase calibration still relies on the assumption that the oscilloscope sampling heads introduce no phase distortion A nonlinear measurement system based on a Microwave Transition Analyser In 1994, a high power on-wafer measurement system was developed based on the microwave transition analyser (MTA) for the complete characterisation of the large signal behaviour of transistors by Markus Demmler, P.J. Tasker and M. Schlechtweg [11]. One key feature of the MTA based measurement setup is that during power sweeps the harmonic behavior can be measured up to 40 GHz. This vector calibrated measurement system allows both the measurement of the fundamental and the higher harmonics, both in magnitude and phase. Figure 1-13 represents a simplified schematic of the measurement setup. The MTA allows the measurement of the phase and amplitude of the fundamental and harmonics present at both input channels. Since it is based on the harmonic mixing principle [3], rather than on equivalent time sampling, the instrument allows data acquisition about 100 times faster than what is possible with a sampling oscilloscope, for the same or even better dynamic range (typically better than 50 db). 20

49 Existing nonlinear measurement techniques CH1 Microwave Transition Analyser CH2 RF GEN Bias Tee DUT Bias Tee FIGURE Simplified measurement setup Instead of using a separate network analyser during calibration, the transition analyser itself is used to do the network analyser measurements that are needed in order to characterise the test set. The major advantage of this setup is the fast data acquisition which takes only one second per measurement. The disadvantage is that still only one excitation signal can be used and that the reflected waves at the output of the DUT can not be measured. Another remaining problem is that the calibration is based on the assumption that the transition analyser introduces no phase distortion when measuring harmonics. The MTA is able to measure phase relations between the harmonics. However, those phase relations are not calibrated and can contain instrument induced phase errors An alternative for the Nonlinear Vectorial Network Analyser based on a Vector Network Analyser In 1998, an alternative to the Nonlinear Vectorial Network Analyser (NVNA) was presented based on a modified classical Vectorial Network Analyser (VNA) by D. Barataud [4]. The proposed measurement system can be seen as a time-domain load-pull system because it allows the control and the measurement of current/voltage waveforms at both ports of the DUT. Figure 1-14 represents a simplified measurement setup. A Vectorial Network Analyser can only provide sequential measurements of complex power wave ratios at each frequency of interest. It does not allow the measurement of the absolute phases of the incident and reflected waves. To overcome this problem, the test set of a 21

50 Nonlinear Vectorial Network Analyser Vectorial Network Analyser Signal Processing a Ref () t b [] 1 () t a [] 1 () t a [] 2 () t b [] 2 () t Phase Ref In RF In 2 P1 P2 RF In 2 S R D RF GEN DUT Load FIGURE Simplified measurement setup conventional VNA has been extended by three extra connections: Phase Ref In, RF In 1 and RF In 2. At both RF inputs (RF In 1 and RF In 2), load networks or RF signal generators can be connected to perform large signal characterisation of nonlinear devices. A phase reference generator is connected to the phase reference input. This phase reference generator is a Step Recovery Diode (SRD) [see Paragraph 3.6.2] fed with the signal coming from the RF source. The output signal of the SRD is fully known by calibrating it with a nose-to-nose procedure [see Paragraph 3.6.2]. The key idea to determine the absolute phases of the four waves impinging the DUT, is to measure the complex power wave ratio between b [ 2] ( ω i ) and the known phase reference signal at each frequency component ω i of interest ( b [ 2] ( ω i ) a Ref ( ω i ) e j ( θ 2( ω i) ϕ Ref ( ω i )) ). The known phase reference signal is generated by the step recovery diode. Out of this relative measurement, the phase relationship between all frequency components of the b [ 2] -spectrum can be determined since the phase relations of the reference signal are known. The waves a[ 1 ], a [ 2] and b [ 1] can then be obtained from the classical power wave ratio measurements. Note however that phase errors induced in the Phase Ref In channel can not be taken into account, since they can not be distinguished from the phase of the reference signal. 22

51 Nonlinear Vectorial Network Analysers The calibration procedure consists of a classical Short-Open-Load-Thru (SOLT) calibration ([17] + Chapter 3), an amplitude calibration performed by a power meter and a phase calibration based on the reference generator. 1.4 Nonlinear Vectorial Network Analysers The Nonlinear Vectorial Network Analyser (NVNA) that will be used throughout this work, was developed in 1993 by a co-operation of the HP-NMDG group and the department ELEC of the Free University of Brussels ([39], [40]). In the following paragraphs the concept of the instrument and the working of the different parts are explained Concept The purpose of the NVNA was to build an absolute wavemeter that allows to capture the whole wave spectrum in a single take. The instrument must be able to measure the absolute magnitude of the waves as well as the absolute phase relations between the harmonics. In other words, the NVNA can be seen as an absolute Fast Fourier Transform (FFT) analyser for microwaves. Figure 1-15 represents a simplified block schematic of a two-port NVNA to perform connectorised continuous wave (CW) measurements. The DUT can be excited at one or both ports by an RF generator. The incident and reflected waves at both ports of the DUT are then measured through couplers, which have a bandwidth from about 500 MHz to 50 GHz. The high frequency content of the signals does not allow to digitize these signals immediately. Therefore, the measured RF spectrum is downconverted to an IF spectrum by using harmonic mixing. This part of the setup is referred to as the downconvertor of the NVNA and is in fact the key component of the instrument. The downconvertor is based on two modified Microwave Transition Analysers (MTA): four fully synchronised RF data acquisition channels are available. The harmonic mixing principle and the operation of the downconvertor will be explained in paragraph Before the waves are downconverted, attenuators can be used to bring the signal level at the input of the downconvertor below -10 dbm. This is necessary to prevent the downconvertor from being 23

52 Nonlinear Vectorial Network Analyser Syncro CLK ADC ADC Sample CLK ADC ADC A A A A Downc Downc Downc Downc b [] 1 a [] 1 a [] 2 b [] 2 ATT ATT IF-Gen ATT ATT RF GEN b [] 1 a [] 1 a [] 2 b [] 2 DUT Port 1 Port 2 RF GEN FIGURE Simplified block schematic of a two-port NVNA pushed into its nonlinear operation region. After downconversion, the measured data can be amplified and digitized by four synchronised analog-to-digital convertor (ADC) cards of type HPE1437 [18]. The ADC cards sample the data at a rate of 20 MHz and have a usable bandwidth of 8 MHz. The four ADC cards, the downconvertor and the RF generator are clocked by a common 10 MHz reference clock in order to obtain a fully synchronised phase coherent measurement instrument. The calibration of a NVNA consists of three steps: a classical S -parameter calibration, a power calibration and a phase calibration. After the S -parameter calibration, the NVNA can be easily used as a classical vectorial network analyser to obtain S -parameter measurements (see Chapter 3). The power calibration is done by using a power meter, while the phase calibration is based on a known reference element which is called a golden diode (Chapter 3). The phase relations between the frequency components of the golden diode signal are assumed to be exactly known. However, since the reference signal of the golden diode is measured by a nose-to-nose procedure (Chapter 3), the variance of these measurements can be used to alleviates the assumption of an exactly known phase reference signal. Measuring the reference signal with the NVNA and comparing the measured phase relations with the known phase 24

53 Nonlinear Vectorial Network Analysers relations, allows to correct the measured signals of a DUT in phase. By the additional power and phase calibration, absolute waves are obtained and nonlinear effects can be measured. Advantages of the NVNA: 1. Since the whole spectrum of the incident and reflected waves is measured in a single take, phase synchronisation problems between the spectral components are circumvented. 2. The phase relation between the measured harmonics can be calibrated and thus absolutely known. As a result, the time waveforms can be perfectly reconstructed. Disadvantages of the NVNA: 1. The signal-to-noise (SNR) ratio of the NVNA ( 60 db) is much smaller than the SNR of a classical Vectorial Network Analyser like the HP8510 ( 100 db). 2. The NVNA consists of a complex hardware infrastructure, which requires a powerful, easy to manage software to be of practical use Harmonic sampling principle A. Principle Before digitizing the measured RF signals, they must be downconverted to a much lower frequency content. Thereto, they are compressed into an IF spectrum. This is done by the downconvertor which is based on the same principle as sampling oscilloscopes: the harmonic mixing principle. In simple words, this means that the RF signals are sampled by a sampling frequency which introduces alias on purpose! In other words, the sampling is done by neglecting the Shannon-Nyquist theorem which states that the used sample frequency f s must be at least twice the maximal frequency of the signal f max : f s 2f max. In time domain, this operation can be seen as if the signals were slowed down. Consider a single sine wave with frequency f RF. The o symbols in Figure 1-16 are the sampling points 25

54 Nonlinear Vectorial Network Analyser which are taken with a repetition rate of f s «f RF. It is then clear that after sampling a slower version of the original signal is obtained with a frequency f IF = f RF f s. 1 Amplitude [mv] ] Time Samples FIGURE Harmonic sampling principle in time domain In the frequency domain, the harmonic sampling principle is easily understood by taking a closer look at Figure f IF 2f IF f s kf s f RF 2kf s 2f RF (a) f IF 2f IF (b) FIGURE Harmonic mixing principle in the frequency domain: (a) the RF spectrum, (b) the IF spectrum 26

55 Nonlinear Vectorial Network Analysers Consider that the measured spectrum contains a fundamental RF frequency f RF and its second harmonic 2f RF. By mixing this spectrum with an appropriate sampling frequency f s which does not obey Nyquist s rule, the RF frequency and its harmonic are respectively downconverted to a much lower frequency f IF = f RF kf s and 2f IF = 2f RF 2kf s. For the NVNA, the sampling frequency f s is chosen between 19 and 20 MHz. B. Hardware setup In practice, the RF signals are sampled by a sampling head which is driven by a local oscillator (Figure 1-18), whose frequency is produced by the setup of Figure sampling head RF input IF output Local Oscillator FIGURE Downconvertor setup SL channel 1 10 MHz reference clock FRACN synthesizer A SL SL channel 2 channel 3 SRD SL channel 4 FIGURE Local Oscillator setup A 10 MHz reference clock is fed to a FracN synthesizer, which is able to synthesize any local oscillator frequency between 10 MHz and 20 MHz with a frequency resolution of better than 1 Hz and with an accuracy determined by the accuracy of the reference clock. Since this 10 MHz reference clock is also used to clock the RF generator of the NVNA, the whole setup 27

56 Nonlinear Vectorial Network Analyser is synchronised and phase coherence between all signals is valid. The step recovery diode (SRD) [16] will transform the nearly sinusoidal output of the FracN into a train of narrow pulses, whose repetition equals the FracN signal frequency. The pulse train is then split into four synchronised pulse trains to drive the sampling heads of the four measurement channels. To obtain four synchronised equally powered pulse trains, the splitting of the pulse train needs to be done symmetrically. The shock lines (SL) present in the four channels will steepen the rising edges of the local oscillator pulses and reduce the steepness of the falling edges. These reshaped local oscillator pulses are then used to drive the sampling heads as can be seen in Figure Full three-port Nonlinear Vectorial Network Analyser One of the major drawbacks of today s Nonlinear Vectorial Network Analysers is their limitation to two-port measurements. As a result, mixers can not be fully characterised by the NVNA since there is always one signal path that can not be measured. A three-port or sixchannel NVNA would solve this problem. However, besides the need for extra hardware, a non-evident hardware adaptation needs to be done: redesigning the local oscillator signal (Figure 1-19). Designing a three-port NVNA requires two extra signals to be downconverted. Therefore, two additional local oscillator signals are needed to drive the additional sampling heads. This means that the local oscillator signal coming from the SRD (see Figure 1-19) needs to be split into six instead of four synchronised signals. As a result, a new symmetrical splitting setup must be designed which is furthermore capable of delivering sufficient power to drive the six sampling heads. 28

57 Conclusion 1.5 Conclusion In this chapter an answer is given to a simple question: Why are nonlinear microwave measurements so important? Thereto, the major differences between a linear and nonlinear system are given. Furthermore, some advantages of knowing the nonlinear behavior of a system are highlighted. It is obvious that if one is able to measure the nonlinear effects of a system, the global behavior of that system will be better understood. Taking the nonlinear effects into account makes modelling more accurate. An overview of earlier nonlinear measurement instrument designs is given. The advantages as well as the disadvantages are stated and compared to the Nonlinear Vectorial Network Analyser (NVNA). Furthermore, the concept and functioning of the used NVNA is fully explained. The key principle of the NVNA, which is the harmonic sampling principle, is worked out in detail. Since the NVNA does not allow more than two-port measurements, mixers can not be fully characterised. The adaptations needed to develop a three-port NVNA are highlighted. 29

58 Nonlinear Vectorial Network Analyser 30

59 CHAPTER 2 VIRTUAL INSTRUMENTATION Abstract: An abstract model based instrumentation framework is proposed. The abstract description of the instrument to realise consists of a network of basic instrumentation functions. The framework allows to obtain instrument interchangeability: whenever an instrument provides the functionality required by the setup, it can be plugged-in and the virtual instrument can be restarted without additional code change. As an example, the implementation of the Nonlinear Vectorial Network Analyser is proposed. Using the abstract instrument description enables transparent integration of a database system into the instrument software. Using databases facilitates the storage of instrument settings and measured data during each measurement step. It is then possible to virtually redo the measurement and process the data off-line without rewriting any code. 31

60 Virtual Instrumentation 2.1 A dreamed measurement instrument In the past, measurement devices were only able to measure one single directly measurable quantity under static conditions. A measurement setup contained a low number of instruments used to gather a low number of data. The main goal was the measurement of one single aspect of reality. However, modern instrumentation allows simultaneous measurements of several dynamically varying quantities under variable operating conditions. The main goal now is to describe or characterise the device totally for its complete operating range. As a consequence, the amount of data available is very large and the setting of the complex measurement setup for different experiments has to occur automatically to avoid errors. Data compression is also a significant issue for the measurement instrument: performant algorithms will be required to convert the huge amount of raw data to a restricted amount of device characteristics. Clearly, software is a key part of the modern instrument. In this chapter, a software instrumentation framework is proposed to create generic abstract instruments. The realisation of such an instrument framework has several aspects: The instrumentation software will be made as independent as possible on hardware details. The ultimate goal here is to be able to switch instruments of equal capability in the virtual instrument without having to change the instrument code. For a complex setup, it is very difficult to retrieve the settings of all the building blocks exactly. However, a lack to do so often results in the need to perform the measurements over and over again to allow sensible post processing. The framework saves the setup of all parts of the instrument transparently during the measurements to free the user of this additional burden. 32

61 Introduction of abstraction in a complex instrument framework For a multiple experiment measurement campaign, it is often very difficult to save the data in a format such as to allow easy retrieval and processing. Integration of a transparent database allows off-line processing of the data and saving of these data in a structured format to preserve an optimal data usage capability Once I had a dream... I dreamed that if I could measure a nonlinear system, with a Nonlinear Vectorial Network Analyser (NVNA), I could concentrate totally on the characterisation of the device. I did not have to care about what type of instrument was used to build up the analyser. Was it an RF source of type HP8648B or HP83650B? I pushed the source setting buttons and the source was set by the same instructions. I did not have to worry about the command syntax necessary to steer the instruments building up the NVNA, could forget about where and how the instruments are connected,... I could take a new modulation source out of my laboratory, include it to the measurement setup and measure again without rewriting or adding any instruction to the code! In fact, the whole measurement system acted as an off-the-shelf instrument, tailored to the needs of the current test. Instrument interaction occurred only through measurement ports. The DUT to characterise is connected to these measurement ports and the interactions take place by a predefined standard set of instructions independent of the instrument type. As a result, my NVNA combined the advantages of a reconfigurable instrument and the interchangeability of the different instrumentation modules. Did the dream come true? 2.2 Introduction of abstraction in a complex instrument framework Building complex virtual instruments requires the abstraction model to meet two different requirements. On one hand, the complexity of hardware and software that does not ease the understanding of the measurement process for the end-user, has to be hidden. This significantly lowers the steepness of the instrument learning curve. On the other hand, hardware details that do not contribute to instrument functionality should be hidden from the 33

62 Virtual Instrumentation instrument programmer. This allows instrument programmers to focus on the implementation of the measurement process itself. How can one implement such a multiple-abstraction level system? The key to the solution exists already for a very long time, and is used all the way long during the hardware design of an instrument. Most instruments are designed using a top-down procedure. First, a system level design is performed, resulting in an instrument block schematic. The knowledge contained in this abstract representation of the instrument is all what the end-user requires to handle the instrument. Next, the separate blocks are designed and built up using a set of existing hardware modules. The knowledge of the required functionality and of the available hardware modules is sufficient for the designer to clear the job. During this design, care is taken to use components that have a second source and can be exchanged with the actually preferred parts. Note here the parallel with the approach proposed for the instrument programmer. Finally, there is a second type of designers designing the components themselves. Their task is to combine physical hardware modules such as to realise a specified functionality. These are the people that write the instrument drivers in the proposed software approach. The task here is somewhat different, since the driver programmer has to tear existing instruments apart to extract and isolate the abstract functionalities that they contain. This is often cumbersome, because the instruments were never designed to be used as a concatenation of several abstract functions. Clearly, the design methodology for hardware can at least partly be leveraged into software design. The different steps in the instrumentation design will now be described in detail below. 2.3 Designing a complex instrument framework through abstraction Making a complex instrument user friendly requires that the complexity is maximally hidden for the user. Using the measurement setup may not be harder than taking a voltmeter out of the lab, connecting the DUT and obtaining the voltage applied to the DUT. Furthermore, performing a complex task is always best done by dividing the task into simple sub-tasks done by different people specialised for only one sub-task. The design of a complex instrument 34

63 Designing a complex instrument framework through abstraction framework will be sliced to allow such an approach. The software model is a layered structure in which each layer introduces a new level of abstraction. Each layer has to perform its own task by using the bare minimum of needed information. Figure 2-1 represents the instrument { User Interaction Layer USER Port API_1: User level Network Layer Abstract Network API_2: Instrument Level Abstract Module Layer Abstract Nodes RF ADC AWG... Instrumentation Layer Transmission Layer { { Abstract Module INST.1 INST.2 INST.3 VISA... API_3: Abstract Function Provider API_4: VISA Driver Programmer Hardware FIGURE 2-1. Abstract Instrument Model 35

64 Virtual Instrumentation model used throughout this work. Note that the shorthand API stands for Application Program Interface. An API is the interface of a function to a higher and a lower abstract layer User Interaction Layer The knowledge required by the user to perform a measurement should be limited to a system level block schematic of the measurement instrument, delivered by the designer. Only the functionalities of the instrument have to be known without knowledge of the internal hardware details or the internal structure of the instrument. To obtain this, the user can only communicate with the instrument through measurement ports. The user connects the DUT to the ports and sends abstract user action requests to the signals connected to these ports. Each measurement port has six signal connections: the excitation signal (GEN), the DUT connection, the measured incident ( α) and measured reflected ( β) wave, the voltage (V) and current (I) signal. α β GEN PORT V I DUT FIGURE 2-2. Port module definition and connections Why those six signal connections? When a user wants to know the characteristics of a DUT, he will apply an excitation to the DUT and measure its response. So it is obvious that the user needs a DUT connection to connect the device with the instrument. To apply an excitation signal to the DUT, a generator is needed and thus an excitation connection. The user knows that he can control the generator through the GEN connection of the port. The source signal flows through the port into the DUT. In general, when performing high-frequency measurements, one is interested in the incident and reflected waves and the biasing control of the DUT. Therefore, two measured wave signal connections ( α, β), a current signal connection (I) and a voltage signal connection (V) are necessary. The position of the arrows 36

65 Designing a complex instrument framework through abstraction towards or out of the port in Figure 2-2 is explained by the signal flow. The generator signal flows towards the port into the DUT while the signals to measure flow out of the ports. API_1: User Level. Requirements: the user actions must be supplied to the instrument through ports. The user must specify which module is to be used, which action the module has to perform and which parameters are needed by the action. Implementation: the calls used to address the instrument are all of the same form: ModuleName_Action(Port, Parameters). The name of the call contains the type of functional block to which the call is addressed and the action to be taken. The first parameter is always the name of the port to which the desired signal is connected. The other parameters depend on the type of call and determine the final result of the action. Example 2.1 Consider a simple low frequency network analyser with the functional block schematic of Figure 2-3. Syncro CLK ADC Sample CLK ADC RF GEN DUT Port 1 Port 2 Load FIGURE 2-3. Functional block schematic If the user wants to set the number of acquisition points of the ADC card connected to port 1, the following call is sent: ACQ_SetBlockSize(Port1,#Points). Thus, the user only needs to know that there is a data acquisition card or module present, what the functionalities are and 37

66 Virtual Instrumentation through which port the ADC signal can be accessed. The user doesn t have to worry about where the ADC cards are located in the setup or what the internal structure of the instrument looks like. Consider for example, that the setup of Figure 2-3 is expanded with two amplifiers in the acquisition paths: Syncro CLK ADC Sample CLK ADC A A RF GEN DUT Port 1 Port 2 Load FIGURE 2-4. Functional block schematic The presence of the amplifiers in the acquisition paths will, however, not change the user interaction request to set the number of acquisition points of the ADC card: ACQ_SetBlockSize(Port1,#Points). The user interaction request is then passed to the network layer. Note that the blocks addressed by the user are not required to be atomic structures. They can very well, in their own turn, contain a setup of different abstract blocks. Using a single abstract block to describe a whole set of modules has the advantage that a setting can be performed on a composite instrument as if it was a single module. More elaborated super blocks, such as a network analyser module or a spectrum analyser module can be constructed, which makes life a lot easier for the user Network Layer The network layer contains the implementation of the user system level block schematic, by a network of nodes, which are interconnected through signal paths. These nodes are standardised 38

67 Designing a complex instrument framework through abstraction elements grouped by functionality. The network layer knows the name of the nodes that are present in the network, but doesn t know their functionalities. The only task that the network layer has to perform, is routing the call sent by the user to the right node. In fact, this layer has only a searching task and does not have to worry about the action contained in the call. This allows for a change in the abstract model structure without any change to the user s code. Since the first parameter of the call contains the port, the network layer knows automatically where to start its search for the required network node. If however, the network layer finds more than one node of the same type connected to the same port, more specific information about which instrument to send the call to is required from the user. This is, however, not counterintuitive, since the system level block schematic will also include more blocks with identical functionality. Consider that 2 generators are present in the block schematic of an instrument. If the user asks to a colleague to set the power of the generator, the colleague would also require additional information to choose between both generators. What happens when a super node is addressed? An additional routing of the commands through the abstract components is then required. Whenever a device is called with a command that it does not understand, the command is propagated to the devices which are connected to it. This allows the abstract devices to inherit the functionality of their sub-devices. Consider for example the super module of a modulated generator MODGen: SIG Out RFGen MOD AWG MODGen CLK Out FIGURE 2-5. Super module MODGen This modulated generator contains three different abstract modules: an RF generator, a modulator and an AWG. To set the carrier frequency of the MODGen the following command is executed: MODGen_SetFrequency(Port1, Frequency). If the MODGen receives this 39

68 Virtual Instrumentation command, the call is passed to the sub-module connected to his signal port. In this case, it is the modulator module (MOD). However, since this module can not set any frequency, the call is passed to his neighbour modules. The RFGen is then the only module which is capable of handling the call. API_2: Instrument Level. Requirements: translate the user action request into an abstract node specific call. Implementation: the name of the call is identical to the name of the user action request, but the port name is now replaced by a node number: ModuleName_Action(Node, Parameters). The functional call of Example 2.1 is thus translated into: ACQ_SetBlockSize(NodeNbr, #Points). This call is then passed to the abstract module layer Abstract Module Layer This layer has the knowledge of all types of nodes. As previously mentioned, nodes are standardised elements sorted by functionality. They can be seen as units which can not be split up. By combining these nodes, a complete measurement setup can be build. Building an instrument using only standardised instrumentation functions has multiple advantages: It makes the instrument code robust to changes in hardware: as long as the added hardware modules provide a functionality that can be made functionally equivalent to the existing one, the system will operate correctly. It eases the programming (hence minimises risk for errors) of the instrument itself. It lowers the steepness of the learning curve for the instrument programmers, as the only function set they have to use is the normalised set consisting of the standardised instrumentation functions. All abstract nodes can be represented by a generic module that has six signal connections: a signal input and output, a clock synchronisation input and output and a trigger signal input and 40

69 Designing a complex instrument framework through abstraction output. The signal input and output define the signal paths in the instrument while the clock and trigger signals define the control flow that is required for the module synchronisation.the connections for the generic module are shown in Figure 2-6: CLK In TRIG In SIG In NODE SIG Out CLK Out TRIG Out FIGURE 2-6. Definition of the generic module and its signal paths Each abstract module has a minimal set of functions which allows to describe the functionality of the module. Combining these functions one builds the desired instrument. Note that the obtained command set for each abstract module is expanded to make sure that all the parameters that can be set, can also be queried. This property minimises the duplication of status data inside the instrument and hence enforces instrument data consistency. The different types of nodes are split up in classes. By taking a closer look to these classes, it becomes clear that combining them allows to create whatever measurement setup needed to completely characterise a device. The different classes are: 1. Dynamic Signal Measurement Acquisition module: this module contains acquisition instruments which are able to measure all sorts of dynamic varying signals. The settings of these acquisition modules are: the number of acquisition points, the coupling mode (AC or DC), the trigger delay in samples, the DC offset in Volts and the acquisition range. 2. Static Signal Measurement Digital current meter module: contains instruments which are able to measure static current signals. Digital voltmeter module: contains instruments which are able to measure static voltage signals. 41

70 Virtual Instrumentation 3. Signal Conditioning Amplifier module: contains instruments which are able to amplify the signals. The amplification factor and a DC offset can be specified. Attenuator module: contains instruments which are able to attenuate the signals. The attenuation factor can be freely chosen within the range of the attenuator. Filter module: this module is able to eliminate undesired frequency components in the signal paths. The passband can be specified. 4. Modulation Amplitude modulation module: this module delivers an AM modulated signal at its output when excited by an RF and IF signal. Phase modulation module: this module delivers a PM modulated signal at its output when excited by an RF and IF signal. Complex modulation module: this module delivers an IQ modulated signal at its output when excited by an RF signal and 2 IF signals. 5. IF Signal Sources: these are the IF sources needed to drive modulators or to design low frequency setups. IF arbitrary waveform generator module: these instruments are able to create low frequency arbitrary waveforms like multisines. The amplitude (given in Volts) can be adapted. The waveform of the signals can be freely chosen. The frequency of the signals is determined by the sampling frequency of its clocking device. IF pulse generator module: these generators only create pulses with adjustable period and width. The amplitude (in Volts) can also be changed. IF sine waveform generator module: these generators deliver an IF sine waveform with adjustable frequency and amplitude. 42

71 Designing a complex instrument framework through abstraction 6. RF Signal Sources: RF generator module: these generators produce high frequency sine waves with adjustable frequency and power (in dbm). 7. Static Signal Generators Digital voltage source module: contains instruments which are able to produce static voltage signals such as DC biasing signals. The DC levels and the maximum allowed current compliance can be set. 8. Synchronisation Clock module: this module produces stable reference signals that can be used to synchronise different parts of the setup. The frequency of the clock signal can be set together with the scaling of this clock frequency. Some clock modules have a zooming function and can be swept. Trigger generator module: digital signal generator that can trigger instruments. 9. Power Measurements Power sensor module: converts the power content of a signal into a DC level. Power meter module: measures the DC level of the power sensor and computes the corresponding power level. The sensitivity of the power meter and the frequency bandwidth to measure can be specified. 10. Downconvertor Module: this module is able to downconvert high frequency signals to intermediate frequency bands. 11. Device Under Test Module: what the user wants to characterise. 12. Switch Module: this module allows to switch between different signal paths, clock paths or trigger paths. This means that the setup becomes dynamic and thus can be adapted during the measurements. 13. Calibration Standard Module: contains all the calibration elements needed to perform a linear S -parameter calibration 43

72 Virtual Instrumentation 14. Tuner Module: allows to tune the output impedance of the instrument The abstract module layer knows which types of nodes exist and what their functionality is. This means that only this layer can check whether the forwarded call contains a correct action for the specified node and whether the parameters satisfy the parameter definition. This layer does not contain any information about hardware details or the data format necessary to communicate with the instrument API_3: Abstract Function Provider. Requirements: translate the node specific functions into instrument specific functions. Implementation: since the abstract module layer knows the hardware type of a specified node, the node specific functions can be translated into instrument specific functions. Here, the node name is translated into an instrument name and the node number is translated into a session reference: InstrumentName_Action(Session, Parameters). The session reference is a unique reference which connects an abstract module with the hardware. This reference must be sufficient to access the device driver without any ambiguity about the hardware. Typically the VISA [48] session ID of the instrument is used as reference. The abstract node function of Example 2.1 is thus translated into: HPE1430_SetBlockSize(session, #Points). This call is then passed to the instrumentation layer Instrumentation Layer The instrumentation layer contains instrument specific drivers. Each and every instrument has its own driver which translates the instrument specific functions coming from the abstract module layer to hardware specific calls. When a message based or register based instrument is delivered with a Plug and Play (PNP) driver [49], this driver can then be used to access the instrument. When a message based instrument has no PNP driver, one can still easily communicate with the instrument through the message base. However, for a register based instrument without PNP driver, the driver must be written from scratch to communicate with the instrument. 44

73 Designing a complex instrument framework through abstraction When the settings of an instrument are used in calculations, it is important to use and hence to be able to query the actual instrument settings. The correct way to recall the instrument state is by sending these setting requests directly to the instrument. The instrument then replies with the actual settings. However, not all instruments are able to respond to these setting requests. Hence, to avoid inconsistencies, the instrument settings need to be saved into instrument variables during instrument setup and recalled whenever the user sends a setting request to the instrument. Note that each time the user changes an instrument setting, the corresponding instrument variables are overwritten with the new setting to preserve the instrument data consistency. Since the instrument can not respond to the queries, instrument roundings of the user settings must be taken into account in the instrument driver. API_4: VISA Driver Programmer. Requirements: translate the hardware specific calls to a set of VISA calls that implement the action. Implementation: the translation of the user command into VISA is done by the instrument drivers through a set of MEX-functions [24]. The instrument drivers are usually written in C++, while the user commands are written in Matlab Transmission Layer The transmission layer, which uses the VISA standard [48], is the only layer that knows how the different measurement instruments are connected to the computer (GPIB, VXI,...). As such, it is able to put the instructions coming from the instrumentation layer in the right format on the right interface and send it to the right instrument, without care for the contents of the instruction. The VISA standard allows to use identical calls for IEEE-488, VXI, ethernet or even serial programmable instruments Hardware Layer The hardware layer contains all sorts of measurement instruments ranging from GPIB programmable devices to VXI modules which are all connected to a computer. The VXI modular instrumentation is here preferred because of its high level of interoperability, its 45

74 Virtual Instrumentation synchronisation capabilities and the availability of high quality ADC and AWG modules. Usage of GPIB devices fills the gap for specific measurement devices. The price to pay for this heterogeneous collection is that the system hardware complexity is very high. 2.4 Instructions for dummies A measurement setup can contain a lot of abstract nodes and each node contains in turn a certain number of functionalities. This results in a maximal instrumental flexibility for the user. The drawback is that the user must be a power user to know all the different instructions of the modules. To overcome this shortcoming while maintaining flexibility, an additional software shell of highly automated high-level measurements is designed. These routines will select sensible instrumentation choices automatically based on a safe operation mode: a suboptimal choice that does the job in any case is preferred over an optimal method that may fail for some settings. This is the only way out to gain the confidence of the users and avoid them writing their own home-made (and potentially wrong) utilities for each case. Note here the parallel with classical instruments. The normal user will operate the instrument by using the predefined instrument buttons. A power user, on the other hand, has the possibility to interact with the instrument using the enclosed software commands. However, the functionality offered by these software commands is nearly always an exact copy of the functionality of the buttons. As a result, the total flexibility of the instrument setup does not increase. This is in high contrast with the virtual instrumentation framework, which indeed allows a power user to design his/her dreamed setup. Consider for example a simple two-port Nonlinear Vectorial Network Analyser measurement with the excitation signal connected to port 1. The complexity of the instrument is maximally hidden for the user and setup choices are made automatically. The only parameters that the user must define to setup the instrument are the frequency and power of the excitation signal. The high-level instruction to setup the instrument then becomes: Meas_Setup(Port1, Freq, Pow). 46

75 Instructions for dummies However, the following instructions would be required if the user had to specify himself all the necessary setup parameters: %// Set the frequency and power of the RF Source RFG_SetPower(Port1,Pow); RFG_SetFreq(Port1,Freq); %// Calculate the downconversion frequency and ADC block length [DownCFreqCalc,NPts] = GetDownConvFreq(Freq); %// Load the calculated downconversion frequency into the downconvertor CLK_SetFreqScale(Port1,DownCFreqCalc); %// Auto scale the ADC range and set the ADC acquisition block size ACQ_SetBlockSize(Port1, NPts); ACQ_SetRange(Port1, 'AUTO'); %// Set the attenuators into a safe operation mode ATT_SetAtt(Port1, AUTO ); To perform a measurement, the following high-level instruction can be used: Measure(Repetition). Here, only the required number of repeated measurements needs to be specified. This high-level call hides the following calls: %// Set the RF source ON RFG_SetRFOn(Port1, ON ); %// Perform the repeated measurements loop for ind = 1:Repetition %// Trigger the ADC cards to perform a measurement ACQ_Measure; %// Read in the data of the ADC card Data(ind) = ACQ_GetData(Port1); end; 47

76 Virtual Instrumentation One can easily see that it is more userfriendly to obtain a quick measurement of the DUT by using the high-level instructions than by executing all individual abstract module instructions. When the need calls, the user can still copy the standard setup and modify it to fit his/her needs exactly, but can start from a working piece of code to do so. 2.5 The Nonlinear Vectorial Network Analyser as implemented for the measurements In this paragraph, the virtual instrumentation software framework is used to implement the Nonlinear Vectorial Network Analyser. The computer platform is a Pentium II Compaq Deskpro under Windows NT 4.0. The VXI card cage is connected through the MXI slot zero controller and a National Instruments PCI-MXI 2000 card. The GPIB instruments are connected on a GPIB-PCI controller also from National Instruments. The abstract function calls are implemented in Matlab, while the PNP drivers and the Matlab glue are coded in C++. The concept is easily implemented in any programming language. All the instrumentation interfaces are driven by the VISA driver. The simplified block schematic representation of the NVNA, represented in Figure 2-7 and its functionality are the only knowledge required by the user to perform nonlinear measurements. Syncro CLK ADC ADC Sample CLK ADC ADC Downc Downc Downc Downc β [ 1] α [ 1] α [ 2] β [ 2] RF GEN IF-Gen β [ 1] α [ 1] α [ 2] β [ 2] DUT Port 1 Port 2 LOAD FIGURE 2-7. Block schematic representation of the NVNA 48

77 The Nonlinear Vectorial Network Analyser as implemented for the measurements Using the definition of the generic modules, the block schematic representation of Figure 2-7 can be translated into the abstract model of Figure 2-8 used by the network layer. Note that CLK GEN CLK Master CLK Slave1 CLK Slave2 CLK Slave3 TRIG ACQ ACQ ACQ ACQ CLK DNCV DOWNCONVERTOR SIGNAL COND A SIGNAL COND B SIGNAL COND C SIGNAL COND D GEN PORT 1 PORT 2 DUT FIGURE 2-8. Abstract model of the NVNA signal conditioning is added at the device signal ports to allow the model to cope with signal levels that have a high dynamic range. An overview of the different implemented modules can be found in Appendix 2.A. For every abstract module, general commands which compose the functionality of the module are implemented using Matlab (Appendix 2.B). Consider that the user wants to set the power of the RF sinewave generator connected to measurement port 1. The only thing that the user needs to know is the port through which the RF signal can be accessed (Port 1) and the abstract module class to which the device belongs (RFGen). The functionalities of the RFGen node are: setting the power level, the carrier frequency and turning the generator on or off. The user specific function to set the power of the RF generator connected to port 1 is then: RFG_SetPower(Port1, Power). The network layer routes this call to the right node in the abstract model network. The call is then translated into 49

