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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 recon- structing 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 97 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.
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