DIGITAL COMPENSATION OF IN-BAND IMAGE SIGNALS CAUSED BY M-PERIODIC NONUNIFORM ZERO-ORDER HOLD SIGNALS

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1 DIGITAL COPENSATION OF IN-BAND IAGE SIGNALS CAUSED BY -PERIODIC NONUNIFOR ZERO-ORDER HOLD SIGNALS Christian Vogel 1, Christoph Krall 2 1 ETH Zurich, Switzerland Signal and Information Processing Laboratory ISI C.Vogel@ieee.org 2 Rohde & Schwarz GmbH & Co. KG Radiocommunications Systems Division Christoph.Krall@ieee.org ABSTRACT In this paper we introduce the design of a digital time-varying filter to compensate the spurious in-band spectra caused by -periodic nonuniform zero-order hold signals. For this purpose, a continuous-time framework to describe and analyze such signals is developed. Based on the error analysis, an equivalent discrete-time framework is derived and used for the design of a discrete-time time-varying precompensation filter. We exemplify the design of the precompensation filter for the two-periodic case through simulations and measurements. Furthermore, we discuss Farrow filters to reduce the design complexity. Keywords: periodic, nonuniform, zero-order hold, compensation, reconstruction. 1 INTRODUCTION Emerging communication systems require high-speed and high-resolution data converters [1]. Since the integration density of digital circuits grows much faster than the one of analog circuits, we have an increasing gap between analog and digital circuits in terms of speed, area, and power consumption. In particular, data converters - as the central interface between analog and digital circuits - are affected by this technological gap [2]. A new design paradigm exploits the technology gap by using digital signal processing to overcome the analog impairments of data converters, e.g. [3]. In this paper we discuss digital-to-analog converters DACs using a zero-order hold ZOH for the signal reconstruction. We investigate the compensation of in-band images caused by ZOHs periodic nonuniform hold signals, i.e., a nonuniform ZOH, as illustrated in Fig. 1. The individual sample instants deviate by r n T from the ideal time instants nt, where T is the nominal sampling period of the DAC and r n is the relative time offset for time n period, i.e., r n = r n+. Beside the typical sin x/ x shaped output spectrum, such nonuniform ZOH signals introduce additional spurious images that significantly reduce the DAC performance [4]. We can find such spurious images in DACs driven by clock signals deterministic jitter [5] and in time-interleaved DACs timing mismatches [6-8]. Although the effects of yt xt T + r T 1T + r 1 T x1t x2t x3t x4t 2T + r 2 T 4T + r 1 T 3T + r T Figure 1: Output of a three-periodic nonuniform ZOH =3 in the time domain. nonuniform ZOHs have been extensively analyzed [4,5,9], their digital compensation has not yet been addressed in the literature. We propose a digital time-varying filter to compensate for the spurious image spectra caused by periodic nonuniform ZOH signals. Therefore we develop a discrete-time model of a nonuniform ZOH in Sec. 3 and use this model in Sec. 4 to derive the ideal frequency response of a timevarying compensation filter. In Sec. 5 we present filter design examples including a Farrow filter design and finally draw our conclusions in Sec. 6. We want to note that preliminary results of the presented work have been presented at two conferences [1,11]. 2 PRELIINARIES Sampling a continuous-time signal x c t at uniform time instances t = nt produces the t Ubiquitous Computing and Communication Journal 1