78 Virtual Instrumentation an abstract node specific call: RFG_SetPower(Node, Power) and passed to the abstract module layer. If the requested action exists and the parameters are syntactically correct, the call is translated into a instrument specific function: HPE83650B_SetPower(session, Power). The obtained instrument specific call can now be passed to the instrument driver and further to the VISA layer. Up to this point, all the calls were implemented in Matlab. To translate the Matlab calls into VISA, a set of MEX functions (Matlab extension functions [24]) is implemented. This is done by the driver which is written in C++ and translates the MEX calls into matlab calls. These calls are used for almost all message based devices. For the register based devices, the Plug and Play standard for functional drivers is used. The call then finally arrives in the right format at the instrument and the power of the RF generator is set. A similar reasoning can be followed to set whatever instrument present in the NVNA. Because of the developed software framework, the NVNA becomes an easy-to-use, reconfigurable and adaptable measurement instrument. 2.6 Databases: the way to fully automated measurements Measuring nonlinear devices does not only require a complex instrumentation setup. Testing a device under test over its whole range of operation, also results in a huge amount of data, because the measurement of each power-frequency combination results in a full-blown measurement. Furthermore, since a complex measurement setup is involved, a lot of settings are needed. Post processing the raw data requires the accurate knowledge of all correct settings for all instruments. How can one save the results without losing the link between the raw data and the instrument settings? Since nonlinear characterisation requires a lot of measurements, it takes also a long measurement time. Hence, the user should be able to measure without actually being present. To obtain this, data and settings must be stored automatically. To ease the post processing of the data, the user should be able to virtually redo the measurements completely or partially, without need for the hardware infrastructure and without a change to the measurement code. The instrument would then operate in a play back mode, presenting the data that were earlier 50

79 Databases: the way to fully automated measurements measured on an on-request basis. How can one make this play back mode transparent for the user? The above questions are easily solved if a transparent database integration into the complex instrument framework is realised. The use of databases will lead the way to fully automated measurements and makes life a lot easier for the user. The requirements of a transparent database measurement system are multiple: The user may not see any difference between a measurement done directly on the hardware or on the database. Thus, the same abstract user instructions must be valid for a hardware measurement or a virtual database measurement. Retrieving data from the database should be done in the same way as obtaining data from a measurement instrument. Saving of the data and settings is done transparently for the user. The structure of the database must be built automatically out of the network lay-out. Integration of a database into the complex instrument framework is based on the same layered model as Figure 2-1 (see Figure 2-9). The instrumentation layer is now extended with a database device for each node. The transmission layer contains the hard disk as a device. Note, however, that the database has a different function during real-world measurements and virtual measurements. Real-world measurements: during a measurement, the setup of the instrument and the measured raw data need to be saved into the database. Thereto, the database structure is automatically built out of the abstract network and contains a table to save the network structure of the instrument, a table to save all state variables of all nodes for every measurement step, and a different table for each measured signal. Due to a set of unique links between the setup table and the data tables, no mistakes can be made when post processing the 51

80 Virtual Instrumentation { User Interaction Layer USER Port API_1: User level Network Layer Abstract Network Abstract Module Layer Abstract Nodes RF ADC AWG... API_2: Instrument Level Instrumentation Layer Transmission Layer { { INST.1 Abstract Module INST.2 INST.3 VISA... DB Device Disk API_3: Abstract Function Provider API_4: VISA Driver Programmer Hardware FIGURE 2-9. Abstract Database Instrument Model measured data. Hence, the data saved in the database is sufficient to redo the measurements afterwards. 52

81 Databases: the way to fully automated measurements When the user wants to save the settings of a module during a measurement, the following call is sent: ModuleName_SaveSettings(Port). When the abstract module layer receives this call, it knows that the only instrument in the instrument layer that can save settings is the database device. Furthermore, the abstract module layer will automatically retrieve all setting parameters of the specified module, so that the call is translated to DB_SaveSettings(Session,Parameters) and send to the database. The same technique is used when saving raw data. Virtual measurements: instead of sending the user request to an instrument in the instrument layer, it is sent to a database device which handles the call. Sending a measurement call, retrieves the required data from the database. When a call that requires an instrument setting is sent, nothing happens, since it considers here database measurements. However, the user can still use the same command set and will see no difference in the results if the same measurement is repeated. It is clear that expanding the instrument framework by a transparent database has a lot of advantages: There is no need for code changes to handle a replay measurement from the database. The user can still use the same functions. Saving data and settings happens transparently for the user. Virtual measurements can be done without hardware present. Measurements can be done off-site and post processing of the data on-site. The data are saved in a structured format that is selected such as to save the full measurement state information and preserving optimal usage possibility for the data. Post processing the data can be done at any time and any place without fear of losing any data. 53

82 Virtual Instrumentation 2.7 Conclusion An abstract model based instrumentation framework is proposed, implemented by a layered structure. The layered structure allows to introduce different levels of abstraction that hide the complexity of the system for the user. The end-user only needs to know the system level block schematic of the measurement setup and its functionality. The use of a network model, based on standardised elements, allows the framework to obtain instrument interchangeability. Using databases is a major step towards fully automated measurements, since the raw data and settings of the very complex nonlinear measurement setup are saved transparently during each measurement step. Furthermore, it allows to play-back the measurement and process the data off-line without rewriting any code. 2.8 Appendices Appendix 2.A : Implemented modules The following list represents the different implemented modules. The Matlab implementation and necessary drivers are available for each specified module. 1. Acquisition modules: HP54121A - HPE HPE HPE HPE Amplifier modules: HP85120/K60 - HP8565E - HP Attenuator modules: HP8565E - HP85120/K60 4. Clock modules: HP85120/K60 - HP54121A - HP83650B - HP8565E - HP8648B - HPE HPE HPE HPE HPE Digital voltage source modules: HP HPE Digital current meter modules: HP HP HPE Downconvertor modules: HP85120/K60 8. Device under test modules: DUT 9. Digital voltmeter modules: HP HP HPE Filter modules: HP54121A - HP8565E - HPE HPE

83 Appendices 11. Switch modules: ATTSW - Calbox - FiUser - IFCalSW - PWRMSW 12. Arbitrary waveform generator modules: HP HPE Pulse generator modules: HP33120A - HP54121A - HP83650B 14. Sine waveform generator modules: HP33120A - HP54121A - HP85120/K60 - HP83650B - HP8648B - HPE Amplitude modulation modules: HP83650B - HP8648B - UserMod 16. Phase modulation modules: HP85120/K60 - HP83650B - HP8648B - UserMod 17. Power meter modules: EPM441A - HP8565E 18. Power sensor modules: HP8487A 19. RF generator modules: HP8340B - HP83650B - HP8648B 20. Calibration standard modules: Load - Open - Short - Thru 21. Trigger generator modules: HP85120/K60 - HP HP54121A - HPE HPE HPE HPE HPE Tuner modules: ATN Notice that by combining different modules, more complex measurement instruments such as a spectrum analyser or network analyser can be made. Appendix 2.B : User commands The general commands which compose the functionality of a module, are given for each abstract module: 1. Acquisition modules: ACQ_ConnectCLK(Port,SourceFreq) ACQ_ConnectTRIG(Port) [NPts] = ACQ_GetBlockSize(Port) [Coupling] = ACQ_GetCoupling(Port) [Data, overload] = ACQ_GetData(Port) 55

84 Virtual Instrumentation [DelaySamples] = ACQ_GetDelay(Port) [Offset] = ACQ_GetOffset(Port) [AcqRange] = ACQ_GetRange(Port) ACQ_GrpInit ACQ_GrpSync ACQ_IndexData ACQ_Init ACQ_MakeMeasTable(DataBase,Port) ACQ_MakeTable(DataBase,Port) ACQ_Measure ACQ_SaveData(DataBase,Port,timDat) ACQ_SaveSettings(DataBase,Port) ACQ_SetBlockSize(Port,NPts) ACQ_SetCoupling(Port,Coupling) ACQ_SetDefault(Port ACQ_SetDelay(Port,DelaySamples)) ACQ_SetOffset(Port,Offset) ACQ_SetRange(Port,Range,Time) 2. Amplifier modules: [GainFactor] = AMP_GetGain(Port) [OffSet] = AMP_GetOffset(Port) AMP_Init(Port) AMP_MakeTable(DataBase,Port) AMP_SaveSettings(DataBase,Port) AMP_SetDefault(Port) AMP_SetGain(Port,GainFactor) 56

85 Appendices AMP_SetOffset(Port,OffSet) 3. Attenuator modules: [attenuation] = ATT_GetAtt(Port) ATT_Init(Port) ATT_MakeTable(DataBase,Port) ATT_SaveSettings(DataBase,Port) ATT_SetAtt(Port, attenuation) ATT_SetDefault(Port) 4. Clock modules: CLK_Generate(Port,ClockInput,ClockFreq) [SourceFreq] = CLK_GetFreq(Port) [MulFac, DivFac] = CLK_GetFreqScale(Port,channel) [Offset] = CLK_GetOffset(Port) [SourcePeriod] = CLK_GetPeriod(Port) [Source] = CLK_GetSource(Port,Type) [SourceSpan] = CLK_GetSpan(Port) CLK_Init(Port) CLK_MakeTable(DataBase,Port) CLK_SaveSettings(DataBase,Port) CLK_SetDefault(Port) CLK_SetFreq(Port,SourceFreq) CLK_SetFreqScale(Port,MulFac, DivFac) CLK_SetOffset(Port,Offset) CLK_SetPeriod(Port,Period) CLK_SetSpan(Port,SpanFreq) 5. Digital voltage source modules: 57

86 Virtual Instrumentation [IMax] = DCV_GetIMax(Port) [Voltage] = DCV_GetV(Port) DCV_Init(Port) DCV_MakeTable(DataBase,Port) DCV_SaveSettings(DataBase,Port) DCV_SetDefault(Port) DCV_SetIMax(Port,IMax) DCV_SetV(Port,Voltage) 6. Digital current meter modules: [Value] = DIM_GetData(Port) [Mode] = DIM_GetMode(Port) DIM_Init(Port) DIM_MakeMeasTable(DataBase,Port) DIM_MakeTable(DataBase,Port) DIM_Measure(Port) DIM_SaveData(DataBase,Port,measID,binNr,dat) DIM_SaveSettings(DataBase,Port) DIM_SetDefault(Port) DIM_SetMode(Port,mode) 7. Downconvertor modules: DNCV_ConnectCLK(Port,SourceFreq) DNCV_ConnectTRIG(Port) DNCV_Init(Port) DNCV_MakeTable(DataBase,Port) DNCV_SaveSettings(DataBase,Port) DNCV_SetDefault(Port) 58

87 Appendices 8. Device under test modules: DUT_Init(Port) DUT_MakeTable(DataBase,Port) [session] = DUT_Open(Port) DUT_SaveSettings(DataBase,Port) DUT_SetDefault(Port) 9. Digital voltmeter modules: [Value] = DVM_GetData(Port) [Mode] = DVM_GetMode(Port) DVM_Init(Port) DVM_MakeMeasTable(DataBase,Port) DVM_MakeTable(DataBase,Port) DVM_Measure(Port) DVM_SaveData(DataBase,Port,measID,binNr,dat) DVM_SaveSettings(DataBase,Port) DVM_SetDefault(Port) DVM_SetMode(Port,mode) 10. Filter modules: [LowFreq,HighFreq] = FILT_GetBand(Port) FILT_Init(Port) FILT_MakeTable(DataBase,Port) FILT_SaveSettings(DataBase,Port) FILT_SetBand(Port,LowFreq,HighFreq) FILT_SetDefault(Port) 11. Switch modules: [InPort, OutPort] = SWFI_Get(Port) 59

88 Virtual Instrumentation SWFI_Init(Port) SWFI_MakeTable(DataBase,Port) SWFI_SaveSettings(DataBase,Port) SWFI_Set(Port, InID, OutID) SWFI_SetDefault(Port) 12. Arbitrary waveform generator modules: IFOn = AWG_GetAWGOn(Port) Offset = AWG_GetOffset(Port) VMaxOut = AWG_GetVMax(Port) AWG_Init(Port) AWG_MakeTable(DataBase, Port) AWG_SaveSettings(DataBase,Port) AWG_SetAWGOn(Port,IFOn) AWG_SetDefault(Port) AWG_SetOffset(Port,Offset) AWG_SetVMax(Port,VMax) AWG_SetWave(Port,ArbWave) 13. Pulse generator modules: [Amplitude] = PUL_GetAmpl(Port) [Offset] = PUL_GetOffset(Port) [PulPeriod] = PUL_GetPeriod(Port) [IFOn] = PUL_GetPulseOn(Port) PulWidth = PUL_GetWidth(Port) PUL_Init(Port) PUL_MakeTable(DataBase,Port) PUL_SaveSettings(DataBase,Port) 60

89 Appendices PUL_SetAmpl(Port,Ampl) PUL_SetDefault(Port) PUL_SetOffset(Port,Offset) PUL_SetPeriod(Port,PulPeriod) PUL_SetPulseOn(Port,IFOn) PUL_SetWidth(Port,PulWidth) 14. Sine waveform generator modules: SIN_ConnectCLK(Port,SourceFreq) [Freq] = SIN_GetFreq(Port) [Power] = SIN_GetPower(Port) [IFOn] = SIN_GetSineOn(Port) SIN_Init(Port) SIN_MakeTable(DataBase,Port) SIN_SaveSettings(DataBase,Port) SIN_SetDefault(Port) SIN_SetFreq(Port,Freq) SIN_SetPower(Port,Ampl) SIN_SetSineOn(Port,IFOn) 15. Amplitude modulation modules: [ModStat] = MODA_GetActive(Port) [IFSrc] = MODA_GetIFSrc(Port) [RFSrc] = MODA_GetRFSrc(Port) MODA_Init(Port) MODA_MakeTable(DataBase,Port) MODA_SaveSettings(DataBase,Port) MODA_SetActive(Port, ModStat) 61

90 Virtual Instrumentation MODA_SetDefault(Port) MODA_SetIFSrc(Port,IFSrc) MODA_SetRFSrc(Port,RFSrc) 16. Phase modulation modules: [ModStat] = MODP_GetActive(Port) [IFSrc] = MODP_GetIFSrc(Port) [RFSrc] = MODP_GetRFSrc(Port) MODP_Init(Port) MODP_MakeTable(DataBase,Port) MODP_SaveSettings(DataBase,Port) MODP_SetActive(Port, ModStat) MODP_SetDefault(Port) MODP_SetIFSrc(Port,IFSrc) MODP_SetRFSrc(Port,RFSrc) 17. Power meter modules: PWRM_Cal(Port) [AvgFactor] = PWRM_GetAvg(Port) [Power] = PWRM_GetData(Port) [DutyCycle] = PWRM_GetDutyCycle(Port) PWRM_Init(Port) PWRM_MakeMeasTable(DataBase,Port) PWRM_MakeTable(DataBase,Port) [Power] = PWRM_Measure(Port,Frequency) PWRM_SaveData(DataBase,Port,measID,binNr,specDat) PWRM_SaveSettings(DataBase,Port) PWRM_SetAvg(Port,AvgFactor) 62

91 Appendices PWRM_SetDefault(Port) PWRM_SetDutyCycle(Port,DutyCycle) 18. Power sensor modules: [CalTableName, FreqTable, CalTable] = PWRS_GetCalTable(Port) PWRS_Init(Port) PWRS_MakeTable(DataBase,Port) PWRS_SaveSettings(DataBase,Port) PWRS_SetDefault(Port) 19. RF generator modules: [CWFreq] = RFG_GetFreq(Port) [CWPower] = RFG_GetPower(Port) [RFOn] = RFG_GetRFOn(Port) RFG_Init(Port) RFG_MakeTable(DataBase,Port) RFG_SaveSettings(DataBase,Port) RFG_SetDefault(Port) RFG_SetFreq(Port,CWFreq) RFG_SetPower(Port,CWPower) RFG_SetRFOn(Port, RFOn) 20. Calibration standard modules: STD_Init(Port) STD_MakeTable(DataBase,Port) STD_SaveSettings(DataBase,Port) STD_SetDefault(Port) 21. Trigger generator modules: TRIG_Generate(Port) 63

92 Virtual Instrumentation [InTrig,OutTrig] = TRIG_GetSource(Port) TRIG_Init(Port) TRIG_MakeTable(DataBase,Port) TRIG_SaveSettings(DataBase,Port) TRIG_SetDefault(Port) 22. Tuner modules: [Imp] = TUN_GetImp(Port) TUN_Init(Port) TUN_MakeTable(DataBase,Port) TUN_SaveSettings(DataBase,Port) TUN_SetDefault(Port) TUN_SetImp(Port, Imp) 64

93 CHAPTER 3 CALIBRATION OF A NONLINEAR VECTORIAL NETWORK ANALYSER Abstract: In this chapter an answer is given to some important calibration issues. Why does a measurement instrument need to be calibrated? Is there a difference between calibrating a Vectorial Network Analyser and a Nonlinear Vectorial Network Analyser? Can a stochastical framework replace the classical S -parameter calibrations and what is it worth? Finally, the advantages of using a database enabled instrument during the different calibration steps are investigated. 65

94 Calibration of a Nonlinear Vectorial Network Analyser 3.1 The need for calibration and calibration standards When performing measurements, systematic errors are often introduced due to the imperfections of the measurement devices. This is especially true for RF and microwave equipment, where the measurement hardware is far from being ideal. To minimize these errors a calibration procedure is required, where known and stable elements (standards) act as measured Devices Under Test (DUT) to determine the measurement error of the instrument. In general, it is very hard - or even impossible - to reconstruct the measurement error of the device based on the measurement of known elements. However, if Assumption 3.1 the measurement device is a linear time invariant (LTI) system, the relation between exact (unknown) and measured waves becomes a linear relation too. Taking a measurement instrument that has n p and measured waves becomes: measurement ports, the relation between exact β() i α() i = T NL () i bi () ai () (3-1) For every angular frequency ω i (here denoted by i ), the value of the error correction matrix T NL () i C 2n p 2n p will differ. Herein, β() i = β [ 1] () β i [ np ] i and bi () α() i = = b [ 1] () b i [ np ] i represent respectively the measured and exact reflected waves; α [ 1] () α i [ np ] i and ai () = a [ 1] () a i [ np ] i represent respectively the measured and exact incident waves. Superscript T denotes the transposed vector. Note that β() i and αi () are considered here to be noisy data. Noise free data will be indicated by a subscript 0, e.g. the measured incident waves: α() i = α 0 () i + n α () i where α 0 () i are the 66

95 The need for calibration and calibration standards noise free incident waves and n α () i N( 0, σ α ) distributed measurement noise on α 0 () i [35]. the additive circular complex normal In the classical Vectorial Network Analyser (VNA), the DUT is a linear time invariant system too. Hence, the S -matrix of the scattering parameters describes the system completely. To calibrate S -parameters, it is sufficient to calibrate wave ratios. As a result, one of the elements of T NL () i can be freely chosen. In the rest of this chapter T NL[1,1] () i will be fixed so that the calibration equations become 1 : β() i α() i = Ki ()Ti bi () ai () (3-2) 1 where Ki () can be everything but 0, and Ti () = T NL () i T NL[1,1] () i For noise free data, relation (3-2) becomes: β 0 () i α 0 () i = K 0 ()T i 0 () i b 0 () i a 0 () i (3-3) This relation can only be exactly known if the measurements are noise free. However, in practice, measurement noise is always present and thus T 0 () i and K 0 () i are unknown. Assuming that, Assumption 3.2 the standards are LTI, 1. Classical approaches fix T NL[3,3] () i. The calibration equations (3-9) to (3-15) becomes then more simple, but symmetry between the equations is lost. Moreover, the calculations for the stochastic approach are easier when T NL[1,1] () i is fixed to a constant value. To be consistent throughout the text, T NL[1,1] () i is chosen to be fixed here also. 67

96 Calibration of a Nonlinear Vectorial Network Analyser the definition relation of the standard as a function of the frequency becomes bi () = Si ()ai with Si () the S -parameters of the standard. For the measurement of the standards, equation (3-1) becomes: β() i α() i = T NL () i Si () I np ai () (3-4) where I np is the n p n p identity matrix. Each standard thus yields n p equations between the unknown elements of T NL. To determine the ( 2n p ) 2 1 complex elements of T NL () i, n p independent experiments are required. Common practice shows that these n p independent experiments can be obtained by exciting each port p = 1 n p separately, or put in a more formal way by selection of aip (, ) = a src ()Z i p with Z p = [ 0 1 ( p 1), 10, 1 ( np p) ] T and a src () i the wave generated by the source connected to the measurement instrument. 0 1 ( p 1) and 0 1 ( np p) represent respectively a 1 ( p 1) and a 1 ( n p p) matrix of zeros. In the next paragraph, the determination of the correction matrix discussed, when the sets of standard measurements Ti () of equation (3-2) will be β( i, p) α( ip, ) (3-5) with exactly known Si () are given. 3.2 Classical S-parameter calibration Before one is able to perform a classical S -parameter calibration, which solves the calibration equations (3-2) analytically, three hypotheses must be made. 68

97 Classical S-parameter calibration Assumption 3.3 The S -parameters of the calibration standards are exactly known. This assumption can have serious implications on the accuracy of the calibration. By assuming, for example that the true impedance of the load is 50Ω, while it is exactly 50.3Ω, a systematic error will be introduced in the calibration, hence also in each measurement. Assumption 3.4 The measurement noise can be neglected. This assumption is reasonable for the VNA, since it is built around a very small bandwidth intermediate frequency (IF) detection (a few Hz). Hence, the signal-to-noise ratio (SNR) of the measurements is high enough to neglect the influence of these stochastic variations on the calibration process. However, by construction the IF detection of the NVNA has a wider bandwidth than that of the VNA (typically a few khz). So, it is reasonable to assume that the measurement noise will influence the calibration of the NVNA significantly. Assumption 3.5 The uncertainty on the reconnections of the standards is neglected. Example 3.1 For simplicity, the classical S -parameter calibration is demonstrated below on a 2-port DUT. Extension of the results to multiple port devices is quite straightforward. β [ 1]0 α [ 1]0 β [ 2]0 α [ 2]0 RF GEN a [ 1]0 a [ 2]0 DUT 50Ω b [ 1]0 b [ 2]0 Port 1 Port 2 FIGURE 3-1. Two-port device under test 69

98 Calibration of a Nonlinear Vectorial Network Analyser Due to Assumption 3.4, the calibration coefficients can be worked out analytically. To further simplify the equations, assume that Assumption 3.6 the crosstalk between the two ports is negligible. The measured waves at port 1 are then only related to the true waves at port 1 and not to the true waves at port 2 and vice versa. As a result only 7 error coefficients of Ti ()(3-2) need to be determined. This assumption is reasonable for connector based measurements. For on-wafer measurements, however, this assumption will result in a degradation of the calibration accuracy. To improve the accuracy a full 16 error terms calibration should be performed. For each frequency i, a linear relation between the measured input and output waves ( α, β) and the true waves ( a, b) exists and is given by equation (3-2). Under the assumption of neglecting the crosstalk between the ports, equation (3-2) becomes β []0 1 () i β [ 2]0 () i α []0 1 () i α [ 2]0 () i = K 0 () i 1 0 T [ 13, ]0 () i 0 0 T [ 22, ]0 () i 0 T [ 24, ]0 () i T [ 31, ]0 () i 0 T [ 33, ]0 () i 0 0 T [ 42, ]0 () i 0 T [ 44, ]0 () i b []0 1 () i b [ 2]0 () i a []0 1 () i a [ 2]0 () i (3-6) Since a linear device can be fully characterised by the S -parameters which are wave ratios, the complex K 0 () i -factor must not be determined for an S -parameter calibration and only 7 complex error correction terms remain to be determined. These 7 complex error correction terms are determined through measurement of calibration standards, which are assumed to be exactly known. The type of used calibration standards determines the name of the calibration methods, for example: SOLT (Short-Open-Load-Thru), TRL (Thru-Reflect-Line), LRM (Line- Reflect-Match),... 70

99 Classical S-parameter calibration Since the SOLT calibration is the most widely used and best known calibration method, this approach will be worked out to solve the calibration equations (3-6). For the SOLT calibration, equations (3-6) result in 1 equation for the load, the open, the short and the thru [17]. Measuring the reflection coefficient Γ() i of the calibration standards at both ports of the measurement instrument results then in a system of 7 complex equations with 7 complex unknowns and thus 14 degrees of freedom: β Γ lx() i []X0 l () i = with l = 12, the port index and X = S(hort), L(oad), O(pen) (3-7) α []X0 l () i β Γ T() i [ 2]T0 () i = for the thru (3-8) α [ 1]T0 () i The diacritical symbol ~ denotes the measured value of the reflection coefficient. To be able to solve these equations analytically, one must remind that the S -parameters of the standards are exactly known (Assumption 3.3), which results in the following exact reflection coefficients: Γ 1O () i = Γ 2O () i = 1, Γ 1L () i = Γ 2L () i = 0, Γ 1S () i = Γ 2S () i = 1, Γ T () i = 1. As a result, the following error correction terms are obtained: T [ 13, ]0 () i = Γ 1L()Γ i ( 1O() i Γ 1S() i ) (3-9) T [ 22, ]0 () i Γ Γ 1O() i Γ 1S() i = T() i (3-10) T [ 24, ]0 () i Γ T()Γ Γ 1O() i Γ 1S() i Γ 2O() i Γ 2S() i = i 2L() i (3-11) T [ 31, ]0 () i = 2Γ 1L() i Γ 1S() i Γ 1O() i (3-12) 71

100 Calibration of a Nonlinear Vectorial Network Analyser T [ 33, ]0 () i = Γ 1O() i Γ 1S() i (3-13) T [ 42, ]0 () i = Γ T() i Γ 1O() i Γ 1S() i ( 2 + ( Γ 2O() i Γ 2S() i )( Γ 2S() i Γ 2L() i )) Γ 2S() 2 i 1 (3-14) T [ 44, ]0 () i Γ Γ 1O() i Γ 1S() i Γ 2O() i Γ 2S() i = T() i (3-15) with l = 2Γ lo()γ i ls() i + Γ ll()γ i lo() i + Γ ls()γ i ll() i. The true waves at the ports of the DUT can be determined through equation (3-6) by using these correction terms and the measured waves. 3.3 How to obtain a better calibration accuracy? The previous simple example (Example 3.1) of an 8 error term two-port calibration shows that the formulae for the error correction terms are quite involved. The major disadvantage of this approach is that for each combination of error terms or measurement ports, the calculations must be done over again. In case of on-wafer measurements crosstalk is often present and a calibration with more than 8 error terms may be required. For differential measurements, two ports are needed for each device port and multiple port standards can be used. Hence, there is a need for a calibration method with more flexibility. The analytical method also makes a large number of assumptions that are only partially met in practice. The standards are assumed to be exactly known (Assumption 3.3), the measurement noise (Assumption 3.4) and even the reconnection uncertainty is assumed to be zero (Assumption 3.5). Departures from these assumptions will introduce systematic errors on all measurements. These are difficult to notice as they do not result in visible noise on the measurements. Noise can deteriorate the calibration significantly, even if the levels are small, 72

101 How to obtain a better calibration accuracy? whenever the numerical conditioning [15] of the calculation is high. This will effectively blow up the noise. However, higher accuracy can be obtained by building in redundancy in the calibration process. State-of-the-art are the self-calibration methods [13] (e.g. TRL), which allow the standards to be only partially exactly known. Only the well known properties of the standards which contain a minimal error and variance are used. A well known example is the characteristic impedance of the transmission line in a TRL calibration. By considering some standards to be only partially ideal and replacing these ideal figures by redundancy or other dependencies that have much lower error (e.g. the reciprocity of the transmission line), redundancy is built in and the calibration becomes more robust. However, the uncertainty on the standards and the measurements is still not taken into account and the nonrepeatability of the reconnection is still neglected. Using a stochastic framework allows to include even more prior knowledge and redundancy. Instead of using the absolute values of all standards, care is be taken to get maximum advantage from using constraints that are much less error prone, such as the time-invariance of one-port standards or the reciprocity of two-port standards. Hereby, the number of standards that has to be absolutely known is reduced to the bare minimum. In most cases, only one value will be required. Next, the uncertainty on the standards will be taken into account. Since the error bounds or the variances on the standards are specified by most manufacturers, this additional knowledge will be used to further reduce calibration errors. The uncertainty on the reconnection can be modelled as a stochastic process, which is user dependent. A user with golden hands or a high connection repeatability will have a much smaller variance on the reconnection than a clumsy user and will so be rewarded by a more accurate calibration with tighter error bounds. The variance or standard deviation on the reconnection must be determined once for every user from repeated connections and can then be used for every calibration performed by the same user. Since a NVNA has a much wider IF detection bandwidth than the VNA (1000 times), the measurement noise can no longer be neglected. It will be taken into account, assuming that it is normal distributed with known variance, by using the variance on the measurements. In practice a very good estimation of the variance or standard deviation on the measurements can 73

102 Calibration of a Nonlinear Vectorial Network Analyser be obtained by performing repeated measurements during each calibration for each measured standard. The stochastic calibration includes prior knowledge of all the noise sources that are present to diminish the bias of the calibration coefficients while minimizing their variability. Besides the measurement noise, the noise framework will be extended to include the uncertainties on the standards and the reconnections. This will increase the overall accuracy of the calibration and will allow to determine a proper uncertainty boundary on the calibration. 3.4 Stochastical based S-parameter calibration A stochastical based calibration is fundamentally different from a deterministic calibration. Instead of determining the error correction terms Ti () exactly, redundancy of measurement information is used to obtain an estimation of the error terms by a numerical optimization. The estimation takes the measurement uncertainty on the S -parameters, the uncertainty on the exact value of the calibration standards and the reconnection uncertainty of these standards into account. Constraints are imposed on the estimator to include the knowledge about the standards. The determination of the constraints depends on the actual calibration method and is not a self-evident exercise. However, it only has to be done once during the design of a calibration method. This altogether shows that the stochastical calibration method is a calibration framework more than a method. On one hand, the framework as a whole is useful to calibration designers as it allows to develop new types of calibration procedures. On the other hand, a particular implementation of one method is useful to the end-user. When the predefined calibration steps are followed, a better calibration, increased accuracy and error bounds on the calibration itself are obtained. The stochastical calibration method is based on reference [42]. In what follows, this technique is extended in several ways. First, a set of rules is given to determine the constraints associated with a general calibration setup. Next, the method is extended to multiple port calibrations. All the steps in the algorithm will be fully explained and commented to make the implementation more accessible to practitioners. Particular types of calibration procedures can then be developed using this framework. 74

103 Stochastical based S-parameter calibration Building the calibration equations From Equation (3-2) it is known that for an between the exact and measured waves is: n p -port measurement instrument the relation β() i α() i = Ki ()Ti bi () ; Ti () C 2n p 2n p and Ki () C (3-16) ai () Since an S-parameter calibration is considered, one entry of the T-matrix is fixed to a non zero value. The complex coefficient Ki () may be fixed to 1. ( 2n p ) 2 1 entries of Ti () remain to be determined. For the ease of notation, the frequency dependency will be omitted in the further explanation of the method. However, one must remind that the calibration is to be repeated for every measured frequency. All necessary calculation steps between two equations are given, but are often put in appendices to make the text more readable. The calibration process consists of connecting n std calibration boxes to the network analyser under calibration (NAUC). A calibration box consists of a set of calibration standards, with one connection for each measurement port. Changing one of these standards, is described as connecting another calibration box to the NAUC. The goal of the calibration procedure consists of determining the matrix T out of the measured values of the S-parameters and a priori knowledge of the calibration standards. The calibration equations will first be derived assuming full deterministic settings: no measurement noise present, standards known without uncertainty and a perfectly repeatable calibration procedure. Assume that the k th calibration box with exact scattering matrix S k0 is connected to the NAUC, then b k0 a k0 = S k0 a I k0 np (3-17) where I np is the n p n p identity matrix and S k0 C n p n p. a k0 C n p 1 and b k0 C n p 1 represent respectively the true incident and reflected waves at the ports of the NAUC when the k th calibration box is connected. Substituting equation (3-17) into equation (3-16), with K 0 () i = 1 results in: 75

104 Calibration of a Nonlinear Vectorial Network Analyser β k0 α k0 = T 0 S k0 I np a k0 (3-18) with α k0 C n p 1 and β k0 C n p 1 respectively the noise free measured incident and reflected waves. Equation (3-18) can then be partitioned into S β k0 I np 0 k0 = np T 0 a k0 I np (3-19) S α k0 0 np I k0 = np T 0 a k0 I np (3-20) where 0 np is an n p n p matrix of zeros. Since the true incident waves a k0 are unknown, they must be eliminated from equations (3-19) and (3-20). Equation (3-20) can be written as S 1 a k0 0 np I k0 = np T 0 αk0 I np (3-21) Substituting equation (3-21) into equation (3-19), results then in S β k0 I np 0 k0 S 1 np T 0 0 I np I k0 = np T 0 αk0 np I np (3-22) On the other hand, the following relation is valid for the measured waves β k0 = S k0 α k0 (3-23) 76

105 Stochastical based S-parameter calibration where S k0 is the noise free measured scattering matrix when a calibration box with exact scattering matrix S k0 is connected to the NAUC. The measured scattering matrix S k0 is obtained from the equations in appendix 3.A. Hence, since equation (3-22) and (3-23) express the same relation for all measured incident waves α k0, one obtains S S k0 I np 0 k0 S 1 np T 0 0 I np I k0 = np T 0 np I np (3-24) Or after some mathematical operations, worked out in Appendix 3.B (1.), I np S k0 T 0 S k0 = 0 ; k = 1 n (3-25) I std np Since the goal here is to calculate the elements T [ ij, ]0 of T 0, this linear set of equations in T 0 will be transformed to the more suitable form AT 0 = 0. Thereto, the vec-operator (Appendix 3.C), which transforms a matrix into a vector and formula vec{ XYZ} = ( Z T X)vec{ Y} (Appendix 3.C) with the Kronecker matrix product (Appendix 3.C) is applied to equation (3-25): vec I np S k0 T 0 S k0 I np T S k0 = I I np S k0 vec{ T 0 } = 0 np (3-26) S T I I k0 np np S k0 tv0 = 0 (3-27) ( where t v0 vec{ T 0 } C 2n p) 2 1 =. Given the true scattering parameters S k0 and the 2 measured quantities S k0, equation (3-27) yields n p linear, complex, homogeneous calibration equations in t v0. Note that the true scattering parameters S k0 herein are still unknown. 77

106 Calibration of a Nonlinear Vectorial Network Analyser Taking measurement noise into account In general, it is impossible to fulfil equation (3-27) with the same vector t v0 for all connected calibration boxes k because the measured S-parameters S k are corrupted by measurement noise S k = S k0 + n. S The noise on the measured S-parameters S k is dealt with by using an estimation procedure. To estimate the unknown model parameters t v, a Weighted Least Squares estimator (WLS) will be used which minimizes the error between the model and the measurements: ε( S k0, t v ) = S T I I k0 S np np k tv C n p 2 1 (3-28) Note that the error vector equations are linear in the parameters minimize hence becomes for k = 1 n std [34]: t v. The cost function to VS ( k0, t v0, t v ) = n std 1 -- ε H 1 S 2 ( k0, t v )C εk ( S k0, t v0 )ε( S k0, t v ) k = 1 (3-29) The superscript H represents the Hermitian transpose. The weighting matrix C εk ( S k0, t v0 ) is the covariance matrix of the residuals of equation (3-27). Herein S k is the only stochastic quantity. These residuals are calculated in the true parameters t v0 and true scattering parameters S k0. Hence C εk ( S k0, t v0 ) = Cov S T I I k0 S np np k tv0 (3-30) Since the measured S-parameters S k are the only stochastic contributions to the weighting matrix C εk ( S k0, t v0 ), it is more convenient to separate these stochastic contributions from the deterministic contributions. After performing the transformations of Appendix 3.B (2.), one obtains 78

