PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

Transcription

1 PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg TÁMOP /2/A/KMR

2 Peter Pazmany Catholic University Faculty of Information Technology ELECTRICAL MEASUREMENTS (Elektronikai alapmérések) Fundamentals of signal processing A jelfeldolgozás alapjai Dr. Oláh András TÁMOP /2/A/KMR

3 Lecture 3 review Deprez instrument, hand instruments Measuring alternating current or voltage RMS (Root Mean Square) Measurement error Measuring very high and very low voltage Digital voltmeter Level measurement Waveform measurement Measuring time philosophical considerations Measuring frequency Measuring time The ELVIS system TÁMOP /2/A/KMR

4 About the decibel Outline Description of signals in transform domain (Fourier and Laplace transformation) The bandwidth of signal Analog-to-Digital Conversion The noise TÁMOP /2/A/KMR

5 About the decibel: definition The decibel is the ratio of two power quantities: db = 10 log P P 2 1 When referring the measurements of field amplitude (voltage quantity) can be consider the ratio of the squares of the quantities (the two resistors are the same value, ie. R 1 = R 2 ): The decibel can depict high range of values on expressive scale. For example the range between 1kV and 1μV means 10 9 :1 ratio, which is only 180dB value. P P 2 1 = db U R 2 / 2 U 2 R db = 10 log = 20 log + 10 log 2 U R U 1 / 1 1 R db U 2 U 2 db = 20 log = U 1 U TÁMOP /2/A/KMR

6 About the decibel: resolution The resolution is a fundamental parameter in measurements (roughly it means the capability of measurement device to differentation of two close values). It can characterizes the relative sensitivity of the measurement: For example, 4000 digits range DVM (Digitális Voltage Meter) has 4000:1 nominal resolution, in decibel scale this resolution is 72 db. An other example: n bit ADC has 2 n different quantization levels, 10lg(2 n /1) =6n, ie. The increasing of the dynamic is 6 db per bit. Comment: the resolution is often measured in percentage (% = 10-2 ), and the excellent resolution is expressed in ppm (parts per million = 10-6 ) TÁMOP /2/A/KMR

7 About the decibel: definition We can convert an absolute power or voltage measure x into db scale: P [db] =10 lg( P/ P ref ) or U [db] =20 lg( U/ U ref ) where x ref is reference value. The used reference can be recognized by the notation: dbv (feszültség egység ): the common voltage reference is U REF = 1V effective value (Root Mean Square) dbfs : FS: Full Scale dbc : c: carrier dbr : r: relative, the application determines the reference value TÁMOP /2/A/KMR

8 About the decibel: some tricks 1:1 0dB 10:1 20 db (obvious conversions: log(1) = 0, log(10)= 1) 2:1 6dB (Note: log(2) 0,3) 4 = = 12 db (log(x y) = log(x) + log(y)) 8 = = 18 db db = 20log (rate) 9 ( between 8 18 db and db by linear interpolation) 19 db 3 ( 9=3 3) 9.5 db 6 = = 15.5 db 5 ( between 4 and 6, by interpolation) 14 db 7 (by interpolation) 17 db arány 1:1 2:1 3 4(=2 2) 5 6(=2 3) 7 8(=2 4) 9 10:1 db TÁMOP /2/A/KMR

9 Signal decomposition In exanimation and description of informatics systems the signals should be treated as the sum of harmonic signals (Fourier analyses). Question: What conditions must be satisfied to compose a signal as the sum of harmonic components? We give the engineering approach to define the Fourier (signal spectrum) and Laplace transformations. According to the signal spectrum we can define the signal (and the system) bandwidth: it is the difference between the upper and lower frequencies in a contiguous set of frequencies TÁMOP /2/A/KMR

10 Limited energy: T 1 lim T T T Limited support: Electrical measurements: Fundamentals of signal processing Categories of analog time signals / 2 x / 2 2 ( t) dt < Ta x( t) = 0 if t T or t T a b Tb Entrant: xt () = 0 if t< 0 Periodicity: x( t) = x( t + kt ) k =... 2, 1,0,1,2, TÁMOP /2/A/KMR