2 sequence x[n] = x c nt, where T denotes the sampling period. The discrete-time Fourier transform of x[n] is Xe j = 1 T 2 p X c j. 1 p= T T Throughout this paper, we assume that the signal x c t is bandlimited, i.e., the continuous-time Fourier transform of x c t satisfies X c j=, for T. 2 With these assumptions, the signal x c t can be recovered from the sequence x[n] by using a sequence-to-impulse train converter and an ideal continuous-time low-pass filter gain T and cut-off frequency /T. Y j = 1 T p= X c j p 2 H j T T = X c j p 2 H j T p= 6 sin T 2 H j = T e jt We realize that the factor T of the ZOH compensates the 1/T -factor introduced by the sampling process. Furthermore, the output signal consists of the 2 /T -periodic extended continuous-time signal X c j weighted by the sincx function given by Eq. 7. yt xt x1t x2t x3t x4t 3 SYSTE ODELS After clarifying the behavior of a uniform ZOH, we introduce a ZOH -periodic nonuniform hold signals and derive a discrete-time model. T 1T 2T 3T 4T t Figure 2: Ideal uniform ZOH. In practice, however, we can neither generate ideal Dirac-delta impulses nor can we implement an ideal continuous-time low-pass filter. Instead a ZOH and a low-order analog low-pass filter are used. Employing an ideal uniform ZOH for the reconstruction process results in the reconstructed signal yt, which is illustrated in Fig. 2 and is given by yt = x t h tt 3 where x t = x c t t nt 4 n= is the sampled continuous-time signal and h t = 1 T ut utt 5 is the 1/T -weighted impulse response of an ideal ZOH ut being the unit step function. The reason to explicitly introduce the T -factor in Eq. 3 can be seen in the frequency domain. Applying the continuous-time Fourier transform CTFT to Eq. 3 results in 3.1 Continuous-time model Figure 1 illustrates the output of a ZOH -periodic nonuniform hold signals in the time domain. In contrast to the uniform case, each sampling instant is time-shifted by r n, where r n is periodic, i.e., r n = r n+. These time offsets of the hold signal can be modeled by the - periodic impulse response h n = h n+ starting at r n and ending at 1 + r n+1 T [4,5], i.e., h n t = 1 T ut r n ut 1 + r n+1 T. 8 Therefore, the output signal y t in Fig. 1 can be modeled as the sum of weighted impulse trains of period T x n t = x c t t mt nt 9 m= convolved impulse responses h n tt, i.e., 1 yt = x n t h n tt. 1 n= The CTFT of Eq. 1 results in Ubiquitous Computing and Communication Journal 2

3 1 Y j = X c j k 2 p= k= T j H k and H k 1 j= 1 H n j n= e jkn 2 sin 1+ r n+1 r n H n j = T 2 Energy Density Spectrum dbc T 2 p 2 T e j 1+r n+1 +r n T Desired Image 1 9 Image 2 Image Normalized frequency Ω/Ω s Figure 3: Output of a DAC a 4-periodic nonuniform ZOH =4 in the frequency domain. Figure 3 illustrates the spectral components given by Eq for = 4. The desired output is the 2 -extended continuous-frequency spectrum weighted by H j, which is the average frequency response over all frequency responses H n j for n =,..., 1. This spectrum is similar to the output spectrum in the uniform case; however, we additionally have 1 spectra, which are modulated and filtered images of the spectrum of a uniform ZOH. As these images appear in the fundamental band, i.e., < /T, they cannot be removed by an analog low-pass filter and significantly degrade the output signal quality. A detailed analysis of the impact on the performance can be found in [4,5]. 3.2 Discrete-time model In order to derive discrete-time compensation filters for the in-band images, we represent the continuous-time output signal given by Eq. 11 for < /T in discrete-time. Therefore, the output in Eq. 11 has to be ideally bandlimited, which can be realized by the filter H id 1, < j= T,. 14 T Using Eq. 14 and Eq. 11, we can express the ideally bandlimited signal as Y b j = 1 T Y jth id j. 15 Because we assume that the underlying continuoustime signal x c t is bandlimited, which is implicitly satisfied if x[n] is a synthetic signal, we can relate [12, Chap. 4] = 1 T X j c T 16 Xe j and H k e j = H k j 17 T for <. Consequently, we can express Eq in discrete-time as Y e j = Xe j k 2 H k e j 18 k= H k e j = 1 1 H n e j n= and e jkn 2 sin 1+ r n+1 r n H n e j = , 19 e j 1+r n+1 +r n Using Eq. 18 and Eq. 1 Y e j it can be verified that Y b j = Y e jt TH id j 21 results in Eq. 15 and the continuous-time and the discrete-time frequency output are related by Y e j = 1 T Y j, <. 22 T Ubiquitous Computing and Communication Journal 3