107 Stochastical based S-parameter calibration C εk ( S k0, t v0 ) = S T k0 I np T 0 T 0 n p I np I np Cov( vec{ S k} ) S T k0 I np T 0 T 0 n p I np H I np (3-31) For the ease of notation, a few shorthands will be introduced. 0 Let D 1 k ( S k0, t v0 ) S T k0 I np TT n = p 0 and consider W k C n p 2 n2 p to be the square root of I np the inverse covariance matrix Cov 1 ( vec{ S k} ): W H k W k = Cov 1 ( vec{ S k} ). After taking the inverse of equation (3-31), one obtains: C 1 εk ( S k0, t v0 ) = ( D k ( S k0, t v0 ) I np ) H W H k W k ( D k ( S k0, t v0 ) I np ) (3-32) By using equation (3-32) the WLS cost function becomes VS ( k0, t v0, t ) v = n std 1 2 v ST k0 I np I S H np k ( D k ( S k0, t v0 ) I np ) H W H k k = 1 W k ( D k ( S k0, t v0 ) I np ) S T I I k0 S np np k tv (3-33) A shorthand notation is then introduced as L k ( S k0, t v0 ) = W k ( D k ( S k0, t v0 ) I np ) S T I I k0 S np np k (3-34) By applying ( X Y) ( P Q) = XP YQ (Appendix 3.C (4.)) with X = D k ( S k0, t v0 ), Y = I np, P = S T k0 I np, Q = I np S k the double Kronecker matrix product of equation (3-34) can be simplified to: L k ( S k0, t v0 ) = W k D k ( S k0, t v0 )S T k0 S k D k ( S k0, t v0 ) I np (3-35) 79

108 Calibration of a Nonlinear Vectorial Network Analyser Taking equation (3-35) into account, the cost function (3-33) can be simplified to VS ( k0, t v0, t v ) = n std 1 -- t 2 H v L H k ( S k0, t v0 )L k ( S k0, t v0 )t v k = 1 (3-36) Or equivalently, after introduction of the 2-norm of a vector: VS ( k0, t v0, t v ) = LS ( 2 k0, t v0 )t v 2 (3-37) where LS ( k0, t v0 ) L 1 ( S k0, t v0 ) L 2 ( S k0, t v0 ) = C 2n p n std ( 2n p ) 2 L nstd ( S k0, t v0 ) (3-38) Since this is an S-parameter calibration, one entry of T NL (3-1) is fixed in this minimization problem. This also avoids to obtain the trivial solution tv = 0. The bias resulting ( 2n p ) 2 1 from noise and nonrepeatability is affected by this choice [12]. To prevent ill-conditioned matrices, it is best to fix an entry on the main diagonal of T NL, especially when dealing with a nearly perfect NAUC (a NAUC for which the measured and the true waves are nearly the same) where the off-diagonal elements are close to zero. The calculations below will be carried out assuming that T NL[ 1, 1] = t v1 [ ] = 1, which eases the computations. Since the first entry of t v is fixed, it will be computational much easier to determine the other parameters by partitioning tv and LSk0 (, tv0 ) as t v = 1 with v C n v 1 (3-39) v and LS ( k0, t v0 ) = ys ( k0 ) FS ( k0, v 0 ) ; ys ( k0 ) C 2n p n std 1 ; FS ( k0, v 0 ) C 2n p n std n v (3-40) 80

109 Stochastical based S-parameter calibration where n v = ( 2n p ) 2 1 is the number of error correction terms to determine. By this choice, one is able to write the estimates of the parameters as a simple function of ys ( k0 ). Taking equations (3-39) and (3-40) into account the cost function (3-36) becomes VS ( k0, v 0, v) 1 2 = -- FS ( 2 k0, v 0 )v y( S k0 ) 2 (3-41) This cost function must be minimized over v to find the estimates vˆ. However, one should notice that in practice the true parameter values v 0 and S k0 are still unknown Minimizing the cost function The problem analysed here is linear in the parameters v and thus the cost function (3-41) is quadratic in v. To find the estimates vˆ ( S k0, v 0 ) of the model parameters v which minimize the cost function (3-41) vˆ ( S k0, v 0 ) = argminvs ( k0, v 0, v), the equations to be solved are: v VS ( k0, v 0, v) = 0 (3-42) v By using a Weighted Least Squares estimator this results in the estimates vˆ ( S k0, v 0 ) of the parameters v [14] as: vˆ ( S k0, v 0 ) = ( F H ( S k0, v 0 )FS ( k0, v 0 )) 1 F H ( S k0, v 0 )ys ( k0 ) (3-43) Note that here FS ( k0, v 0 ) is non-square. Computing the matrix product F H ( S k0 )FS ( k0 ) (which indeed results in a square matrix) before taking the inverse is numerically inefficient as the number of required significant digits is hereby doubled. A standard solution is to replace the normal equations (3-43) by the following overdetermined system of equations [14]: FS ( k0, v 0 )vˆ ( S k0, v 0 ) = ys ( k0 ) (3-44) This can be computed by using the QR-decomposition of FS ( k0, v 0 ) [15], 81

110 Calibration of a Nonlinear Vectorial Network Analyser FS ( k0, v 0 ) = QS ( k0, v 0 ) RS ( k0, v 0 ) 0( 2np n std n v ) n v (3-45) = Q 1 ( S k0, v 0 ) Q 2 ( S k0, v 0 ) RS ( k0, v 0 ) 0( 2np n std n v ) n v (3-46) = Q 1 ( S k0, v 0 )RS ( k0, v 0 ) (3-47) QS ( k0, v 0 ) is a unitary matrix ( QS ( k0, v 0 ) H = QS ( k0, v 0 ) 1 ), partitioned into Q 1 ( S k0, v 0 ) C 2n p n std n v that spans the regular space and Q 2 ( S k0, v 0 ) C 2n p n std ( 2n p n std n v ) that spans the null space of FS ( k0, v 0 ). RS ( k0, v 0 ) C n v n v is a regular upper triangular matrix. By substituting equation (3-47) into equation (3-44) one obtains the estimated parameters vˆ ( S k0, v 0 ) = R 1 ( S k0, v 0 )Q H 1 ( S k0, v 0 )ys ( k0 ) (3-48) Taking the nonrepeatability into account Note that to obtain the estimates vˆ ( S k0, v 0 ) (3-48), the S-parameters S k must be known exactly. This, however, is not possible due to measurement noise and poor connection repeatability. Or in other words, connecting the same calibration box twice will result in two slightly different S -parameter matrices, depending on the connector repeatability of the user. This nonrepeatability is modelled as a stochastic process. Consider the connection uncertainty on a standard to be Gaussian distributed. Two realisations from this distribution will result in two different outcomes, but will have the same mean or expected value [38]. How can one include this additional a priori knowledge about standards that are not or partially known in the calibration process? To illustrate this, consider a simple example. Consider the reflection coefficient of a short, connected to port 1 in calibration box k and to port 2 in another calibration box l. Since the standard has to be reconnected to go from calibration box k to l, two different realisations of the same stochastic process with the same distribution appear. The 82

111 Stochastical based S-parameter calibration exact value of the standards is hence unknown due to the perturbation of the reconnection. To state this formally, two different parameters θ si are introduced in the parametrisation of the calibration box instead of a single one to model the short: S k ( θ s ) θ s1 0 = and S l ( θ s ) = θ s2 The parameter θ s1 can be modelled as θ s1 = θ sshort + n θs1 with θ sshort the true and unknown S-parameter value of the short. n θs1 is the noise contribution due to the reconnection uncertainty with the following properties: E { nθs1 } = 0 and E { n θs1 n * θs1 } = σ2 s. Similar: θ s2 = θ sshort + n θs2 where n θs2 has the same properties as n θs1. Out of these models for θ s1 and θ s2, one can conclude that θ s1 θ s2 = n θs1 n θs2. The mean and variance of this noise contribution are known: E { n θs1 n θs2 } = E { n θs1 } E { n θs2 } = 0 and E { ( n θs1 n θs2 )( n θs1 n θs2 ) * } = 2σ2 s. The contribution of these noisy values θ s1 and θ s2 to the cost function results in an additional error equation ( θ s1 θ s2 ) 2σ s weighted according to its uncertainty. A similar reasoning can be done for each calibration box used during calibration. Taking the uncertainty on the S -parameters of the calibration standards into account by extending the cost function with additional error equations, increases the overall accuracy of the calibration. A few guidelines to introduce additional information on the calibration standards are: If all the S -parameters of the calibration standard are exactly known, no additional θ s - parameters nor additional constraints must be introduced. The exact value of the S - parameters must be filled in into S k ( θ s ). If the exact S -parameters of the standard θ si0 are known up to an uncertainty, one wants to express that the expected value of the calibration standard is known. This requires the introduction of an additional θ s -parameter and a constraint to the cost function: ( θ si θ si0 ) σ. 83

112 Calibration of a Nonlinear Vectorial Network Analyser If the same calibration standard is used in more than one calibration box and is not reconnected, then only 1 θ s -parameter is introduced for all the boxes, since it considers here only 1 realisation of the stochastic process. If the standard is reconnected, one additional θ s -parameter must be introduced for each reconnection. A constraint is then imposed to express the equality of the expected values of the different realisations taken from the same stochastic process. In order to determine the estimates of the error correction terms t v and the unknown standard parameters θ s the following rule must be valid: the total number of calibration measurements must be larger or equal to the difference between the total number of unknown parameters and additional constraints #( θ si ) + #( t v ) #(constraints) #(S-parameter measurements) Applying those guidelines to all the calibration boxes of a calibration process results in an additional set of n eq linear complex error equations among the θ s -parameters, which is here denoted by: Eθ s r (3-49) with E the matrix of constraint equations and r the known values. These additional error equations are added to the cost function (3-41). Example 3.2 Let s work out a simple calibration method such as a SOLT type calibration as an example to clarify the method of adding constraints. Consider a two-port NAUC. Seven calibration standards will be used during calibration: an open M/F, a short M/F, a load M/F and a thru. The superscript M/F denotes the type of connector respectively as Male and Female. Suppose the following calibration boxes are used ( n ): thru, load M -short F, open M -short F, open M std = 7 - load F, short M -load F, short M -open F and load M -open F. As a result a total of 16 calibration measurements must be performed. The impedance standard is obtained by assuming that the S -parameters of the load are known, up to the measurement uncertainty. Taking the order in 84

113 Stochastical based S-parameter calibration which the calibration manipulations are performed into account, one obtains the following parametrisation: Thru: the thru is assumed to be perfectly symmetric and perfectly matched up to the measurement errors: S 1 ( θ s ) = θ s1 θ s3 θ s3 θ s2 The estimated parameter θ s1 can be modelled as: θ s1 = θ srefl + n θs1, where θ srefl is the exact value of the S-parameter and n θs1 a noise contribution due to reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs1 } = 0 and E { n θs1 n * θs1 } = σ2 r. Due to the assumption of a perfectly matched thru, the value of θ srefl is 0. The contribution of the noisy value θ s1 to the cost function hence becomes: ( θ s1 0) σ r. Using the same reasoning, the contribution of θ s2 is: ( θ s2 0) σ r. Reconnection uncertainty is herein assumed independent of the connector type (M/F). The model for θ s3 is θ s3 = θ strans + n θs3. Due to the assumption of a perfect thru, the exact value θ strans is 1 and the contribution of θ s3 hence is: ( θ s3 1) σ t with σ2 t = E { n θs3 n * θs3 }. load M -short F : no crosstalk is assumed between the measurement ports, so that S 212 [, ] = 0 and S 221 [, ] = 0. S 2 ( θ s ) = θ s4 0 0 θ s5 The estimated parameter θ s4 can be modelled as: θ s4 = θ sload + n θs4, where θ sload is the exact value of the reflection coefficient of the load and the noise contribution due to reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs4 } = 0 and E { n θs4 n * θs4 } = σ2 l. However, the exact reflection coefficient of the load θ sload is known, up to the measurement noise: θ sload = Γ load + n θsload. The mean and variance of the measurement noise are assumed to be known: n θs4 85

114 Calibration of a Nonlinear Vectorial Network Analyser E { n θsload } = 0 and E { n θsload n* θsload } = σ2 a. To take the influence of both measurement uncertainty and reconnection uncertainty into account, two additional contributions to the cost function are hence introduced: ( θ sload Γ load ) σ a and ( θ s4 θ sload ) σ l. The estimated parameter θ s5 can be modelled as: θ s5 = θ sshort + n θs5, where θ sshort is the exact value of the S-parameter and n θs5 a noise contribution due to the reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs5 } = 0 and E { n θs5 n * θs5 } = σ2 s. However, the exact value of the reflection coefficient of the short θ sshort is unknown. So that at this point no extra contributions to the cost function are introduced. open M -short F : no crosstalk is assumed between the measurement ports. The short is not reconnected since the previous calibration box, so that no additional parameter nor contribution is needed. S 3 ( θ s ) = θ s6 0 0 θ s5 The estimated parameter θ s6 can be modelled as: θ s6 = θ sopen + n θs6, where θ sopen is the exact value of the S-parameter and n θs6 a noise contribution due to the reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs6 } = 0 and E { n θs6 n* θs56 } = σ2 o. Again, the exact value of the reflection coefficient of the open θ sopen is unknown. As a result no extra contributions to the cost function are introduced. open M -load F : no crosstalk is assumed between the measurement ports. The open is not reconnected since the previous calibration box, so that no additional parameter nor contribution is needed. S 4 ( θ s ) = θ s6 0 0 θ s7 86

115 Stochastical based S-parameter calibration Similar to parameter θ s4, the model for θ s7 can be written as: θ s7 = θ sload + n θs7, where θ sload is the exact value of the reflection coefficient of the load and n θs7 the noise contribution due to the reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs7 } = 0 and E { n θs7 n * θs7 } = σ2 l. Note that the reconnection uncertainty is herein assumed independent of the connector type (M/F). The contribution of the noisy value ( θ s7 θ sload ) σ l. θ s7 to the cost function hence becomes: short M -load F : no crosstalk is assumed between the measurement ports. The load is not reconnected since the previous calibration box, so that no additional parameter nor contribution is needed. The short however is reconnected compared to previous used calibration boxes ( S 2 ( θ s ), S 3 ( θ s )) and requires an additional parameter θ s8. S 5 ( θ s ) = θ s8 0 0 θ s7 Similar to parameter θ s5, the model for θ s8 can be written as: θ s8 = θ sshort + n θs8, where θ sshort is the exact but unknown value of the reflection coefficient of the short and n θs8 the noise contribution due to the reconnection uncertainty. The mean and variance of the noise contribution are assumed to be known: E { n θs8 } = 0 and E { n θs8 n * θs8 } = σ2 s. Note that the reconnection uncertainty is herein assumed independent of the connector type (M/F). Out of the model for θ s5 and θ s8 follows that: θ s5 θ s8 = n θs5 n θs8. The mean and variance of this noise contribution are known: E { n θs5 n θs8 } = E { n θs5 } E { n θs8 } = 0 and E { ( n θs5 n θs8 )( n θs5 n θs8 ) * } = 2σ2 s. The contribution of the noisy values θ s5 and θ s8 to the cost function now becomes: ( θ s5 θ s8 ) 2σ s. short M -open F : no crosstalk is assumed between the measurement ports. The short is not reconnected since the previous calibration box, so that no additional parameter nor contribution is needed. The open is reconnected compared to previous used calibration boxes ( S 3 ( θ s ), S 4 ( θ s )) and requires an additional parameter θ s9. S 6 ( θ s ) = θ s8 0 0 θ s9 87

116 Calibration of a Nonlinear Vectorial Network Analyser Using the same reasoning as for the short, the contribution of the noisy values θ s6 and θ s9 to the cost function is: ( θ s6 θ s9 ) 2σ o. load M -open F : no crosstalk is assumed between the measurement ports. The open is not reconnected since the previous calibration box, so that no additional parameter nor contribution is needed. The load is reconnected compared to previous used calibration boxes ( S 2 ( θ s ), S 4 ( θ s ), S 5 ( θ s )) and requires an additional parameter θ s10. S 7 ( θ s ) = θ s θ s9 Using the same reasoning as for the parameter θ s7, the contribution of the noisy value θ s10 to the cost function becomes: ( θ s10 θ sload ) σ l. These additional contributions to the cost function can be written as Eθ s r with E = σ 1 r σ 1 r σ 1 t σ 1 l σ 1 l ( 2σ s ) ( 2σ s ) ( 2σ o ) ( 2σ o ) σ 1 l σ 1 l σ 1 l 1 l σ 1 a ; 88

117 Stochastical based S-parameter calibration θ s1 θ s2 θ s3 θ s4 0 0 σ 1 t θ s θ s5 = θ s6 and r θ s7 θ s8 θ s9 = Γ load σ a θ s10 θ sload The number of calibration boxes n std and the constraints on the calibration elements must be selected by the calibration developer. The set of additional constraints is fixed for a certain type of calibration. A calibration user only has to use this fixed matrix to complement the set of equations to get the calibration coefficients. Since the exact values S k0 are unknown, they are replaced by the parameters θ s to be estimated. As a result the cost function (3-41) becomes a function of θ s. Furthermore, the additional contributions Eθ s r are taken into account: V( θ s, v 0, v) = 1 -- F( θ 2 s, v 0 )v y( θ s ) Eθ 2 2 s r 2 2 (3-50) Substituting the results for the parameters vˆ ( θ s, v 0 ) (3-48) into this cost function gives: V( θ s, v 0, vˆ ( θ s, v 0 )) = 1 -- F( θ 2 s, v 0 )R 1 ( θ s, v 0 )Q H 1 ( θ s, v 0 )y( θ s ) y( θ s ) Eθ 2 s r 2 (3-51) Note that cost function (3-51) still depends on the true parameter values v 0. As a result these parameters must be known to minimize the cost function. In practice, however, these values are unknown. Since the v parameters have a physical meaning, they will hence be replaced by 89

118 Calibration of a Nonlinear Vectorial Network Analyser their ideal values. In all subsequent steps, the estimated parameters v obtained in the iteration step n are then used to calculate the weighting matrix in iteration step n+ 1. This technique is known as the Iterative Quadratic Maximum Likelihood (IQML) principle [31], [8]. Cost function (3-51) hence becomes independent of the true parameter values v 0 and can be simplified as shown in Appendix 3.B (3.) yielding: V( θ s, vˆ ( θ s )) = -- 2 Q H + -- Eθ 2 ( θ s )y( θ s ) 2 s r = εθ ( 2 s ) 2 (3-52) Q H with εθ ( s ) 2 ( θ s )y( θ s ) =. Eθ s r This cost function is non quadratic in the parameters θ s and can be minimized using the Gauss-Newton technique [14]. By iteration on θ s, it will converge to a local minimum of the ( n) cost function. In the Gauss-Newton method, a guess θˆ s is improved to ( n + 1) ( n) θˆ s = θˆ s + d ( n + 1), where d ( n + 1) is the solution of the normal equation [14]: d ( n + 1) ( ) H ( n) ( J T ( θˆ sn )J T ( θˆ s )) 1 ( n) ( n) = J T ( θˆ s )ε( θˆ s ) (3-53) with J T ( θˆ s ( n) ) = J( θˆ s ( n) ). E ( n) J( θˆ s ) is the Jacobian matrix of the first part of the cost function (Appendix 3.B (4.)) and E is the Jacobian matrix of the additional constraints: ( Eθ s r) θ sl = E. ( n) H ( n) Computing the matrix product J T ( θˆ s )J T ( θˆ s ) before taking the inverse is numerically inefficient as the number of required significant digits is hereby doubled. Instead, the following non-square overdetermined system of equations is considered: J( θˆ s ( n) ) d ( n) = εθˆ ( s ( n) ) E (3-54) Since (3-54) is an overdetermined set of linear equations, it is easily solved in the least squares sense by using the QR- decomposition [15] 90

119 Stochastical based S-parameter calibration J( θˆ s ( n) ) E = X 1 ( θˆ s ( n) ) X 2 ( θˆ s ( n) ) Y( θˆ s ( n) ) 0 ( 2np n std n v + n eq n θs ) n θs (3-55) with X a unitary matrix; spans the 1 ( θˆ s ( n) ) X 2 ( θˆ s ( n) ) X 1 θˆ s ( n) ( 2n p n std n v + n eq ) n θs ( ) C regular space and X 2 θˆ s ( n) ( 2n p n std n v + n eq ) ( 2n p n std n v + n eq n θs ) ( ) C spans the null space of J( θˆ s ( n) ). Y θˆ s ( n) n θs n θs ( ) C is an upper triangular regular matrix. The least squares E solution to (3-54) is then found by solving the upper-triangular set Y( θˆ s ( n) )d ( n + 1) X H 1 ( θˆ s ( n) ) Q 2 H ( θˆ s ( n) )y( θˆ s ( n) ) = Eθˆ s ( n) r (3-56) ( n + 1) The estimate for v or t v is then found from θˆ s using (3-48) and (3-39). The iteration over n is stopped when the 2-norm of d ( n + 1) becomes smaller than the numerical precision (i.e. d ( n + 1) 2 2 < 10 8 ). However, if this threshold is not reached within 10 iteration steps, the iteration is stopped. Note that the maximum number of iterations is chosen arbitrary. By comparing the obtained value of the cost function to the expected value, one can decide whether the estimated parameters are acceptable or not. The quality of a calibration can be verified by performing some additional measurements on a well-known DUT. If the measured characteristic well matches the expected characteristic of the DUT, the calibration is considered to be of good quality. Instead of performing additional measurements, the quality of other calibration process can then be determined by comparing the minimal value of its cost function to the cost function of a good quality reference calibration. Hence, the additional measurements to obtain the reference calibration only needs to be done once. When doing this verification after each calibration, one can easily detect whether there is a problem with the measurement instrument or when something went wrong during calibration, like connecting the wrong calibration elements. 91

120 Calibration of a Nonlinear Vectorial Network Analyser Accuracy bounds After estimating the parameters t v, one wants to know how good the estimated values approximate the true values of the parameters. The variability of the estimated parameters can be obtained by determining the covariance of the estimated parameters. Consider that the estimated parameters can be written as an explicit function of the noisy measurements. The perturbation analysis is then based on a linearisation of the explicit function by taking the derivative of this explicit function to all noisy parameters. From this linearisation the covariance of the estimated parameters can then easily be obtained. However, in case of the cost function used during the calibration process (3-49): V( θ s, v) = 1 -- F( θ 2 s )v y( θ s ) Eθ 2 2 s r 2 2 (3-57) one can not write the parameters as an explicit function of the noisy measurements, but only as an implicit function V( θ s, v). For the ease of notation, the error correction terms v and the S-parameters θ s are stacked in T the parameter vector Θ = v θ s. The estimation of the parameters Θ is based on noisy measurements of the S -parameters. The noise on the measurements is assumed to be zero mean, circular complex normal distributed. Let n be a vector containing the noise and nonrepeatability outcomes of a particular experiment. The cost function (3-57) can then be rewritten as V( Θ, n) = 1 --g 2 H ( Θ, n)g( Θ, n) (3-58) F( θ with g( Θ, n) s )v y( θ s ) = (3-59) Eθ s r Remember here that g( Θ, n) is a function of the measured S-parameters. The vector g( Θ, n) is, hence, perturbed by the noise through the contributions of S k = S k0 + n Sk in F( θ s ) and y( θ s ). 92

121 Stochastical based S-parameter calibration Define Θ 0 as the noise free parameters, such that the vector g( Θ 0, 0) is zero. As a result, the noise free cost function V( Θ 0, 0) is zero in the parameters Θ 0. Let Θˆ = Θ 0 + n Θ be the estimated parameters which minimize the cost function V( Θ, n) in the presence of noise: V( Θ, n) Θ Θˆ = g H ( Θ, n) g( Θ, n) Θ Θˆ (3-60) Since g( Θ, n) is an analytical function of both Θ and n; and g( Θ 0, 0) = 0, one can now linearise equation (3-60) by writing the first order truncated Taylor series 1 (Appendix 3.D) of (3-60): V( Θ, n) V( Θ, n) g H ( Θ, n) g( Θ, n) ( Θˆ Θ Θ Θ Θ Θ 0 ) + Θˆ Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 n = 0 g H ( Θ, n) g( Θ, n) n 0 (3-61) Θ n Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 The first term of (3-61) minimal in Θ 0. V( Θ, n) Θ Θ = Θ 0 n = 0 is zero since the noise free cost function must be Hence, Θˆ g Θ H ( Θ, n) g( Θ, n) g H ( Θ, n) g( Θ, n) n Θ Θ Θ n Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 n = 0 n = 0 (3-62) Since for spectral data, the noise n is zero mean circular complex normal distributed [35], this first order analysis shows that there is no bias on the estimates: E { Θˆ Θ 0 } = It may well be that the higher order moments of a function θˆ do not exist. However, for sufficiently high SNR this function can be approximated by a function θ which is close to θˆ and for which the higher order moments do indeed exist. By performing the noise analysis of θ, one can get a good idea of the accuracy of θˆ. Furthermore, when the SNR of the measurements is high enough, the higher order terms in the noise contributions may be neglected. 93

122 Calibration of a Nonlinear Vectorial Network Analyser The covariance matrix of the estimated parameters (3-62) is given by: g Cov( Θˆ Θ 0 ) H ( Θ, n) g( Θ, n) g H ( Θ, n) Θ Θ Θ Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 n = 0 g( Θ, n) Cov n g( Θ, n) g H ( Θ, n) g( Θ, n) n Θ Θ Θ Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 n = 0 n = 0 (3-63) Note that in (3-35) L k ( θ s, t v ) = W k D k ( θ s, t v )S T ( θ, the scaling k s ) D k ( θ s, t v ) I np S k with W k and D k ( θ s, t v ) was introduced to make the covariance of the upper 4n std entries of g( Θ, n) (=[ F( θ s )v y( θ s )]) equal to the unit matrix. The remaining n eq entries (=[ Eθ s r] ) are assumed to have uncorrelated, unit variance residuals that are uncorrelated with the 4n std upper entries. This can be achieved by proper scaling the additional constraints Eθ s r. Hence, by construction g( Θ, n) Cov n n Θ = Θ 0 n = 0 = I( 4nstd + n eq ) (3-64) So that covariance Cov( Θˆ Θ 0 ) becomes g Cov( Θˆ Θ 0 ) H ( Θ, n) g( Θ, n) Θ Θ Θ = Θ 0 Θ = Θ 0 n = 0 n = 0 (3-65) In practice, the noise free parameters Θ 0 are unknown, so that the covariance (3-65) must be evaluated at the estimates using the noisy data: g Cov( Θˆ ) H ( Θ, n) g( Θ, n) Θ Θ Θ = Θˆ Θ = Θˆ (3-66) Since the calibration procedure consists of determining the error correction terms v, one is particularly interested in the variability of the estimated parameters vˆ or, in other words in 94

123 Stochastical based S-parameter calibration Cov( vˆ ), which is a submatrix of (3-66). The covariance Cov( vˆ ) can be found as calculated in Appendix 3.B (5.): Cov( vˆ ) = R 1 ( θ s )( I + J K ( θ s )J H K ( θ s ))R H ( θ s ) (3-67) with J K ( θ s ) = Q H 1 ( θ s )J( θ s )Y 1 ( θ s ), Y( θ s ) is the upper triangular matrix of the QRdecomposition of the Jacobian J( θ s ) E H and Q 1 ( θ s ) spans the signal space of F( θ s ) The algorithm implemented The implementation of the calibration can be visualised by the flow chart in Figure 3-2. Starting values Building the cost function V( θˆ s ( n 1), tˆv ( n 1) ) Eliminate + solve for ( ) tˆvn n = n + 1 Solve the cost function for θˆ s ( n) Yes d ( n) 2 2 > 10 8 No Solution θˆ s ( n) ( ) tˆvn FIGURE 3-2. Flow chart representing the calibration program 95

124 Calibration of a Nonlinear Vectorial Network Analyser As an example the complete algorithm to design a two-port, 16-error term calibration is worked out. S -parameter 1. Starting values Set the iteration counter n = 0 and let θˆ s ( 0) and tˆv ( 0) be the initial guesses of respectively the parameters that describe the calibration boxes and the error correction terms. Since these parameters have a physical meaning, their initial guesses can be replaced by the ideal values of the parameters. 2. The cost function Define the matrices M 1k ( θˆ s ( n), tˆv ( n) ), M 2k ( θˆ s ( n), tˆv ( n) ) C 4 4 as M 1k ( θˆ s ( n), tˆv ( n) ) M 2k ( θˆ s ( n), tˆv ( n) ) = W k D k ( θˆ s ( n), tˆv ( n) ) I 2 S k (3-68) with k = 1 n std. W k can be found from W H k W k = Cov 1 ( vec{ S k} ). The matrix D k ( θˆ s ( n), tˆv ( n) ) can be found from D 1 k ( θˆ s ( n), tˆv ( n) ) S T, (3-69) k ( θˆ s ( n) ) I Tˆ ( n)t 0 2 = k = 1 n 2 std I 2 If n = 0, then D( 0) k = I 2. This can be obtained by making Tˆ ( 0) = I 4 and substituting in equation (3-69). Now form L k ( θˆ s ( n), tˆv ( n) ) = S k[ 11, ] ( θˆ s ( n) )M 1k + S k[ 12, ] ( θˆ s ( n) )M 2k S k[ 21, ] ( θˆ s ( n) )M 1k + S k[ 22, ] ( θˆ s ( n) )M 2k T (3-70) M 1k M 2k Stack the L k s on top of each other to find L as in (3-38). Partition L into y and F as in (3-40) ( n) L( θˆ s ) = ( n) ( n) y( θˆ. s ) F( θˆ s ) 96

125 Stochastical based S-parameter calibration 3. Eliminate the error correction terms t v Since the cost function is a function of 2 different types of parameters tv or v and θs, one of ( n) ( n) T these parameters will be eliminated. The parameters vˆ ( θ s ) ( tˆv = ( n) 1 vˆ ( θ ) are s ) eliminated and a solution is found by solving the upper triangular set of linear equations (3-48) vˆ ( θ s ) = R 1 ( θ s )Q H 1 ( θ s )y( θ s ). The matrices Q 1 ( θ s ) and R( θ s ) can be obtained by determining the QR-decomposition of ( n) R( θ F( θ s ): F( θ s ) Q 1 ( θ s ) Q 2 ( θ s ) s ) = (3-45). Find then the error correction terms 0 ( n + 1) ( n) tˆv from vˆ ( θˆ s ). 4. Solve the cost function for the S -parameters θ s Determine the Jacobian matrix The l the column of the Jacobian matrix (3-131) ( n) J kl [] ( θˆ s ) Q H ( n) 2 ( θˆ s ) L θˆ ( n) k( s ) ( n + 1) = is obtained by taking the derivative of θ tˆv L k ( θ s ) sl with respect to the l th entry of the parameter vector θ s : ( n) L k ( θˆ s ) θ sl = T ( n) ( n) S k[ 11, ] ( θˆ s ) S k[ 12, ] ( θˆ s ) M θ 1k M sl θ 2k sl ( n) ( n) S k[ 21, ] ( θˆ s ) S k[ 22, ] ( θˆ s ) M θ 1k M sl θ 2k sl (3-71) 0 0 T The error correction vector tˆv can be written as tˆv = with. So tˆ1 tˆ2 tˆ3 tˆ4 tˆm C 1 4 that one obtains for k = 1 n std 97

126 Calibration of a Nonlinear Vectorial Network Analyser ( n) L k ( θˆ s ) ( n + 1) j kl = θ tˆv = sl ( n) ( n) S k[ 11, ] ( θˆ s ) ( n + 1) S k[ 12, ] ( θˆ s ) ( n + 1) M θ 1k tˆ M sl θ 2k tˆ1 + sl ( n) ( n) S k[ 21, ] ( θˆ s ) ( n + 1) S k[ 22, ] ( θˆ s ) ( n + 1) M θ 1k tˆ M sl θ 2k tˆ2 sl (3-72) The complete Jacobian matrix can then be composed as J = QH 2 j. Determine the least squares solution Perform the QR-decomposition of the Jacobian matrix (3-55) ( n) ( n) J( θˆ s ) = ( n) ( n) Y( θˆ X and solve the upper triangular set of 1 ( θˆ s ) X 2 ( θˆ s ) s ) E 0 equations (3-56) to find d ( n + 1) d ( n + 1) Y 1 ( θˆ s ( n) )X H 1 ( θˆ s ( n) ) Q 2 H ( θˆ s ( n) )y( θˆ s ( n) ) = Eθˆ s ( n) r Add d ( n + 1) to θˆ s ( n) to obtain θˆ s ( n + 1). 5. Stopping criterion If d ( n + 1) 2 2 > 10 8, increment n and return to step 2. The final parameter estimate is θˆ s ( n + 1). The error correction matrix Tˆ ( n + 1) ( + 1) is found from tˆvn ( tˆv = vec{ Tˆ }). 6. Variability of the parameters t v Compute the estimate of variability for the error correction terms v as: Cov( vˆ ) = R 1 ( θˆ s) ( I + J K ( θˆ s)j H K ( θˆ s) )R H ( θˆ s) with J T ( θˆs ) = Q H 1 ( θˆs )J( θˆs )Y 1 ( θˆs ) Properties of the estimator When developing an estimator, one wants to have an idea of how good the estimator is. The quality of the estimator is reflected in its asymptotic properties. A good estimator satisfies the following properties (under the assumption that the measurement noise is circular complex normal distributed): 98

127 Stochastical based S-parameter calibration 1. Correct: when replacing the noisy measurements by the true noise free values, the exact parameters are obtained. 2. Consistency: if the number of measurements k goes to infinity, the difference between the estimated value and the true value becomes infinitely small or the estimates come closer to the exact values when the number of experiments rises. 3. Convergence Rate: the difference between the estimated and the true parameters converges to zero when the number of measurements k goes to infinity at a speed equal to k 12 /. 4. Asymptotic Normality: the distribution of the estimates is a normal distribution with a mean value equal to the true parameter value and a covariance matrix given by the inverse of the Fisher information matrix [34]. 5. Asymptotic Efficiency: the uncertainty on the estimates reaches its minimal value, which is the Cramér-Rao bound for an unbaised estimator [34]. Does the Weighted Nonlinear Least Squares (WNLS) estimator, used in the stochastical S - parameter calibration, satisfy all these properties? The WNLS estimates minimize the following cost function V( θ s, t v ) over both θ s and t v : V( θ s, t v ) = n std 1 -- t 2 H v H H k ( θ s )C 1 1 εk ( θ s, t v )H k ( θ s )t v + -- ( Eθ 2 s r) H ( Eθ s r) k = 1 (3-73) Since the weighting matrix C εk ( θ s, t v ) is a function of the parameters θ s and t v, the cost function (3-73) is not a quadratic function in both θ s and t v. In general, this WNLS estimator is then inconsistent and, moreover, moments of second order and higher will not exist. So that it becomes very difficult to discuss the properties of the estimator. Some simplifying assumptions must be made to be able to tell something about the properties of the estimator. A. The S-parameters θ s are exactly known In an attempt to simplify things, it will be considered first that 99

128 Calibration of a Nonlinear Vectorial Network Analyser Assumption 3.7 the weight C εk does not depend on the parameters θ s, but is calculated in the true unknown parameter values θ s0. Furthermore, assume that Assumption 3.8 the parameters θ s are perfectly known and normally distributed. The cost function (3-73) can then be rewritten as: Vt ( v ) = n std 1 -- t 2 H v H H k C 1 εk ( t v )H k t v k = 1 (3-74) and is quadratic in the measurements and the parameters t v. Since this cost function is 2 quadratic in the measurements, the first and second order moments and exist. The t v t v estimates which minimizes the cost function (3-74) can be found by using a Weighted Least Squares estimator or Markov estimator [35]. The Markov estimator only requires the knowledge of the noise covariance diagonal matrix C, which can be obtained from S k independent repeated measurements of the S-parameters of a calibration box k. The Markov estimator has some well-known properties [35]: 1. Consistency: the Markov estimator is strongly consistent for the parameters t v. This means that lim P[ tˆv t v0 > δ ] = 0, δ >0. Or in other words, if the number of k measurements k goes to infinity the difference between the estimated value and the true value becomes infinitely small or the estimates come closer to the exact values when the number of experiments rises. 100

