Jerri, A.J. (1977). The Shannon sampling theorem -- Its various extensions and applications: A tutorial review. Proceedings of the IEEE, 65, 1565-1596.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approximation. The paper is organized as follow. In section II the WKS approach is developed. In section III the the control parameters are stated, section IV gives some design examples, with conclusions given in section V. II. THE WKS APPROACH It is well known that Shannon s sampling theorem [4] is used to construct bandlimited functions from their samples, provided that the samples are taken sufficiently close. The construction is carried out by introducing at the specified frequency samples pulse like interpolating function (Drichlet kernels) scaled to the value of the specifications ....

A. Jerri," The Shannon Sampling Theorem-Its Various Extensions arid Applications: A Tutorial", Proc. IEEE, Vol.6, No.11, PP 1565-96, Nov. 1977.

.... These are well documented subjects: Shannon sampling by itself is the subject of the four books [15, 19, 20, 27] and of a forthcoming volume by Higgins and Stens; some of the chapters of [1] are concerned with sampling and related issues for example, 5] There are several review papers [2 4, 14, 18]; the conference proceedings [16, 17] also contain many relevant works. The band limited interpolation and extrapolation problems have also been exhaustively studied. The Papoulis Gerchberg iteration [12,23] can be readily adapted to the discrete context. This was studied in [6] a work that ....

A. J. