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Generalizations of the Sampling Theorem: Seven Decades After Nyquist P 1094 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 2001 Generalizations of the Sampling Theorem: Seven Decades After Nyquist P. P. Vaidyanathan, Fellow, IEEE Abstract—The sampling theorem is one of the most basic and then established for the cases of bandlimited as well as nonban- fascinating topics in engineering sciences. The most well- known dlimited sampling in Sections IV-B and C. The connection be- form is Shannon’s uniform-sampling theorem for bandlimited sig- tween this kind of stability and functional subspaces is reviewed nals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection in Section IV-D. The role of nonuniform sampling in MR theory between such extensions and the theory of filter banks in DSP has is explained in Section V. We show in particular that the MR been well established. This paper presents some of the less known coefficients can often be calculated using FIR filtering opera- aspects of sampling, with special emphasis on non bandlimited sig- tions without the use of oversampling (which is often resorted nals, pointwise stability of reconstruction, and reconstruction from to). Sampling theorems for non bandlimited discrete time signals nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed. are discussed in Section VI. These results are based on well es- tablished concepts in multirate digital signal processing. Results Index Terms—Bandlimited signals, FIR reconstruction, multi- resolution, nonuniform sampling, sampling theorems. on FIR reconstructibility in this context are established in Sec- tions VI-B–D. Certain basic properties satisfied by bandlimited signals are summarized in the Appendices for convenience. I. INTRODUCTION B. Notations HE sampling theorem is one of the most basic and fas- T cinating topics in engineering sciences. The most well- Unless mentioned otherwise, all notations are as in [23]. The known form is the uniform sampling theorem for bandlimited term -BL refers to signals that are bandlimited to signals, due to Nyquist and Shannon [9], [13]. This has also (i.e., Fourier transform is zero outside). We use the notations been attributed to Whittaker and Cauchy (see [6]). It is the fun- and to denote the decimated version damental tool that allows the processing of real signals using and its -transform. The expanded version digital signal processors (DSP). Extensions of this to bandpass mul of signals and multiband signals, and to nonuniform sampling are otherwise also well-known [6], [22]. The connection between such exten- sions and the theory of filter banks in DSP has also been well is similary denoted by and its -transform established (e.g., see Chap. 10 in [23]). Further novel extensions denoted by . In situations where the -transform does can be found in [5]. This paper is a review of some of the less not exist in the conventional sense (e.g., ideal filters), the nota- known aspects of sampling, with special emphasis on non ban- tion stands for so that is the frequency response dlimited signals, stability of reconstruction, and reconstruction . The type 1 and type 2 polyphase representations of from nonuniform samples. [24]–[27]. with respect to an integer are given by [23] A. Outline (type 1 polyphase) In Section II, we consider sampling theorems for non ban- (type 2 polyphase). dlimited signals. These are often referred to as multiresolution The type used is usually clear from the context and is therefore (MR) or wavelet sampling theorems. An application in digital often not mentioned. interpolation is reviewed in Section III, and demonstrated with BIBO stability stands for bounded input bounded output sta- -spline interpolation of images. Stability of the reconstruction bility [10]. A sequence is an sequence if process is defined in Section IV and addressed in considerable and it is an sequence if . A linear time in- detail there. We explain the importance of pointwise stability as variant filter is BIBO stable if and only if the impulse response opposed to stability in terms of energy. Such pointwise stability is is in . For continuous time signals similar definitions ( and ) apply with sums replaced by integrals. Note that Manuscript received April 16, 2000; revised March 23, 2001. This work was but we do not have . supported in part by the National Science Foundation under grant MIP-0703755, in part by the Office of Naval Research under grant N00014-99-1-1002, and in part by Microsoft Research, Redmond, WA. This paper has been presented in II. NONBANDLIMITED SIGNALS parts at conferences This paper was recommended by Associate Editor M. O. Ahmad. If we have the apriori information that a signal is