129 Stochastical based S-parameter calibration Convergence Rate: tˆv converges in probability at the rate O P ( n std ) to t v0. This means that the difference tˆv t v0 converges to zero when the number of standards n std goes to infinity at a speed equal to n 12 / std. 3. Asymptotic Bias: since it considers here a minimization problem that is linear in the parameters, the truncated Markov estimator is asymptotically unbiased. This means that when the number of measurements k goes to infinity, the expected value of the estimates equals the true parameter value: lim E { tˆv} = t v0. It can be shown that the bias error k decreases as the square of the signal-to-noise ratio: SNR 2. This means that when the noise level is divided by 2, the bias will be divided by Asymptotic Normality: due to the presence of noise on the measurements, the estimates tˆv are stochastic variables. The distribution of these estimates tˆv is a normal distribution with a mean value equal to the true parameter value t v0 and a covariance matrix given by C tv = Fi 1. Fi is herein the Fisher information matrix [34]: Fi = E { ( lnv( t v ) t v )( lnvt ( v ) t v ) H }, where Vt ( v ) is the loglikelihood function. 5. Asymptotic Efficiency: since it considers here a Markov estimator of the type signal model [35], the estimator is asymptotic efficient ([35] Theorem 17.4). B. Uncertainty on the S-parameters θ s In practice, however, there is an uncertainty on the standards and the reconnection. This means that one can not presume to known the values of θ s exactly. Therefore the values of the S - parameters θ s also need to be estimated: V( θ s, t v ) n std 1 -- t 2 H v H H k ( θ s )C 1 1 = εk H k ( θ s )t v + -- ( Eθ 2 s r) H ( Eθ s r) k = 1 (3-75) The uncertainty on the standards is given by the manufacturer, while the reconnection uncertainty can be determined for every user through repeated reconnections. If the uncertainty on the standards is high, the estimator will be inconsistent. However, since calibration measurements are considered here, and these measurements have to be of high quality with a large SNR for the calibration to be reliable at all, the uncertainty on the standards will be small 101

130 Calibration of a Nonlinear Vectorial Network Analyser in practice. The bias on the estimates, which is inversely proportional to the square of the SNR, will hence also be very small Why using a stochastic calibration? Why should one do the effort of implementing a stochastic based calibration process when the classical S -parameter calibration gives satisfying results? In this paragraph the advantages and disadvantages of both methods are compared. A. Stochastic S-parameter calibration During a stochastic calibration process, the error correction terms are determined by a Weighted Nonlinear Least Squares estimator. By a good selection of the standards, the prior knowledge regarding the standards can be reduced to a single quantitatively known impedance reference and a qualitative law for the other characteristics. The information concerning the characteristics of the standards is inserted in the estimator through constraints. Determining these additional constraints is a hard task for the calibration developer. The user, however, does not need to worry about it and simply has to apply the algorithm as developed in the design stage. Furthermore, using a stochastic calibration framework allows to take all the uncertainty sources of the calibration process into account: the uncertainty on the calibration standards, the uncertainty on the measurements and the reconnection uncertainty of the standards. All these uncertainties can be experimentally determined using sample variances or are supplier specified: the measurements uncertainty is obtained from repeated measurements, the reconnection uncertainty is determined from one set of repeated connections for each user and the uncertainty on the standards is delivered by the supplier. As a consequence, one of the major advantages of a stochastic S -parameter calibration is that a quality control is built in. Through the knowledge of the noise sources, the expected value of the cost function associated with a particular calibration matrix can be predicted. By inspecting the value of the cost function of the calibration estimates, one can verify whether the right calibration standards were connected or not. This avoids monday morning calibrations. 102

131 Comparing real-world performance of stochastic and deterministic S-parameter calibrations Furthermore, by keeping track of the cost function in time, degradation of the quality of the measurement instrument can be detected in an early phase. In general, the stochastic S-parameter calibration will take a longer time than a classical calibration, but will be more accurate and less error prone. The extra effort that the calibration designer needs to deliver, is rewarded by an accurate and user-friendly calibration process. B. Classical S-parameter calibration During a classical calibration process, the error correction terms are determined by solving the calibration equations analytically. All the standards are assumed to be perfectly known, which does not allow to take any uncertainty on the standards into account. Furthermore, the measurement uncertainties are neglected. The quality control is absent and human errors are not detected. However, a classical calibration method takes less time then the stochastical process and is easier to design. 3.5 Comparing real-world performance of stochastic and deterministic S-parameter calibrations In this paragraph, the robustness to measurement and reconnection noise of the deterministic and stochastic S -parameter calibration is tested and compared. It is investigated which method will win the race to accuracy and if it is indeed necessary to use stochastic frameworks when calibrating a Nonlinear Vectorial Network Analyser (NVNA). The comparison is performed on the NVNA (NNMS-HP85120A-K60). By performing repeated calibrations the statistical properties of both methods are determined Determining the statistical properties The error correction matrix Ti ()(see equation (3-2)): β() i α() i = Ki ()Ti bi () ai () (3-76) 103

132 Calibration of a Nonlinear Vectorial Network Analyser is determined twice: once by a deterministic and once by a stochastic calibration. To ease the notations, the frequency dependency i will not be further written. For both calibration methods, the measurement of each calibration box is repeated N times, i.e. N error correction matrices are calculated from the N repeated measurements of each calibration box and this for the same settings of the measurement instrument. Out of these N error correction matrices the statistical properties of the calibration are determined. The sample mean N 1 [ m] T [ k, l] = --- T with N [ kl, ] kl, = 1 2n p (3-77) m = 1 and the sample standard deviation N [ m] 2 T s [ k, l] T [ kl, ] 2 [ kl, ] = with kl, N 1 = 1 2n p (3-78) m = 1 of each element T [ k, l] of the error correction matrices T over the realisations m = 1 N is taken to asses stochastic performance Comparing the deterministic and stochastic calibration First, the NVNA is calibrated by a classic 8-error term two-port correction. In a next step, the NVNA is calibrated by an 8-error term two-port stochastic based method. For the stochastical calibration, the a priori knowledge regarding the value of the standards is reduced to the knowledge of the load. It is set to a fixed reflection coefficient of Both methods use a SOLT calibration and are repeated 100 times for the same instrument settings. The sample mean and standard deviation of the error correction matrices are calculated as in (3-77) and (3-78). For an 8-error term calibration the error correction terms T [ 12, ], T [ 14, ], T [ 21, ], T, [ 23, ] T, [ 32, ] T, [ 34, ] T [ 41, ] and T [ are zero. 43, ] 104

133 Comparing real-world performance of stochastic and deterministic S-parameter calibrations Figure 3-3 shows the magnitude of the mean of the remaining 8 error correction terms, while Figure 3-4 shows the standard deviation of the error correction terms. Note that the error correction term T [ 11, ] is fixed to 1, since it considers here an S -parameter calibration where one of the error correction terms may be freely chosen T[1,1] 1 Ñ Ñ Ñ Ñ Ñ T[1,3] Frequency [MHz] Frequency [MHz] T[3,1] 0.01 T[3,3] Frequency [MHz] Frequency [MHz] 0.05 T[2,2] 1.03 T[2,4] Frequency [MHz] Frequency [MHz] Ñ 0.02 Ñ T[4,2] 0.01 T[4,4] 0.96 Ñ Ñ Ñ Frequency [MHz] Frequency [MHz] FIGURE 3-3. Magnitude of the mean of the error correction terms (* = Deterministic Calibration, = Stochastic Calibration) 105

134 Calibration of a Nonlinear Vectorial Network Analyser 1E Ñ Ñ Ñ T[1,1] T[1,3] Ñ Ñ Frequency [MHz] Frequency [MHz] T[3,1] T[3,3] Frequency [MHz] Frequency [MHz] T[2,2] T[2,4] Frequency [MHz] Frequency [MHz] T[4,2] T[4,4] Frequency [MHz] Frequency [MHz] FIGURE 3-4. Standard deviation of the error correction terms (* = Deterministic Calibration, = Stochastic Calibration) As expected, the mean for T [ 13, ], T [ 24, ], T [ 31, ], T [ 42, ] is close to zero, while the mean for T [ 22, ], T [ 33, ] and T [ 44, ] is close to 1 for both methods. Due to the fact that the reflection 106

135 Comparing real-world performance of stochastic and deterministic S-parameter calibrations coefficient of the load is slightly different from 0 for the stochastically based calibration, the values for the mean are slightly different for both methods. Figure 3-4 clearly shows that for all error correction terms, the standard deviation for the stochastically based calibration is approximately 50% smaller than for the deterministic calibration. This is of great importance when nonlinear modulated measurements are performed with the Nonlinear Vectorial Network Analyser, because of the lower SNR of these measurements. By using a stochastical calibration method, the influence of noise on the measurements is seriously reduced Experimental verification Measurements on a 40 db attenuator are performed using the NVNA. An input wave at a frequency of 800, 900, 1000, 1100 and 1200 MHz and a power level of 0 dbm is applied to the attenuator. The incident and reflected wave spectra at both ports of the attenuator are measured and corrected by the deterministic and stochastical calibration method. These measurements are repeated 100 times. The magnitude of the mean as well as the standard deviation of the corrected waves for both methods are shown in Figure 3-5 where: * = Mean of the data corrected by deterministic calibration = Mean of the data corrected by stochastic calibration + = Standard deviation of the data corrected by deterministic calibration o = Standard deviation of the data corrected by stochastic calibration The y-axis (logarithmic scale) represents the value of the mean and the standard deviation of the corrected waves in Volts. For all incident and reflected waves, the SNR is higher for the data corrected by the stochastically based calibration. 107

136 Calibration of a Nonlinear Vectorial Network Analyser α 2 α 1 1E+1 Ñ Ñ Ñ Ñ Ñ 1E-1 1E-3 1E-5 1E Frequency [MHz] 1E+0 1E-2 1E-4 Ñ Ñ Ñ Ñ Ñ β 1 β 2 1E-1 Ñ Ñ Ñ Ñ Ñ 1E-3 1E-5 1E Frequency [MHz] 1E-1 Ñ Ñ Ñ Ñ Ñ 1E-3 1E-5 1E Frequency [MHz] 1E Frequency [MHz] FIGURE 3-5. Mean and standard deviation of the corrected data Conclusion The comparison between the statistical properties of a deterministic and a stochastical S - parameter calibration shows that the standard deviation for the stochastical calibration is 50% lower than for the deterministic calibration. The wideband IF detection of the NVNA results in a significant degradation of the SNR of the measurements. By using the stochastical calibration, the further decrease of the SNR, after the measurements are calibrated, is prevented. A stochastically based calibration is therefore preferred if wideband IF measurements are performed with the Nonlinear Vectorial Network Analyser. 3.6 Nonlinear calibration Since S -parameters do not fully describe a nonlinear system, the knowledge of the separate incident and reflected waves are required when measuring nonlinear devices. As a result it will no longer be sufficient to calibrate wave ratios as done by a relative S -parameter calibration. 108

137 Nonlinear calibration Hence, the calibration for nonlinear devices must be extended by two additional calibration steps: a power and a phase calibration. This is illustrated in the following simple example: Example 3.3 Consider the measurement of the incident and reflected waves of a one-port DUT: β [] 1 α [] 1 k b k a RF GEN Port 1 a [] 1 b [] 1 DUT FIGURE 3-6. One-port measurement of a DUT The RF generator excites the DUT with a wave of power P src. For simplicity, the measurement instrument is considered to have only scaling errors k a () i and k b () i, dependent on the frequency i. For a linear DUT the true relation between the incident and reflected waves, as a function of frequency i, is: b [ 1] () i = S [ 11, ] ()a i [ 1] () i (3-79) The goal is to determine the reflection coefficient waves β [ 1] () i and α [ 1] () i : S [ 11, ] () i of the DUT out of the measured S [ 11, ] () i β [ 1] () i k b ()b i [ 1] () i k b () i = = = S α [ 1] () i k a ()a i [ 1] () i k a () i [ 11, ] () i (3-80) 109

138 Calibration of a Nonlinear Vectorial Network Analyser If the ratio k a () i k b () i is known, the true reflection coefficient S [ 11, ] () i can be determined from the measured waves. This means that a relative measurement, hence a relative calibration, is sufficient to fully characterise a linear DUT. For a nonlinear DUT, the true relation between incident and reflected waves can, for example, be written as: b [ 1] () i = ηi ()a3 [ 1] () i (3-81) Again, one is interested in knowing the true reflection coefficient η() i. This can be obtained from the measured waves as: η () i β [ 1] () i k b ()b i [ 1] () i = = α3 [ 1] () i k 3 = a ()a i 3 [ 1] () i k b () i η k 3 () i a () i (3-82) From equation (3-82) it is now clear that it is no longer sufficient to know the ratio k a () i k b () i. The scaling factors k a () i and k b () i must be known separately to determine η() i. This means that an absolute calibration is needed instead of a relative calibration. To determine the complex calibration factor Ki () Ki () e jφ K = () i of equation (3-2), which contains two degrees of freedom (dof), the magnitude and phase relation over the frequencies must be determined separately. This requires two additional measurements: the absolute value Ki () is obtained by a power calibration, while the phase φ K () i is obtained by a phase calibration Power calibration To determine the absolute value Ki (), the power flowing in the DUT, hence out of the excitation port of the NVNA, needs to be known absolutely. To do so, a power meter is connected to port 1 of the NVNA. The source is then exciting the DUT with a sine wave of 110

139 Nonlinear calibration constant power P cal for every frequency i. The source power P cal is then measured by the power meter and the NVNA. ADC ADC Downc β Downc α [1] [1] β [] 1 α [] 1 a [] 1 Power meter α P RF GEN Port 1 b [] 1 Γ P FIGURE 3-7. Power calibration In Figure 3-7 the absolute value of α P () i represents the power measured by the power meter. Γ P () i is the reflection coefficient of the power meter sensor. α [ 1] () i and β [ 1] () i are the waves measured by the NVNA. From Figure 3-7 and equation (3-2) one can see that α P () i = a [ 1] ()1 i ( Γ P () i ) = ( 1 Γ P () i ) α Ki () [ 1] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i β T [ 31, ] () i [ 1 ] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i (3-83) How can equation (3-83) be interpreted? Assuming that Assumption 3.9 the input impedance of the power meter is perfectly matched to the impedance of the NVNA, 111

140 Calibration of a Nonlinear Vectorial Network Analyser the reflection coefficient of the power meter Γ P () i = 0 and the reflected wave b [ 1] () i = 0. Equation (3-83) then becomes: α P () i = α Ki () [] 1 () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i β T [ 31, ] () i [] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i (3-84) Since by assumption b [ 1] () i = 0, and if the couplers of the NVNA are assumed to be of good quality (i.e. are able to split the incident and reflected waves accurately), the error correction terms T [ 13, ] () i and T [ 31, ] () i will be much smaller than T [ 33, ] () i ( T [ 13, ] T [ 31, ] «T [ 33, ] ) and T [ 33, ] () i 1. As a result: 1 α [ 1] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i β [ 1] () i T [ 31, ] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i = = α [ 1 ] () i Equation (3-84) can then be simplified as: α P () i = α [ 1] () i Ki () (3-85) Ki () = α [ 1] () i α P () i (3-86) For a good quality NVNA and a good matched power meter, the absolute value of the calibration factor Ki () can be found as the difference between the source power measured by the NVNA and the power meter. Indeed, this is what one would expect intuitively. However, in practice the NVNA is not ideal and the power meter not perfectly matched. Then, equation (3-83) must be used for the power meter calibration of the NVNA. This equation contains one correction factor for the mismatch of the power meter and one for the nonideality of the NVNA. To use this equation, the reflection coefficient Γ P () i of the power meter must be known. Example 3.4 shows how the reflection coefficient of the power meter can be measured by using a VNA. The error correction terms of equation (3-83) are determined by performing a linear calibration on the NVNA as is demonstrated in Example

141 Nonlinear calibration Example 3.4 The VNA (HP8510) is used to measure the reflection coefficient of the power meter (HP EPM- 441A). A one-port SOLT calibration is performed using the same 2.4 mm calkit as the one used to calibrate the NVNA. The reflection coefficient is measured in a frequency range from 45 MHz to 3 GHz with 801 frequency points. By performing repeated measurements (10 repetitions), the uncertainty bounds on the measurements as induced by measurement noise only can be determined. The measured reflection coefficient of the power meter and the standard deviation on the measurements are given in Figure Return loss dB 75dB Frequency [MHz] FIGURE 3-8. Measured reflection coefficient of the power meter: mean ( deviation ( ) ) and standard Since, the dynamic effects in the behaviour of the measured reflection coefficient are only present at low frequencies, Figure 3-8 only represents the frequency range from 45 MHz to 3 GHz. Note that it has been experimentally verified that the power meter is well matched in the frequency range from 0.5 to 26.5 GHz and that the behaviour of the power meter between 3 GHz and 26.5 GHZ is identical to its behaviour between 0.5 GHz and 3 GHz. The SNR on the measurements is ± 25 db. For low frequencies ( < 500 MHz) the mismatch of the power meter is apparently larger. This is partially due to the couplers, which have a bandwidth from 500 MHz to 50 GHz and thus a bad directivity at frequencies below 500 MHz and partially by the capacitive signal coupling of the power sensor which causes the peak at 333 MHz in the 113

142 Calibration of a Nonlinear Vectorial Network Analyser S 11 -curve. The noise influence becomes more important for low frequencies as can be seen from the standard deviation. Example 3.5 The error correction terms of equation (3-83) are determined by performing a stochastic oneport SOLT S -parameter calibration on the NVNA. This calibration uses a 2.4 mm calkit and is done for a frequency range from 800 MHz to 1200 MHz. This frequency range is chosen, as it considers the range where most of our measurements are done. Figure 3-9 shows the result for the magnitude of the error correction terms and T [ 33, ] () i. T [ 13, ] i ()T, [ 31, ] () i T[1,3] T[3,1] Frequency [MHz] Frequency [MHz] 1 T[3,3] Frequency [MHz] FIGURE 3-9. Error correction terms One can conclude that by making the assumption of a good quality NVNA ( T [ 13, ] ()T i, [ 31, ] () i «T [ 33, ] () i ), only small errors will be introduced by using the following equation to determine Ki (): Ki () = ( 1 Γ P () i )α () i [ 1] α P () i (3-87) 114

143 Nonlinear calibration Phase calibration Determining the phase of the calibration factor Ki () requires a special calibration element: the reference generator. The concept of the reference generator will be explained and its use will be demonstrated. A. The reference generator One can think of the reference generator as a device that, when exited by a single tone input frequency f in, delivers at its output all harmonics of this input frequency up to about 26 GHz. Assume that the phase relation between those harmonics is known. This means that the phase distortions introduced by the NVNA can then be determined by measuring the phase relations between the harmonics with the NVNA and comparing them to the known phase relations. What does a reference generator looks like? Figure 3-10 gives a schematic representation of the reference generator: RF source A attenuator 3dB SRD attenuator 6dB attenuator 20dB a R () i Reference Generator FIGURE Block schematic of the reference generator The key component of the reference generator is the step recovery diode (SRD-Herotek GC1050A) which is a comb (harmonic) generator [16]. This means that the output spectrum of an SRD contains all the harmonics of the single tone input frequency (see Figure 3-11). In time domain this results in a very sharp pulse as can be seen from Figure To obtain Figure 3-11 and Figure 3-12 a single tone input frequency of 900 MHz was applied to the SRD. The SRD is driven by an RF source whose output power is amplified to obtain sufficient input power for the SRD. The SRD is matched at the input to 50Ω, over a small frequency bandwidth. Putting an attenuator (HP8490D-3 db (DC-50 GHz)) between the amplifier and the SRD will result in a better matching over a larger bandwidth. This prevents that reflected 115

144 Calibration of a Nonlinear Vectorial Network Analyser Magnitude [dbm] Spectral Component FIGURE Spectral domain representation of the SRD output (frequency resolution: 225 MHz) 10 Amplitude [mv] Time Samples [ µsec] FIGURE Time domain representation of the SRD output waves from the SRD will cause nonlinearities in the amplifier and the RF source and prevents disturbances of the constant output power of the RF source. The use of an attenuator to improve the match between two devices is called padding. The purpose of putting an attenuator of 26 db (HP 8490D (DC-50 GHz)) at the output of the SRD is twofold. On one hand, it reduces the signal level of the SRD output signal to a safe level. This prevents that the SRD output signal would cause nonlinear distortions when measured by the NVNA. On the other hand, the effect of padding is used again to create a better match between the reference generator and the NVNA. 116

145 Nonlinear calibration Using this reference generator allows to calibrate the phase relations between a fundamental and its harmonics for a carrier frequency range from 600 MHz to 1200 MHz, with a resolution of 2 MHz. How can the phase relations between the harmonics of the SRD be known? The true phase relations between the harmonics are obtained measuring the output signal of the reference generator with a calibrated broadband sampling oscilloscope. The calibration of the oscilloscope is done by a nose-to-nose procedure ([46],[47]), which is an accurate method to determine the impulse response of broadband sampling oscilloscopes. In the nose-to-nose calibration procedure two oscilloscopes are connected by their input connector. When the sampling oscilloscope uses a non-symmetric sampler, such as a one-diode sampler, a small part of the sampling pulse leaves the sampling oscilloscope through the input connector. This is called the kick-out pulse. This kick-out pulse can be assumed to be proportional to the impulse response of the oscilloscope. This kick-out pulse is then measured by the second sampling oscilloscope. As a result the convolution of the impulse responses of both oscilloscopes is obtained. Deconvolution then yields the characteristic of the instrument. Measuring the output signal of the reference generator with a calibrated sampling oscilloscope returns the phase relations between all harmonics. For the reference generator of the Nonlinear Vectorial Network Analyser, the phase relations between each fundamental and 32 higher order harmonics are known. B. Performing the phase calibration Determining the phase φ K () i of the calibration factor Ki () is done by connecting the reference generator to, for example, port 1 of the NVNA. The RF input port of the NVNA is terminated with a 50Ω load impedance. The reference generator is excited by a single tone at frequency i coming from the RF source. The output signal of the reference generator is then measured by the NVNA and compared to the known frequency comb of the reference generator at the same input frequency i. The phase difference between known and measured phase relations results then in an estimate of the phase distortions introduced by the NVNA. 117

146 Calibration of a Nonlinear Vectorial Network Analyser ADC ADC Downc β [1] Downc α [1] 50Ω β [] 1 α [] 1 Port 1 a [] 1 Γ R a R b [] 1 RF GEN FIGURE Phase calibration In Figure 3-13 a R () i represents the wave generated by the reference generator. Γ R () i is the reflection coefficient of the reference generator. α [ 1] () i and β [ 1] () i are the waves measured by the NVNA. a [ 1] () i and b [ 1] () i are the true waves at port 1 of the NVNA. Under the assumption that the reference generator and the NVNA are well matched, one can see from Figure 3-13 that: b [ 1] () i = a R () i (3-88) Taking equation (3-2) into account then results in: Ki () = a R () i β [ 1] ()T i [ 33, ] () i α [ 1] ()T i [ 13, ] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i (3-89) Since one is only interested in the phase of Ki (), equation (3-89) can be written as: φ( Ki ()) φ( a R () i ) φ β [ 1] ()T i [ 33, ] () i α [ 1] ()T i [ 13, ] () i = T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i (3-90) 118

147 Nonlinear calibration If the couplers of the NVNA are of good quality, i.e. are able to split the incident and reflected waves accurately, the error correction terms T [ 13, ] () i and T [ 31, ] () i will be much smaller than T [ 33, ] () i ( T [ 13, ] T [ 31, ] T ) and [ 33, ] [ 33, ] () i = 1. As a result: T [ 33, ] () i β [ 1] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i α [ 1] () i T [ 13, ] () i T [ 33, ] () i T [ 13, ] ()T i [ 31, ] () i = = β [ 1 ] () i Equation (3-84) can then be simplified as: φ( Ki ()) = φ( a R () i ) φ( β [ 1] () i ) (3-91) For a good NVNA quality and a well matched reference generator, the phase of the calibration factor Ki () can be found as the difference in phase between the known and the measured reference generator signal Noise analysis of the nonlinear calibration From paragraph 3.1 one knows that, if the measurement instrument has n p measurement ports, the relation between exact and measured waves becomes: β() i α() i = T NL () i bi () ai () (3-92) with i denoting the frequency ω i and T NL () i Ki ()Ti C 2n p 2n p = the error correction matrix. β() i = β [ 1] () β i [ np ] () i T and bi () = b [ 1] () b i [ np ] () i T represent respectively the measured and exact reflected waves; α() i = α [ 1] () α i [ np ] () i T and ai () = a [ 1] () a i [ np ] () i T represent respectively the measured and exact incident waves. The exact or corrected waves can be obtained from the measured waves and the error correction terms, by solving equation (3-92): 119

148 Calibration of a Nonlinear Vectorial Network Analyser bi () ai () = T 1 () i β() i Ki () α() i (3-93) Note that since noisy data are considered here and the noise is additive, the previous equations can also be written as: b 0 () i a 0 () i + n b () i n a () i = 1 K ( 0 () i + n K () i T () i + n 0 T () i ) 1 β 0 () i + α 0 () i n β () i n α () i (3-94) where subindex 0 denotes the noise free value and ni () the noise on the quantity. To have an idea of how good the exact waves bi () and ai () approximate the noise free values b 0 () i and a 0 () i, a noise analysis must be performed. In practice, this means that one is interested in how large the noise n b () i and n a () i on the exact waves is. By determining the covariance matrix of n b ()n i a () i T a good idea of the variability of the exact waves bi () and ai () is obtained. For the ease of notation, the frequency dependency will not be written explicitly in the further explanation of the noise analysis. Under the assumption, that the noise on the exact values is small 1, the noise analysis of the incident and reflected waves a and b can be linearised. This can be done, in theory, by determining the 1st order truncated Taylor series (Appendix 3.D) of (3-93) in the noise free values b 0 and a 0. In practice, these noise free values are unknown. However, since the noise is assumed to be small, the exact values can be replaced by the noisy measured data. To determine the 1st order truncated Taylor series, the derivative of the product in equation (3-94) to all noisy parameters K, β, α and T must be determined. By assuming that the noise is 1. This assumption is valid, since it considers here calibration measurements, which are high precision measurements. 120

149 Nonlinear calibration small, taking the derivative can be replaced by the linearisation of the different factors of the product: n K K 0 + n K K 0 K 0 ( T 0 + n T ) 1 ( I T 1 n 0 T )T 1 0 (3-95) (3-96) Substituting equations (3-95) and (3-96) into equation (3-94) and neglecting higher order terms in the noise contributions, results in: n b T 1 n K 0 a 0 n β n α n K b 0 T K b 1 0 n 0 T 0 a 0 a 0 (3-97) n In what follows the noise matrix b will be denoted as n. n a The variability of the calibration is obtained by determining the covariance of the noise matrix n : Cov( n). The noise n contains a measurement noise and a reconnection noise contribution. Since both noise contributions are assumed to be circular complex normal distributed (see Appendix 3.E), the covariance matrix can be calculated as: Cov( n) = E { nn H } (3-98) Or, Cov( n) 1 = E { T 1 K 0 0 n β 1 n H T 1 β n K 0 } + α 0 n α n K b nk b H 0 E { K 0 a K 0 0 a } + 0 E { T b 1 0 n 0 T a 0 T b H 1 0 n 0 T } a 0 (3-99) 121

150 Calibration of a Nonlinear Vectorial Network Analyser To determine equation (3-99) the measurement noise ( n β, n α ), the noise on the error correction terms ( n T ), as well as the noise on the calibration factor K ( n K ) must be known. The measurement noise n β, n α can be determined out of repeated measurements. The noise on the error correction terms n T is known from paragraph where the accuracy bounds on the error correction terms are determined. From the nonlinear calibration (Paragraph 3.6), it is clear that the calibration factor K is a function of the following stochastic quantities: K = f( T, [ 13, ] T, [ 31, ] T, [ 33, ] α p, β p, α, P Γ, P α ph, β ph, a R ) (3-100) with T, [ 13, ] T, [ 31, ] T the error correction terms from the linear calibration; β p and α p [ 33, ] the power calibration measurements of the NVNA, α P the power meter measurements, Γ P the reflection coefficient of the power meter; α ph and β ph the phase calibration measurements of the NVNA and a R the known reference generator wave. As a result, the noise n K on the calibration factor K depends on the following noise contributions: n K = fn ( T, n [ 13, ] T, n [ 31, ] T, n [ 33, ] α p, n β p, n αp, n ΓP, n α ph, n β ph, n ar ) (3-101) The noise contributions n can be easily obtained out of repeated α p, n β p, n αp, n α ph, n β ph measurements. n ΓP and n ar are properties of respectively the power meter and reference generator. The noise on the error correction terms nt, n is known from [ 13, ] T, n [ 31, ] T[ 33, ] paragraph where the accuracy bounds on the error correction terms are determined. The noise n K can be obtained by performing the calculations of Appendix 3.F. Hence, the covariance matrix can be calculated as follows Cov( n) = T Cov K 0 n T0 H β n α K* 0 b Re( n a K k ) 2 Im( n K ) 2 1 E { + } b H K* 0 a 0 T b T 0 T 1 b H 0 Cov vec n a ( { T }) 0 T a 0... (3-102) 122

151 A database driven calibration where Re( n k ) and Im( n K ) can be calculated from Appendix 3.F. However, the noise free values are unknown, so that equation (3-102) must be calculated using the measured data and estimated parameters. Note that this is only valid if the noise on the measured data is small. Cov b T a K Cov n β T H b n α K * Re( n a K k ) 2 Im( n K ) 2 1 = E { + } b K * a T b a T 1 Cov ( vec { nt }) T b a T 1 H H... (3-103) Out of equation (3-103), one gets an idea of the variability of the incident and reflected waves. 3.7 A database driven calibration A classical DUT measurement consists of two steps. First, the instrument is calibrated. Next, the actual device is measured. Care is taken that the settings of the instrument are identical in both steps. If the instrument has several different hardware settings, all settings need to be calibrated to obtain calibrated measurements. For the NVNA, the number of settings is extremely high: 7 attenuator settings and 10 ADC ranges are present for each channel, resulting in 70 combinations per channel. Furthermore, in the context of nonlinear measurements, this technique has another major disadvantage: one has to know beforehand, which spectral components will be significant for all the measurements at different frequencies and powers. Since the number of significant harmonics depends on the frequency and power settings, the number of significant harmonics to be calibrated has to be guessed before the measurement. To avoid that the user has to specify a number of harmonics for each port/ frequency/power combination, the number of calibrated harmonics will be mostly fixed to a certain value for all measurements. If the number of significant harmonics is too large, the calibration procedure will take (much) more time than necessary. On the other hand, useful frequency components may be omitted when this number is underestimated. 123

152 Calibration of a Nonlinear Vectorial Network Analyser These problems can easily be solved if measurements of the device are performed before the calibration is undertaken. At that moment, a selection algorithm can be used to discriminate between noise and signal lines (Chapter 4) during each measurement step. One has the freedom to decide which component is worth while calibrating after the measurements are done. As a result, at the end of a measurement loop, the frequencies which need to be calibrated are perfectly known. Furthermore, the actual instrument settings can be retrieved from the saved measurement data, since both instrument settings and raw data are saved in the database. So that only the required instrument settings are calibrated. To implement this post-factum calibration, a database oriented solution is selected. This design decision depends mainly on two factors. First, it has already been shown that measuring a nonlinear device means automatically that a huge amount of experiments is needed. A database allows for a well-organised save and retrieve structure. Next, the state of all instruments at each measurement has to be recallable to allow the calibration to determine the instrument states that require calibration. Again, the structured organisation of the database gives a significant advantage in this context. During calibration, one goes through the list of frequencies to calibrate stored in the database and an error correction table is set up. At the end of a measurement-calibration cycle, the database not only contains the raw and corrected waves but also the linear error correction terms, the power and phase calibration factors, the calibration type, the used calibration standards,... The database oriented calibration is structured as follows: A. Preparation 1.Determine the spectral lines to calibrate for each measurement. 2.Determine the instrument state to calibrate for each measurement. 3.Build a complete calibration list containing all frequency-state combinations to calibrate. B. Calibration 4.Determine the unique frequency-state combinations out of the complete list. 124

153 Conclusion 5.Do a linear calibration on all unique combinations. 6.Do a power calibration on all unique combinations. 7.Build a list of phase reference values that is linked to the unique list. 8.Do a phase calibration on the unique list. C. Calculation 9.Calculate the error correction terms. D. Correction 10.Retrieve the measurement data from the database. 11.Get the associated instrument state. 12.Correct the data using the associated error correction terms. Besides the ability of performing a calibration after the measurements, using a database driven calibration has many other advantages: A unique automated link exists between the raw waves, the associated error correction terms and the corrected data. As a result, human mistakes like using the wrong error correction terms to correct a wave are omitted. Even after a long time, one can still remind what type of calibration was used, what were the settings of the instrument,... This allows also to detect possible errors during calibration. One can perform a posteriori virtual calibrated measurements without needing the hardware. Measurement data can be published over the internet. Using a database is a key component on the way to fully automated measurements. 3.8 Conclusion Calibration is one of the most important supporting beams of high quality network analyser measurements. Hence, it is important to use an accurate calibration process. This is especially true in a modelling context, where any discontinuity in phase/amplitude is exacerbated during the model fitting. The analytical calibration methods have some shortcomings because of the noise free operation assumption. A stochastic calibration process accounts for all types of 125

154 Calibration of a Nonlinear Vectorial Network Analyser errors: measurements are not only compensated for the systematic errors induced by the measurement instrument, but compensation is weighted such as to include the effect of the uncertainty on the standards, on the measurements and on the reconnection of elements. As a result, more accurate calibrated measurements are obtained in comparison to a classical calibration together with error bounds on the error correction coefficients. To acquire absolute calibrated waves, the linear calibration of the Nonlinear Vectorial Network Analyser needs to be extended by two additional calibration steps: a power and a phase calibration. The implemented database driven calibration is a major step forward to fully automated measurements for nonlinear systems. Furthermore, it moves the delicate selection of the significant number of harmonics away from the user. This gives a certain level of comfort to the user and robustifies the calibration procedure for user mistakes (such as not remembering all frequencies to calibrate, not doing things in the right sequence,...). 3.9 Appendices Appendix 3.A : Measured scattering matrix To determine the measured S-matrix S k of a calibration box k connected to the measurement instrument, n p independent measurements are to be carried out. One possibility to do this, is to excite each port p = 1 np of the measurement instrument in turn. Since the termination and the couplers in the Nonlinear Vectorial Network Analyser are not ideal, the incident waves α, [ 1] α, and α [ l 1] [ will not be zero when port is excited. To avoid l + 1 ], α, [ np ] l degeneration in the excitation matrix α, it will be assumed that the NAUC is terminated on all the non-excited ports by an impedance close to the characteristic impedance Z 0 = 50Ω of the NVNA. In this case, the excitation matrix α will be diagonal-dominant with a non-zero diagonal, hence of full rank. Moreover, when exciting the different ports, the analyser changes because the source and the termination are not perfectly matched and differ from port to port. One can obtain the S-parameters S k() i by solving the following system of equations: 126