11 Signal decomposition basic idea x(t) What are the signal characteristics? What frequencies contained in the signal? What kind of amplifier bandwidth should be used etc.? t From this representation can not be answered s k (t) Basic signal: s ( t) = A sin( 2πkf0 t) k = 0,1,2,... k k t Amplitude Frequency x( t) s ( t) = A sin 2πkf0 t k = 0,1,2 k k k k ( ),... We get answers for all technical questions!!! TÁMOP /2/A/KMR

12 Signal decomposition Signal Decomposition (transformation) Meaningful representation for the given engineering task Technical specification Design of signal processing etc. What are the basic signals??? TÁMOP /2/A/KMR

13 Advantages Basic signal 1 Const 1 basic signal 1 Signal Basic signal 2 Linear System Const 2 basic signal 2 Basic signal n Const n basic signal n Physically difficult to interpret Physically easy to interpret The effect of linear system can be easily interpreted Characteristics of linear system: const 1, const 2,., const n TÁMOP /2/A/KMR

14 Choice of base signals ( j2πkf t) A k exp 0 B k exp H ( j( 2π kf t + ϕ )) A exp 0 k ( j2πkf t) k k 0 ( j2πkf t) A k exp 0 Eigenfunction of a linear system exp( j2π kf t) Const exp( j2πkf t) A k 0 System A k TÁMOP /2/A/KMR

15 δ(t) Dirac-delta impulse signal Mathematical discussion Impulse response function h(t)=φ(δ(t)) x(t) t Lin. inv. system h(t) y(t) y ( t) t ( t τ ) x( τ ) dτ = h( τ ) x( t τ ) = h dτ Convolution j 2 kf0t sk() t = Ak e π ( τ) ( τ) τ = () h s t d const s t k k?? j2 πkf0( t τ) j2πkf0t j 2πkf0τ j 2πkf0t j 2πkf0τ j 2πkf0t h( τ) Ake dτ = h( τ) Ake e dτ = Ake h( τ) e dτ = const Ake!! const ( τ ) j2πkf0τ = H ( kf0 ) : = h e dτ TÁMOP /2/A/KMR

16 Signals in the spectral domain j 2πkf0t Can we composite x(t) as the sum of s ( t) = e? jk2π 0 If x(t) is periodic signal, then x(t) f : 1 T xt () = 1 = T k k= ce T jk 2π f t c : x t e dt = () 0 0 k 0 k f t t x(t) ck FOURIER SERIES TÁMOP /2/A/KMR

17 Consequence x(t) Lin. inv. system h(t) y(t) xk Lin. inv. system H yk y ( t) ( t τ ) x( τ ) = h dτ ( kf ) x k y k = H 0 y = Hx const ( τ ) j2πkf0τ = H ( kf0 ) : = h e dτ H H (0) 0 = H ( f ) 0 0 H (2 f 0 0 ) H (3 f 0 ) TÁMOP /2/A/KMR

18 X Problem: not all signal is periodic j π ft = x ( t ) e 2 dt ( f ) : FOURIER TRANSFORMATION Time domain Frequency domain dx ( t) j2π fx ( f ) dt t 0 x( u) du x 2 ( t) dt 1 j2π f X ( f ) X 2 ( f ) df FT basic properties: Linearity, Translation, Modulation, Convolution, Scaling, Parseval's theorem TÁMOP /2/A/KMR

19 Signal s spectrum Example: x(t)=u(t)e -αt X(ω)=1/(α+jω) X ( ω) = α + ω arcx Problems: 1. The Dirac delta function has not FT 2. Contstans signal has not FT 3. FT of periodic signals { δ ( t) } { } ( ωt) F =? F const. =? Fsin =? { } ( ω) ω = arctan α TÁMOP /2/A/KMR

20 x ε () t { ()} () Spectrum of rectangular signal 1 if t < ε = ε 2 0 otherwise ε ε 2 -jωt 2 -jωt -jωt 1 e F xε t = x t e dt = e dt = ε = -jω ε 2 ε 2 - ε/2 -jωε 2 jωε 2 -jωε 2 jωε 2 jωε 2 -jωε 2 1e e 2 e e 2 e e = = = = ε -jω εω -j2 εω j2 2 sin( ωε 2) X(ω) = sin ( ωε 2 ) = εω ω ε 2 X ( f ) = x(t) ε/2 ( ωε ) sin / 2 ωε /2 t ω TÁMOP /2/A/KMR