bandlimited The author is with the Department of Electrical Engineering, California In- to a known region, we can recover it from appropriately-spaced stitute of Technology, Pasadena, CA 91125 USA (e-mail: [email protected] tech.edu). samples by filtering. If a signal is not bandlimited, can we still Publisher Item Identifier S 1057-7122(01)07720-0. recover it from samples? The answer depends on what other 1057–7122/01$10.00 © 2001 IEEE VAIDYANATHAN: GENERALIZATIONS OF THE SAMPLING THEOREM: SEVEN DECADES AFTER NYQUIST 1095 Fig. 1. A nonbandlimited signal. apriori information we have. For example, suppose we have the knowledge that has the form (signal model) (1) Fig. 2. Two non Nyquist choices of 0@tA. where is a known function. Denoting the continuous time If for all we can write Fourier transforms (FT) of and as and , . That is, we can identify from using and the discrete time FT of as , we equivalently have . As a first example, assume that (7) is a time-limited signal as demonstrated in Fig. 1. In this example, the samples of at integer points are . where is the of convolutional inverse of i.e., its So we can trivially reconstruct from its samples using the Fourier transform . Recovery of for formula all can then be done using (1). In summary, we have recovered from the samples . (2) Return now to the examples in Fig. 2. We see that for the first function the nonzero samples of are whereas where . More generally, this holds if has the for the second function these are . In the first case zero-crossing property , i.e., whereas in the second case, at .So is for other integers (3) reconstrucible from for the first case, but not the second. We conclude this section with a few remarks. Thus reconstruction from samples has been possible inspite of aliasing due to nonbandlimitedness. 1) Discrete time model. The preceding discussion also shows A function satisfying the zero-crossing property (3) is this: given a discrete time signal and an arbitrary also referred to as a Nyquist(1) function in the literature. The function we can almost always assume that argument “(1)” signifies that the zero crossings are separated by can be written in the form (5) for appropriate choice of one unit of time. A special case is the example where is the the only theoretical condition being that sine funtion for all . In particular, we can regard as sam- ples of a continuous time signal of the form (4) . 2) Lack of shift invariance. For fixed , let denote Since this is -BL, the sum (1) is also -BL. The sine function the space of all signals which can be represented as in is Nyquist(1), so the reconstruction formula (2) holds. This cor- (1) for appropriate finite-energy . When is the responds to the familiar Shannon sampling and reconstruction. sine function we know that any shifted version of If a function can be represented as in (1) where is not (e.g., ) also belongs to the space because Nyquist (Fig. 2), can we still reconstruct from samples time-shift does not affect bandlimitedness. For arbitrary ? The answer is yes for a large class of as we now however, even though reconstruction from samples show.1 From (1) we see that the samples of are given by is often possible, the shifted versions of do not in general belong to the same space . This is readily ver- (5) ified with examples. For example shift in Fig. 2 to obtain . This result cannot be expressed as a linear combination of the integer shifts . which is nothing but a discrete-time convolution equation. De- noting the discrete time Fourier transforms of the sequences 3) Undersampling a wideband signal. As a special case of , , and by , and we get the signal model, suppose is -BL, that is, bandlim- ited to . Then is also -BL and the (6) Shannonsampling rate would be , implying the sample spacing . Assuming for sake of argument that 1Notice that the sample spacing is aI, and that the sampling phase is such that t aHis included. For a different sampling phase, the reconstruction is real and positive in ,wesee conditions and equations have to be worked out again. that (aliased version of ) is nonzero for 1096 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 9, SEPTEMBER 2001 Fig. 3. Reconstruction of x@tA from samples. Fig. 5. MR decomposition based on ideal filters. Fig. 6. Interpolation of a signal x@nA with digital filters. The signal is assumed to have a continuous time model x@tAa @kA0@t kA. Fig. 4. Nyquist(1) or zero-crossing property of the reconstruction filter s@tA. III. APPLICATION IN INTERPOLATION all and the reconstruction (7) is valid. Thus the model As explained in Section II, a discrete time signal can be (1) allows a smaller sampling rate, that is, wider sample viewed as a sampled version of .
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