155 Appendices β k ( i, 1) S k () i 0 n 0 p np α k ( i, 1) β k ( i, 2) β k ( in, p ) = 0 np S k () i 0 n p 0 np 0 np S k () i α k ( i, 2) α k ( in, p ) (3-104) where β k ( ip, ) = β [ 1]k ( ip, ) β[ np ]k ( ip, ) T are the measured reflected waves at all ports when port p is excited, α k ( ip, ) = α [ 1]k ( i, p) α[ np ]k ( ip, ) T are the measured incident waves at all ports when port p is excited, S k() i C n p n p is the S -parameter matrix of the kth calibration box. This results in a system of n p n p equations and n p n p unknown S - parameters that must be solved to determine the measured S -parameters out of the measured waves. To make this somewhat more concrete, consider the simple case of a two-port calibration. A forward (port 1 excited) and a reverse (port 2 excited) measurement must be performed to determine the S -parameters of a calibration box connected to both ports. The system of equations to be solved then becomes (for the ease of notation the frequency dependency i is not written): β [ 1]Fk β [ 2]Fk β [ 1]Rk = S k[ 11, ] S k[ 12, ] 0 0 S k[ 21, ] S k[ 22, ] S k[ 11, ] S k[ 12, ] α [ 1]Fk α [ 2]Fk α [ 1]Rk (3-105) β [ 2]Rk 0 0 S k[ 21, ] S k[ 22, ] α [ 2]Rk where superscripts F and R refer to a forward and reverse measurement respectively. This leads to the classical relations for the measured S -parameters: 127

156 Calibration of a Nonlinear Vectorial Network Analyser S k[ 11, ] = β [ 1]Fk β [ 1]Rk α [ 1]Fk α α [ 2]Fk [ 2]Rk α [ 1]Fk α [ 2]Fk 1 α α [ 1]Rk [ 1]Fk α [ 2]Rk (3-106) S k[ 12, ] = β [ 1]Rk β [ 1]Fk α [ 2]Rk α α [ 1]Rk [ 1]Fk α [ 2]Rk α [ 2]Fk 1 α α [ 1]Rk [ 1]Fk α [ 2]Rk (3-107) S k[ 21, ] = β [ 2]Fk β [ 2]Rk α [ 1]Fk α α [ 2]Fk [ 2]Rk α [ 1]Fk α [ 2]Fk 1 α α [ 1]Rk [ 1]Fk α [ 2]Rk (3-108) S k[ 22, ] = β [ 2]Rk β [ 2]Fk α [ 2]Rk α α [ 1]Rk [ 1]Fk α [ 2]Rk α [ 2]Fk 1 α α [ 1]Rk [ 1]Fk α [ 2]Rk (3-109) Appendix 3.B : Mathematical computations 1. Paragraph , page 77 S S k I np 0 k0 S 1 np T 0 0 I np I k0 = np T 0 np I np (3-110) S S k 0 np I k0 S np T 0 I I np 0 k0 np T 0 = 0 np I np (3-111) S k 0 n I p np I np 0 np S k0 T0 = 0 I np (3-112) 128

157 Appendices 0 np S k I np 0 np S k0 T0 = 0 I np (3-113) I np S k T 0 S k0 = 0 ; k = 1 n (3-114) I std np 2. Paragraph page 79 C εk ( S k0, t v0 ) = Cov S T I I k0 S np np k tv0 (3-115) = Cov T S k0 I np I np S k vec{ T 0 } Using equation (3-147) of Appendix 3.C (3.) results in: C εk ( S k0, t v0 ) = Covvec I np S k T 0 S k0 I np (3-116) Since S k = 0 np S k + I np 0 np = S k 0 n I and the p np + I np 0 np I np covariance of a deterministic quantity ( I np 0 np ) is zero, equation (3-116) equals C εk ( S k0, t v0 ) Cov vec S k 0 S n I k0 = p np T 0 I np Application of equation (3-147) vec{ XYZ} = ( Z T X)vec{ Y} X = I np, Y S S = k and Z 0 np I k0 = np T 0 yields I np of appendix 3.C with 129

158 Calibration of a Nonlinear Vectorial Network Analyser S T C εk ( S k0, t v0 ) Cov 0 np I k0 = np T 0 I I np vec { S k} np (3-117) Note that the first factor is purely deterministic, so that C εk ( S k0, t v0 ) = S T k0 I np T 0 T 0 n p I np I np S T k0 Cov( vec{ S k} ) I np T 0 T 0 n p I np H I np (3-118) 3. Paragraph page 89 For simplicity the second term of the cost function (3-51) will not be written and dependency on the parameters θ s is omitted. Note that in the following calculations the exact parameter values v 0 are already replaced by starting values. V( θ s, vˆ ( θ s )) = 1 -- FR 2 1 Q H 2 1 y y 2 (3-119) = 1 -- ([y 2 H [Q 1 R H F H I][FR 1 Q H 1 I]y] ) (3-120) QH Since Q 1 Q 2 is a unitary matrix, the matrix Q 1 Q 1 2 = I 2np n can be QH std inserted in equation (3-121): 2 V( θ s, vˆ ( θ s )) 1 -- y 2 H Q 1 R H F H QH = ( I) Q 1 Q 1 (FR 1 2 Q H 1 I )y QH = E 2 H w E w (3-121) The factor E w can thus be written as: E w = Q H 1 FR 1 Q H 1 QH 1 y Q H 2 FR 1 Q H 1 QH 2 (3-122) 130

159 Appendices Applying the QR-decomposition of F( θ s ) (equation (3-47)) and knowing that Q H 1 Q 1 = I ; that the signal and noise space of a QR decomposition are orthogonal Q H 2 Q 1 = 0 and that RR 1 = I results in E = 0 w QH y 2 (3-123) So that when written in the 2-norm and taking the dependencies on θ s cost function becomes into account, the V( θ s, vˆ ( θ s )) = Q H 2 ( θ s )y( θ s ) Eθ 2 s r 2 (3-124) Stacking the two contributions to one error vector leads to V( θ s, vˆ ( θ s )) Q H with εθ ( s ) 2 ( θ s )y( θ s ) =. Eθ s r = εθ ( 2 s ) 2 (3-125) 4. Paragraph page 90 J( θˆ s ( n) ) is the Jacobian matrix whose l th column is the partial derivative of Q H 2 ( θˆ s ( n) )y( θˆ s ( n) ) with respect to the l th entry of θ s : J [] l ( θˆ s ( n) ) Q H 2 ( θˆ s ( n) ) y ( θˆ s ( n) ) Q H ( θˆ s ( n) ) = y( θˆ s ( n) ) θ sl θ sl (3-126) Since the dependency of QH 2 on θ s is not known, one can not determine the derivative Q H 2 ( θˆ s ( n) ) Therefore the derivative of Q in equation (3-126) is replaced by a θ H 2 ( θˆ s ( n) ) sl known derivative. Thereto, some assumptions needs to be made. Remember that the error vector εθˆ ( s ( n) ) can be written as ε θˆ s ( n) F( θˆ s ( n) )v( θˆ s ( n) ) y( θˆ s ( n) ) ( ) = (3-41). If the Eθˆ s ( n) r 131

160 Calibration of a Nonlinear Vectorial Network Analyser error vector εθˆ ( s ( n) ) is small, the first term of ε( θˆ s ( n) ) is also close to zero: y( θˆ s ( n) ) F( θˆ s ( n) )v( θˆ s ( n) ). As a result Q H 2 ( θˆ s ( n) )y( θˆ s ( n) ) Q H 2 ( θˆ s ( n) )F( θˆ s ( n) )v( θˆ s ( n) ). Since F( θˆ s ( n) ) = Q 1 ( θˆ s ( n) )R( θˆ s ( n) ) and by orthogonality of Q 1 ( θ s ) and Q 2 ( θ s ), one thus obtains Q H 2 ( θˆ s ( n) )Q 1 ( θˆ s ( n) )R( θˆ s ( n) )v( θˆ s ( n) ) = 0. The derivative with respect to the l th entry of the parameter vector θ s then becomes: Q H 2 (( θˆ s ( n) ) ( n) )Q 1 ( θˆ s ( n) )R( θˆ s ( n) )v( θˆ s ( n) ) = 0 θ sl (3-127) Equation (3-127) leads to Q H 2 ( θˆ s n) ) Q θ 1 ( θˆ s n) )R( θˆ s n) )v( θˆ s n) ) Q H 2 θˆ s n) ( ) Q 1( θˆ s n) )R( θˆ s n) ) v( θˆ s n) ) sl θ sl Q H 2 ( θˆ s n) )Q 1 ( θˆ s n) )R θˆ s n) ( ) v ( θˆ s n) ) θ sl = 0(3-128) Since, by orthogonality of Q 1 ( θˆ s ( n) ) and Q 2 ( θˆ s ( n) ), the third term in equation (3-128) is zero and knowing that Q 1 ( θˆ s ( n) )R( θˆ s ( n) )v( θˆ s ( n) ) = y( θˆ s ( n) ), equation (3-128) results in: Q H 2 ( θˆ s ( n) ) y( θˆ s ( n) ) Q θ H 2 θˆ s ( n) ( ) F ( θˆ s ( n) ) = v ( θˆ s ( n) ) sl θ sl (3-129) By substituting equation (3-128) into equation (3-126), the l th column of the Jacobian matrix can be written under the following form J [] l ( θˆ s ( n) ) = Q H 2 ( θˆ s ( n) ) y( θˆ s ( n) ) θ sl F( θˆ s ( n) ) θ sl 1 vˆ ( θˆ s ( n) ) (3-130) J [] l ( θˆ s ( n) ) Q H 2 ( θˆ s ( n) ) L ( θˆ s ( n) ) = θ tˆv ( θˆ s ( n) ) sl (3-131) 132

161 Appendices where tˆv θˆ s ( n) 1 ( ) =. vˆ ( θˆ s ( n) ) 5. Paragraph page 95 The covariance of the estimates of the error correction terms will be determined. For the ease of notation, the parameter dependencies of the different functions will not be written. Equation (3-66) partitioned as: Cov vˆ ( ) g Cov( Θˆ ) H ( Θ, n) g( Θ, n) Θ Θ Θ = Θˆ Θ = Θˆ will be Cov vˆ θˆ s = g H g v v g H g θ s v g H v g θ s g H g θ s θ s (3-132) Using the matrix inversion lemma Appendix 3.C (5.) results in: Cov vˆ ( ) g H g 1 g H g 1 g H g = v v v v v θ s g H g g H (3-133) θ s θ g g H s θ s v g 1 g H g v v g H v θ g g H s θ v s g 1 v v To obtain a more symmetric form, the previous equation can be written as Cov vˆ ( ) = g H g 1 g H v v v κ g g H v g 1 v v (3-134) g g with κ I H g g I H. θ s θ s v g 1 g H g 1 g = H v v v θ s θ s In what follows the different derivatives of worked out. g( Θ, n) in the previous equation will be F( θ By (3-59) g( Θ, n) s )v y( θ s ) g = the derivative becomes: Eθ s r v 133

162 Calibration of a Nonlinear Vectorial Network Analyser g v F( θ s ) = = 0 neq n v Q 1 ( θ s )R( θ s ) 0 neq n v (3-135) The last step is a substitution of (3-47) F( θ s ) = Q 1 ( θ s )R( θ s ). As a result, g H g 1 g H v v = R v 1 ( θ s )Q H 1 ( θ s ) 0 nv n eq (3-136) By using the derivatives (3-135) and (3-136), one obtains g I 2nθs g H g 1 g n std + n H = I eq v v v v Q 1 ( θ s )R( θ s ) 0 neq n v R 1 ( θ s )Q H 1 ( θ s ) 0 nv n eq = I Q 1 ( θ s )R( θ s )R 1 ( θ s )Q H 1 ( θ s ) 0 2nθs n std n eq 0 neq 2n θs n 0 std neq = I 2nθs n Q std 1 ( θ s )Q H 1 ( θ s ) 0 2nθs n std n eq 0 neq 2n θs n I std neq (3-137) Since Q( θ s ) = Q 1 ( θ s ) Q 2 ( θ s ) is a unitary matrix, the following equation is valid: Q 1 ( θ s ) Q 2 ( θ s ) Q H 1 ( θ s ) Q H 2 ( θ s ) = Q 1 ( θ s )Q H 1 ( θ s ) + Q 2 ( θ s )Q H 2 ( θ s ) = I 2nθs n std (3-138) By using equation (3-138), equation (3-137) can be written as g I 2nθs g H n std + n eq v g 1 g H v v = v Q I QH I (3-139) 134

163 Appendices As a result equation (3-134) becomes Cov vˆ ( ) g H g 1 g H v v v κ g g = H v g 1 v v g with κ I g H Q QH g g H = θ s θ s 0 I θ 0 I s θ s (3-140) Furthermore, the derivative of g with respect to the parameters θ s equals g θ s = F( θ s ) y( θ v s ) θ s θ s E (3-141) By using the derivative of g θ s one obtains QH 2 0 neq g 0( 2np n std n v ) ( 2n p n std ) I θ neq s = QH 2 F ( θ s) v QH y ( θ s) + θ s θ s E (3-142) Equations (3-130) J [] l ( θ s ) Q H y( θ 2 ( θ s ) s ) F( θ s ) 1 = and (3-55) θ sl θ sl vˆ ( θ s ) J( θ s ) E T = X 1 ( θ s )Y( θ s ), hence result in QH 2 0 neq g = X 0( 2np n std n v ) ( 2n p n std ) I θ 1 Y neq s (3-143) So that equation (3-140) becomes Cov( vˆ ) R 1 Q H g 1 0 I ( Y θ H X H 1 X 1 Y) 1 g = H Q 1 s θ s 0 R H (3-144) Or, 135

164 Calibration of a Nonlinear Vectorial Network Analyser Cov( vˆ ) = R 1 ( θ s )( I + J K ( θ s )J H K ( θ s ))R H ( θ s ) (3-145) with J K ( θ s ) = Q H. 1 ( θ s ) g( Θ, n) Y θ 1 ( θ s ) = Q H 1 ( θ s )J( θ s )Y 1 ( θ s ) s Appendix 3.C : Matrix algebra 1. The vec -operator: let X C m n : X [:,1] vec{ X} = X [:,2]. X [:,n] 2. Kronecker matrix product : Let X C m n and Y be general matrices X [ 11, ] Y X [ 1, n] Y X Y = (3-146) X [ m, 1] Y X [ m, n] Y 3. Let X, Y and Z be matrices of compatible dimensions, i.e. all occurring products below exist, then [9] vec{ XYZ} = ( Z T X)vec{ Y} (3-147) 4. Let X, Y, P and Q be matrices of compatible dimensions, i.e. all occurring products below exist, then [9] ( X Y) ( P Q) = XP YQ (3-148) 5. Let X C n n, Y C n m and Z C m m, then the n n upper left block of 1 X Y equals Y H Z X 1 + X 1 YZ ( Y H X 1 Y) 1 Y H X 1 (3-149) 136

165 Appendices Appendix 3.D : Taylor series Definition 3.10 Let f( x) and its n + 1 first derivatives be continuous in an interval around x = d. Then in this interval: f( x) = f '( d) fd ( ) f ''( d) ( x d) ( x d) 1! 2! f n) ( d) ( x d) n! + R n + 1 ( n) (3-150) x where R n + 1 ( n) = ( x t) n f with between and. n! f () t dt ( ξ) = ( ( n + 1)! d ) n 1 ξ d x a If R n ( x) 0 as n then the Taylor series can be written as f( x) = f k ( d) ( x d) k! k k = 0 (3-151) The Taylor s series for a function of two variables x and y, equals: f( xy, ) = fd ( + hq, + l) = h l---- k k! x y fxy (, ) x = d k = 0 y = q (3-152) Appendix 3.E : Properties of the reconnection noise To benefit from the good properties of the estimator, the assumption that the measurement as well as the reconnection noise on the measured waves is circular complex normal distributed, must be valid. While the measurement noise on complex spectra is already proven to be circular complex normal distributed [35], the distribution of the reconnection noise requires further analysis. For a noise contribution to be circular complex normal distributed, the standard deviation of the real and imaginary parts of the measured quantities must be equal and independent [35]. To verify if the reconnection noise is circular complex normal distributed, reconnection 137

166 Calibration of a Nonlinear Vectorial Network Analyser measurements are needed. These reconnection measurements are done on a 2.4 mm male short in a frequency range from 600 MHz to 6 GHz in steps of 50 MHz. The short is reconnected 50 times and during each reconnection measurement, 5 periods of the waves are measured and averaged to reduce the influence of the measurement noise. Out of the measured incident and reflected waves, the parameter S [ 11, ] = β [ 1] ( ω) α [ 1] ( ω) is calculated. Note that the measured S -parameter is not only disturbed by measurement noise but also by reconnection noise. How can one prove that the reconnection noise is circular complex normal distributed? A. Circular complex distributed For the reconnection noise to be circular complex distributed, the standard deviation of the real and imaginary parts of the measured parameter S [ 11, ]( ω) = β [ 1] ( ω) α [ 1] ( ω) must be equal and independent. To simplify notations, the frequency dependency will not further be written. 1. Equal The 95% confidence bounds on the sample variance of a normalised N( 01, ) distribution, for 50 independent experiments are: 0.6 σ This means that using the sample variance as an estimate for the exact (unknown) variance results in an uncertainty of 40% of the normalised variance. To decrease this uncertainty to 10%, 800 independent experiments are needed. However, this is very harmful for the calibration elements and thus not advisable. Hence, the standard deviation of the real and imaginary parts of the equal if the following relation is valid [38]: σ R σ I σ R σ I S [ 11, ] -parameters are (3-153) where σ R and σ I are respectively the standard deviation of the real and imaginary part of the S [ 11, ] -parameter. The theoretical derivation of the boundaries can be found in reference [38]. Note that in practice, the sample standard deviations s R and s I will be used to verify relation (3-153): 138

167 Appendices s R s I s R s I (3-154) with N 2 1 [ k] k s R = S N 1 ( R11 [, ] S R11 [, ] )( S [ ] R11 [, ] S R11 [, ] ) * k = 1 N 2 1 [ k] k s I = S N 1 ( I11 [, ] S I11 [, ] )( S [ ] I11 [, ] S I11 [, ] ) * k = 1 (3-155) (3-156) Herein, N is the number of reconnections, S R11 [, ] and S I11 [, ] are respectively the real and imaginary part of the measured S [ 11, ] -parameter. S R11 [, ] and S I11 [, ] are respectively the sample mean of S R11 [, ] and S I11 [, ] and can be calculated as follows: N 1 [ k] S R11 [, ] = --- S N R11 [, ] k = 1 N 1 [ k] S I11 [, ] = --- S N I11 [, ] k = 1 (3-157) (3-158) Figure 3-14 shows the experimentally obtained ( s R s I ) ( s R s I ) with their 95% confidence bounds for all measured frequencies. The dashed lines represent the boundaries. 139

168 Calibration of a Nonlinear Vectorial Network Analyser 0.5 s R s I s R s I Frequency [GHz] FIGURE ( s R s I ) ( s R s I ) for all measured frequencies Since 91% of all measured frequencies lie between the two boundaries, one can conclude that the real and imaginary part of the S [ 11, ] -parameter can be considered equal. 2. Independent The standard deviation of the real and imaginary parts of the independent if: tanh 1 3 s RI tanh N 3 s R s N 3 I S [ 11, ] -parameters are (3-159) 2 1 with s RI, the correlation between the real and imaginary parts. N N 1 e [ k ] [ k] = R ei k = 1 [ k] [ k] Herein, N is the total number of reconnections and e R, e I are respectively the real and imaginary parts of e [ k] [ k] 1 N [ k] = S [ 11, ] --- with the measured - N S [ 11, ] S k = 1 [ 11, ] S [ 11, ] parameter. s R and s I are defined as in the previous paragraph. The theoretical derivation of the boundaries can be found in reference [38]. Figure 3-15 represents a plot of the correlation for all measured frequencies. The dashed lines represent the boundaries. 140

169 Appendices 0.5 Correlation Frequency [GHz] FIGURE Correlation s RI ( s R s I ) for all measured frequencies Since 92% of all measured frequencies lie between the two boundaries, one can conclude that the real and imaginary part of the S [ 11, ] -parameters can be considered to be independent. Note that due to the random behaviour of the correlation between the confidence bounds, it is acceptable to state that the variables are totally uncorrelated. However, proving this statement would require a very large number of experiments, since the uncertainty bounds on the correlation only tend very slowly to zero as a function of the number of experiments. Since the real and imaginary parts of the S [ 11, ] -parameters are equal and independent, the noise on the measured S [ 11, ] -parameters is circular complex distributed. B. Normal distributed To test whether a distribution is normal or not, one can use the Kolmogorov-Smirnov test ([38],[29]). This test is very simple and powerful. It states that when a band of a certain known width is drawn around the sample distribution function, there is a known probability that the exact cumulative density function lies totally within this band. This test allows to set confidence levels on the continuous cumulative density function as a whole. Psup S N ( x) F( x) > d x α = α (3-160) 141

170 Calibration of a Nonlinear Vectorial Network Analyser Herein, S N ( x) is the cumulative sample distribution function and F( x) is the theoretical cumulative distribution function. According to [38], d α for a confidence level of 99% ( α = 0.01 ) can be approximated by: d α = N (3-161) with N the number of realisations. The theoretical cumulative distribution function is found to be: F() t x 2 t = e dx = 2π t + --erf (3-162) with erf the generalised error function [1]. Figure 3-16 represents the cumulative sample distribution function of the real and imaginary parts of the S [ 11, ] -parameter for all frequencies, while Figure 3-17 represents the theoretical cumulative distribution function. 1 Px ( < x N ) Norm. variable value FIGURE Cumulative sample distribution function 142

171 Appendices 1 Px ( < x N ) Norm. variable value FIGURE Theoretical cumulative distribution function By calculating the residual between the theoretical and sample distribution function, Figure 3-18 is obtained for all measured frequencies. The dashed lines delimit the 99% probability region of the Kolmogorov test. Note that 100 realisations are available since both the real and imaginary parts are used to calculated the distributions. 0.3 Residual Realisations FIGURE Residual between theoretical and sample distribution function Since for all realisations and frequencies, the residual stays between the Kolmogorov confidence bounds, one can conclude that the measured S -parameters are normal distributed. Hence, the noise on the measured S [ 11, ] -parameters is circular complex normal distributed. However, care must be taken! Consider Figure 3-19 (a) which represents the standard deviation of the S [ 11, ] -parameter over the repeated measurements for one single reconnection. This figure gives an idea of the importance of the measurement noise. Figure 3-19 (b) 143

172 Calibration of a Nonlinear Vectorial Network Analyser represents the standard deviation of the S [ 11, ] -parameter over the reconnections and thus represents the RMS (Root-Mean-Square) sum of the measurement noise and the reconnection noise Std [db] -53 Std [db] Frequency [GHz] (a) Frequency [GHz] (b) FIGURE Standard deviation of the S [ 11, ] -parameters over 5 repeated measurements (a) and 50 reconnections (b) By comparing both figures, one can conclude that, for this user, the reconnection noise is not larger than the measurement noise and will have no influence on the accuracy of the measurements. Hence, for each user who has a connector repeatability comparable to this example, the influence of the reconnection noise on the calibration can be neglected. In an attempt to make the reconnection noise visible and thus the measurement noise smaller, 200 periods of the waves were measured and averaged during each reconnection. Figure 3-20 represents the standard deviation of the S [ 11, ] -parameter over the reconnections and hence represents the RMS sum of the measurement noise and reconnection noise. Std [db] Frequency [GHz] FIGURE Standard deviation of the S [ 11, ] -parameters over 50 reconnections for 200 repeated measurements 144

173 Appendices After a few calculations, one can conclude that at this point the reconnection noise becomes almost visible, but still more repeated measurements are needed to reduce the measurement noise below the reconnection noise level. However, the reconnection measurement campaign with 200 repeated measurements took already more than one day. The interruption of the measurements over the night, has a serious influence on the measured results. Figure 3-21 shows for one single frequency the complex S [ 11, ] -parameters of the reconnection experiment in case of 5 repeated measurements (a) and 200 repeated measurements (b). Two states are clearly visible in case of 200 repeated measurements, which correspond with the two days measurement campaign. 9.5E-3 6.7E-3 Imaginary 8.0E-3 Imaginary 5.7E-3 6.5E-3-4.4E-2-4.3E-2-4.1E-2 Real (a) 4.7E E E E-1 Real (b) FIGURE Complex S [ 11, ] -parameter for 5 (a) and 200 (b) repeated measurements. These two clusters make the circular complex normal distribution test fail. However, performing the test on both clusters separately results in circular complex normal distributions. Out of this test, one can conclude that depending on how good the connector repeatability of the user is, the reconnection noise must or must not be taken into account. In our case the influence of the reconnection noise can be neglected. Appendix 3.F : The noise n K on the calibration factor K From the nonlinear calibration (Paragraph 3.6), it is clear that the calibration factor K can be written as: 145

174 Calibration of a Nonlinear Vectorial Network Analyser K = 1 Γ P α P ph ph β Im a [] 1 T[ 33, ] α [] 1 T[ 13, ] R T [ 33, ] T [ 13, ] T [ 31, ] jatan ph ph β p p Re a [] 1 T[ 33, ] α [] 1 T[ 13, ] α [ 1] β [ 1] T R [ 31, ] T [ 33, ] T [ 13, ] T [ 31, ] e T [ 33, ] T [ 13, ] T [ 31, ] (3-163) with T, [ 13, ] T, [ 31, ] T the error correction terms from the linear calibration; β p p [ 33, ] [] and α 1 [] 1 the power calibration measurements of the NVNA, α P the power meter measurements, Γ P the ph ph reflection coefficient of the power meter; α [] 1 and β [] 1 the phase calibration measurements of the NVNA and a R the known reference generator wave. To obtain the noise n K on the calibration factor K, equation (3-163) will be approximated by its 1st order truncated Taylor series (Appendix 3.D): K K K K 0 + n K K n Γ ΓP n P α αp K n p α p K K = = n P 0 0 α 1[] p β p n [] 1 β [] 1 T T + [ 33, ] 33, [] K K K K K n (3-164) T T n [ 31, ] [ 31, ] T T n [ 13, ] [ 13, ] a ar n ph α ph n R α [] 1 ph β ph [] 1 β [] 1 [] [ ] The notation. 0 denotes that after taking the derivative the noise free values of all the parameters must be filled in. The noise contributions n can be easily αp, n [] 1 β p, n αp, n [] 1 α ph, n [] 1 β ph [] 1 obtained out of repeated measurements. n ΓP and n ar are properties of respectively the power meter and reference generator. The noise on the error correction terms nt, n [ 13, ] T, n [ 31, ] T[ 33, ] is known from paragraph where the accuracy bounds on the error correction terms are determined. Since K (3-163) is not an analytical function, the derivatives of equation (3-164) can not be calculated. However, it is possible to take the derivatives of K towards the real and imaginary parts of the variables separately. Giving a single expression for Re( n K ) and Im( n K ) would result in a very complicated formula. The real and imaginary parts of the different terms contained in the noise n K will be determined separately instead. Thereto only the derivatives 146

175 Appendices of K towards the different noise contributions will be given. Some shorthand notations will be introduced to ease the understanding of the equations. Equation (3-163) can be written as: K = B I jatan B R A e (3-165) p p ph ph 1 Γ P α where A [ 1] β [ 1] T[ 31, ] β = , B a [] 1 T[ 33, ] α [] 1 T =. α P T [ 33, ] T [ 13, ] T R [ 13, ] [ 31, ] T [ 33, ] T [ 13, ] T [ 31, ] The subscript R and I denotes respectively the real and imaginary part: B I = Im( B) and B R = Re( B). B K A * p p I Re α [ 1] β [ 1] T[ 31, ] jatan B R = e Γ PR 2 AA * α P T [ 33, ] T [ 13, ] T [ 31, ] (3-166) and p p K Im A* α [ 1] β [ 1] T[ 31, ] jatan B R = e Γ PI 2 AA * α P T [ 33, ] T [ 13, ] T [ 31, ] B I (3-167) B K A * p p I 1 Γ Re P α [ 1] β [ 1] T jatan [ 31, ] B R = e α PR 2 AA * 2 α T [ 33, ] T [ 13, ] T [ 31, ] P (3-168) and K 2Im A* p p 1 Γ P α [ 1] β [ 1] T jatan [ 31, ] B R = e α PI 2 AA * 2 α T [ 33, ] T [ 13, ] T [ 31, ] P B I (3-169) 147

176 Calibration of a Nonlinear Vectorial Network Analyser K p 2Re A * jatan Γ P B R = e α [ 1]R 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) B I (3-170) and K p 2 Im A* jatan Γ P B R = e α [ 1]I 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) B I (3-171) K p 2Re A * jatan ( Γ P 1)T [ 31, ] B R = e β [ 1]R 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) B I (3-172) and K p 2 Im A* jatan ( Γ P 1)T [ 31, ] B R = e β [ 1]I 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) B I (3-173) K A * Re T Γ p p P α[ 1] + β [ 1] T jatan [ 31, ] B R = [ 33, ]R 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B Im( M 2 2 T ) I Re( M B R + B [ 33, ] 2 2 T ) [ 33, ] I B R + B I (3-174) and 148

177 Appendices K 2Im A* T Γ p p P α[ 1] + β [ 1] T jatan [ 31, ] B R = [ 33, ]I 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B I Re( M 2 2 T ) Im( M B R + B [ 33, ] 2 2 T ) I B R + B [ 33, ] I (3-175) ph ph T with M [ 13, ] T [ 31, ] β [ 1] α [ 1] T T = [ 13, ]. [ 33, ] ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 K A * Re T ΓP α p p [ 1] T[ 13, ] β [ 1] T jatan [ 33, ] B R = [ 31, ]R 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B Im( M 2 2 T ) I Re( M B R + B [ 31, ] 2 2 T ) [ 31, ] I B R + B I (3-176) and K 2Im A* T ΓP α p p [ 1] T[ 13, ] β [ 1] T jatan [ 33, ] B R = [ 31, ]I 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B I Re( M 2 2 T ) Im( M B R + B [ 31, ] 2 2 T ) I B R + B [ 31, ] I (3-177) ph 2 ph α with M [ 1] T[ 13, ] β [ 1] T[ 13, ] T T = [ 33, ]. [ 31, ] ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 149

178 Calibration of a Nonlinear Vectorial Network Analyser K A * Re T ΓP α p p 2 [ 1] T[ 31, ] β [ 1] T jatan [ 31, ] B R = [ 13, ]R 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B Im( M 2 2 T ) I Re( M B R + B [ 13, ] 2 2 T ) [ 13, ] I B R + B I (3-178) and K 2Im A* T ΓP α p p 2 [ 1] T[ 31, ] β [ 1] T jatan [ 31, ] B R = [ 13, ]I 2 AA * α P ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 e + B I B I jatan B R Ae B R B I Re( M 2 2 T ) Im( M B R + B [ 13, ] 2 2 T ) I B R + B [ 13, ] I (3-179) ph ph α with M [ 1] T[ 33, ] β [ 1] T[ 33, ] T T = [ 31, ]. [ 13, ] ( T [ 33, ] T [ 13, ] T [ 31, ] ) 2 B I jatan K B R = Ae a RR and B R B I 2 2 B R + B I (3-180) B I jatan K B R = Ae a RI B R + B I 2 2 B R + B I (3-181) B I jatan K ph Ae B R = α [ 1]R B R B ImM I 2 2 ph ReM B R + B α 2 2 ph [ 1] I B R + B α [ 1] I (3-182) 150

179 Appendices and B I jatan K ph Ae B R = α [ 1]I B R B ReM I 2 2 ph B R + B α ImM 2 2 ph [ 1] I B R + B α [ 1] I (3-183) T with M ph = [ 13, ]. α[ 1] T [ 33, ] T [ 13, ] T [ 31, ] jatan K ph Ae B R = β [ 1]R and B I B R B ImM I 2 2 ph B R + B β ReM 2 2 ph [ 1] I B R + B β [ 1] I (3-184) B I jatan K ph Ae B R = β [ 1]I B R B ReM I 2 2 ph B R + B β ImM 2 2 ph [ 1] I B R + B β [ 1] I (3-185) T with M ph = [ 33, ]. β[ 1] T [ 33, ] T [ 13, ] T [ 31, ] Using equation (3-164) and the derivatives (3-166) to (3-185), results in an expression for Re( n K ) and Im( n K ). Calculations are easily performed with any mathematical CAD software, such as Matlab. 151

180 Calibration of a Nonlinear Vectorial Network Analyser 152

181 PART II MEASUREMENT TECHNIQUES FOR RF COMPONENTS AND SYSTEMS 153

182 154

183 CHAPTER 4 AN AUTOMATIC HARMONIC SELECTION SCHEME Abstract - Nonlinear measurements result in huge amounts of data, since extended power and frequency sweeps are required. However, are all of these data relevant? One of the major issues in nonlinear network analysis is, hence, data reduction. A method is presented to automatically select the significant harmonics in measured spectra. This stochastic selection criterion, which is based on the Student t 2 -distribution, allows to save and calibrate all and only significant spectral components, without the need for wild guessing the number of signal containing harmonics in the measured data. The usefulness of the method is proven on amplifier measurements performed by a Nonlinear Vectorial Network Analyser. 155

184 An Automatic Harmonic Selection Scheme 4.1 Introduction For a nonlinear system excited with periodic signals, the output spectra can contain spectral lines at all linear combinations of the input frequencies. However, not all of these possible frequencies are always significantly excited, or in other words, they do not always contain a signal contribution. To avoid the measurement of a large number of these spectral components that only contain noise, it is hence vital to discriminate between noise and signal carrying harmonics. To save and calibrate all and only significant spectral components, an automatic selection procedure for signal carrying harmonics is needed. Till now, two methods were available for such a selection. A first method is to guess the number of significant harmonics a priori. In the best case, this creates a lot of overhead because many noise-only contributions are measured. Worst case, a number of significant harmonics are overlooked. Feature extraction or accurate modelling then become impossible or error prone since inputs are only partially known. A second option is to examine the raw data by hand and to take only those spectral lines into account that seemingly rise above the noise floor. This method however is both extremely time consuming due to the large number of measurements and error prone because of the user interaction. To obtain a fully automated measurement and calibration process, an automatic spectral line selection procedure is required. Using stochastic significance as a selection criterion, the decision can be made whether only noise or noise and signal are present in the measured spectral lines, at the cost of a low number of repeated measurements. The proposed test uses the distribution of the ratio of the sample mean and the sample variance, which is a Student t 2 - distribution [38], to test the signal presence hypothesis. The usefulness of this method is illustrated on real-world measurements of an amplifier in saturation performed by a Nonlinear Vectorial Network Analyser (NVNA) [39]. 156

185 Selection criterion 4.2 Selection criterion Stochastic model Consider N complex samples, i.e. repeated measurements x [ k] with k = 1,, N, of a spectral line x carrying both a deterministic signal component x 0 and a noise contribution n x : x [ k] = x + n [ k ] 0 x (4-1) Assume that, Assumption 4.1 [ k] the n x noise samples are taken from a stochastic variable which is assumed to be zero mean µ nx = E { n x } = 0, circular complex normal distributed [34] with variance 2 = E { n n * } : x x σ nx n[ k] x N( 0, σ nx ) (4-2) where * denotes the complex conjugate. In the considered application, these complex samples x [ k] are computed by the DFT. Assumption 4.1 is hence fulfilled whenever the time domain noise entering the DFT has a time correlation which vanishes over time [27],[33]. The complex samples x [ k] [ k] [ k] can be split up in a real x R and imaginary part x I : x [ k] [ k] [ K] = x R + jx I = ( x R0 + n[ k] xr ) + jx ( I0 + n[ k] xi ) (4-3) with j 2 = 1 and subindex R and I denote respectively the real and imaginary part. Taking Assumption 4.1 into account, the complex samples distributed: x [ k] N ( µ x, σ x ) = Nx ( 0, σ nx ) x [ k] are found to be normal (4-4) 157