21 Solution: x ε Signal s spectrum: the Dirac delta Rectangular approximation () t 1 if t < ε = ε 2 0 otherwise - ε/2 x(t) ε/2 t X(ω) X ( f ) = ( ωε ) sin / 2 ωε / δ F ( t) = lim xε ( t) δ ( ) ε 0 { const. } = const. δ ( f ) { } { ( )} F t = lim F x t = 1 ε 0 ε ω TÁMOP /2/A/KMR

22 Solution: Electrical measurements: Fundamentals of signal processing Signal s spectrum: sine wave 3. Sine wave in the frequency domain { () ( )} () j ω0t () -j ω0t e e 1 F x t sin ω0t = F x t x t = X ( ω ω0) X ( ω+ ω0) 2j 2j 2j F 1 2j { 1sin ( ω0t) } = δ ( ω ω0) δ ( ω+ ω0) X(ω) Fourier Transformation Modulation -ω 0 ω 0 ω TÁMOP /2/A/KMR

23 Bandwidth of a signal: the concept It is desirable to classify signals according to their frequency-domain characteristics (their frequency content): Low-frequency signal: if a signal has its spectrum concentrated about zero frequency High-frequency signal: if the signal spectrum concentrated at high frequencies. Bandpass-signal: a signal having spectrum concentrated somewhere in the broad frequency range between low frequencies and high frequencies.

24 Bandwidth of a signal: the concept (cont ) The quantative measure of the range over which the spectrum is concentrated is called the bandwidth of signal. We shall say that a signal is bandlimited if its spectrum is zero outside the frequency range f B, whereb is the absolute bandwith. The absolute bandwidth dilemma: Bandlimited signals are not realizable! Realizable signals have infinite bandwidth! (No signal can be time-limited and bandlimited simultaneosuly.)

25 Bandwidth of a signal: the concept (cont ) In the case of a bandpass signal (f min f f max ), the term narrowband is used to describe the signal if its bandwidth B= f max f min, is much smaller than the median frequency (f max + f min )/2. Otherwise, the signal is called wideband. There are many bandwidth definitions depending on application: noise equivalent bandwidth 3 db bandwidth η% energy bandwidth

26 The noise equivalent bandwidth It is definied as the bandwidfth of a system with a rectangular transfer funtiuon that receives as much noise as the system under consideration S ( f ) White noise PSD B f

27 The 3 db bandwidth Is the bandwidth at which the absolute value of the spectrum (energy spectrum or PSD) has decreased to a value that is 3 db below its maximum value. 2 ( ), ( ), ( ) X f X f S f ( ) Xmax = max X f f ε X max ε = 0.5 B ε f

28 The η% energy bandwidth Is the bandwidth that contains η % of total emitted. 2 ( ), ( ) X f S f 90% B 90% f

29 Biological Signals Seismic signals Electromagnetic signals Electrical measurements: Fundamentals of signal processing Frequency ranges of some natural signals Type of Signal Electroretinogram 0-20 Pneumogram 0-40 Electrocardiogram (ECG) Electroenchephalogram (EEG) Electromyogram Sphygmomanogram Speech Seismic exploration signals Eartquake and nuclear explosion signals Radio bradcast 3x10 4-3x10 6 Frequency Range [Hz] Common-carrier comm. 3x10 8-3x10 10 Infrared 3x x10 14 Visible light 3.7x x10 14

30 Time domain Electrical measurements: Fundamentals of signal processing y ( t) The convolution The Fourier transform translates between convolution and multiplication of functions. ( t τ ) x( τ ) = h dτ j 2πft = dτe dt Frequency domain j2πft Y ( f ) y( t) e dt = h( τ ) x( t τ ) j j2 π fτ 2πft j2πf ( t τ ) ( t τ ) e dtdτ = h( τ ) x( t τ ) e = h( τ ) x e dtdτ = = = h( τ ) x j 2πfu j 2πfτ j 2πfτ j 2πfu ( u) e e dudτ = h( τ ) e dτ x( u) e du = H ( f ) X ( f ) Y ( f ) = H ( f ) X ( f ) TÁMOP /2/A/KMR