4 4 DERIVATION OF THE COPENSATION FILTER In this section we propose a time-varying filter to compensate the in-band images. We derive the design equations and analytically solve them for the two-periodic case. Eq. 26 gives Se j = where 1 G k e j k 2 k= Xe j k Design equations To compensate for the spurious images, we propose an -periodic time-varying filter g n [l], which relates the input x[n] to the output s[n] as s[n] = g n [l]x[n l]. 23 l= The precompensation filter and the discrete-time model of the nonuniform ZOH are illustrated in Fig. 4 for the two-periodic case. G k e j = 1 1 G n n= e j e 2 jkn 28 is the DTFT of Eq. 25. Because of the timevarying filter g n [l], the input signal into the ZOH is s[n] instead of x[n] and Eq. 18 becomes 1 Y e j = Se j k 2 H k e j. 29 k= x[n] G n e jω Ğ e jω discrete-time model ZOH s[n] H e jω y[n] Substituting the output of the filter given by Eq. 27 in Eq. 29 results in Ğ 1 e jω 1 n 1 n H 1 e jω Figure 4: Precompensation filter in front of the discrete-time model of a nonuniform ZOH =2. The time-varying filter g n [l] should modify the signal x[n] in such a way that the signal s[n] driving the nonuniform ZOH does not generate image signals in the fundamental band. Since g n [l] is periodic, we can represent it as the inverse discrete-time Fourier series DTFS [13, Chap. 4] g n [l] = 1 k= g k [l] e jkn 2 the related DTFS g k [l] = 1 1 n= g [l] n e 24 2 jkn. 25 After substituting Eq. 24 in Eq. 23, we can write s[n] = 1 g k [l]e jkn x[n l] l= k= 2 1 = g k [l]x[n l] e jkn 2. k= l= 26 = Ye j 1 1 H k1 k 1 = k= X e j k 1 +k e j 2 G k e j k 1 +k. 2 With l = k 1 + k we can simplify Eq. 3 to 1 1+k Ye j = H lk e j G k e j l 2 k= l =k X e j l = 1 1 k= l = 2 H lk X e j l e j G k e j l where we have exploited the periodicity of H lk e j. Thus, the overall transfer function can be expressed as Ye j = 1 F l e j l 2 Xe j 2 l 32 l = The discrete-time Fourier transform DTFT of Ubiquitous Computing and Communication Journal 4

5 F l e j = 1 H lk e j +l 2 k= G k e j 33 as it is shown in Fig. 5 for the two-periodic case. x[n] F e jω F 1 e jω 1 n y[n] Figure 5: Overall transfer function of the cascaded system for =2. From the theory of multi-rate systems we know that a system is defined as perfect reconstruction system if the output is a scaled and delayed version of the input [14]. In our compensation problem we want to maintain unity gain of the overall system and therefore require that F l e j l 2 = e j, for l =, for l = 1,2,K, 1 34 where is the delay of the system. If Eq. 34 is fulfilled all image spectra are cancelled and the sinx/x distortions in the fundamental band are equalized. The time-varying filter g n [l] can be efficiently implemented as an -channel maximally decimated multi-rate filter bank [14]. 4.2 Two-periodic nonuniform ZOH We will exemplify the filter design procedure for the two-periodic case, i.e., = 2. By expressing Eq. 33, and Eq. 34 in matrix notation we obtain e j H e j = H 1 e j + H 1 e j G e j H e j + G 1 e j Solving the matrix equation leads to 35 H e j + G e j G 1 e j = P e j H 1 e j + e j 36 P e j P e j = H 1 e j j + H 1 e H e j j + H e. 37 Applying the inverse DTFS as defined in Eq. 24 to Eq. 36 and substituting the relation Eq. 19 for H k e j we obtain the transfer functions of the time-varying filter g n [l] that are H 1 e j + G e j G 1 e j = Pe j H e j + Pe j Pe j e j 38 = 1 2 H e j 1 H e j + H e j j + H 1 e. 39 Analytically solving the matrix equations for larger orders of is infeasible. Instead the matrix has to be numerically solved for each frequency used in the filter design procedure. 5 FILTER DESIGN EXAPLES In the following we will discuss the design of a two-periodic time-varying FIR filter, the compensation of a two-channel time-interleaved DAC, and the design of a Farrow filter. 5.1 FIR filter design For the design example we have assumed a two-periodic ZOH signal time offsets of r = and r 1 =.391. To approximate the ideal frequency responses given by Eq. 38 we use causal finite-impulse response FIR filters transfer functions L G a n e j = g a jl n [l]e 4 l= and approximate the ideal frequency responses in the minimax Chebychev sense, i.e., min G n e j G a n e j D 41 where D is the definition domain of and frequencies above D belong to the don't care band. We have designed filters of order L = 17, = 8.5, and D =.7,K,.7 by solving the approximation problem the atlab optimization toolbox CVX [15]. Ubiquitous Computing and Communication Journal 5