186 An Automatic Harmonic Selection Scheme So that the real and imaginary part of x [ k] are distributed as follows: [ k] σ nx x R Nx R0, (4-5) [ k] σ nx x I Nx I0, (4-6) Notice that both real and imaginary parts have the same standard deviation and are statistically independent for the DFT lines. In the ideal case, the distribution of x is exactly known. A distinction between signal and noise-only contributions in the measured spectral data then boils down to a comparison of the distance of the mean value of a spectral line to zero. The mean value or expected value of a spectral line is: E { x } = x 0 + µ nx (4-7) where µ nx is the mean value of the noise which is zero by assumption (Assumption 4.1) and x 0 is the deterministic signal value of the spectral line. Hence, for signal carrying lines the expected value E { x } = x 0 will significantly differ from zero. For noise-only lines, the expected value E { x } = x 0 will be zero. To select automatically when a signal contribution is present in a spectral line, it is hence sufficient to test whether the mean value of the spectral line equals zero or not. In practice, an infinitely large number of measurements are needed to know the exact distribution of x, or in other words, to know the mean value of a spectral line. Since only a limited number of measurements is available, one can get an estimate of the mean value µ x and variance σ2 x of a spectral line, by using the sample mean x and sample variance s2 x as an estimation [34]. When only x and s2 x are known, the right tool to verify whether the mean value µ x of a spectral line equals zero or not, is statistical hypothesis testing. To compare mean values a single-sided t-test is best used, which is based on the Student t-distribution [38]. 158

187 Selection criterion Automatic detection algorithm To obtain a measurement based estimation of the mean µ x and standard deviation σ x of a spectral line x, the sample mean x and sample variance s2 x of a set of repeated measurements x k] of is used: x = N x N [ k] k = 1 (4-8) N s2 1 x = x N 1 ( [ k] x) ( x [ k] x) * k = 1 (4-9) The goal now is to test if the unknown mean value µ x using the sample mean and sample variance. of the spectral line is zero or not by First, consider for simplicity that the samples x k] are real. Statistical theory shows that to test the hypothesis µ x = 0, a single-sided t -test is required [38]. If the hypothesis µ x = 0 is true, then relation T x µ = x = s x N x s x N (4-10) will be a Student t-distribution (Appendix 4.A) with N 1 degrees of freedom (dof). This will be denoted as: t N 1. In other words, if the hypothesis is true and an infinite number of T values is evaluated for different experimental measurements of the same quantity x, then the probability that T is smaller than a certain level can be determined. In practice, only one T value is obtained. It is then possible to assess if T is drawn from a distribution that fulfils the probabilistic bound only up to some preset significance level ρ : PT ( t N 11, ρ ) = 1 ρ (4-11) 159

188 An Automatic Harmonic Selection Scheme Note that t N 11, ρ stands for the value of a Student t-distribution with N 1 dof and for a significance level ρ. The value of a Student t-distribution t N 11, ρ can be found from statistical tables in the literature [6]. Equation (4-11) leads to the following decision rule: if T t N 11, ρ then the hypothesis µ x = 0 can be considered to be true with a certainty of 1 ρ %. Thus, the significance level ρ is the probability of wrongly rejecting the hypothesis. Note that the choice of ρ is critical. If ρ is small, then all noisy lines will be rejected, but the sensitivity of the algorithm is then also very small. If ρ is large, the algorithm is very sensitive, but many noise lines will be wrongly selected as signal lines. Since, for spectral data, the samples x [ k] are not real but complex, a Student t 2 -distribution must be used instead of a Student t-distribution to obtain real values for T 2. The hypothesis µ x = 0 will then be true, if the relation T 2 = xx * s x N (4-12) is a Student t 2 -distribution with 2( N 1) degrees of freedom: t2 2( N 1). As for the case of real samples, it is possible to assess if T 2 is drawn from a distribution that fulfils the probabilistic bound only up to some preset significance level: PT ( 2 t2 2( N 1), 1 ρ ) = 1 ρ (4-13) Note that the values of t2 2( N 1), 1 ρ can be found from the statistical tables of the Fisher s F - distribution, since the Student t 2 -distribution is only a special case of the more general Fisher s F-distribution (Appendix 4.B): t2 2( N 1), 1 ρ = F 22N, ( 1), 1 ρ. As a result, equation (4-13) leads to the following decision rule: if, for a given F 22N, ( 1), 1 ρ, T 2 F 22N, ( 1), 1 ρ, then the hypothesis can be assumed to be true with 1 ρ % certainty and thus, the spectral line only contains noise. If, on the other hand, T 2 > F, a significant harmonic has been found with ρ 22N, ( 1), 1 ρ % uncertainty. 160

189 Simulations To implement the above test for a certain value of N and ρ, one should know the values of F 22N, ( 1), 1 ρ for every N and ρ. As a result, the statistical tables of the F -distribution must be included in the implementation of the selection criterion. To avoid this, the sample distribution F S of a Student t 2 -distribution (Appendix 4.C) can be calculated instead [38]. FS T2 The above test can then be implemented by calculating the sample distribution value ( ) for every spectral line x in the measured data. The spectral lines whose sample distribution value satisfies equation (4-14) F S ( T 2 ) 1 ρ (4-14) with ρ the level of significance, are classified as noise lines, while the spectral lines whose sampling distribution value satisfies F S ( T 2 ) > 1 ρ (4-15) are considered to contain signal information and to be significant harmonics. The correctness of the proposed stochastic selection criterion is shown on simulations, while its applicability will be shown on real-world measurements. 4.3 Simulations Consider a record of circular complex normal distributed samples with zero mean µ x = 0 and standard deviation σ x = 1. The experiment is repeated 11 times ( N = 11 ). For all these samples the value for T 2 is determined through equation (4-12). If the selection T 2 criterion is correct, the number of samples for which this -value is smaller than a given F ν1, ν with and, must be. can be 2, 1 ρ ν 1 = 2 ν 2 = 2( N 1) 100 ( 1 ρ )% F ν1, ν, 1 ρ 2 obtained from tables of the Fisher s F -distribution in the literature [6]. Table 4-1 shows the obtained percentages from simulations using a significance level ρ of 0.1, 0.05, 0.025, 0.01, and

190 An Automatic Harmonic Selection Scheme Theoretical Experimental 100( 1 ρ)% 100( 1 ρ)% TABLE 4-1. Experimental obtained fraction for ρ These results are in good agreement with the theoretically predicted probabilities and fall within the expected confidence intervals. 4.4 A practical example based on NVNA measurements To prove the correctness and usefulness of the proposed stochastic selection criterion, measurements are performed on a power amplifier of type MRFIC2006 (Motorola). The excitation signal consists of a single tone at a fundamental frequency of 900 MHz. The input power is swept from -14 dbm to 10 dbm in steps of db. The power amplifier has a supply voltage of 4V and is terminated in a 50 Ω load impedance. The measurements of the incident and reflected wave spectra at both ports of the amplifier are performed by a Nonlinear Vectorial Network Analyser (NNMS-HP85120A-K60) (Figure 4-1). β [ 1] α [ 1] β [ 2] α [ 2] RF GEN a []0 1 a []0 2 Amplifier 50Ω b []0 1 b []0 2 FIGURE 4-1. Measurement setup 162

191 A practical example based on NVNA measurements When the input power of the amplifier rises, the output stage of the amplifier will gently enter into compression. Due to this nonlinear saturation phenomenon, multiple harmonics are created which rise above the noise floor. Since one can not predict the number of signal containing spectral lines in the measured spectra, the stochastic selection criterion will be used. To apply this stochastic test, repeated measurements ( N = 5 ) are performed. The level of significance is chosen to be ρ = This means that on average 1 out of 1000 spectral lines may be wrongly selected for every measured wave. However, the used procedure combines the incident and reflected waves, so that on average a total of 4 spectral lines out of 1000 may be wrongly selected. Figure 4-2 to Figure 4-5 show the raw measured output spectra β [ 2] of the amplifier at an input power of -14 dbm, 0.25 dbm, 5.5 dbm and 10 dbm. The harmonics selected through the stochastic selection criterion are pin-pointed with an o -sign. The upper right corner of each plot shows a zoom of the significant harmonics β [] Spectral Component FIGURE 4-2. Measured β [ 2] -wave at an input power of -14 dbm 163

192 An Automatic Harmonic Selection Scheme β [] Spectral Component FIGURE 4-3. Measured β [ 2] -wave at an input power of 0.25 dbm β [] Spectral Component FIGURE 4-4. Measured β [ 2] -wave at an input power of 5.5 dbm 164

193 A practical example based on NVNA measurements β [] Spectral Component FIGURE 4-5. Measured β [ 2] -wave at an input power of 10 dbm Table 4-2 summarises the spectral components or harmonics selected for the different input powers: Input Power Selected Spectral Component [dbm] TABLE 4-2. Spectral components selected by the harmonic selection criterion (the underlined values are not harmonically related) These experimental results show that all the significant harmonics which rise above the noise floor are automatically selected. However, Table 4-2 also shows that some selected spectral components are not harmonically related to the fundamental (see underlined values) and hence are not signal carrying lines. This is a consequence of the stochastic selection procedure which allows that 4 out of 1000 spectral lines may be wrongly selected as being signal containing lines. To solve this problem a combined selection criterion will be used. 165

194 An Automatic Harmonic Selection Scheme 4.5 Combined selection criterion To obtain a reasonable sensitivity of the previous algorithm to newly appearing and significant lines, the significance level ρ must be large enough. However, to exclude all noisy lines ρ must be small. A fair trade-off between these contradictory constraints is to allow that on average 1 spectral line may be wrongly selected for each experiment. The corresponding value for ρ is then 1 n, with n the number of measured spectral lines in one experiment. During a typical NVNA measurement, that consists of a long series of experiments, many noisy lines will hence be selected. As a result the measurement and calibration time becomes unnecessarily large. To solve this problem, a combined selection method is proposed. For every harmonic selected by the previous selection criterion, the largest number of consecutive experiments where this harmonic appears ( m), is determined. Since the occurrence of an error at spectral line x is independent for 2 consecutive experiments, the combined probability that a spectral line would be wrongly selected in m consecutive experiments is then ρ m ( ρ is the probability of selecting a noisy line in a single experiment). Thus, the probability that a line is wrongly selected in m consecutive measurements becomes very small. Hence, if the line is selected in more than m consecutive measurements, it can be considered to be a significant line. If a maximal error probability P ε of wrongly selecting a noise contribution is desired, the number of successive realisations where the spectral contribution must be present in order to be considered significant, is: P ε ρ m log( P ε ) m log( ρ) (4-16) Measurements of the amplifier of Paragraph 4.4 are used to demonstrate the difference between the simple and the combined selection method. The error probability ρ for one experiment is 4/1000. If one chooses that the harmonic must appear in five or more successive experiments to be considered significant, then the probability that a noisy line would be selected, becomes very small: = 1e 12. The spectral components selected by this combined method are 5, 9, 13, 17, 21, 25, 29, 33, 37 and 41. When comparing these spectral components with those in Table 4-2, the underlined (and wrongly selected) spectral 166

195 Conclusion components disappear. In Figure 4-6 the harmonics selected by the simple selection method are pinpointed with a o -sign, those selected by the combined method are pinpointed with a + -sign. β [] Spectral Component FIGURE 4-6. Measured β [ 2] -wave at an input power 10 dbm: single selection method (o), combined selection method (+) At the cost of a few repeated measurements this technique hence allows to select accurately and fully automatically the significant harmonics present in the measured data 4.6 Conclusion The proposed method allows, based on the (almost-always-fulfilled) hypothesis that the noise is zero mean, to discriminate signal carrying spectral lines from noise-only spectral contributions at the cost of a low ( >5 ) number of repeated measurements. Up till now, selection of the significant spectral contributions in NVNA measurements was mainly a task of trial and error. The proposed method enables the NVNA technology with a key component on the way to an automatic measurement and calibration procedure, freeing the user from a selection decision that seems to be arbitrary and is strongly device dependent. Usefulness of the method was proven on real-world measurements and simulations. 167

196 An Automatic Harmonic Selection Scheme 4.7 Appendices Appendix 4.A : Student t- and t 2 -distribution 1. Introducing the notation Consider Z k samples independent and standard normal distributed, then: N X = Z2 k with Z k N( 01, ) k N (4-17) k = 1 is χ2 N -distributed with N degrees of freedom: X χ2 N. Consider the stochastic variable X to be χ 2 -distributed with N degrees of freedom (dof) and Z standard normal distributed and independent of X. The ratio T: T = Z --- X N (4-18) with Z N( 01, ) and X χ2 N, is then Student t-distributed with N dof: T t N For a Student t 2 -distribution, equation (4-18) becomes: T 2 = Z X N (4-19) and is t 2 -distributed with N dof: T 2 t2 N. It can be proven that the t 2 -distribution is a special case of the more general Fisher s F -distribution (Appendix 4.B). 168

197 Appendices ( x µ 2. Is equation (4-10) T x ) = a Student t -distributed with 2( N 1) degrees of s freedom? x ( N) Consider N complex samples x [ k] with k = 1,, N which are normal distributed N ( µ x, σ x ). On one hand, it can be proven that [38]: Nx ( µ x ) N( 01, ) σ x (4-20) is a standard normal distribution with x = 1 N --- N x[ k] k = 1 the sample mean. On the other hand, it can be proven that [38]: ( N 1)s x χ2 σ2 2( N 1) x (4-21) is χ 2 -distributed with 2( N 1) dof and s2 1 N x = the sample N 1 ( x [ k] x) ( x [ k] x) * k = 1 variance. The superscript * denotes the complex conjugate. Since both hypothesis needed for a Student t -distribution are hence fulfilled, equation (4-10) ( x µ T x ) = is indeed Student t -distributed with 2( N 1) degrees of freedom. It is then s x N easy to pass from a Student t-distribution to a Student t 2 -distribution. Appendix 4.B : The Fisher s F-distribution Consider two independent stochastic variables X χ2 M and Y χ2 N. Then the ratio F X ---- = ---- M Y --- N (4-22) is a Fisher s F-distribution with M dof in the numerator and N dof in the denominator: F F MN,. 169

198 An Automatic Harmonic Selection Scheme Is a Student t 2 -distribution a special case of the Fisher s F -distribution? Nx ( µ x ) From Appendix 4.A it is known that N( 01, ). So that the hypothesis of a χ - σ 2 x distribution is fulfilled and: Nx ( µ x )( x µ x ) * χ2 σ2 2 x (4-23) is X 2 -distributed with 2 dof. Furthermore, (Appendix 4.A) ( N 1)s x χ2 σ2 2( N 1) x (4-24) is also χ 2 -distributed with 2( N 1) dof. As a result by taking into account the definition relation of an F -distribution: F Nx ( µ x )( x µ x ) * σ x ( x µ x )( x µ x ) * = = = T 2 F ( N 1)s x s2 22N, ( 1) ---- x σ2 2( N 1) N x (4-25) one can conclude that the Student t 2 -distribution is no more than a special case of the Fisher s F-distribution. So that the statistical tables of the F-distribution can be used when working with a t 2 -distribution. Appendix 4.C : Sampling distribution of the t 2 -distribution The t 2 -distribution has a known sample distribution F S, describing the behaviour of (4-12) for a finite number of samples, given by: 170

199 Appendices ν 2 F S 1 I ξ ν 1 =, (4-26) where ν 1 represents the degrees of freedom of Nxx * σ2 x (= 2) and ν 2 the degrees of freedom ( N 1)s 2 of x (= 2(N-1)). I is the Incomplete Beta function [38]: σ2 ξ x ν 2 I ξ ν ν 1, ξ = r 2 ( 1 r) 2 ν ν dr 1 0 r 2 ( 1 r) 2 dr 0 ν 2 (4-27) with ν 1 ξ T ν 2 1 = 2 (4-28) 171

200 An Automatic Harmonic Selection Scheme 172

201 CHAPTER 5 CONTINUOUS WAVE AND MODULATED MEASUREMENTS Abstract: Nothing is more practical than a good theory. Up to this chapter, the theoretical side of the Nonlinear Vectorial Network Analyser measurement setup has been described and worked out. To ease the understanding of the theoretical aspects, a practical measurement and its interpretation will be performed. The potentials of the NVNA are illustrated through continuous wave and modulated measurements of an amplifier in the telecommunication band (900 MHz base frequency). After this chapter, it will become more clear that the Nonlinear Vectorial Network Analyser is the appropriate tool to gain insight in the operation of a nonlinear system. 173

202 Continuous Wave and Modulated Measurements 5.1 Measurement cook book If I want to make a pie, I open my cook book, follow the recipe and at the end I may get a delicious result. Wouldn t it be nice if we could do the same thing when measuring the behaviour of a system? We take our measurement cook book, choose a type of measurement, follow the procedure and at the end we obtain the desired characteristics... Unfortunately, designing a measurement cook book is a very difficult task. The diversity of measurement types is very large and the setup depends on the type of measurements. However, this chapter can provide the user with some useful guidelines, for measurements with the Nonlinear Vectorial Network Analyser (NVNA) Measurement guidelines The following guidelines are written for the NVNA, but can also be used for other measurement setups. 1. Selecting the NVNA hardware The hardware setup of the NVNA must be adapted to the number of Device Under Test (DUT) ports. If the DUT contains more ports than the available measurement ports, the user must select the most important DUT ports to measure and terminate or excite the other ports using impedances or power levels that are representative for the targeted application. 2. Selecting the excitation signal The excitation signal must be chosen to match the application conditions of the measured device as closely as possible (modulated or CW,...). 3. Experiment design Measurement range: characterising the behaviour of a DUT requires to setup multiple sweeps: * Frequency sweep: depending on the application of the DUT * Power sweep: depending on the operation limits of the DUT * Bias sweep: depending on the operation limits of the DUT 174

203 Measurement cook book First, a coarse grid of frequency and power points can be scanned, to get a rough idea of the DUT s behaviour and to detect in which area the important phenomena occur. In a second step, a more specific, dense space can be scanned. This way of working avoids long measurement campaigns with unsatisfying results. Measurement points: when determining the number of acquisition points, 2 constraints must be taken into account. *The number of acquisition points determines the signal-to-noise ratio (SNR) of the measured signal after taking the Discrete Fourier Transform (DFT). The SNR normally rises as a NPts -law, where NPts denotes the number of acquisition points [33]. *To convert the measured spectra to a Fast Fourier Transform (FFT) grid, the number of measured points NPts must satisfy the following rule: NPts = 2 N with N N. By experience one knows that a number of acquisition points equal to 1024 ( = 2 10 ) gives good results and results in a SNR of 60 db. Hence, when measurements with a SNR of X SNR db are required, the following rule allows to determine the order of magnitude of the number of 2 acquisition points: NPts = 1024( X SNR 3600). Repeated measurements: Repeated measurements reduce the measurement noise. It can be proven that 6 repeated measurements are sufficient [33] to extract sensible information about the measurement noise. 4. Experiment constraints The limited amount of available measurement time, calibration time and data storage space puts constraints on the following experiment settings: * Number of frequencies: #F * Number of power points: #P 175

204 Continuous Wave and Modulated Measurements * Number of bias settings: #B * Number of acquisition points: #NPts * number of repeated measurement: #Rep Measurement time The total required measurement time can be calculated by the following rule: T meas = #Rep #F #P #B ( T NPts + T reg ) (5-1) where T NPts is the acquisition time needed to measure NPts measurement points and T reg is the time needed by the measurement instrument to be in steady-state after changing an instrument setting. For a continuous wave (CW) measurement done with the NVNA, T NPts + T reg is about 3 seconds, while for modulated measurements T NPts + T reg equals about 8 seconds. The difference in time between CW and modulated measurements is explained by the fact that for modulated measurements the modulation waveform has to be loaded in the arbitrary waveform generator. Calibration time The required time T cal to perform a linear calibration is determined by the following rule: T cal = #Std #Rep #SC #Pts ( T NPts + T reg ) (5-2) with #Std the total number of calibration standards, #Pts the number of measurement ports and #SC the total number of spectral components that needs to be calibrated. Note that for modulated signals, the total number of spectral components can be very high and hence has a serious influence on the calibration time. The time required to perform a power calibration scaling law: T calpow obeys the following T calpow = #F #Rep ( T NPts + T reg + T pow ) (5-3) where T pow is the measurement time of the power meter. The phase calibration time T calphase can be calculated by: 176

205 Measurement cook book T calphase = #FundF #Rep ( T NPts + T reg ) (5-4) with #FundF the number of fundamental frequencies to measure between 600 MHz and 1200 MHz. Data storage space The total required storage space to save the measured data can be approximated by the following rule: #Rep #F #P #B #Pts #NPts 8bytes (5-5) with #Pts the number of measurement ports. Note that 8 bytes are needed to save 1 real number. 5. Downconversion requirements The downconversion settings are taken care off by the measurement instrument so that leakage is avoided. 6. Calibration procedure For a linear calibration, the user must specify the calibration approach, the calibration type and the number of error correction terms (Chapter 3). *Calibration approach: the classical analytical approach or the stochastic based calibration. *Calibration type: a Short-Open-Load-Thru (SOLT) calibration, a Thru-Reflect-Line (TRL) calibration or a Line-Reflect-Match (LRM) calibration. *Error terms: 8, 12 or 16 error correction terms. To characterise the nonlinear behaviour of the DUT, a phase and power calibration are necessary. 177

206 Continuous Wave and Modulated Measurements 5.2 Nonlinear continuous wave measurements of an amplifier Since one of the most commonly used components in RF designs is an amplifier, it is a good example to take a closer look at. Here, the continuous wave (CW) behaviour of a microwave amplifier is analysed Device Under Test The Device Under Test (DUT) is an amplifier of the MRFIC amplifier class of Motorola [26]. The MRFIC2006 is an integrated power amplifier for linear operation in the 800 MHz to 1 GHz frequency range. Applications for the MRFIC2006 include CT-1 cordless telephones, remote controls, video, low cost cellular radios and ISM band transmitters. The main characteristics of the amplifier are: Freq. Range (MHz) Gain (db) 900 MHz Max. Input Power (dbm) 1 db Comp. Point (dbm) DC Power Current (ma) Voltage (V) 46 4 TABLE 5-1. Properties of the MRFIC2006 amplifier Measurement setup The nonlinear characteristics of the amplifier are measured by the setup shown in Figure 5-1: Syncro CLK ADC ADC Sample CLK ADC ADC A A A A Downc Downc Downc Downc β α [] 1 [] 1 α [] 2 β [] 2 ATT ATT IF-Gen ATT ATT RF GEN β [] 1 α [] 1 α [] 2 β [] 2 DUT Port 1 Port 2 50Ω FIGURE 5-1. CW measurement setup of an amplifier 178

207 Nonlinear continuous wave measurements of an amplifier The amplifier is excited by an RF source and terminated in a 50Ω load impedance. The incident and reflected waves at both ports of the amplifier are measured through couplers and downconverted to a much lower frequency by the harmonic samplers (see Chapter 1). Finally, the measured waves are digitized by four synchronised analog-to-digital convertor (ADC) cards. The whole measurement setup is phase coherent to the common 10 MHz reference clock of the RF source Experiment design To measure the behaviour of the DUT in its operating range, a frequency sweep and a power sweep are needed: Input power sweep: Frequency sweep: 14 dbm 5 dbm 600 MHz 1370 MHz in 96 steps of 0.2 db in 78 steps of 10 MHz The number of acquisition points taken by the ADC cards is Repeated measurements are used to somewhat reduce the noise variance on the measurements and to get uncertainty bounds. The figures shown below are obtained by taking the mean of 5 repeated measurements. Instead of taking 5 separate measurements of 1024 measurement points each, 5*1024 measurement points are taken at once and split up in 5 repetitions of 1024 measurement points afterwards. It can still be assumed that the periods are independent, even if they are measured all together. As a result, the 5 repetitions are always synchronised, so that one can easily compute the mean Downconversion requirements Before digitizing the measured waves, they must be downconverted to a lower frequency band. Thereto, the frequency of the IF generator or the FracN of the downconvertors must be correctly set, so that the spectral lines are converted to the frequencies of the FFT lines in the IF spectrum (see Chapter 1) Calibration The linear calibration consists of a two-port Short-Open-Load-Thru calibration with 8 error correction terms and is extended by a power and phase calibration. As explained in paragraph 3.7 of Chapter 3, the calibration procedure is done after the measurements are performed. 179

208 Continuous Wave and Modulated Measurements Measurement results 1. Uncalibrated spectra Figure 5-2 shows the uncalibrated measured spectra of the incident and reflected waves when the DUT is excited by a sinusoidal signal of 900 MHz and an input power of -5 dbm. In the upper right corner of each plot, a zoom of the first 60 spectral components is shown. The most significant harmonics are present in this region. As can be seen from the measured output wave β [ 2], harmonics rise above the noise level when the amplifier goes into compression. Note that when comparing the measured incident wave α[ 1 ] and the output wave β [ 2], one would conclude that the amplifier has no significant amplification factor. However, due to the large signal level of β [ 2], an attenuator (10 db) was switched on in the b [ 2] channel of the NVNA (Figure 5-1). Taking in consideration that it considers here uncalibrated data, explains why the amplification factor is seemingly small. The drop of the noise floor at the end of the frequency band is caused by the anti-aliasing filters of the ADC-cards α [ 1 ] -100 β [ 1 ] Spectral Component Spectral Component α [ 2 ] -100 β [ 2 ] Spectral Component Spectral Component FIGURE 5-2. Uncalibrated spectra of the incident and reflected waves 180

209 Nonlinear continuous wave measurements of an amplifier This type of measurements and representation allows to get a quick idea of the spectral content of the measured signals at one pair carrier frequency - power level. The user can easily see which harmonics are significant and thus, worth to be considered further on. 2. Harmonics behaviour When characterising a nonlinear system, the behaviour of the higher order harmonics becomes important. Figure 5-3 represents the power behaviour of the fundamental and 4 harmonics of the incident and reflected waves as a function of the input power and for a carrier frequency of 900 MHz. These figures are obtained from repeated and calibrated measurements. The figures on the right side represent the noise floor of the measured data. Note the jumps present in the measured characteristic of the harmonics of the incident wave switchings of signal paths in the generator. α [ 1]. These are caused by Some conclusions concerning the behaviour of the amplifier can be drawn from Figure 5-3: The input as well as the output match of the amplifier is poor. This can cause problems when cascading the amplifier with other devices and emphasises the importance of a model that is able to cope with additional source and load mismatches. Especially for the output wave β [ 2], the higher order harmonics become significant. The noise level of the fundamental tone rises with the input power. This means that some additive and multiplicative noise is present and should be taken into account. 181

210 Continuous Wave and Modulated Measurements α [ 1 ] [dbm] β [ 1 ] [dbm] β [ 2 ] [dbm] α [ 2 ] [dbm] Input Power [dbm] Input Power [dbm] Input Power [dbm] Input Power [dbm] Fund Harmon 1 Harmon 2 Harmon 3 Harmon 4 α [ 1 ] [db] Std Std β [ 1 ] [db] Std α [ 2 ] [db] β [ 2 ] [db] Std Input Power [dbm] Input Power [dbm] Input Power [dbm] Input Power [dbm] FIGURE 5-3. (a) Power characteristic of the fundamental and 4 harmonics for the incident and reflected waves. (b) Standard deviation of the measured data for the incident and reflected waves. 182

211 Nonlinear continuous wave measurements of an amplifier These types of plots give an idea of the behaviour of the nonlinearity at one single frequency. However, what happens from one frequency to the other? 3. Exploring the nonlinearity When a system is to be designed, it is important to know how the system behaves as a function of frequency and power. The behaviour of a linear system is fully characterised by the location of its poles and zeros [34], which results in the linear transfer function. However, for a nonlinear system this is no longer the case. Other mechanisms are necessary to discover and understand the behaviour of the nonlinearity. To characterise a NICE system, it is important to find the dominating energy transport channels which describe the behaviour of the nonlinearity. Or, in other words, how can the output energy of a system be explained by contributions of the input energy? Consider for example Figure 5-4. The input energy is divided over the fundamental tone and 2 harmonics (here symbolised by a circle containing the harmonic number). The same energy containers are present for the output signal. How is the energy transported from the input to the output? Is there a linear relationship? Or, do different components of the input spectrum combine to yield an output tone? Input Signal 1-1?? 1-1 Output Signal FIGURE 5-4. Energy transport mechanism Finding the energy transport mechanisms which describe the behaviour of the system can be done in different ways: 183

212 Continuous Wave and Modulated Measurements 1. Making all combinations of the input tones allows to find the most important contributions to the output tones. However, a lot of combinations are possible and hopefully only a few of these combinations will contain a dominant contribution. Thus, the number of combinations must be limited. 2. If a nonlinearity can be described in the least squares sense by a Volterra series (see Chapter 6), the total number of combinations can be reduced by only considering these Volterra combinations. Consider for example the Volterra expansion of the fundamental output tone β [ 2] ( ω) as a function of the fundamental input tone α [ 1] ( ω) : β [ 2] ( ω) = H 1 α [ 1] ( ω) + H 3 α [ 1] ( ω)α [ 1] ( ω)α [ 1] ( ω) +. The Volterra kernels H 1 and H 3 will then give the most important contributions. Note, however, that one is not interested in the value of the kernels, but only wants to know how the system transports the energy from the input to the output Input Signal 1-1 H 1 H Output Signal FIGURE 5-5. Volterra energy contributions for the fundamental tone 3. The Volterra combinations can further be reduced by omitting all combinations in which sub-combinations result in DC components. Consider for example the following combination: H 3 α [ 1] ( ω)α [ 1] ( ω)α [ 1] ( ω), where the sub-combination α [ 1] ( ω)α [ 1] ( ω) results in a DC component. It can be proven that these contributions have a variable amplitude behaviour, but a coherent phase behaviour. As a result, their contribution to the global behaviour of the system can not be distinguished from other contributions. 184

213 Nonlinear continuous wave measurements of an amplifier Not all of these Volterra contributions will be significant to describe the energy transport from the input to the output. The contributions can be divided in three classes: 1. Dominant contributions: these contributions will describe the global behaviour of the system. 2. Small coherent contributions: these will have a minor influence on the global behaviour of the system. 3. Incoherent contributions: these have no influence at all on the behaviour of the system and can be neglected. The difference between the dominant and only small coherent contributions can be made based on plots of the contributions as a function of frequency and input power. Let s try to discover the nonlinear behaviour of the MRFIC amplifier, by detecting the major energy transports for the fundamental tone and the second harmonic. Fundamental tone Consider that the output fundamental tone of the amplifier can be written as the following Volterra series: 2 β [ 2] ( ω) = H 1 α [ 1] ( ω) + H 3 α [ 1] ( ω)α [ 1] ( ω) (5-6) Which of these contributions is the most important and which ones can be neglected? The results obtained in this paragraph are based on calibrated measurements. The behaviour of a VIOMAP kernel H m can be obtained by considering this kernel as the dominant one and hence, by neglecting all other contributions in the Volterra series. For example, out of equation (5-6) the behaviour of H 3 is in general obtained by: β H [ 2] ( ω) H 3 = α [ 1] ( ω)α [ 1] ( ω) α [ 1] ( ω)α [ 1] ( ω) (5-7) However, if H 3 is considered to be a dominant contribution, a good approximation of the VIOMAP kernel is obtained by neglecting all contributions in equation (5-6) which do not contain H 3. This approximation will be used to determine the behaviour of the kernels in this paragraph. 185

214 Continuous Wave and Modulated Measurements Figure 5-6 represents the magnitude of the amplifier gain for the fundamental tone as a function of carrier frequency and power (magnitude of H 1 ). This is the equivalent of the linear transfer function of the amplifier. It is clear that when the input power rises, the gain decreases, or in other words, the amplifier goes into compression. From the behaviour of the amplifier gain as a function of frequency, one can conclude that the amplifier is optimized for a small bandwidth around 900 MHz. The phase behaviour of the amplifier gain for the fundamental tone as a function of frequency and power is shown in Figure 5-7 (phase of H 1 ). The behaviour as a function of frequency is smooth and will not introduce envelope distortions for narrowband signals. For broadband signals, however, nonlinear phase distortions will be introduced, especially in the passband, which will distort the time signals. β [ 2 ] α [ 1 ] [db] FIGURE 5-6. Magnitude of β [ 2] ( ω) α [ 1] ( ω) of the fundamental as a function of carrier frequency and input power 186

215 Nonlinear continuous wave measurements of an amplifier β [ 2 ] α [ 1 ] [deg] Input Power [dbm] FIGURE 5-7. Phase of β [ 2] ( ω) α [ 1] ( ω) of the fundamental as a function of frequency and input power Figure 5-8 and Figure 5-9 represent respectively the magnitude and phase of 2 β [ 2] ( ω) α [ 1] ( ω)α [ 1] ( ω) (= H 3 ) as a function of carrier frequency and power. From these figures, it is clear that the contribution of H3 will not influence the global phase behaviour of the amplifier and has only a small influence on the amplitude behaviour. From Figure 5-6 to Figure 5-9, one can conclude that both contributions H 1 and H 3 are coherent, but that H 1 is the dominant contribution. Since H 3 has only a small contribution, it will not be necessary to take higher order Volterra contributions into account during modelling. 187

216 Continuous Wave and Modulated Measurements Magnitude H 3 [db] 2 FIGURE 5-8. Magnitude of β [ 2] ( ω) α [ 1] ( ω)α [ 1] ( ω) Phase H 3 [deg] Input Power [dbm] 2 FIGURE 5-9. Phase of β [ 2] ( ω) α [ 1] ( ω)α [ 1] ( ω) 188

217 Nonlinear continuous wave measurements of an amplifier A second important question is to know if the higher order input harmonics have a contribution to the output fundamental tone. Consider for example that the second and third input harmonic contribute to the output fundamental tone: β [ 2] ( ω) = H 3 α [ 1] ( 2ω)α [ 1] ( ω) + H 5 α [ 1] ( 3ω)α [ 1] ( 2ω) (5-8) Are these coherent contributions? The behaviour of a VIOMAP kernel H 3 can be obtained by considering this kernel as the dominant one and hence, by neglecting all other contributions in the Volterra series. Figure 5-10 and Figure 5-11 represent respectively the magnitude and phase of β [ 2] ( ω) α [ 1] ( 2ω)α [ 1] ( ω) as a function of frequency and input power. In this contribution, a large signal component α [ 1] ( ω) is combined with a small signal component α [ 1] ( 2ω). Both figures show incoherent contribution results, which have no influence at all on the global system behaviour. Magnitude H 3 [db] FIGURE Magnitude of β [ 2] ( ω) α [ 1] ( 2ω)α [ 1] ( ω) 189

218 Continuous Wave and Modulated Measurements Phase H 3 [deg] FIGURE Phase of β [ 2] ( ω) α [ 1] ( 2ω)α [ 1] ( ω) Figure 5-12 and Figure 5-13 represent respectively the magnitude and phase of β [ 2] ( ω) α [ 1] ( 3ω)α [ 1] ( 2ω) as a function of frequency and input power. In this contribution 2 small signal components α [ 1] ( 3ω) and α [ 1] ( 2ω) are combined. This is also an incoherent contribution, which has no influence at all on the global system behaviour. Magn. H 5 [db] FIGURE Magnitude of β [ 2] ( ω) α [ 1] ( 3ω)α [ 1] ( 2ω) 190

219 Nonlinear continuous wave measurements of an amplifier Phase H 5 [deg] FIGURE Phase of β [ 2] ( ω) α [ 1] ( 3ω)α [ 1] ( 2ω) The above figures show that when modelling the amplifier for the fundamental tone, the most important contribution is delivered by the linear transfer function. However, higher order combinations of the input fundamental tone can not be neglected at higher input power levels. Combinations of other input harmonics will not influence the global behaviour and hence, will not further optimize the modelling procedure. Second harmonic The same methodology can be used to detect the major energy transport channels for the second harmonic output tone. Consider that the second harmonic output energy is obtained through the following contributions: β [ 2] ( 2ω) = H 2 α [ 1] ( 2ω) + H 4 α [ 1] ( ω)α [ 1] ( ω). (5-9) Figure 5-14 and Figure 5-15 represent respectively the magnitude and phase of the amplifier gain for the second harmonic: β [ 2] ( 2ω) α [ 1] ( 2ω) (= H 2 ). From both figures, it is clear that there is nearly no correlation between β [ 2] ( 2ω) and α [ 1] ( 2ω) and thus, α [ 1] ( 2ω) does not contribute to the energy of β [ 2] ( 2ω). The energy of β [ 2] ( 2ω) can not be explained by a linear contribution, but only by a nonlinear contribution. 191