31 Consequence x(t) Lin. inv. system h(t) y(t) X ( f ) Y ( f ) Lin. inv. system H(f) y ( t) ( t τ ) x( τ ) = h dτ H ( f ) : ( τ ) j2 τ = π f h e dτ Y ( f ) = H ( f ) X ( f ) Frequency response Impulse response function TÁMOP /2/A/KMR

32 Problem: not all signal is absolutely integrable If x(t) entrance and then I o x t) ( dt < not satisfied, but ( x t ) e ): = x ( t ) e e dt = x αt ( t) e dt 0 αt x( t) e has Fourier Transform α t α t j 2π ft ( α + j 2π f ) t ( x ( t ) e dt 0 st = x( t) e dt s : α + j2πf X ( s) LAPLACE TRANSFORM 1 x ( t) = X ( s) e st ds 2πj G 0 = complex frequency < There are a lot of algebraic methods available for inverse transform TÁMOP /2/A/KMR

33 Advantage of Laplace transformation X F : = x( t) : x( t) dt 0 < X : = x( t) : x( t) e dt 0 L αt < F X X L We extend algebraic apparatus to broader function class. The (complex) frequency lost the direct physical content TÁMOP /2/A/KMR

34 Consequence x(t) Lin. inv. system h(t) y(t) X (s) Y (s) Lin. inv. system H(s) y ( t) ( t τ ) x( τ ) = h dτ H ( s) : ( τ ) js = h e τ dτ Y ( s) = H ( s) X ( s) Transfer function Impulse response function TÁMOP /2/A/KMR

35 Representation Summary Computation of output signal Properties Time domain impulse response function ( t τ ) y ( t) = h x( τ ) dτ Frequency domain Y ( jω) = H ( jω) X ( jω) Complex frequency domain H ( jω) : j2πft = h( t) e dt ( s) X ( ) Y ( s) = H s H st ( s) : = h( t) e dt 0 Not intuitive, complicate mathematical apparatus (convolution integral) Intuitive, simple mathematical apparatus Not intuitive, but simple mathematical apparatus Comment: In math see integral transformation t 2 Fourier, Laplac, Hilbert, Poisson, etc. y( p) = K( p, t) x( t) dt In 2 dimension: walsch, wavelet, etc. t1 Comment: Calculation of Fourier Transform for discrete signal is DTFT, in practice DFT (FFT) [ see Signal Processing course] TÁMOP /2/A/KMR

36 Characterization of linear invariant systems Input signal Linear Invariant system (eg.: filter) Output signal H ( f ) : ( τ ) j2 τ = π f h e dτ H ( s) : ( τ ) js = h e τ dτ TÁMOP /2/A/KMR

37 Signal manipulation in frequency domain x(t) FT X ( f ) t IFT Lowpass filter H ( f ) f y ( t) ( t τ ) x( τ ) = h dτ Y( f ) = H( f ) X ( f ) f t f TÁMOP /2/A/KMR

38 Signal manipulation in frequency domain x(t) FT X ( f ) t IFT Highpass filter H ( f ) f y ( t) ( t τ ) x( τ ) = h dτ Y( f ) = H( f ) X ( f ) f t f TÁMOP /2/A/KMR

39 Analog-to-Digital Conversion Signal analysis and processing is engaged with studying the different phenomena of nature and draw conclusions about how the observed quantities are changing in time. All applications have one thing in common, signals are studied as a function of time and the analysis is carried out by a computer. However, computers can only process digital sequences, thus the analog signal must first be converted into a binary sequence. analog signal, x(t) binary sequence, c n Analog to Digital Conversion

40 Notations The underlying notation is summarized by the following table: Signal Time Voltage Analog signal x(t) Continuous Continuous Sampled signal x(n) or x(nt) Discrete Continuous Quantized signal x ˆk Discrete Discrete Coded signal c n Discrete Binary