6 E e jω db Energy Density Spectrum dbc Normalized frequency Ω/Ω s Figure 6: Deviation from the ideal overall frequency response Normalized frequency Ω/Ω s Figure 8: Output of a nonuniform ZOH period = E 1 e jω db Energy Density Spectrum dbc Normalized frequency Ω/Ω s Figure 7: Attenuation of the spurious images. For the specified problem we have obtained a maximum approximation error of 61.1 db in the filter design of G a e j and 6.4 db for G 1 a e j. In Fig. 6 we see the impact of the approximation error on E e j = F e j e j 42 i.e., the difference between the average frequency response and an ideal delay, and in Fig. 7 on the residual images E 1 e j = F 1 e j. 43 Within the definition band D, the approximation error of an ideal delay is less than -98 db and the spectral images are attenuated by at least -6 db. To further test the filter design, a discrete-time multitone signal frequencies [.581,.1162,.1743,.2324,.295,.3486] 1/T has been used as an input to the nonuniform ZOH. Without any compensation we have obtained the energy density spectrum shown in Fig. 8. We see the sinx/x Normalized frequency Ω/Ω s Figure 9: Precompensated output of a nonuniform ZOH period =2. output spectrum and the additional image spectra due to the nonuniform ZOH signal. The largest unwanted spur in the fundamental band, i.e., the band between and.5, is about -29 dbc. The energy density spectrum compensation is shown in Fig. 9. Within the fundamental band the unwanted spectral images are considerably reduced. The largest spur is about -61 dbc, which is an improvement of 32 db compared to the uncompensated case. A higher attenuation of the images is achievable by increasing the filter order. When comparing the out-of-band energy of Fig. 8 and Fig. 9 we recognize that the out-of-band energy in Fig. 9 is slightly increased. This minor amplification of the out-of-band energy, however, should not change the design requirements of the analog reconstruction filter significantly. 5.2 Time-interleaved DAC We have implemented a two-channel timeinterleaved digital-to-analog converter that produces two-periodic nonuniform ZOH signals [8]. Therefore, we have used a XILINX Virtex-4 FPGA evaluation platform [16] to generate the digital signal and two high-speed DACs from Analog Ubiquitous Computing and Communication Journal 6

7 Devices AD9734 [17] to convert the digital signals into the analog domain. An 8 Hz differential clock generator has generated the 18 degrees phase-shifted clock signals for the two DACs, where the differential outputs have been used as single ended clock signals. Thus, the overall sampling rate of the twochannel time-interleaved DAC is 16 S/s. The two analog output signals have been combined a power combiner and measured an Agilent 54855A 6 GHz scope a sampling frequency of 2 GS/s. For the presented measurements we have generated a single sinusoid a frequency of 3 Hz and have identified the time offsets as r = and r 1 =.12. Along the identified time offsets a filter, which compensates the digital signal as depicted in Fig. 4, has been designed according to Eq. 41. The digitally compensated and converted signal has been measured the highspeed sampling scope. The results of the measurements are shown in Fig. 1. The mirror images at 5 Hz have been reduced by about 2 db. Due to uncertainties in the time offset estimation, a better reduction has not been possible. Energy Density Spectrum dbc Energy Density Spectrum dbc uncompensated Frequency GHz compensated Frequency GHz Figure 1: easurements for the uncompensated and the precompensated sinusoid. 