Reconstruction of Nonuniformly Sampled Bandlimited Signals

RECONSTRUCTION OF NONUNIFORMLY SAMPLED BANDLIMITED SIGNALS USING TIME-VARYING DISCRETE-TIME FIR FILTERS Håkan Johansson and Per Löwenborg Division of Electronics Systems Department of Electrical Engineering SE-581 83 Linköping University, Sweden

Abstract – This paper deals with reconstruction of non- (a) xa(t) (b) xa(t) uniformly sampled bandlimited continuous-time signals using time-varying discrete-time FIR filters. The points of departures are that the signal is slightly oversampled as to the average sampling frequency and that the sampling t t t t t t t instances are known. Under these assumptions, a representa- 0 T4T2T3T 5T t 0 1 2 3 4 5 tion of the reconstructed sequence is derived that utilizes a (c) xa(t) time-frequency function. This representation enables a proper utilization of the oversampling and reduces the reconstruction problem to a design problem that resembles an ordinary filter design problem. Furthermore, for an d0 2T+d0 4T+d0 t important special case, corresponding to a certain type of d1 2T+d1 4T+d1 periodic nonuniform sampling, it is shown that the recon- igure 1. (a) Uniform sampling. (b) Nonuniform sampling. struction problem can be posed as a filter-bank design prob- c) Periodic nonuniform sampling. lem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions.

1. INTRODUCTION cut-off frequency. Although different weights (windows) Nonuniform sampling occurs in many practical applications can improve the situation, it is recognized that other design either intentionally or unintentionally [1]. An example of the methods are usually preferred [8]. When designing filters, latter is found in time-interleaved analog-to-digital convert- one does normally not attempt to approximate a desired ers (ADCs) where static time-skew errors between the dif- function over the whole frequency range ωT ≤ π , ωT ferent subconverters give rise to a class of periodic being the frequency variable. Instead, one allows certain nonuniform sampling [2], see Fig. 1(c). region(s) referred to as the transition region(s) where no Regardless whether the continuous-time (CT) signal, say requirements are stated. In this way, one can in the remain- xa(t), has been sampled uniformly [Fig. 1(a)], producing the ing frequency regions fully control the filter performance. It sequence x(n) = xa(nT) or nonuniformly [Fig. 1(b)], produc- is therefore reasonable to take the same action when recon- ing the sequence x1(n) = xa(tn), it is often desired to reconstructing nonuniformly sampled signals. That is, instead of struct x (t) from the generated sequence of numbers. Thus, approximating perfect reconstruction (PR) in the whole fre- a ω ≤ π in the nonuniform-sampling case, it is desired to retain xa(t) quency range T , in which case one is doomed to face from the sequence x1(n). This can, in principle, be done in problems, one should a priori assume a small oversampling two different ways. The first way is to reconstruct x (t) factor in which case PR needs to be approximated only in a ω ≤ ω < π directly from x1(n) through CT reconstruction functions. the region T 0T . In this way, one can fully con- Although it is known how to do this in principle (see e.g. trol the reconstruction by properly designing the reconstruc- [3]–[7]), problems arise when it comes to practical imple- tion system. For example, the method introduced in [9] mentations. In particular, it is very difficult to practically employs causal interpolation functions and it was observed implement CT functions with high precision. It is therefore experimentally that the reconstruction deteriorates when the desired to do the reconstruction in the digital domain, i.e., to bandwidth approaches π, which further motivates the bene- first recover x(n). One then needs only one conventional dig- fits of allowing a slight oversampling. ital-to-analog converter (DAC) and a CT filter to obtain Although oversampling itself is undesired since it gener- xa(t), which are much easier to implement than general com- ates more samples than necessary for reconstruction, plicated CT functions. according to the Nyquist sampling theorem, it is known that Recovering x(n) from x1(n) in the digital domain can in a slight oversamling is required for practical implementation principle be done by digital reconstruction functions of conventional ADCs and DACs. It is therefore conjectured