220 Continuous Wave and Modulated Measurements Magn. β [ 2 ] 2ω ( ) α 1 [ ] ( 2ω ) [db] Input Power [dbm] FIGURE Magnitude of β [ 2] ( 2ω) α [ 1] ( 2ω) as a function of frequency and input power Phase β [deg] [ 2 ] ( 2ω ) α [ 1 ] ( 2ω ) FIGURE Phase of β [ 2] ( 2ω) α [ 1] ( 2ω) as a function of frequency and input power 2 The first nonlinear contribution verified, is β [ 2] ( 2ω) α [ 1] ( ω). Figure 5-16 and Figure show the magnitude and phase of β [ 2] ( 2ω) α [ 1] ( ω) as a function of frequency and input 192

221 Nonlinear continuous wave measurements of an amplifier power. From these figures, which have an identical behaviour as Figure 5-6 and Figure 5-7, a high correlation between β [ 2] ( 2ω) and α [ 1] ( ω) is observed. This means that the energy of β [ 2] ( 2ω) is mainly obtained from the fundamental tone α [ 1] ( ω). Combinations of other input harmonics have a negligible influence on the global behaviour of the second harmonic output tone. [db] ( ω ) [ ] Magn. β [ 2 ] ( 2ω ) α FIGURE Magnitude of β [ 2] ( 2ω) α [ 1] ( ω) as a function of frequency and input power Phase β [deg] [ 2 ] ( 2ω ) α [ 1 ] ( ω ) 2 Input Power [dbm] 2 FIGURE Phase of β [ 2] ( 2ω) α [ 1] ( ω) as a function of frequency and input power 193

222 Continuous Wave and Modulated Measurements Searching the energy transport channels of a system, allows to obtain a good insight in the system. Hence, the measurement data can be properly preprocessed before modelling the device. Modelling also becomes a lot easier: non parametric models are obtained nearly directly from the figures. 194

223 Nonlinear AM-modulated measurements of an amplifier 5.3 Nonlinear AM-modulated measurements of an amplifier Device Under Test The Device Under Test (DUT) is a MAR amplifier class of Mini-Circuits [25]. The MAR amplifiers are MMIC broadband amplifiers based on a Darlington transistor pair. The MAR-6 IC is selected for his low noise figure and large gain factor: Freq. Range (MHz) Gain (db) 1GHz Max. Input Power (dbm) 1 db Comp. Point (dbm) Noise Figure (db) DC DC Power Current (ma) Volt TABLE 5-2. Properties of the MAR-6 amplifier Measurement setup The characteristics of an amplifier under narrowband AM modulated excitation can be measured using the setup of Figure 5-18: Syncro CLK ADC ADC Sample CLK ADC ADC A A A A Downc Downc Downc Downc β α [] 1 [] 1 α [] 2 β [] 2 ATT ATT IF-Gen ATT ATT RF GEN β [] 1 α [] 1 α [] 2 β [] 2 DUT Port 1 Port 2 50Ω AWG GEN FIGURE Measurement setup of an amplifier under AM excitation The amplifier is excited by a narrowband AM modulated signal, obtained through mixing of an RF signal and an IF signal. The output of the amplifier is terminated by a 50Ω load 195

224 Continuous Wave and Modulated Measurements impedance. The incident and reflected waves at both ports of the amplifier are measured through couplers and downconverted to a much lower frequency by the harmonic samplers (see Chapter 1). Finally, the measured waves are digitized by four synchronised ADC cards. The whole measurement setup is synchronised by the common 10 MHz reference clock of the RF source Experiment design To measure the behaviour of a DUT under modulated excitation, a frequency sweep, a carrier power sweep as well as a modulation power sweep is needed: Carrier input power sweep: Carrier frequency sweep: 20 dbm 0 dbm 1000 MHz 1300 MHz in 21 steps of 1 db in 7 steps of 50 MHz Modulating signal: this signal consists of a narrowband multisine composed of six equally spaced spectral components with a frequency resolution of 5 khz and having a random phase relation. 1 Norm. Amplitude 0.5 FIGURE Amplitude characteristic of the modulating signal Modulation amplitude sweep: Frequency [khz] 0.2V 0.8V in 2 steps The selection of the number of acquisition points to be taken by the ADC cards is more complicated here. The number of acquisition points must be chosen such that the measured spectrum lies on FFT grid lines kf s N with k N, f s the ADC sampling frequency and N the number of acquisition points [10]. Furthermore, the FFT grid lines of the arbitrary waveform generator (AWG) signal must coincide with the FFT grid of the ADC cards to avoid leakage. One period of the AWG signal contains 4000 sampled data points. The number of 196

225 Nonlinear AM-modulated measurements of an amplifier acquisition points is also chosen to be 4000, to obtain identical frequency resolution in the acquisition and generator subsystem. Since repeated measurements reduce the noise variance on the measurements but also give uncertainty bounds, the figures shown below are obtained by taking the mean of 3 repeated measurements. Instead of taking 3 times 4000 measurement points, 3*4000 measurement points are taken at once and split up in 3 repetitions of 4000 measurement points afterwards. As a result the 3 repetitions are always synchronised, so that one can easily take the mean without concern about synchronisation of the different measurements Downconversion requirements Before digitizing the measured waves, they must be downconverted to a lower frequency band. Thereto, the IF generator or the FracN of the downconvertors must be correctly set, so that the spectral lines are converted to frequencies that lie on the FFT grid lines in the IF spectrum (see Chapter 1). The downconversion of modulated measurements requires additional considerations to avoid leakage. The frequency resolution f R of the Fast Fourier Transform used to transform the modulated measured waves to the frequency domain is chosen such that both the downconverted carrier and all the modulation components lie on a common grid kf R ( k N ) Calibration The linear calibration consists of a two-port Short-Open-Load-Thru calibration with 8 error correction terms and is extended by a power and phase calibration. The reference generator, used to perform the phase calibration, only allows to calibrate the phase relation between a carrier and its higher order harmonics (see Chapter 3). Thus, it is not yet possible to calibrate the phase relation between a carrier and the modulation components. However, since a narrowband modulation is considered here, the phase of the modulation components can be calibrated using the phase correction term of the carrier. Note that for wideband modulation, this can lead to severe errors. As explained in paragraph 3.7 of Chapter 3, the calibration procedure is done after the measurements are performed. 197

226 Continuous Wave and Modulated Measurements Measurement results 1. Uncalibrated spectra Figure 5-20 shows the uncalibrated measured spectra of the incident and reflected waves when the DUT is excited by a modulated wave with a carrier frequency of 1 GHz and a carrier input power of 0 dbm. Note that at this input power level, the amplifier is in compression. In the upper right corner of each plot, a zoom of the spectral components around the carrier frequency is shown. As can be seen from the measured waves β [ 1] and β [ 2], harmonics rise when the amplifier goes into compression. Furthermore, when a modulated signal passes through a weakly nonlinear circuit like an amplifier in compression, the signal bandwidth is broadened by odd-order nonlinearities. This is caused by the generation of mixing products between the individual frequency components of the spectrum and is called the spectral regrowth of the amplifier. The spectral regrowth is clearly visible in the zoom around the carrier frequency of the output wave β [ 2]. The drop of the noise floor at the end of the frequency band is caused by the anti-aliasing filters of the ADC-cards. This type of measurements and representation allows to get a quick idea of the spectral content of the measured signals at one carrier frequency. The user can easily see which harmonics are significant and if spectral regrowth is present. 198

227 Nonlinear AM-modulated measurements of an amplifier α [ 1 ] β [ 1 ] Spectral Component Spectral Component α [ 2 ] β [ 2 ] Spectral Component Spectral Component FIGURE Uncalibrated spectra of the incident and reflected waves 2. Modulation components and spectral regrowth When using a modulated excitation, it is important to know how the modulation components are treated by the nonlinear system. Figure 5-21 represents the power characteristic of the carrier tone and the 12 modulation components around the carrier for the incident and reflected waves as a function of the input power. The carrier frequency is set to 1 GHz and the modulation amplitude to 0.8V. These figures are obtained from repeated and calibrated measurements. The black curve in Figure 5-21 represents the behaviour of the carrier, the coloured curves represent the modulation components. Since one is only interested in the 199

228 Continuous Wave and Modulated Measurements global behaviour of the modulation components and not in the behaviour of one specific modulation component, no legend is put on Figure 5-21 in order not to overload this figure. α [ 1 ] [dbm] Input Power [dbm] β [ 1 ] [dbm] Input Power [dbm] α [ 2 ] [dbm] β [ 2 ] [dbm] Input Power [dbm] Input Power [dbm] FIGURE Power characteristic of the modulation components as a function of input power The input modulation components have all equal magnitude as can be seen from the input wave α [ 1]. Hence, one would expect equal magnitudes for all modulation components at the output of the amplifier. However, this is only true in the linear operation region of the amplifier as can be seen from the output wave β [ 2]. When compression rises, the modulation components are treated differently by the nonlinearity. Both the input and output match of the amplifier are poor. This can cause problems when cascading the amplifier with other devices. This emphasises the importance of a model that is able to cope with additional source and load mismatches. Figure 5-22 shows the behaviour of the spectral regrowth components around the fundamental frequency as a function of input power. The carrier frequency is set to 1 GHz and the 200

229 Nonlinear AM-modulated measurements of an amplifier modulation amplitude to 0.8V. As in Figure 5-21, the black curve represents the carrier frequency and the coloured curves represent the first 12 spectral regrowth components. α [ 2 ] [dbm] α [ 1 ] [dbm] Input Power [dbm] Input Power [dbm] β [ 2 ] [dbm] β [ 1 ] [dbm] Input Power [dbm] Input Power [dbm] FIGURE Power characteristic of the spectral regrowth as a function of input power From the measured output wave β [ 2], it is clear that the spectral regrowth components rise significantly above the noise floor when the amplifier goes into compression. Hence, when modelling the amplifier, these components will probably influence the model. These types of plots give an idea of the behaviour of the nonlinearity at one single carrier frequency. The next question that rises is to know what happens from one frequency to the other. 3. Exploring the nonlinearity As in paragraph 5.2.6, we will try to discover and understand the behaviour of the nonlinearities by searching for the major energy transport channels. This is done by testing the Volterra contributions and looking for the dominant ones. The results in this paragraph are based on calibrated measurements. 201

230 Continuous Wave and Modulated Measurements Fundamental tone Consider that the fundamental tone of the output wave the following Volterra series: β [ 2] of the amplifier can be written as β [ 2] ( ω) = H 1 α [ 1] ( ω) (5-10) The higher order contributions are experimentally verified to be very small and will not be shown here. Furthermore, combinations of higher order harmonics will not result in coherent contributions to the fundamental tone. The most important contribution is delivered by the linear transfer function H 1. Figure 5-23 represents the magnitude of the amplifier gain for the fundamental tone as a function of carrier frequency and power (= magnitude of H 1 ). This is equivalent to the linear transfer function of the amplifier. When the input power rises, the amplifier goes into compression. The amplifier has a linear behaviour as a function of the carrier frequency. The phase behaviour of the amplifier gain for the fundamental tone as a function of frequency and power is shown in Figure 5-24 (phase of H 1 ). The behaviour as a function of frequency is linear and will not introduce phase distortions in the specified frequency band. Magn. H Magn. B2/A1 1 [db] [db] Frequency [MHz] Input Power [dbm] FIGURE Magnitude of β [ 2] ( ω) α [ 1] ( ω) 202

231 Nonlinear AM-modulated measurements of an amplifier 150 Phase. H [deg] Phase B2/A1 1 [deg] Frequency [MHz] Input Power [dbm] FIGURE Phase of β [ 2] ( ω) α [ 1] ( ω) Second harmonic As for the amplifier in continuous wave regime, the major energy transport channel for the second harmonic of the fundamental tone is: 2 β [ 2] ( 2ω) = H 4 α [ 1] ( ω) (5-11) The linear contribution H 2 α [ 1] ( 2ω) is an incoherent contribution. Hence, the energy of β [ 2] ( 2ω) can only be explained by a nonlinear contribution. Figure 5-25 and Figure 5-26 show respectively the magnitude and phase of 2 β [ 2] ( 2ω) α [ 1] ( ω) as a function of the carrier frequency and input power. For high input powers, the magnitude behaviour of H 4 is coherent and the phase behaviour quite smooth and linear. 203

232 Continuous Wave and Modulated Measurements Magn. H Magn. B2/A1square 2nd 4 [db] harm. [db] Frequency [MHz] Input Power [dbm] 0 2 FIGURE Magnitude of β [ 2] ( 2ω) α [ 1] ( ω) Phase. H Phase B2/A1square 4 [deg] 2nd harm [deg] Frequency [MHz] Input Power [dbm] 2 FIGURE Phase of β [ 2] ( 2ω) α [ 1] ( ω) Modulation When using a modulated excitation signal, one is particularly interested in the energy transport mechanisms for the modulation components. Does the nonlinearity treat them in the same way as the fundamental tone? Is there only a linear contribution or do higher order terms influence the global behaviour? 204

233 Nonlinear AM-modulated measurements of an amplifier Consider the first modulation component around the carrier at frequency ω+ with the frequency resolution of the modulating multisine. The major energy transport channels which describe the behaviour of this component are given by the following Volterra series: β [ 2] ( ω+ ) = H 1 α [ 1] ( ω+ ) + H 3 α [ 1] ( ω + 2 )α [ 1] ( ω)α [ 1] ( ω ) (5-12) Figure 5-27 and Figure 5-28 represent respectively the magnitude and phase behaviour of the equivalent linear transfer function contribution for the first modulation component as a function of the carrier frequency and carrier input power. When comparing these figures with Figure 5-23 and Figure 5-24, one can see an identical behaviour. Hence, the linear contributions of the modulation components H 1 and of the fundamental tone H 1 have the same behaviour. All modulation components are compressed when the input power of the fundamental rises. The contribution is close to be frequency independent, only a small linear delay is present. 15 Magn. H 1 [db] Magn. B2/A1 1ste mod [db] Frequency [MHz] Input Power [dbm] 0 FIGURE Magnitude of β [ 2] ( ω+ ) α [ 1] ( ω+ ) 205

234 Continuous Wave and Modulated Measurements Phase B2/A1 1ste mod [deg] Phase. H 1 [deg] Frequency [MHz] Input Power [dbm] 0 FIGURE Phase of β [ 2] ( ω+ ) α [ 1] ( ω+ ) Figure 5-29 and Figure 5-30 represent respectively the magnitude and phase behaviour of β [ 2] ( ω+ ) α [ 1] ( ω + 2 )α [ 1] ( ω)α [ 1] ( ω ) as a function of the carrier frequency and carrier input power. This is only a small contribution, which varies linearly as a function of the carrier frequency and input power. All other possible combinations of input modulation components and the fundamental tone which result in the same output modulation component β [ 2] ( ω+ ), have an identical behaviour. The dominating contribution in the energy transport of the modulation components is the linear contribution. This also implies that to model the modulation components, a linearisation can be used. This nicely illustrates that a nonparametric model can easily be put together starting from these measurement results. 206

235 Nonlinear AM-modulated measurements of an amplifier -30 Magn. H [db] Magn. B2/A1 1ste 3 mod [db] Frequency [MHz] Input Power [dbm] 0 FIGURE Magnitude of β [ 2] ( ω+ ) α [ 1] ( ω + 2 )α [ 1] ( ω)α [ 1] ( ω ) Phase. H Phase B2/A1 1ste 3 [deg] mod [deg] Frequency [MHz] Input Power [dbm] FIGURE Phase of β [ 2] ( ω+ ) α [ 1] ( ω + 2 )α [ 1] ( ω)α [ 1] ( ω ) Spectral Regrowth Is the linear transfer function also the most import contribution when describing the energy transport at the frequencies of the spectral regrowth components? Intuitively one is tempted to discard this possibility, since no external energy is exciting the frequencies at which the spectral regrowth components are present in the input wave. However, a source pull effect could inject unwanted energy there. 207

236 Continuous Wave and Modulated Measurements Consider that the major energy transport channels for the first spectral regrowth component at frequency ω + 7 can be approximated by the following Volterra series: β [ 2] ( ω + 7 ) = Ĥ 1 α [ 1] ( ω + 7 ) + Ĥ 3 α [ 1] ( ω + 6 )α [ 1] ( ω+ )α [ 1] ( ω) + Ĥ 5 α [ 1] ( ω + 4 )α [ 1] ( ω + 3 )α [ 1] ( ω) (5-13) Note that this Volterra series can be extended by higher order contributions when necessary. Figure 5-31 and Figure 5-32 represent respectively the magnitude and phase behaviour of the linear transfer function Ĥ 1 as a function of the carrier frequency and input power. From both figures, it is clear that this is a totally uncorrelated behaviour and confirms intuition. Since the linear contribution is not significant, other mechanisms will be responsible for creating the spectral regrowth components. 60 Magn. Ĥ Magn. B2/A1 1ste 1 [db] SR [db] Frequency [MHz] Input Power [dbm] FIGURE Magnitude of β [ 2] ( ω + 7 ) α [ 1] ( ω + 7 ) 0 208

237 Nonlinear AM-modulated measurements of an amplifier 500 Phase. Phase B2/A1 Ĥ 1ste 1 [deg] SR [deg] Frequency [MHz] Input Power [dbm] 0 FIGURE Phase of β [ 2] ( ω + 7 ) α [ 1] ( ω + 7 ) Figure 5-33 and Figure 5-34 represent respectively the magnitude and phase behaviour of β [ 2] ( ω + 7 ) α [ 1] ( ω + 6 )α [ 1] ( ω+ )α [ 1] ( ω) (= Ĥ 3 ) as a function of the carrier frequency and input power. This results in a coherent contribution. The phase behaviour is smooth and linear. For high input powers, the magnitude of Ĥ 3 has a smooth behaviour. For low input powers, however, the magnitude of Ĥ 3 rises and becomes very noisy. This can be explained by the small signal-to-noise ratios of these low input power measurements. -70 Magn. B2/A1 Ĥ 1ste 3 SR [db] Frequency [MHz] Input Power [dbm] 0 FIGURE Magnitude of β [ 2] ( ω + 7 ) α [ 1] ( ω + 6 )α [ 1] ( ω+ )α [ 1] ( ω) 209

238 Continuous Wave and Modulated Measurements 400 Phase. Ĥ Phase B2/A1 1ste 3 [deg] SR [deg] Frequency [MHz] Input Power [dbm] 0 FIGURE Phase of β [ 2] ( ω + 7 ) α [ 1] ( ω + 6 )α [ 1] ( ω+ )α [ 1] ( ω) Figure 5-35 and Figure 5-36 represent respectively the magnitude and phase behaviour of β [ 2] ( ω + 7 ) α [ 1] ( ω + 4 )α [ 1] ( ω + 3 )α [ 1] ( ω) (= Ĥ 5 ) as a function of the carrier frequency and input power. This is also a coherent contribution and is similar to the contribution of Ĥ 3. By this type of plots, it can be experimentally verified that the energy of the output spectral regrowth components is obtained from intermodulation products. Hence, to model spectral regrowth components, one will need to model the intermodulation products. The linear terms can be left out. Since all contributions obtained by combining the input modulation components are similar, this means that these Volterra kernels are static in the modulation frequency. Hence, a first sensible guess to model spectral regrowth components is to use a static nonlinear model, independent of the modulation frequency. 210

239 Nonlinear AM-modulated measurements of an amplifier -70 Magn. Ĥ Magn. B2/A1 1ste 5 [db] SR [db] Frequency [MHz] Input Power [dbm] FIGURE Magnitude of β [ 2] ( ω + 7 ) α [ 1] ( ω + 4 )α [ 1] ( ω + 3 )α [ 1] ( ω) 300 Phase. Ĥ Phase B2/A1 1ste 5 [deg] SR [deg] Frequency [MHz] Input Power [dbm] 0 FIGURE Phase of β [ 2] ( ω + 7 ) α [ 1] ( ω + 4 )α [ 1] ( ω + 3 )α [ 1] ( ω) Searching the energy transport channels of a system, allows to obtain a good insight in the behaviour of that system to modulated excitations.this will make the modelling of a system in modulated regime a lot easier: non parametric models are obtained nearly directly from the figures. 211

240 Continuous Wave and Modulated Measurements 5.4 Conclusion Good measurements are a major step towards good modelling! However, besides these measurements, modelling a system also requires a good insight in the system behaviour. Do harmonics rise above the noise floor? Must they be taken into account during modelling? Is the system essentially dynamic or static? An answer to the above questions lies in the preprocessing of the measured data as obtained by the Nonlinear Vectorial Network Analyser. This technique makes nonlinear effects more visible and allows to understand the energy transport mechanisms behind the device operation. In this chapter, two essential issues for modelling are discussed. First, a technique to obtain high quality measurements is proposed. Easy to handle criteria are given to guide the user in the selection of the appropriate measurement settings. Next, a method is proposed to convert the huge amount of measured data to knowledge and to understand the system behaviour. Even if simple spectrum-versus-power plots allow to assess the level of spectral distortion, they are of limited help when it comes to gain insight in the system itself. Based on - but not limited to - Volterra models, a graphical method is used to discriminate between the energy transport mechanisms that the system uses and the ones that are not relevant to the produced output. In a second stage, dominant energy transport channels can be located. This visual method also plays a very important role during model quality assessment. Comparing the energy transport channels described by the model with those extracted from measurements yields a simple visual model quality assessment. 212

241 PART III MODELLING OF RF COMPONENTS AND SYSTEMS 213

242 214

243 CHAPTER 6 MEASUREMENT BASED NONLINEAR MODELLING UNDER CONTINUOUS WAVE EXCITATION Abstract: To continue the present trend of miniaturization and increasing functionality of the telecommunication equipment, the characterisation and understanding of the nonlinear behaviour of radio-frequency systems and subcomponents is of critical importance. Although RFICs are designed to behave linearly, they are often used in - or close to - their nonlinear operation region where nonlinearities become important. Even though currently available white-box models are accurate at the component level, they mostly fail to yield simple quantitative descriptions of the operation of a circuit as a whole. In this chapter, black-box models based on the Volterra theory are introduced, which describe the linear as well as the nonlinear behaviour of the RFICs. The models are developed based on measurements done by the Nonlinear Vectorial Network Analyser. 215

244 Measurement Based Nonlinear Modelling under Continuous Wave Excitation 6.1 Finding the right tool for the job In general, when modelling a system, one has the choice between a white-box model or a black-box model. A white-box model is based on physical laws and the internal structure of the system. It is very useful if one wants to get insight in the internal behaviour of the system. However, since it is not based on measurements, it can not take external factors such as crosstalk, parasitic coupling or other side effects into account. Black-box models, on the other hand, do not require any system knowledge at all. As the word says, the user can not know the internal structure of the system, but only measures the input and output signals. Thus, a blackbox model is used to describe the measured output behaviour of a system as a function of the measured input signals. Since our goal is to build measurement based models for a class of nonlinear systems, black-box models will be developed in this chapter. Until today, many black-box models for nonlinear microwave systems have been extracted from simulations of white-box models. In a sense, this is cheating! Starting from simulations means that one has already an idea of the internal structure of the component and of the applied excitation signal. Thus, black-box models of known systems are developed. Why then not use measurements instead of simulations? Characterising nonlinear microwave systems requires more than knowing the amplitude spectrum of the waves. If the phase relation between the harmonics is not absolutely known, it is impossible to reconstruct the corresponding shape of the time signals (see Chapter 1, Paragraph 1.2), to which nonlinearities are very sensitive. Furthermore, not knowing the phase distortions introduced by the measurement instrument means that the behaviour of the measurement instrument will also be part of the model. Present measurement setups, however, are not able to fully characterise the incident and reflected multi-frequency waves of a nonlinear system in both amplitude and phase. Developing real black-box models is therefore impossible. However, thanks to the Nonlinear Vectorial Network Analyser, one has the knowledge of the absolute phase relations between the measured harmonics. Let s use it! Can we tackle any nonlinear system? No, we can not. In general, all systems which do not behave as a linear system are considered to be nonlinear systems. However, hard 216

245 Finding the right tool for the job nonlinearities such as chaotic systems or systems which create subharmonics will not be considered. Only the subclass of soft nonlinearities, which transform a periodic input signal into a periodic output signal of equal period, is modelled. The linear system theory allows to fully understand, describe and design a linear system. The understanding and description can be obtained by measuring the transfer function at a discrete set of frequencies. This approach is called non parametric, as it describes the system behaviour using a set of values, rather than using a closed mathematical expression. Put in an RF context, this model is the S -parameter representation as measured by a Vectorial Network Analyser. The designing and a part of the understanding of a linear system is obtained through parametric models. In a parametric model, the linear system is represented as a closed mathematical form, i.e. a rational expression in the Laplace variable s. The roots of the numerator and denominator polynomial are the poles and zeros of the system. A linear time invariant system is fully characterised by the position of its poles and zeros, which are the quantities used in the design to shape the system response. In what follows, we will try to find a nonlinear theory which allows to fully describe, understand and design a class of nonlinear systems. 1. Static Nonlinear Theory [32] In this case, the linear system is expanded towards a nonlinear behaviour without memory. The output signal yt () at time instant t is assumed to be a polynomial function of the input signal(s) ut () at the same time instant only: yt () = Put ( ()) (6-1) In the frequency domain this implies that the system has an infinite bandwidth. Furthermore, the behaviour of the system is frequency independent. 2. Volterra Theory [32] Whenever the nonlinearity depends on the frequency content of the applied input, the system will contain memory effects. The relation between the input signals ut () and output signals yt () then becomes: yt () = Puτ ( ( )) with < τ < t (6-2) 217

246 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Systems with fading memory, i.e. the influence of past events diminishes as time goes on, belong to the class of Volterra systems. Volterra systems extend the linear timeinvariant system class in the sense that besides the linear time-invariant transfer function, they also include higher order transfer functions. For a system excited by a pure sinewave at frequency ω 0, the Volterra response becomes [32]: Y( ω 0 ) = H 1 ( ω 0 )U( ω 0 ) + H 3 ( ω 0, ω 0, ω 0 )U( ω 0 )U( ω 0 )U( ω 0 ) + (6-3) with U( ω 0 ) and Y( ω 0 ) respectively the input and output spectral components, H m ( m = 12,,, ) are the Volterra kernels, which are very hard to determine in general for all frequencies. It can be proven that a kernel function H m can be synthesized as a combination of multipliers and linear time invariant systems [32]. An interesting capability of the Volterra models is that they are able to approximate soft and hard nonlinearities (without bifurcations) with an arbitrary precision in mean-square sense. This means that the Volterra model and the system characteristic will become arbitrary close for increasing order of the Volterra model. The Volterra model will thereto wrap itself around the device characteristics. However, one has to be careful with this result. Even if the model and the system can get arbitrary close, it remains true that the physical model of the system is not exactly a Volterra model. Hence, derived quantities, such as derivatives can be very different for the model and the system. 6.2 Polynomial Volterra models For soft nonlinearities, polynomial Volterra models are able to describe the physical behaviour of a system. These models have low orders and thus, a low complexity. Practical experience shows that for many components or systems operating beneath the 1 db compression point, this is often the case. A polynomial Volterra model for a periodic input signal can be obtained using a Volterra Input- Output Map (VIOMAP) [45]. The VIOMAP can be seen as a multi dimensional transfer function evaluated in 1 dimension and is a direct extension of the Laplace model for linear time-invariant systems to soft nonlinearities. In some sense, Volterra kernels in the frequency 218

247 Rational Volterra models domain are similar to S -parameters: the VIOMAP kernel H M ( ω 1, ω 2, ω, M ) describes the conversion from a combination of input spectral lines U( ω 1 ) U( ω M ) to an output spectral contribution Y( ω ω M ). The complete polynomial VIOMAP equation becomes: m M Y( ω) = H m ( ω 1, ω, m )U( ω 1 ) U( ω m ) m = 1 (6-4) where ω k = ω and M the maximum degree of nonlinearity. k = 1 Herein, U( ω k ) and Y( ω k ) are measured, H m is to be estimated. The Volterra map is hence a kind of power series expansion of the nonlinearity. 6.3 Rational Volterra models Polynomial Volterra models give good performance for mild nonlinearities. For strong compression, i.e. above the 1 db compression point, however, the polynomial Volterra models may need much higher degree contributions to somewhat describe the behaviour of the system. This is inherent to a polynomial model. It is not suited to describe strong saturation effects, because any non zero polynomial tends to infinity when its free variable increases to infinity. A possible approach to circumvent this problem is to use a rational Volterra model, which is a division of two polynomial Volterra models [7]. The rational VIOMAP consists of a quotient of two polynomial VIOMAPs, with the same combinations of spectral input components, but different values of kernels and is formulated as follows Y( ω) = M H m ( ω 1, ω, m )U( ω 1 ) U( ω m ) m = 1 M m = 1 H m ( ω 1, ω, m )U( ω 1 ) U( ω m ) ) (6-5) where H m ( ω 1, ω, m ) and H m ( ω 1, ω, m ) are different values of VIOMAP kernels. ) 219

248 Measurement Based Nonlinear Modelling under Continuous Wave Excitation A rational VIOMAP can very well describe the behaviour of a system in the measurement points. However, between these measured points, the model has a lot of freedom and can behave very wild. This can be easily seen from equation (6-5). If the denominator becomes zero, the model tends to infinity. One can try to reduce this flexibility by only allowing positive functions in the denominator of the model. As a result, the denominator can never be zero and the model can not tend to infinity between the measured points. This extension allows a better inter sample behaviour of the model. 6.4 Implicit Volterra models The problem of saturating systems is, from the modelling point of view, mainly an ill-posed problem. If one takes a saturating nonlinear system, but switches the roles of the input and output signals, a strongly expanding nonlinear system results. This newly obtained nonlinear system can be described much more easily by a polynomial model. Hence, from the modelling point of view, it is quite foolish to make a difference between input and output ports. The implicit model formulation results: FYω ( ( ), U( ω k )) = 0 (6-6) m where. is one of the measured output spectral components and k 1 ω k = ω Y( ω) U( ω = k ) the measured input spectral components at frequency ω k. Solving equation (6-6) means that a system of polynomial equations must be solved. This is a difficult task and results in multiple solutions. Due to these multiple solutions, the model has enough flexibility to even model bifurcations. However, care should be taken, since these multiple solutions also allow to model the noise. Consider for example, a tanh function that is modelled by an implicit VIOMAP. For noise free data, as can be seen in Figure 6-1 (a), the implicit VIOMAP will have 2 different paths (red and green curve), which coincide in the linear part of the tanh function (blue curve). However, if measurement noise is present, both modelling paths will split up and thus, give the model the ability to describe the noise in the linear part of the curve (see Figure 6-1 (b)). This very powerful model hence needs to be handled with care! 220

249 Proposed estimator Input [V] (a) Input [V] (b) FIGURE 6-1. Implicit model of a tanh for (a) noise free data and (b) noisy data 6.5 Proposed estimator In the above Volterra models, the unknown VIOMAP kernels H m must be estimated. For all types of Volterra models, the most appropriate estimator is the Weighted Least Squares estimator [34], which weights each measurement point with its measurement uncertainty. This is an errors-in-variables-like approach [34]. Since this technique gives good results when modelling a linear transfer function, the estimator will be applied to model nonlinearities without worrying too much about the theoretical properties of the estimator. Note however that since the Volterra models contain products of input spectra UU * and the expectation value of this term will be UU * 2 E { } = U 0 + σ U, a bias on the results is introduced in any case. The superscript * denotes the complex conjugate Polynomial VIOMAP In general, a polynomial VIOMAP can be written in matrix notations as Y = F( U)H (6-7) where the matrix Y C N 1 contains the output spectral components at N different power levels. H C M 1 contains the VIOMAP kernels H m and FU ( ) C N M is the matrix containing the spectral products of the input components U C N 1. To simplify the notations, the frequency dependency is not written. 221

250 Measurement Based Nonlinear Modelling under Continuous Wave Excitation To estimate the unknown VIOMAP kernels H m, the estimator minimizes the error between the model and the measurements: εθ ( ) = Y F( U)θ (6-8) with θ = H = [ H m ] C M 1 the model parameters to estimate. Note that the error vector equations are linear in the parameters θ. Since it is not possible to minimize a whole function, some norm of this function will be minimized instead. In this case, the 2-norm, resulting in a Least Squares estimator, is minimized: V N ( θ) = ε( θ) H εθ ( ) 2N (6-9) with εθ ( ) the error vector and N the number of measurement points, which equals the total number of measured input powers in case of a Volterra model. The superscript H represents the Hermitian transpose. To find the estimates θˆ of the model parameters θ which minimize cost function (6-9) θˆ = argminv N ( θ), the stationary equations to be solved are: θ V N ( θ) θ 1 = ---ε H ( θ) εθ ( ) = 0 (6-10) N θ Since εθ ( ) is parameter independent, a linear Least Squares estimator results. The estimates θ θˆ of the parameters θ are hence the solution of the normal equations [34]: θˆ = J H 1 H ( J) J εθ ( ) (6-11) with J = εθ ( ) the Jacobian matrix. Hence, no iteration is needed to obtain the estimates θˆ. θ Computing the matrix product J H J before taking the inverse is numerically inefficient as the number of required significant digits is hereby doubled. A standard solution is to replace the normal equations by the following overdetermined system of equations [14]: Jθˆ = ε( θ) (6-12) which can be solved by 222

251 Proposed estimator θˆ = J + εθ ( ) (6-13) where J + is the Moore-Penrose pseudo-inverse of the Jacobian matrix J [5]. This can be computed by using the singular value decomposition of the Jacobian matrix [15]. Equation (6-13) hence results in an estimate of the VIOMAP kernels H m Rational VIOMAP A rational VIOMAP can be written in matrix notation as: Y = ( P( U)H) ( QU ( )H ) ) (6-14) where represents the element-wise division of the vector elements and is called the Hadamard-Schur division. The matrix Y C N 1 contains the output spectral components at N different power levels. H and H, both elements of C M 1, represent the VIOMAP kernels of respectively the numerator and denominator. PU ( ) C N M and QU ( ) C N M ) are the matrices containing the input spectral products. To simplify the notations, the frequency dependency is left out below. For the rational VIOMAP, the same Least Squares estimator approach as above can be used. The error vector can then be written as: εθ ( 1, θ 2 ) = Y ( P( U)θ 1 ) ( QU ( )θ 2 ) (6-15) with θ 1 = H = [ H m ] C M 1 and θ 2 = H = [ H m ] C M 1 the model parameters to estimate. Note that the error vector equations are not linear in the parameters θ 1 and θ 2. This implies that an iterative procedure is required to determine the estimates θˆ 1 and θˆ 2 which minimize the following cost function: ) ) V N ( θ 1, θ 2 ) = ε( θ 2N 1, θ 2 ) H εθ ( 1, θ 2 ) (6-16) To start this procedure, a good set of initial θ 1 and θ 2 parameter values is required. These initial values can be determined using an approximate linearised version of the cost function, following the reasoning as proposed by Levi [20]: 223