41 Analog-to-Digital Conversion ADC has three main steps: sampling when sample the value of the signal x(t) at certain discrete time instants obtaining a sequence x k ; quantization when the values of the samples x k are rounded to some allowed discrete levels (referred to as quantization levels) and having a finite set of these levels they can then easily be represented by binary codewords. coding when quantization symbols are mapped into binary codewords Sampling Quantization x(t) T ΔT x(nt) x(n) ˆx( n) Coding Optimal representation Compressing c n

42 The challenge of ADC Question: Is there any loss of information in the course of the conversion? What is the optimal representation of signals by binary sequences (in terms of length etc.)? Fundamental challenges of sampling and of quantization: choosing proper sampling frequency and quantization levels. ADC is fully characterized by the sampling frequency (denoted by f s ); the number of quantization levels (N), and the rule of quantization. Optimizing ADC means that we seek the optimal values of these parameters in order to obtain efficient binary representation of signals with minimum loss of information.

43 Sampling Sampling is carried out by a switch and temporary we assume that the switch is ideal (i.e. the holding period is zero). Analog signal Sampling Real sampled signal x(t) T Δt x s (t)

44 Sampling Sampling (cont ) x(t) T ΔT x(nt)? x(t) Sampling switch Analog signal Sampled signal Reconstructed analog signal Can analog signal be reconstructed from their samples without any loss?

45 The sampling theorem (Shannon Kotelnikov 1949) If a bandlimited signal x(t) (the band is limited to B) issampled with sampling frequency f s 2B then x(t) can be uniquely reconstructed form its samples as follows: where () ( ) ( ) x t = x nt h t nt () ht n= = 2T sin 2 ( π Bt) 2π Bt

46 Phenomena of aliasing If the sample frequency is not chosen to be high enough (i.e. frequency f s 2B), then X s (f) then there is an overlap in the spectrum, which implies that X (f) cannot be regained from X s (f). Aliasing

47 Summarizing of sampling In the case of practical sampling first we obtain x s (t) fromx(t) and then from x s (t) the original signal x(t) can be regained by letting x s (t) pass through a lowpass filter. Sampling Filtering x(t) T ΔT x s (t) H(f) x(t) f Sampling switch Lowpass filter Analog signal Real sampled signal Reconstructed analog signal

48 Quantization We assume that the signal is already sampled and we deal with samples x(n). Since each sample has continuous amplitude, quantization is concerned to mapping x(n) into xn ˆ( ) which may have only a finite number of values. Sampled signal x( n) R Quantization ˆx( n) Quantified signal ( ) = { } ˆ,,...,, x n Q α1 α2 α N

49 Quantization (cont ) Quantization always entails loss of information due to the rounding process. The design of a quantizer is concerned with two parameters: number of quantization levels; location of quantization levels (uniform or non-uniform); The quality of quantization is described by a Signal-to- Quantization Noise Ratio (SQNR) where the average signal power is compared to the noise power resulting from the quantization error: average signal power SQNR : = average noise power due to quantization [ db] ( SQNR : = 10log SQNR)

50 Signal value is rounded off to predefined thresholds called as quantization values which are equidistantly placed. Notations: the sample range is [-C,C] the distance between the thresholds is, the number of quantization level is N =2C/ =2 n, where n represents the number of bits by which the quantized signal can be represented. the error signal is Uniform quantization ε : = x xˆ and - /2 ε - /2. The quantization characterictics and the quantization error function

51 Uniform quantization (cont )

52 Modeling the quantization noise Since the nature of errors are random the specific value of ε depends on the value of the current sample, thus ε is regarded as a random variable subject to uniform probability density function, and the average noise power is Δ /2 Δ / Δ E( ε ) = u pε ( u) du = u du = Δ 12 Δ/2 Δ/2

53 SQNR of the uniform quantization In the case of full-scale sine wave (with amplitude C ): 2 2 C /2 34C n SQNR : = = = N = 2 [ db] 2 2 ( SQNR : = 6.02n ) Δ /12 2 Δ 2 2 In the case of random input variable subject to uniform probability density function over the interval [-C,C]: ( ) C /12 4C 2 2n [ db] SQNR : = = = N = 2 ( SQNR : = 6.02 n) 2 2 Δ /12 Δ In the case of sine wave with amplitude A (in normal operation i.e. A<C) [ db] ( ) SQNR : = 6.02n log C / A