obtained through sampling of a corresponding CT recon- that a slight oversampling is also required in order to practi- struction function. However, the CT reconstruction func- cally implement a device that reconstructs nonuniformly tions are generally noncausal (two-sided) functions which sampled signals. This is of course not a new paradigm but it therefore must be truncated in order to make the correspond- seems that it is often unsatisfactorily handled when it comes ing digital reconstruction system practically implementable. to analysis, design, and implementation of practical recon- This truncation causes reconstruction errors that are not eas- struction algorithms. ily controlled and one should therefore seek for other tech- This paper deals with reconstruction through time-vary- niques. One faces a similar situation when designing digital ing FIR filters. The point of departure is a bandlimited CT filters. It is well known that filters designed through trunca- signal that is nonuniformly sampled and slightly oversam- tion of infinite-length impulse responses (referred to as win- pled as to the average sampling frequency, the reason for the dowing techniques) exhibits large errors (referred to as latter assumption being as outlined above. It is further Gibb’s phenomenon) in the frequency domain around the assumed that the sampling instances are known. Under these

assumptions, a representation of the reconstructed sequence (a) ω is derived that utilizes a time-frequency function. This repre- Xa(j ) sentation enables a proper utilization of the oversampling 1 and reduces the reconstruction problem to a design problem that resembles an ordinary filter design problem. Further- –π/T –ω ω π/T ω [rad/s] more, in an important special case, corresponding to a cer- 0 0 tain type of periodic nonuniform sampling, it is shown that (b) X(ejωT) the design problem can be posed as a filter-bank design 1/T problem, thus with requirements on a distortion transfer function (that should approximate one) and a number of π π ω ω ω π ω π ω aliasing transfer functions (that should approximate zero). –2 –2 + 0T – 0T 0T 2 – 0T 2 T [rad] A FB formulation of the problem has been done earlier Figure 2. Spectra of a bandlimited signal xa(t) and the sequence in [7], but there is a major difference between the formula- x(n) = x (nT). (Uniform sampling). tion in that paper and the one in this paper. In [7], it is a observed that the nonuniformly sampled signal can be expressed with the aid of a regular decimated analysis filter 1 Xe()jωT =--- X ()jω , –π ≤ ωT ≤ π (4) bank with analysis filter frequency responses that are fixed T a and determined by the sampling instances. It is then shown that x(n) can be retained using a synthesis filter bank with (see also Fig. 2(b)). Equation (4) implies that xa(t) can be ideal non-causal multilevel filters. The issue of using practi- recovered from x(n). In practice, this is done using a DAC cal causal synthesis filters approximating the ideal ones was followed by an CT reconstruction filter. We also note that ω π ⁄ not treated though. That is, it is not known how well a “prac- xa(t) is oversampled unless 0 = T . tical version” of that solution will behave. In this paper, the FB formulation contains adjustable analysis filters but fixed 3. NONUNIFORM SAMPLING and trivial synthesis filters (pure delays). Here, the problem Throughout this paper, it is assumed that the nonuniform is thus to properly design the analysis filters which, for sampling of the CT signal x (t) is done in such a way that a () many sampling patterns, can be done as good as desired the so obtained sequence, say x1 n , is given by with an acceptable filter order, due to the fact that a slight () () oversampling is used. Oversampling is not utilized in [7] x1 n = xa tn (5) which means that one most likely will face problems when designing practical causal synthesis filters approximating where the ideal multilevel filters. Oversampling was utilized in the t = nT + ε T (6) special class of synthesis FB proposed in [10], in which case n n PR can be approximated as close as desired. with ε T representing the distance between the “nonuni- Following this introduction, the paper first recapitulates n form sampling instance” tn and the “uniform sampling uniform sampling in Section 2, the reason being that the instance” nT . The average sampling frequency is thus still reconstruction here aims at retaining x(n) from x1(n), not 1/T. It is also assumed that the sampling instances are dis- xa(t) directly. Section 3 considers nonuniform sampling and tinct, i.e., t ≠ t , nm≠ , and that t < t , nm< . reconstruction using time-varying FIR filters. Section 4 n m n m Given x1(n), a new sequence, y(n), is formed through studies the special case of periodic nonuniform sampling some reconstruction formula. It is desired to achieve y(n) = and shows how the design problem can be posed as a FB () x(n) because, then, xa t can due to (4) be recovered using design problem. Finally, Section 5 concludes the paper. conventional reconstruction methods for uniformly sampled signals. The equality y(n) = x(n) corresponds in the fre- 2. UNIFORM SAMPLING quency domain to Ye()jωT = Xe()jωT . If these equations In uniform sampling, the sequence x(n) is obtained by sam- hold, the reconstruction system is said to be a perfect recon- pling the CT input signal xa(t) uniformly at the time struction system. instances nT, for all integers n, i.e., In this paper, the reconstruction is performed using a () (), …,,,,,,… time-varying FIR filter characterized by the impulse xn = xa nT n = –2 –1 012 (1) () responses hn k . It is assumed here that the order of the FIR where T is the sampling period and f = 1/T is the sam- filter is 2N and thus even. It is convenient and possible to let sample the FIR filter be noncausal which implies that h ()k in this pling frequency. The Fourier transforms of x(n) and xa(t) are , n,,… related according to Poisson’s summation formula as even-order case is non-zero for kN= – – N + 1 N . In a practical implementation, the corresponding causal filter is ∞ obtained by simply introducing a delay of N samples. In the 1 2πr , ,,… Xe()jωT = --- ∑ X jω – j------(2) odd-order case, one has instead kN= – – N + 1 N – 1 T aT or kN= – + 1, – N + 1,,… N , but that will not change the r = –∞ principles dealt with in this paper. Henceforth, only the Since the spectrum of x(n) is periodic with a period of 2π even-order case is therefore considered for the sake of sim- ( π ) ω plicity. 2 -periodic with respect to T, it suffices to consider () Xe()jωT in the interval –π ≤≤ωT π . Throughout this Under the above assumptions, yn is now formed according to paper, it is assumed that xa(t) is bandlimited according to X ()jω = 0, 0 < ω ≤ ω , ω ≤ π ⁄ T (3) N a 0 0 () ()() yn = ∑ x1 nk– hn k (7) (see also Fig. 2(a)). That is, the Nyquist criterion for sampling with a sampling frequency of 1/ T without aliasing is kN= – () () () fulfilled. In this case, It is desired to select hn k so that yn approximates xn

98 -3 as close as possible (in some sense). To see how to choose x 10 () () 20 hn k , x1 n is first written in terms of the inverse Fourier transform of xa(t) by which we obtain, due to (3), (5) and (6), 15 ω 10 0 )-1| [dB] T ω 1 jωT ε ω j 5 () n ()ω j Tn ω (e x1 n = ------π ∫ e Xj e d (8) n 2 |H 0 –ω 0 0 0.2 π 0.4 π 0.6 π 0.8 π 0.9 π ωT [rad] Inserting (8) into (7), and interchanging the summation and integration, one obtains Figure 3. Illustration (for one value of n) of reduced reconstruction error when the filter order 2N increases. Solid line: 2N = 16. ω 0 Dashed line: 2N = 32. () 1 ()jωT ()ω jωTn ω yn = ------π ∫ Hn e Xj e d (9) ing to some criteria. The rationale behind this is that, regard- 2 jωT ω less of Xe() , one can generally say that the closer – 0 ()jωT Hn e is to one the closer e(n) is to zero, with “close” where being interpreted in a wide sense. By utilizing the representation of y(n) in (11), this design problem resembles an ordi- N ω ()ε nary filter design problem. One difference is however that ()jωT ()– j Tk– nk– ()jωT Hn e = ∑ hn k e (10) the functions Hn e are unconventional in the sense kN= – that they make use of non-integer delays [see (10)]. (The non-integer delays are merely a consequence of the nonuni- When (4) holds, (9) can equivalently can be written as form sampling and the problem formulation; thus, they are ω T not actually implemented which of course would cause 0 problems.) Another difference is, of course, that a new 1 yn()= ------∫ H ()e jωT Xe()jωT e jωTnd()ωT (11) design has to be done for each n. However, in an important 2π n special case, where the sampling is periodically nonuniform, ω – 0T it suffices to design only a few filters. In this case, the design Equation (11) represents y(n) with the aid of the functions problem can be conveniently posed as a FB design problem ()jωT which eases the design and analysis. This is the topic of the Hn e which can be viewed either as an infinite set of frequency functions or one time-frequency function. Fur- next section. ther, xn() can be expressed in terms of its inverse Fourier Figure 3 illustrates that the reconstruction error reduces as the filter order increases. The sampling instances are here transform according to ε as in (6), with nT being randomly chosen numbers in the ω 1 ω π 0T interval (–0.5T, 0.5T) , whereas 0T = 0.9 . The filters 1 have been designed using a least-squares approach where xn()= ------Xe()jωT e jωTnd()ωT (12) () 2π ∫ each hn k is analytically computed through matrix inver- –ω T sion. Due to the limited space, design details are omitted in 0 this paper but will instead be treated in another paper. Comparing (11) with (12), it is seen that perfect reconstruction is obtained if 4. PERIODIC NONUNIFORM SAMPLING An important special case of nonuniform sampling occurs ()jωT , ω ∈ []ω , ω Hn e = 1 T – 0T 0T (13) when the distances between the “nonuniform sampling instance” ε T exhibits periodicity according to for all n. n ε ε Defining the error e(n) as e(n) = y(n)–x(n) one obtains nT = nM+ T (15) from (11) and (12) that ω for all n, with M being the period. In this case, the sampling 0T is said to be periodic nonunifom sampling with period M. () 1 ()()jωT ()jωT jωTn ()ω An example is given in Fig. 1(c) with period M = 2, en = ------π ∫ Hn e – 1 Xe e d T (14) ε ε ε ε 2 d0 ==0T 2mT and d1 ==1T 2m + 1T , for all ω – 0T integers m. When ε Th satisfies (15), it is obvious that ()k and Apparently, e(n) = 0 in the PR case since, then, ()jωT n n ()jωT ()jωT Hn e exhibits the same periodicity, i.e., Hn e = 1 . In practice, Hn e can only approximate one in the frequency range of interest. The goal is then h ()k = h ()k (16) () n nM+ to determine the coefficients hn k so that the error e(n) is minimized according to some criterion. A problem is that and e(n) does not only depend on h ()k but also on Xe()jωT n ()jωT ()jωT which means that one generally must have knowledge about Hn e = HnM+ e (17) () the input signal spectrum in order to determine hn k in the best possible way. If one does not have complete knowledge for all n. This makes it possible to pose the design problem as about Xe()jωT , which often is the case in practice, one has a FB design problem. To see this, consider first the case to accept a suboptimum solution instead. where the output y(n) is obtained by applying x(n) to a time- The simplest way to obtain a suboptimum solution is to ω () ()j T ε determine each hn k separately so that each Hn e 1. In the literature, one sometimes finds the assertion that | nT|<0.25T must in (10) approximates one in the frequency region be fulfilled to enable reconstruction but this is generally not required. In ωT ∈ []–ω T, ω T , ω T < π , as good as possible accord- particular, when a slight oversampling is allowed, one circumvents many of 0 0 0 the problems associated with reconstruction of critically sampled signals.

99 (a) reconstruction using time-varying filters can be conveniently x(n) y(n) g (k) = g (k) gn(k) n n+M represented by the FB in Fig. 4(b) with analysis filters as given by (23) and (10). The design problem can in this case be posed as a FB design problem where the goal is to deter- (b) mine the M impulse responses h ()k , n = 0, 1, ..., M–1, so Analysis filter bank Synthesis filter bank ()jωT ()jωT n that V 0 e and V m e = 0 , m = 1, 2, ..., M–1 G0(z) 1 M M 1 approximate one and zero, respectively, as good as desired according to some criteria. Here, the problem is thus to properly design the analysis filters since the synthesis filters x(n) –1 y(n) G1(z) z M M z are fixed to pure delays. As already discussed in the introduction, this is different from the FB formulation in [7] where the analysis filters are fixed and the synthesis filters are to be designed. Another difference is, as mentioned ear- M–1 –(M–1) GM–1(z) z M M z lier, that the analysis filters are here unconventional in the sense that they make use of non-integer delays [see (10)]. Finally, it is stressed that the FB in Fig. 4(b) is used here Figure 4. (a) Time-varying filter. (b) Equivalent FB representation. only with the purpose of easing the analysis and design of the reconstructing system. That is, y(n) is not obtained by () varying filter described by the impulse responses gn k , thus implementing the FB which is obvious because that assumes that we already have available the “uniform samples” x(n) N which are precisely the samples we want to recover from the () ()() yn = ∑ xn– kgn k (18) “nonuniform samples” x1(n). The output y(n) is of course still obtained from (7) whereas the FB in Fig. 4(b) is a con- kN= – venient way of representing the reconstruction in the case of () () () Assume now that gn k satisfies gn k = gnM+ k . By periodic nonuniform sampling. making use of the properties of downsamplers and upsam- plers [11], one can in this case readily establish that y(n) 5. CONCLUDING REMARKS in (18), and thus Fig. 4(a), is identical to the output y(n) in The representation of the reconstructed sequence derived in the maximally decimated FB shown in Fig. 4(b). That is, the this paper enables a proper utilization of the oversampling output is obtained as the output of a maximally decimated assumed and reduces the reconstruction problem to a design n FB with the analysis filters z Gn(z), with Gn(z) being the problem that resembles an ordinary filter design problem. transfer function of gn(k), and with the trivial synthesis filters This representation is thus a useful starting point for the anal- –n Fn(z) = z . In the frequency domain, the relation between ysis and development of design techniques of different the input and output can therefore be expressed as [11] classes of input signals and nonuniform sampling patterns. M – 1 Due to the limited space, design issues were not included in this paper but will be considered in future papers. However, ()jωT ()jωT ()j()ωT – 2πmM⁄ Ye = ∑ V m e Xe (19) for one type of sampling pattern, it was illustrated by means m = 0 of an example how the reconstruction error reduces as the filter order increases, using a least-square design technique, where details of which will be published elsewhere. M – 1 1 REFERENCES V ()e jωT = ----- ∑ e– j2πmn⁄ MG ()e j()ωT – 2πmM⁄ (20) m M n [1] F. Marvasti (Ed.), “Nonuniform Sampling: Theory and Practice”, n = 0 Kluwer, NY, 2001. ω [2] W. C. Black and D. A. Hodges, “Time interleaved converter arrays,” ()j T IEEE J. Solid-State Circuits, vol. SC–15, no. 6, pp. 1022–1029, Dec. with Gn e being the Fourier transform of gn(k), thus satisfying 1980. [3] J. L. Yen, “On nonuniform sampling of bandlimited signals,” IRE G ()e jωT = G ()e jωT (21) Trans. Circuit Theory, vol. CT–3, pp. 251–257, Dec. 1956. n nM+ [4] A. J. Jerri, “The Shannon sampling theorem–its various extensions for all n. The term V ()e jωT is the distortion function and applications: A tutorial review,” Proc. IEEE, vol. 65, no. 11, pp. 0 ()jωT 1565–1596, Nov. 1977. whereas the remaining V m e , m = 1, 2, ..., M–1, are [5] A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Cir- aliasing functions. Perfect reconstruction is obtained when cuits, Syst., vol. CAS–24, no. 11, pp. 652–654, Nov. 1977. ()jωT ()jωT V 0 e = 1 and V m e = 0 , m = 1, 2, ..., M–1. [6] H. Choi, D. C. Munson, “Analysis and design of minimax optimal interpolators,” IEEE Trans. Signal Processing, vol. 46, no. 6, pp. Further, utilizing the inverse Fourier transform, the out- 1571–1579, June 1998. put y(n) in Fig. 4(b) can be expressed as [7] Y. C. Eldar and A. V. Oppenheim, “Filterbank reconstruction of band- ω limited signals from nonuniform and generalized samples,” IEEE 0T Trans. Signal Processing, vol. 48, no. 10, pp. 2864–2875, Oct. 2000. 1 [8] T. Saramäki, “Finite impulse response filter design,” in Handbook for yn()= ------G ()e jωT Xe()jωT e jωTnd()ωT (22) 2π ∫ n Digital Signal Processing, eds. S. K. Mitra and J. F. Kaiser, New ω York: Wiley, 1993, ch. 4, pp. 155–277. – 0T [9] H. Jin and K. F. Lee, “A digital-background calibration technique for minimizing timing-error effects in time-interleaved ADC’s,” IEEE Comparing (22) with (11), it is seen that the outputs in the Trans. Circuits Syst. II, vol. 47, no. 7, pp. 603–613, July 2000. two cases are identical provided that [10] H. Johansson and P. Löwenborg, “Reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay fil- ()jωT ()jωT Gn e = Hn e (23) ters,” IEEE Trans. Signal Processing, vol. 50, no. 11, pp. 2757–2767, Nov. 2002. Hence, when the sampling is periodically nonuniform, the [11] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Englewood Cliffs, NJ: Prentice-Hall, 1993.

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