5.3 Farrow filter design As shown in the last section, we can design a time-varying filter consisting of two impulse responses to compensate two-periodic nonuniform ZOH signals given deviations. In some applications it is necessary to redesign the compensation filter from time to time, since the time offsets r n are subject to temperature changes, aging effects, and voltage supply variations. Therefore, the previously introduced filter design process might become too complex for certain applications. For the two-periodic case, the solution to this problem is the adaptation of a Farrow filter [18], which reduces the redesign complexity significantly. By defining r n+1 r n = 1 n and r n+1 + r n =, we can rearrange Eq. 2 as H n e j = A n e j j 1+,e n 1 2 sin 1+ 1 A n e j, = With these definitions, the frequency responses A n e j, depend on one additional spectral parameter as required for Farrow-based filter designs. Furthermore, all frequency responses H n e j share a common factor exp j 1+/2. Hence, we can conclude from Eq. 18 and Eq. 2 that the factor exp j 1+/2 shifts the entire input signal by 1 +/2 in time, but does not produce spurious images. Since we are only interested in compensating the spurious images, we can modify our perfect reconstruction condition given by Eq. 34 for the two-periodic case as F l e j l 2 = e j , for l =, for l = 1 46 By using Eq. 44 and Eq. 46 and following the derivations of the last section we obtain the compensation filters as G e j, G 1 e j, = A e Pe j j +, Pe j j + A 1 e, e j. 47 The ideal time-varying compensation filter depends on the frequency and on an additional spectral parameter. Hence, these filters can be designed and implemented as Farrow filters. A Farrow filter has an impulse response of P1 g a n [l, ] = c n,p [l] p 48 p= L G a n e j,= g a n [ l,] e jl. 49 l= Ubiquitous Computing and Communication Journal 7

8 In general, we have to design two sets of Farrow coefficients c,p [l] and c 1,p [l] for our twoperiodic compensation filter. However, we can further simplify the filter design by exploiting the even symmetry regarding, i.e., A e j,= A 1 e j, 5 which can be verified Eq. 45. In consequence, we can show that Pe j,= Pe j, 51 and can conclude that the transfer functions of the two-periodic time-varying filter are even functions regarding the spectral parameter G e j,= G 1 e j,. 52 compensation filter will exhibit symmetrical or anti-symmetrical impulse responses, the number of coefficient multipliers can be basically halved [11, Chap. 6]. For the design example we have chosen an FIR Farrow filter of order P = 3 subfilters of order L = 17 and a delay of = 8.5, i.e., subfilters 9 multipliers if we exploit the linear-phase property, D =.7,K,.7. The range of possible time offsets has been D =.5,K,.5. The approximation problem was solved in the Chebyshev sense L norm, i.e., min G e j,g a e j, D, D 55 where D is the definition domain of the spectral parameter, by again using atlab and the optimization toolbox CVX. Design procedures can be found for example in [19]. This property can be exploited for the filter design by choosing a definition domain for D that is symmetric around zero, i.e., D = max,k, max. For such a choice, the filter design will lead to two transfer functions related by G a e j,= G a 1 e j, 53 E e jω db λ=±.5 λ=±.391 λ=±.278 λ=±.167 λ=±.56 and can be described in general as G n a e j,= G a e j, n Accordingly, we only have to design one set of filter coefficients c,p [l] and can efficiently implement the compensation filter as shown in Fig. 11. x[n] G a ne jω Normalized frequency Ω/Ω s Figure 12: Deviation from the ideal overall frequency response for different λ=±.5 λ=±.391 λ=±.278 λ=±.167 λ=±.56 c,2 [l] c,1 [l] c, [l] s[n] E 1 e jω db n λ 1 n λ Figure 11: Proposed modified Farrow filter for P=3. Furthermore, G e j, consists of a real zerophase response A e j +,/Pe j and a linearphase response exp j, which can be chosen arbitrarily. By choosing a delay such that the Normalized frequency Ω/Ω s Figure 13: Attenuation of the spurious images for different. For the specified problem we have obtained a maximum approximation error of 55.3 db in the filter design of G a e j,. In Fig. 12 we see the Ubiquitous Computing and Communication Journal 8

