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Classical Sampling Theorems in the Context of Multirate and Polyphase Digital Filter Bank Structures

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Classical Sampling Theorems in the Context of Multirate and Polyphase Digital Filter Bank Structures 1480 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING, VOL. 36, NO. 9, SEPTEMBER 1988 Classical Sampling Theorems in the Context of Multirate and Polyphase Digital Filter Bank Structures Abstract-The sampling theorem has been generalized in several di- frequency is at least equal to the Nyquist frequency 8 rections since its introduction during the first half of this century. Some 2a. One of the earliest extensions of this theorem was of these include derivative sampling theorems and nonuniform sam- pling theorems. Both of these have also been reinterpreted in terms of stated by Shannon himself in his 1949 paper [l], which a multichannel sampling framework. In the world of digital signal pro- says that if x, (t)and its first M - 1 derivatives are avail- cessing, multirate systems have gained substantial attention during the able, then uniformly spaced samples of these, taken at the last 1: decades. A system of this type is the maximally decimated anal- reduced rate of 8/M, are sufficient to reconstruct x,(t). ysislsynthesis filter bank, widely used in subband coding, voice privacy This result will be referred to as the derivative sampling systems, and spectral analysis. Some of the generalizations of the sam- pling theorem can be understood by relating the sampling problem to theorem in this paper. A different type of extension called the continuous-time version of the analysis/synthesis filter bank system, the nonuniform sampling theorem, which was stated by as was done by Papoulis and Brown. Basically, the recovery of a signal Black [7, p. 411 (and attributed to Cauchy, 1841!), has from “generalized samples” is a problem of designing appropriate lin- also been reviewed recently [4]-[6] in various contexts. ear filters called reconstruction (or synthesis) filters. Statements of this extension can be found in [4]-the es- In this paper, this relation is first reviewed and explored further. New sampling theorems for the subsampling of sequences are derived sential message being that nonuniformly spaced samples by direct use of the digital filter bank framework. These results are of x, (t)can be used to reconstruct x, (t)to any desired related to the theory of perfect reconstruction in maximally decimated accuracy, as long as the “average” sampling rate is at digital filter bank systems. One of these pertains to the subsampling of least equal to 8. In accord with the traditionally used lan- a sequence and its first few difirences, and subsequent stable recon- guage [4], these will be collectively referred to as the struction atfinite cost with no error. The reconstruction filters turn out to be multiplierless and of the FIR (finite impulse response) type. These “folk theorems.” A very simple example of this type of ideas are extended to the case of two-dimensional signals, by use of a result can be found in [40, pp. 557-5581. The use of “al- Kronecker formalism. This paper also considers the subsampling of most periodic functions” in the interpretation of sampling band-limited sequences. A sequence x (n) whose Fourier transform theorems is also worth noting in this context [28]. Some vanishes for I o I 5 Ln/M, where L and M are integers with L < M, results on spectral representation of nonuniformly sam- can in principle be represented by reducing the data rate by the amount MIL. Even though techniques are known to accomplish this reduction, pled waveforms can be found in [38]. this paper discusses a different method which consists in retaining only An excellent tutorial review of the sampling theorem L out of M successive samples and discarding the rest (attractive when and several generalizations, both for one-dimensional and signal processing prior to transmission is prohibitive). The reconstruc- multidimensional signals, can be found in [4], along with tion problem in this context is addressed. The digital polyphase frame- an extensive bibliography from which proofs can be in- work is used in this paper as a convenient tool for the derivation as well as mechanization of the sampling theorems. fered. The derivative sampling theorem and nonuniform sampling theorems were given a new and lucid interpre- tation in the fairly recent work by Papoulis [5], [39], by I. INTRODUCTION formulating a unified framework in terms of linear filter- HE sampling theorem, which was introduced several ing operations prior to sampling. The Papoulis approach Tdecades ago [1]-[3], has been extended by many au- has a striking resemblance to the framework of Jilter thors into various generalized versions [4]-[6]. The sim- banks, which is a discipline independently developed in plest version of the theorem states that a continuous-time the discrete-time domain