54 Non-uniform quantization Uniform quantization suffer from one bottleneck: if the sample to be quantized does not exploit the full range of quantization (i.e. [-C,C] the interval) then SNR can deteriorate severly. As result a user having smaller dynamic range suffers a drop in Quality of Service (QoS). Non-uniform quantization is way to compensate this effect: smaller dynamic range there are plenty of quantization levels (to help the users with smaller dynamics) whereas in the case of large dynamic signal there are less quantization levels

55 Non-uniform quantization (cont ) Probability density function of samples in the case of small and large dynamics

56 Non-uniform quantization (cont ) The implementation of nonlinear quantization can be reduced to applying an equidistant quantizer preceded by a proper nonlinear distortion function l(x).

57 The optimal non-uniform quantization The optimal characteristics l(x) can be found by solving the following problem: l opt ( x):max l( x) C 2 u px C C C l 1 ( x) 2 ( ) p u du x ( ) x dx This optimization is a hard problem itself ( solved in the domain of functional analysis), but it is made more difficult by the fact that real life processes are nonstacionary (the sample p.d.f. p(x) is changing in time) and as result this problem must be solved again and again in order to adopt to the changing nature of the process.

58 The logarithmic quantization To circumvent the difficulties of optimization, we are satisfied by choosing an l opt (x) subject to a modified objective function which guarantees uniform C SQRN: 2 u p u du ( x) ( ) C lopt ( x):max = const. l( x) C 1 p 2 x ( x) dx l C x One can easily see that if x 2 ~1/l (x) 2, then indeed the SNR is constant and independent of p x (u). Thus l (x) ~1/x, from which l(x) ~ log(x), which entails logarithmic quantization.

59 The logarithmic quantization (cont ) Characteristics of logarithmic guantizer

60 x( n) Electrical measurements: Fundamentals of signal processing The logarithmic quantization (cont ) Non-Uniform quantization ( ) Quantization 1 ( ) y( n) ŷ n ( ) Compression l x x Expansion l x x ˆx( n) The real compressor l(x) is chosen differently in Europe ( A-law ) or in the US and Far East ( μ-law ).

61 Quantization errors: zero drift

62 Quantization errors: gain error

63 Quantization errors: integral nonlinearity

64 Quantization errors: differential nonlinearity

65 AD converters and main performances Many various AD converters have been designed and developed. However, currently on the market there are only a few main types of them: successive approximations register SAR, pipeline, deltasigma, flash and integrating converters.

66 AD converters and main performances (cont ) We can see that there is no one universal AD converter the converters of high speed are of the poor resolution and vice versa accurate (large number of bits) converters are rather slow. The most commonly used are the SAR (Successive Approximation Register) and Delta-Sigma converters. SAR converters are very accurate, operate with relatively high accuracy (16-bit) and wide range of speed up to 1 MSPS. The Delta-Sigma converters (16-bit and 24-bit) are used when high accuracy and resolution are required. Recently, these converters are still in significant progress.

67 Successive Approximation Register (SAR) The principle of operation of the SAR device resembles the weighting on the beam scale. Successively the standard voltages in sequence: U ref /2, U ref /4, U ref /8... U ref /2 n are connected to the comparator. These voltages are compared with converted U x voltage. analogue signal U x U ref SH U com p + - Controlled voltage source Controlled voltage source register digital signal

68 SAR (cont ) If the connected standard voltage is smaller than the converted voltage in the register this increment is accepted and the register sends to the output 1 signal. If the connected standard voltage exceeds the converted voltage the increment is not accepted and register sends to the output 0 signal. U comp U ref /2 U ref /4 U ref /8 U ref/16 U ref/32 U x time

69 Performance trade-offs of ADC In the realization of the ADC converters improving the sample rate and the resolution at the same time are conflicting requirements.