9 impact of the approximation error due to the filter design on realizing an ideal delay, i.e., E e j = F e j e j and in Fig. 13 on the residual images as in Eq. 43 for various values of. Within the definition band, the approximation error of an ideal delay is less than -7 db and the spectral images are attenuated by at least -63 db. These approximation errors are comparable to the values for the fixed FIR filter; however, for the given specification, the flexible Farrow-based solution needs at least 3 times more multipliers in its implementation. Hence, we can avoid the online filter design and gain flexibility, but have to pay for it higher implementation costs. Finally, we have simulated two-periodic ZOH signals time offsets r = and r 1 =.391 driven by a digitally generated multi-tone signal as in Sec The energy density spectra of the uncompensated and the compensated output of the ZOH are shown in Fig. 8 and Fig. 14, respectively. In the compensated spectrum, the spurs are reduced to -63 dbc. Figure 14: Output of a two-periodic ZOH precompensated by a Farrow filter. 6 CONCLUDING REARKS We have introduced the design of a discrete-time time-varying filter to compensate the spurious inband image spectra caused by -periodic nonuniform ZOH signals. A general filter design procedure is derived and the filter design for the two-channel case is demonstrated by two examples. Furthermore, for the two-channel case, we have extended the design procedure to Farrow filters, which allows for compensating a wide range of possible time offsets by the simple adjustment of some multipliers. ACKNOWLEDGEENT Christian Vogel was supported by the Austrian Science Fund FWF's Erwin Schrödinger Fellowship J279-N2. 7 REFERENCES [1] F. aloberti: High-speed data converters for communication systems, IEEE CircuitsSyst. ag., vol. 1, no. 1, pp , January 21. [2] B. urmann, C. Vogel, and H. Koeppl: Digitally Enhanced Analog Circuits: System Aspects, Proceedings of the 28 IEEE International Symposium on Circuits and Systems, Seattle USA, pp , ay 28. [3] C. Vogel and H. Johansson: Time-Interleaved Analog-to-Digital Converters: Status and Future Directions, Proceedings of the 26 IEEE International Symposium on Circuits and Systems, Kos Greece, pp , ay 26 [4] U. Seng-Pan, S. Sai-Weng, and R. artins,: Exact spectra analysis of sampled signals jitter-induced nonuniformly holding effects, IEEE Trans. Instrum. eas., vol. 53, no. 4, pp , August 24. [5] L. Angrisani and. D Arco: odeling timing jitter effects in digital-to-analog converters, IEEE Trans. Instrum. eas., vol. 58, no. 2, pp February 29. [6] D. Domanin, U. Gatti, P. alcovati, and F. aloberti: A multipath polyphase digital-toanalog converter for software radio transmission systems, The 2 IEEE International Symposium on Circuits and Systems, vol. 2, pp , ay 2,. [7] C.-K. Yang, V. Stojanovic, S. odjtahedi,. Horowitz, and W. Ellersick: A serial-link transceiver based on 8-GSamples/s A/D and D/A converters in.25-m COS, IEEE J. Solid-State Circuits, vol. 36, no. 11, pp , November 21. [8] C. Krall, C. Vogel, and K. Witrisal: Timeinterleaved digital-to-analog converters for UWB signal generation, 27 IEEE International Conference on Ultra Wideband, September 27. [9] Y.-C. Jenq: Digital-to-analog D/A converters nonuniformly sampled signals, IEEE Trans. Instrum. eas., vol. 45, no. 1, pp , February [1] C. Vogel and C. Krall: Compensation of distortions due to periodic nonuniform holding signals, Symposium on Communication Systems, Networks and Digital Signal Processing, Graz Austria, pp , July 28. Ubiquitous Computing and Communication Journal 9

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