252 Measurement Based Nonlinear Modelling under Continuous Wave Excitation ε ( θ 1, θ 2 ) = Y Q( U)θ 2 ) PU ( )θ 1 (6-17) where represents the element-wise multiplication of the vector elements and is called the Hadamard-Schur product. Similarly to the polynomial VIOMAP a linear least squares estimator results. Hence, by minimizing the following cost function, the initial estimates of the T parameters θ I = θ 1 θ 2 are obtained: V N ( θ I ) = ε ( θ 2N I ) H ε ( θ I ) ) ) (6-18) with ε ( θ I ) the error vector and N the number of measured input-output pairs, which equals ) the total number of measured input powers in the case of a Volterra model. The initial estimates of the model parameters θ I are then obtained as follows: θˆ I = J + ε ( θ I ) ) (6-19) with J + ε ( θ is the Moore-Penrose pseudo-inverse of the Jacobian matrix J I ) = θ I T In a second step, the estimates θˆ of the model parameters θ = θ 1 θ 2 which minimize the nonlinear cost function (6-16), can be obtained by solving the following equation system: ) V N ( θ) θ 1 = ---ε H ( θ) εθ ( ) = 0 N θ (6-20) εθ ( ) where the Jacobian J( θ) = is now a function of the parameters θ. Hence, an iterative θ procedure is required. In general, this system of equations can be solved by the Newton procedure, which is based on the complete Hessian: 2 V N ( θ) θ 2 = J H ( θ)j( θ) ε( θ) 2 εθ ( ) N θ 2 (6-21) This Hessian contains a positive definite part J H ( θ)j( θ) and an indefinite part ε( θ) εθ ( ). θ 2 Due to this indefinite part, the Hessian is not guaranteed to be semi positive definite. This can have an adverse influence on the convergence and can even narrow the convergence region. 224

253 Proposed estimator When one gets close to the solution, the error vector εθ ( ) becomes small and equation (6-21) can be approximated by: 2 V N ( θ) θ ( J H ( θ)j( θ) ) N (6-22) This approximation is called the Gauss-Newton procedure [14], which has a larger convergence region than the Newton method. The original Hessian is replaced by a matrix J H ( θ)j( θ) which is always semi-positive definite. By iteration on θ, convergence to a local minimum of the cost function will eventually be obtained, if the initial value of the parameters is close enough to the minimum. In the Gauss-Newton method as used here, a guess θˆ [] i is improved to θˆ [ i + 1] = θˆ [] i + θˆ [ i + 1] (6-23) where θˆ [ i + 1] is the solution of the normal equations [34]: θˆ [ i + 1] = ( J H ( θˆ [] i )J( θˆ [] i )) 1 J H ( θˆ [] i )ε( θˆ [] i ) (6-24) with i the number of iteration step. J( θ) = εθ ( ) is the Jacobian matrix. θ Computing the matrix product J H ( θˆ [] i )J( θˆ [] i ) before taking the inverse is numerically inefficient as the number of required significant digits is hereby doubled. However, by using the singular value decomposition of the Jacobian, equation (6-24) can be transformed in the following overdetermined system of equations: θˆ [ i + 1] = J + ( θˆ [] i )ε( θˆ [] i ) (6-25) where J + ( θ) is the Moore-Penrose pseudo-inverse of the Jacobian matrix J( θ). Combination of equations (6-15), (6-23) and (6-25) hence results in an estimate of the VIOMAP kernels H m and H m after iteration. ) Implicit VIOMAP An implicit VIOMAP can be written in matrix notation under the following form: 225

254 Measurement Based Nonlinear Modelling under Continuous Wave Excitation FUY (, )H = 0 (6-26) where H C M 1 contains the VIOMAP kernels H m and FUY (, ) C N M is the matrix containing the spectral products of the input U C N 1 and output Y C N 1 components. To simplify the notations, the frequency dependency is not written. The error vector is thus equal to: εθ ( ) = FUY (, )θ (6-27) with θ = H = [ H m ] C M 1 the model parameters to estimate. Note that the error vector equations are linear in the parameters θ. To obtain estimates of the parameters θ, the following cost function needs to be minimized: V N ( θ) = ε( θ) H εθ ( ) 2N (6-28) with εθ ( ) the error vector and N the number of measured input-output pairs, which equals the total number of measured input powers in case of a Volterra model. The estimation can be further performed exactly in the same way as for the polynomial VIOMAP case Introducing the weight The set of methods as described above theoretically solve the estimation problem. However, there are important issues that remain unsolved. On one hand, the numerical precision is assumed to be infinite. On the other hand, all measurements are assumed to be of equal quality. 1. Scaling the parameters In many cases, the Volterra model must be developed for a wide power range. The dynamic range of the measured signals can thus be very large. This causes a wide numerical dynamic range in the elements of the Jacobian matrix, which can even reach the numerical precision and hence, causes this Jacobian matrix to be ill-conditioned. This problem can somewhat be countered by scaling the columns of the Jacobian matrix to norm 1. However, it can be proven that this scaling method is based on heuristics and can fail in some cases to improve the numerical conditioning [15]. 226

255 Proposed estimator 2. Scaling the equations In the earlier defined estimators, the weight associated to a measurement, i.e. the influence of this measurement point on the modelling result, is determined by the norm of the row in the Jacobian matrix associated to this measurement. Since the Jacobian matrix contains entries that are products of input spectra, the rows associated with a high power level will also have much higher weight. Finite numerical precision effects can then reduce the influence of the measurements associated with low power levels, to almost nothing and a bad fit of the characteristic at low power levels can result. On the other hand, measurement points with a high uncertainty will have equal influence on the cost function as accurate measurements. As a result, the estimator will put a lot of effort in correctly modelling the noisy data. This trend can be counteracted if a proper weighting, which takes the uncertainty on the measurement equations into account, is designed. This results in a Weighted Least Squares estimator [34]. For a polynomial, rational and implicit VIOMAP the weighted cost function to minimize is: V N ( θ) = εθ ( )H εθ ( ) N ε 2 2 E { ( θ)} 2 (6-29) where E { ε 2 ( θ)} is the variance of the error vector εθ ( ). For the linearised rational VIOMAP, which determines the initial parameter values, the weighted cost function to minimize is ([30],[31]): V N ( θ) = ε ( θ)h ε ( θ) N 2 QU ( )θ 2 2 ) ) (6-30) The weighting is determined by the uncertainty on the output as well as the input signals. Any higher order moments of the noise, that contain power terms such as UU *, will result in bias terms that perturb the weighting function. Removal of these bias terms is very tedious: it can make the weighting negative, because the estimation of these bias terms is very difficult and inaccurate for a low number of repeated measurements. Instead of calculating the sample 227

256 Measurement Based Nonlinear Modelling under Continuous Wave Excitation 2 2 variance of the input signals s U and the output signals s Y separately and then evaluating the variance of the error vector εθ ( ) through a theoretical expression, the sample variance of the 2 error vector s εθ ( ) will be determined directly from the experimental data for each measurement. Note however, that this sample variance depends on the parameters to estimate. Hence, an iteration procedure is required to calculate the estimates. To avoid that the cost function becomes nonlinear in the parameters, the weight will be evaluated with the already obtained estimates to calculate the new ones (IQML [31], [8]). A series of cost functions results, that will eventually converge to a cost function that approximates the nonlinear cost. If the variation of the parameters drops below a certain threshold, the iteration is stopped and an estimation for the VIOMAP kernels is found. In the following paragraphs, a polynomial, rational and implicit Volterra model of an amplifier under continuous wave excitation will be estimated. Nonlinear Vectorial Network Analyser measurements are performed on the device. Measurements of systems with a high amplitude dynamic range, such as an amplifier in compression, require switching of attenuators or amplifiers to maintain measurement accuracy over the sweep range and preserve the linearity of the measurement device. It can be experimentally verified that the switching of the attenuators/amplifiers is not a perfectly repeatable process (see Appendix 6.B). In other words, after switching the attenuators/amplifiers, the value of the attenuation/amplification is always slightly different. These small variations cannot be neglected because they induce (small) jumps in the measured device characteristics. When these measurements are used to model a device by means of a smooth model, poor results are obtained even if the jumps are only hardly visible on a measurement graph. The derivative in a jump tends to infinity, while a smooth model, such as a polynomial function, has a limited derivative and, therefore, is not able to describe fast changes. This results in a larger model error, since the model puts a lot of effort in following the jump. 228

257 Modelling in the presence of switching uncertainties 6.6 Modelling in the presence of switching uncertainties One can deal with these jumps by modelling them as additional parameters, which are independent of the additive noise and only depend on the state of the attenuator or amplifier. Based on a statistical description of the jumps, the modelling method is able to virtually remove the discontinuities Proposed estimator Consider a static nonlinear system with the following exact input-output characteristic: y 0 () i = fu ( 0 ()a i, 0 ) + b 0 (6-31) with u 0 () i and y 0 () i respectively the noise free input and output signals depending on i which can be time, frequency or input power. a 0 and b 0 are the true model parameters. To model the jumps in the data, it will be assumed that b 0 is perturbed by a switching noise source. Each time some part of the hardware is switched, a new realisation of the switching noise occurs. In-between each switching, many measurements can be performed that do not require the hardware to be switched. For all these measurements, the switching noise has the same value. Such a set of measurements associated with one switching state is called a switching set from now on. Let n s ( k) denote the switching noise, depending on the switching set k. Each switching set contains a number of measurements i k : N k 1 < i k N k (6-32) for set k. If a total amount of N measurements were taken and K switching sets are present in the data, N 0 = 0 and N K = N. When relation (6-31) is measured at N different input values u 0 () i, one obtains the following input-output characteristic: yi ( k, θ) = fu ( 0 ( i k ), a 0 ) + b 0 + n y ( i k ) + n s ( k) (6-33) 229

258 Measurement Based Nonlinear Modelling under Continuous Wave Excitation where yi ( k, θ) C N 1 represents the measured output signal as a function of the true model T ( parameters θ a 0 b 0 R n θ + 1) 1 = with a 0 R n θ 1 and b 0 R. u 0 ( i k ) C N 1 is the input signal which is assumed to be noise free; i k represents a measurement point in the switching set k, containing the measurements i k for which: N k 1 < i k N k, k = 1,, K, with N 0 = 0 and N K = N (see Figure 6-2). y n s(k+1) y 0 = f(u 0,a 0 )+b0 k k+1 u(nk-1+1) n(k) s u(n k ) u(n k+1) u(n k+1 ) u FIGURE 6-2. Input - output characteristic Note that the perturbation on the output signal y 0 ( i k ) consists of 2 stochastic contributions: n y () i + n s ( k) (6-34) where n y () i is the additive measurement noise, depending on i; and n s ( k) is the switching noise, depending on the instrument settings k. All the points in one switching set k share the same realisation of the switching noise n s ( k). This switching noise results in a (small) number of jumps in the input-output characteristic. Both noise sources are mutually independent and independent real or circular complex normal distributed [35]: n y () i N( 0, σ y ) n s ( k) N( 0, σ s ) (6-35) (6-36) N( 0, σ) represents a normal distribution with zero mean and standard deviation σ. To obtain an estimate for the parameters a 0 and b 0, 2 different approaches can be considered, based on the Maximum Likelihood estimator or the Bayes estimator. 230

259 Modelling in the presence of switching uncertainties A. Maximum Likelihood Estimator The Maximum Likelihood estimates θˆ ML of the model parameters θ ab T ( = R n θ + 1) 1 are obtained by minimizing the cost function V N ( θ) [34]: θˆ ML = argminv N ( θ) = θ ε( θ) 2N H 1 argmin C ε εθ ( ) θ (6-37) where N is the total number of measurements and εθ ( ) C N 1 the error vector: εθ ( ) = ỹ( i k ) yi ( k, θ) = ỹ( i k ) fu ( 0 ( i k ), a) b (6-38) with ỹ( i k ) C N 1 the measured output signal. In this approach, the switching noise is accounted for as a regular noise source, that is totally correlated for all measurements within the same interval as the noise is exactly the same in 2 successive measurements where no switching occurs. Hence, the covariance matrix of this noise source will be a block diagonal one matrix (consisting only of ones) containing one block of rank 1 for each switching set. Each block is a one matrix of size ( N k N k 1 ) ( N k N k 1 ), which is the total number of measurement points within the switching set k : C ns 1 C ns σ s 2 = 1 1 (6-39) The covariance matrix C ny of the measurement noise n y is a diagonal matrix since the measurement noise is assumed independent from measurement point to measurement point: 2 C ny = σ yi ( Nk N k 1 ) (6-40) with I( Nk N the identity matrix. k 1 ) ( N k N ) k 1 ( N N k k ) 1 The covariance matrix C ε of the error vector is hence equal to: 231

260 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Cov( ε) = C ε = C ny + C ns R N N (6-41) where C ny is the covariance matrix of the measurement noise n y and C ns the covariance matrix of the switching noise n s. Note that the cost function V N ( θ) is quadratic in the parameters. Hence, the Maximum Likelihood estimator boils down to a linear Least Squares estimator and the estimates θˆ ML can be calculated without iteration [34]: θˆ ML = ( J H C 1 J ε ) 1 JH C 1 εθ ( ) ε (6-42) with J = ε( θ) θ the Jacobian matrix C N ( n θ + 1) and n θ + 1 is the total number of model parameters. εθ ( ) is the error vector and C ε the weighting matrix or covariance matrix of the error vector εθ ( ). The Maximum Likelihood estimates for a and b are inconsistent when the amount of measurements N increases to infinity, while keeping the number of switching sets K constant (see Appendix 6.B). Indeed, if K is kept constant then no switching noise reduction occurs in the cost function (6-37) as N tends to infinity. The only way to get consistent estimates is to increase also K to infinity as N, consistency is regained when K increases fast enough, K = O( N) (see Appendix 6.B). In practice, this is not always possible due to the limited life time of expensive switching devices and measurement time constraints. B. Bayes Estimator Better results are obtained using a Bayes estimator. To keep the influence of the switching noise n s ( k) on the measurements small, an estimated value for n s ( k) will be determined. This is possible since the value of n s ( k) is constant for the measurements that belong to the same switching set. However, n s ( k) can not be distinguished from the noise free value b 0, it acts as a bias on b 0 as it is equal for all the consecutive measured points within a switching set k. Hence, K different b values are to be estimated, while in the previous method only 1 parameter b was estimated: 232

261 Modelling in the presence of switching uncertainties b k = b 0 + n s ( k) with k = 12,,, K (6-43) When the expectation value with respect to n s ( k) is taken, b k is a stochastic parameter with mean value b 0 and variance σ 2 s ( k). Under the normal assumption (6-36), the probability density function (pdf) of the stochastic parameters is known, so that a Bayes estimator can be constructed [34]. In a Bayes estimator, an estimate θˆ B of the model parameters θ is obtained by maximizing the following conditional probability: f n [ y ( a, β) ]f β [ β] (6-44) where f n [ y ( a, β) ] is the probability density function (pdf) of the noise on the measurements and f β [ β] is the pdf of the unknown parameters β = [ b 1,, b K ] T. Using a Bayes estimator results in the following cost function V N : K N k N k = 1i k = N k V N ( θ) 1 = εθ ( ) 2N H εθ ( ) = (ỹ( i k ) fu ( 0 ( i k ), a) b k ) 2 ( b k b 0 ) 2 2σ y ( i k ) 2σ 2 s ( k) (6-45) with K the total number of switching sets, θ a T T ( = β IR n θ + K) 1. N k represents again the positions in the measurement array at which the switching occurs. σ y is the standard deviation of the measurement noise, while σ s is the standard deviation of the switching noise. b 0 is the true value of the parameters b k and can be obtained through minimization of the cost 1 function (6-45) with respect to b 0 : bˆ 0 K K = k = 1 b k = b k. As a result, the cost function can be calculated as: N K k N k = 1i k = N k V N ( θ) 1 = εθ ( ) 2N H εθ ( ) = (ỹ( i k ) fu ( 0 ( i k ), a) b k ) 2 K ( b k b k ) 2 2σ y ( i k ) 2σ 2 s ( k) k = 1 (6-46) 233

262 Measurement Based Nonlinear Modelling under Continuous Wave Excitation This means that the estimation is split up in the different parts, one associated with each switchings set. The jumps are then afterwards reassembled to deliver one smooth model with a b value equal to b k. The cost function is quadratic in the parameters a and b k and by using the Bayes cost (6-46), estimates θˆ B are obtained [34]: θˆ B = ( J H C 1 J ε ) 1 JH C 1 εθ ( ) ε (6-47) with J = ε( θ) θ the Jacobian matrix C ( N+ K) ( n θ + K), ε( θ) C ( N+ K) 1 the error vector and C ε the weighting matrix or covariance matrix of the error vector ε : C ε = Cov(ε) R ( N+ K) ( N+ K). This results in a model that is able to take the mean of the jumps and deliver a characteristic without discontinuities, while maintaining consistent a values (see Appendix 6.B) Interpretation Consider the simplified case where no measurement noise is present: σ y = 0. The Maximum Likelihood estimator will reduce all the equations within an interval to 1 single equation. This happens because of the special form of the covariance matrix consisting of ones only. Taking 2 intervals will hence yield only 2 equations, and there is no averaging of the switching noise. Hence, the Maximum Likelihood estimator returns inconsistent estimates for the parameters when the number of switching sets does not go to infinity. In the case of the Bayes estimator, the approach is different in that sense that an additional parameter is used to remove the step induced by the attenuator switching. Hence noise averaging will occur as usually on all the measurement points. This will yield consistent estimates for a even though the number of switching sets remains finite (Paragraph 6.6.3). Note that the input signals are assumed to be noise free. However, in practice measurement noise is always present on the input signals. Furthermore, the nonlinear models contain spectral contributions of the form UU * whose expected value contains one contribution for the signal and a non zero contribution for the noise power. Hence, the estimates of the 234

263 Modelling in the presence of switching uncertainties parameters a will always be inconsistent, even if no switching occurs. The influence of the added b parameters, in case of a Bayes estimator, on the consistency of the original a estimates can hence only be checked if noise free input data is assumed. This consistency test reveals that introducing switchings will not further deteriorate consistency. Hence, a consistent estimator will remain consistent even if switchings are present Simulations The properties of the Maximum Likelihood and Bayes method are first illustrated on a very simple example of a static linear system: yi ( k, θ) = a 0 u 0 ( i k ) + b 0 + n y ( i k ) + n s ( k) (6-48) where the quantities are defined as above. As shown in Figure 6-3, two switching sets are present: N 0 = 0, N 1 = 25 and N 2 = Output y [mv] Input u [mv] FIGURE 6-3. Input - output characteristic The results of the Bayes estimator (Figure 6-4) which splits up the characteristic in two different parts and reassembles them afterwards are compared with the results of the Maximum Likelihood estimator (Figure 6-5), where the characteristic is considered as a whole. In contrast with the Maximum Likelihood estimator, the Bayes estimator is able to remove the discontinuity and to preserve the right slope. 235

264 Measurement Based Nonlinear Modelling under Continuous Wave Excitation 200 Output y [mv] DDDDDDDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDDDDDDD EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE Input u [mv] FIGURE 6-4. Estimated (o) and exact (+) characteristic for the Bayes Estimator Output y [mv] E E E E E E E E E E E E E E DDDDDDDDDDDDDDDDDDDDDDDDD E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E DDDDDDDDDDDDDDDDDDDDDDDDD E E E E Input u [mv] FIGURE 6-5. Estimated (o) and exact (+) characteristic for the Maximum Likelihood Estimator To verify the theoretical and statistical properties of the proposed estimation technique, simulations are performed on a static nonlinear, second order system: yi ( k, θ) = a 10 u 2 0 ( i k ) + a 20 u 0 ( i k ) + b 0 + n y ( i k ) + n s ( k) (6-49) The input signals u 0 ( i k ) are noise free (50 points, equally spaced between 1-16mV). The output yi ( k, θ) is perturbed by two noise sources. The noise n y ( i k ) is normal distributed with 2 zero mean and variance σ y ( ik ) = 0.01( mv) 2 for any i k. The switching noise n s ( k) is also 2 normal distributed with zero mean but variance σ s = 100(mV) 2. The measurement characteristic contains only 2 switching sets N 0 = 0, N 1 = 25 and N 2 =

265 Modelling in the presence of switching uncertainties As a result, the system of equations to be solved by the Bayes estimator becomes: -for the first interval k = 1, 1 i 25 2 yi ( 1, θ) = a 10 u 0 ( i1 ) + a 20 u 0 ( i 1 ) + b 0 + n y ( i 1 ) + n s ( 1) (6-50) -for the second interval k = 2, 26 i 50 2 yi ( 2, θ) = a 10 u 0 ( i2 ) + a 20 u 0 ( i 2 ) + b 0 + n y ( i 2 ) + n s ( 2) (6-51) b The parameters a 1, a 2 and b 1 + b = with b and are 2 1 = b 0 + n s ( 1) b 2 = b 0 + n s ( 2) estimated by using the Bayes estimator as shown above. Theoretically, the Bayes estimator will result in consistent, unbiased estimates of the parameters. Hence, it is important to verify if the estimates of the parameters a 1, a 2 and b are unbiased in practice. Thereto, the bias between the estimated and exact parameter values must be much smaller than the standard deviation on the estimated parameters. Table 6-1 represents the exact parameter values θ 0 as well as the sample standard deviation s θ of the estimated parameters and the bias on the estimates θ θ 0, with θ the sample mean of the estimated parameters: θ θ 0 θ θ s 0 θ a 1 a e-7 8.2e e-6 1.5e-3 b 5 2.6e TABLE 6-1. Exact parameters and Bayes estimation These results are obtained by averaging Monte Carlo simulations. Out of Table 6-1 one can conclude that the bias on the estimates is much smaller than the standard deviation on the estimated parameters. The estimated parameters will asymptotically reach the true parameter values, which confirms the theory of the unbiased estimator. 237

266 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Figure 6-6 shows the results for the exact and estimated characteristic for one experiment. One can see that the proposed estimation technique is able to remove the jump and still preserves the right slope. Output y [mv] D D D E D D E E D D E D D E D D E E DDDDDDDDDDD D D E E E E E E D EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE DDDDDDDDDDDDDDDDDDDDDDDDD Input u [mv] FIGURE 6-6. Estimated (o) and exact (+) characteristic The presence of outliers and the standard deviation of the normal distribution of the parameters θ can be assessed, evaluating the fraction of the simulation runs in the 95% uncertainty interval: θ 0 2σ θ < θˆ i < θ 0 + 2σ θ (6-52) where θ 0 are the exact parameters, θˆ i are the estimated values of the parameters and σ θ the standard deviation of the estimated parameters. If the number of θˆ i s between those 2 boundaries is 95%, the model variance is accurate and the bias is negligible. The next table shows the results for the parameters a 1, a 2 and b : θ a 1 a 2 b # θˆ i 95.4% 95.6% 95.3% TABLE 6-2. Simulated 95% uncertainty region fraction Conclusion When switching phenomena are present in the signal paths of the measurement device, jumps occur in the measured input-output characteristics. By modelling these jumps as an additional 238

267 Logarithmic estimator stochastic noise contribution on the measurements, which only depends on the instrument state, the method is able to remove virtually the discontinuities and return a smooth model. 6.7 Logarithmic estimator In the measurement data of the Nonlinear Vectorial Network Analyser, however, the switching noise is not mainly additive, but multiplicative: y = y 0 n where n N( 1, σ n ). For the Bayes estimator, equation (6-31) then becomes: b k y( i k, θ) = fu ( 0 ( i k ), a) (6-53) If one takes the logarithm of equation (6-53), the switching noise is again additive and the previous theory still holds. The costfunction (6-46) then becomes: N k 1 V N ( θ)= 1 K N --- k = 1i k = N k This is equivalent with cost function (6-46). K log( ỹ( i k )) log( fu ( 0 ( i k ), a) ) log( b k ) 2 ( b k b k ) σ2 logy ( i k ) 2σ 2 s ( k) k = 1 (6-54) Note that this logarithmic estimator can be used for the polynomial as well as the rational Volterra model to deal with the multiplicative switching noise. For the implicit Volterra models, however, the logarithmic estimator can not be used, since in this case equation (6-53) becomes: f( b k y( i k, θ), u ( 0 i k), a) = 0 (6-55) This means that the cost function contains not only a linear contribution in the switching noise b k, but also higher order contributions. Solving this problems requires further research. 6.8 Comparison of the attainable model flexibility When the free variables of a polynomial function - the power in our case- increases to infinity, the function itself will also tend to infinity. However, a system in compression has an 239

268 Measurement Based Nonlinear Modelling under Continuous Wave Excitation horizontal asymptote. Hence, a polynomial model is not suited to describe strong compression over an infinite power range, but will give good results in a restricted amplitude range. If additional model flexibility is required, the nonlinear polynomial models can be extended by introducing the rational models. Since a rational function can easily describe an horizontal asymptotic behaviour when the numerator and denominator tend to infinity at the same speed, the rational Volterra models are able to accurately predict hard compression phenomena over a much wider amplitude range. The difference in flexibility between a polynomial and rational Volterra model will be shown on the simple simulation example of a tanh function, since this characteristic often appears approximately in transistor based systems. Simulations are used, since real-world systems in perfect compression do not exist. Consider the following setup: ut () = Acosωt tanhut () yt () FIGURE 6-7. tanh function Since the nonlinearity is perfectly static, the frequency of the input sine wave does not change the result. Consider an input power sweep ranging from -30 dbm to 5 dbm. The output power characteristic of the tanh then becomes: 240

269 Comparison of the attainable model flexibility 10 tanh [dbm] Input Power [dbm] FIGURE 6-8. Output power characteristic of a tanh function Figure 6-9 represents the modelling results of a tanh by a polynomial VIOMAP with a degree of nonlinearity equal to 5 (= 3 VIOMAP kernels). 10 tanh [dbm] Input Power [db] FIGURE 6-9. Exact power characteristic (-) and polynomial Volterra model ( ) of the power characteristic of the tanh From the above figure, it is clear that the polynomial model tends to infinity when compression rises. The equation error of Figure 6-10 also proves the disability of the polynomial model to describe strong compression and shows that the model wraps itself around the characteristic to model. 241

270 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Equation Error [db] Input Power [db] FIGURE Equation error of the polynomial model Figure 6-11 represents the modelling results of a tanh by a rational VIOMAP with a degree of nonlinearity equal to 3 (= 4 VIOMAP kernels). 10 tanh [dbm] Input Power [db] FIGURE Exact power characteristic (-) and rational Volterra model ( ) of the power characteristic of the tanh Figure 6-11 undoubtedly proves that a rational Volterra model describes hard compression very accurately. This can also be concluded from the equation error (Figure 6-12), which is very small over the whole band of power. 242

271 Model of an amplifier under continuous wave excitation Equation Error [db] Input Power [db] FIGURE Equation error of the rational model This very simple example proves that the flexibility of a rational Volterra model to model saturation is much better than of a polynomial Volterra model. For the polynomial model to accurately describe this high level of compression, a 17th degree of nonlinearity is needed. Hence, 9 VIOMAP kernels need to be estimated, while the rational Volterra model only needs 4 VIOMAP kernels. A rational Volterra model is able to accurately describe hard compression with less parameters than a polynomial Volterra model and has much better extrapolation capability for higher power levels. 6.9 Model of an amplifier under continuous wave excitation A black-box model for a power amplifier (Motorola MRFIC2006 [25]) based on the polynomial, rational and implicit Volterra Input-Output Map (VIOMAP) is designed Measurement setup The measurements of the power amplifier are performed by the Nonlinear Vectorial Network Analyser (NVNA - HP85120A-K60) [40]. The amplifier is excited by a single tone at a frequency of 890 MHz. A powersweep from -14 dbm to 5 dbm is performed in 96 steps. The power amplifier has a supply voltage of 4V and is terminated with a 50Ω load impedance. An absolute calibration is done to correct for systematic errors [40]. 243

272 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Soft compression When applying an input power of -3.5 dbm to the amplifier, the 2 db compression point is reached. This compression point is on the edge between soft and hard compression. Below this compression point, the amplifier has a mild nonlinear behaviour. Going beyond this compression point results in hard nonlinear phenomena. Note that no jumps are present in the measured data since no attenuators have been switched in the NVNA. A. Polynomial VIOMAP A black-box model for the fundamental output tone β [ 2] ( ω 1 ) of the amplifier in mild compression is determined by using a third degree polynomial VIOMAP: 2 β [ 2] ( ω 1 ) = H 1 ( ω 1 )α [ 1] ( ω 1 ) + H 3 ( ω 1, ω 1, ω 1 )α [ 1] ( ω 1 )α [ 1] ( ω 1 ) (6-56) Note that only contributions of the fundamental input tone α [ 1] ( ω 1 ) and no combinations of higher order harmonics are taken into account. Hence, 2 VIOMAP kernels are estimated. Figure 6-13 represents the measured and modelled power and phase characteristic of the β [ 2] - wave of the amplifier. [dbm] β [ 2 ] êêêêêêêêê êêêêêêêêêê êêêê êêêêêêêêêêêêêêêêêêêêêêêêêêê Phase β [ 2 ] [deg] Carrier Input Power [dbm] Carrier Input Power [dbm] (a) (b) FIGURE Measurements (-) and polynomial Volterra model (... ) of the power (a) and phase (b) of the amplifier output wave Figure 6-14 represents the normalised equation error for all measured power levels: 244

273 Model of an amplifier under continuous wave excitation 2 β [ 2]meas β [ 2]model σ εθ ( ) (6-57) with β [ 2]meas and β [ 2]model respectively the measured and modelled output wave of the amplifier. σ εθ ( ) is the standard deviation of the error vector εθ ( ) = β [ 2]meas β [ 2]model. Norm. Residual Carrier Input Power [dbm] FIGURE Normalised equation error Since this residual behaves more or less as white noise, it can be concluded that the model, containing only 2 VIOMAP kernels, captures the amplifier behaviour in mild compression adequately. Furthermore, by remembering the measurement results of Chapter 5, one knows that the modelling procedure will not get any better by taking combinations of higher order input harmonics into account. B. Rational VIOMAP Instead of using a polynomial VIOMAP, a rational VIOMAP of third degree can be used to model the fundamental output tone β [ 2] ( ω 1 ) of the amplifier in mild compression: β [ 2] ( ω 1 ) 2 H 1 ( ω 1 )α [ 1] ( ω 1 ) + H 3 ( ω 1, ω 1, ω 1 )α [ 1] ( ω 1 )α [ 1] ( ω 1 ) = H 1 ( ω 1 )α [ 1] ( ω 1 ) + H 3 ( ω 1, ω 1, ω 1 )α [ 1] ( ω 1 )α [ 1] ( ω 1 ) ) ) (6-58) Note that only contributions of the fundamental input tone α [ 1] ( ω 1 ) and no combinations of higher order harmonics are taken into account. Hence, 4 VIOMAP kernels are estimated. 245

274 êêê Measurement Based Nonlinear Modelling under Continuous Wave Excitation Figure 6-15 represents the measured and modelled power and phase characteristic of the β [ 2] - wave of the amplifier. β [ 2 ] [dbm] êêêêêêêêê êêêêêêêêêê êêê ê êê ê ê ê ê êê ê Carrier Input Power [dbm] Carrier Input Power [dbm] (a) (b) FIGURE Measurements (-) and rational Volterra model (... ) of the power (a) and phase (b) of the amplifier output wave êêêê êêêêêêêêêêê êêêêê êêêêêêêêê êêêê êêêêêêêêêêêêêêêêêêêêêêêêêêê Phase [o] êê Figure 6-16 represents the normalised equation error for all measured power levels: 2 β [ 2]meas β [ 2]model σ εθ ( ) (6-59) with β [ 2]meas and β [ 2]model respectively the measured and modelled output wave of the amplifier. σ εθ ( ) is the standard deviation of the error vector εθ ( ) = β [ 2]meas β [ 2]model. Norm. Residual Carrier Input Power [dbm] FIGURE Normalised equation error Since this residual again behaves as white noise, it can be concluded that this model also captures the amplifier behaviour in mild compression adequately. 246

275 Model of an amplifier under continuous wave excitation Note that for the rational VIOMAP the equation error is smaller than for the polynomial VIOMAP. However, as expected, both modelling techniques can accurately describe a system in mild compression. C. Implicit VIOMAP The smallest error on the estimates of a system in compression is obtained by using an implicit Volterra model. A black-box model for the fundamental output tone β [ 2] ( ω 1 ) of the amplifier in mild compression is determined by using an implicit VIOMAP of third degree: 2 H 11 ( ) α + H [ 1]1 12 ( ) β + H [ 2]1 31 ( ) α α [ 1]1 [ 1] 1 β [ 2]1 + H 32 ( ) α α [ 1]1 [ 1] H 33 ( ) α β [ 1]1 [ 2] 1 + H α β 34 ( ) [ 1]1 [ 2]1 β [ 2] 1 + H 35 ( ) β α [ 2]1 [ 1] 1 + H 36 ( ) β β [ 2]1 [ 2] 1 = 0 (6-60) where α [ 1]1, β [ 2]1 and β [ 2] 1 are shorthand notations for respectively α [ 1] ( ω 1 ), β [ 2] ( ω 1 ) and β [ 2] ( ω 1 ). Equation (6-60) contains 8 VIOMAP kernels to be estimated. To obtain a plot of the modelled β [ 2]1 output wave out of equation (6-60), the following polynomial nonlinear equation must be solved: 2 β [ 2]1 β [ 2] 1 ( H 36 ( ) ) β [ 2]1 β ( [ 2] 1 H 34 ( ) α ) β [ 1]1 [ 2 ]1 ( H 35 ( ) α ) + [ 1] 1 2 β [ 2] 1 ( H 33 ( ) α ) + β ( [ 1]1 [ 2]1 H 12 ( ) + H 31 ( ) α α [ 1]1 [ 1] 1 ) + 2 H 11 ( ) α + H [ 1]1 32 ( ) α α [ 1]1 [ 1] 1 = 0 (6-61) 2 Note that the above equation is a polynomial function in β [ 2]1 and β [ 2]1, which requires 2 special techniques to be solved. The equation error εθ ( ) can however be determined as εθ ( ) = Jθˆ (6-62) εθ ( ) with θˆ the estimated parameters and J = the Jacobian matrix. θ 247

276 Measurement Based Nonlinear Modelling under Continuous Wave Excitation Figure 6-17 represents the normalised equation error for all measured power levels: 2E-5 Norm. Residual Carrier Input Power [dbm] FIGURE Normalised equation error From this equation error, which behaves as white noise and has an extremely small value, one can conclude that the implicit Volterra model describes a system in compression perfectly. In order to show the potentials of the implicit Volterra model, equation (6-61) can be simplified by neglecting all contributions in β [ 2] ( ω 1 ): 2 β [ 2]1 ( H 35 ( ) α ) + β ( [ 1] 1 [ 2]1 H 12 ( ) + H 31 ( ) α α [ 1]1 [ 1] 1 ) 2 H 11 ( ) α + H [ 1]1 32 ( ) α α [ 1]1 [ 1] 1 = 0 (6-63) The above second degree polynomial equation can be easily solved and results in 2 solutions, which are presented in Figure

277 Model of an amplifier under continuous wave excitation β [ 2 ] [dbm] Carrier Input Power [dbm] (a) êêêêêêêêêêêêêêêêêêê êêêêêêêêêêêêêêêêêêêêêêêêêêêêêêê Carrier Input Power [dbm] (b) FIGURE Measurements and simplified implicit Volterra model of the power (a) and phase (b) of the amplifier output wave In Figure 6-18, the full line represents the measured characteristic, the dotted ( ) and dashed (--) lines represent the 2 solutions of the polynomial equation (6-63). The power as well as the phase of the output wave β [ 2] are accurately modelled by one of the solutions. This solution is represented by the dotted line and coincide with the measured characteristic. The second solution, represented by the dashed line, deviates completely from the measured characteristic and can be omitted. In this case, there is no doubt about which of both solutions is the correct one and no bifurcation points are present. However, when modelling a tanh function it may well be that both solutions partly coincide, which results in bifurcation points. Finding the right solution becomes then a hard task. Phase [o] 59 êêêêêêêêêêêêêêêêêêêêêêêêêêêêêê êêêêêêêêêêêêêêêêêêê Since in equation (6-63) the contributions in β [ 2] ( ω 1 ) are left out, the model will describe the measurements less accurate than the complete model. This can be seen from the normalised equation error, represented in Figure 6-19, which is a factor 35 larger than the normalised equation error of the full model (see Figure 6-17) but is still extremely small. 249