70 Available ADC on the market Part Type Bits Sampling rate Manufacturer Price, $ ADC180 Integration ms Thaler 210 ADS1256 Delta-sigma kHz Texas 9 AD7714 Delta-sigma 24 1kHz AD 9 AD1556 Delta-sigma 24 16kHz AD 27 MAX132 Integration 18 63ms Maxim 8 AD7678 SAR kHz AD 27 ADS8412 SAR 16 2MHz AD 23 MAX1200 Pipeline 15 1MHz Maxim 20 AD9480 pipeline 8 500MHz AD 200 MAX105 Flash 6 800MHz Maxim 36

71 Characteristics of ADC per application Application Architecture Resolution Sampling rate Audio SAR Delta-sigma bits bits khz 48-50kHz Medical SAR Delta-sigma 8-16 bits 16 bits khz 192 khz Automatic control SAR Delta-sigma 8-16 bits 16 bits khz 250Hz Wireless comm. SAR Delta-sigma 8 bits 13 bits 270kHz

72 The noise In signal processing the noise can be considered unwanted data without meaning, in other words the noise is an error or undesired random disturbance of a useful information signal. The measurement signals are usually accompanied by some noises and interferences, sometimes of the level comparable to the level of the measured. The typical interference signals are generated by the electric power lines, electrical machines, lighting equipment, commutating devices, radio communication transmitters, atmospheric discharges or cosmic noises. There are also internal sources of noises resistors and semiconductor devices (thermal Noise, shot noise, etc.) TÁMOP /2/A/KMR

73 Classification according to power spectral density The noise is by definition derived from a random signal, we can describe it by its statistical properties (mean, variation, correlation, etc.) What does spectral analysis mean for a random signal? (We know for deterministic signal: Fourier Transformation) Correlation function: E{ [ xt ( + τ ) μ] [ xt ( ) μ] } R( τ ) = 2 σ For stationary stochastic signal R(τ) is constant, the power spectral density is by definition its Fourier Transformation: jωτ S R e d ( ) ( ) ω = τ τ TÁMOP /2/A/KMR

74 Colors of noise Blue noise f Grey noise Violet noise f 2 White noise Pink noise, 1/f Brownian noise 1/f TÁMOP /2/A/KMR

75 Thermal noise (Johnson Nyquist noise) Phenomenon: it is an electronic noise inside an electrical conductor at equilibrium regardless of any applied voltage. Cause: the thermal agitation of the charge carriers. Description : white Gaussian distribution with variance per hertz of bandwidth: 2 vn = 4 kb T R where k B is the Boltzmann s constan, T is the resistor's absolute temperature in kelvins, and R is the resistor value in ohms. For a given f bandwidth (eg.: R=1kΩ, T=300K, vn= 4.07 nv/ Hz) v = v Δ f = 4 k T R Δf n n B (eg.: Δf=10kHz, vn= 400 nv) TÁMOP /2/A/KMR

76 Thermal noise (Johnson Nyquist noise) (cont ) (R=50Ω) -133 dbm 3 db 10 dbm Power Spectral density: S ( f ) = 2 R h f e hf k T B 1 f kt h B << some THz ( ) 2 B S f R k T v = 4 k T R Δf n B TÁMOP /2/A/KMR

77 Shot noise Phenomenon: detectable statistical fluctuations in current measurement. Cause: the current being carried by discrete charges (electrons) whose number per unit time fluctuates Description: individual Poisson processes, together (law of large numbers!) normal distribution with standard deviation: σ i = 2 q I Δf where q is the elementary charge, Δf is a the bandwidth in Hertz over which the noise is measured, and I is the average current through the device (eg.: I=100mA, Δf=1Hz, σ i =0.18nA) TÁMOP /2/A/KMR

78 The spectrum of thermal and shot noise TÁMOP /2/A/KMR

79 Other noises Flicker noise Burst noise (or popcorn noise) Interference noise Quantization noise TÁMOP /2/A/KMR

80 Summary The decibel is logarithmic function of the ratio of two power (or voltage) quantities. The Fourier transform is the operation that decomposes a signal into its constituent frequencies. Computer based measurements can only process digital sequences, thus the analog signal must first be converted into a binary sequence. The noise is by definition derived from a random signal, we can describe it its statistical properties Next lecture: Positioning systems TÁMOP /2/A/KMR