during the last decade. The 1981 signal x,(t), band-limited to I W I < a, can be recon- paper by Brown [6], which reinterprets the Papoulis structed from its equally spaced samples, if the sampling framework, brings this resemblance closer, and also shows that the reconstruction of x,( t)from “generalized Manuscript received October 13, 1987; revised March 5, 1988. This samples” is essentially a problem of desjgning a synthesis work was supported in part by the National Science Foundation under Grant filter bank. DCI 8552579, by the matching funds provided by Pacific Bell and General Electric Co., by Caltech’s Programs in Advanced Technology grant spon- Recent advances in signal processing, which evolved in sored by Aerojet General, General Motors, GTE, and TRW, and by the a manner independent of the above developments, have National Science Foundation under Grant MIP 8604456. resulted in the discipline of multirute digital signal pro- The authors are with the Department of Electrical Engineering, Cali- fornia Institute of Technology, Pasadena, CA 91 125. cessing. A key feature here is the change of sampling rates IEEE Log Number 8822365. at various stages of the system (hence the name multi- 0096-3518/88/0900-1480$01.OO 0 1988 IEEE VAIDYANATHAN AND LIU: CLASSICAL SAMPLING THEOREMS 1481 rate), in the form of decimation and interpolation of dis- crete-time sequences. Advantages and applications of such processing have been well understood [9]-[ 121, and will not be elaborated further. The purpose of this paper is to exploit further the close analogy between sampling-theo- rem frameworks and multirate $iter banks, which involve the downsampling of sequences at various stages, fol- Fig. 1. The M-band multirate analysis/synthesis system. lowed by subsequent reconstruction. Fig. 1 is a typical example of a multirate signal pro- cessing system, called the analysis/synthesis system. Here, the digital filters Hk(z) are called analysis filters, and split the signal x (n)into M subbands in the frequency domain. The downgoing arrows in the figure represent the decimation operation by an integer factor of M. The M decimated sequences are typically encoded and transmit- ted. At the receiver end, each of these decimated subband signals is “interpolated” (the upgoing arrows represent this; see the next section for formal definitions), and then recombined with the help of the synthesis filters Fk(z)to form the reconstructed signal f (n). The main application of this system is in the subband coding of signals and im- -2nC3 0 2x13 ages [IO], [13]-[16]. The system also provides a model Fig. 2. A band-limited sequence, and a procedure to decimate it by the for short-time Fourier transforms as can be seen from the noninteger factor 3/2. results of [17] (with the number of subbands not neces- sarily equal to the decimation ratio M ). Our interest here, however, is in the mathematical framework offered by the same as those in [37], the purpose and problem statement system of Fig. 1. We shall see first that, if suitably in [37] are completely different. adopted into the continuous time domain, it can be used As a second example, let x (n) be a sequence band-lim- for obtaining a simplified understanding of several ver- ited to the region I w 1 < 2a/3 [Fig. 2(a)]. We know that sions of the sampling theorem. Moreover, a direct use of we can obtain a lower-rate signal with the same amount the discrete time structure of Fig. 1 gives rise to new of information simply by creating a continuous time sig- downsampling theorems (or simply “sampling theo- nal (by low-pass filtering) and resampling at two-thirds of rems”) for sequences, some of which can be generalized the original rate, resulting in an obvious data-rate reduc- to two (and higher)-dimensional sequences by further ex- tion. A direct discrete-time method for performing the tensions of Fig. 1. same data-rate reduction is also well known [IO] and is Several interesting questions can be posed in connec- shown in Fig. 2(b). Here, the signal rate is first increased tion with the reconstructibility of a sequence from various by a factor of 2 [by inserting a zero-valued sample be- subsampled versions. These questions are closely analo- tween every two adjacent samples of x(n)],and then fil- gous to the continuous time counterparts, but the answers tered to ensure that the “image” created by the zero-in- often have certain qualitative differences. As an example, sertion operation is eliminated. The resulting sequence let us consider the derivative sampling theorem for a sig- x,(n), which is band-limited to I w 1 < n/3, can be de- nal xu( t).In order to reconstruct exactly the value of x, ( t ) cimated by a factor of 3 without causing any aliasing. The for a given t from the samples of xu ( t)and X, ( t ), one has output x2(n)has the same spectrum [Fig. 2(c)] as that of to perform an infinite amount of computation (because this x(n),except that it is stretched by the factor 3/2.
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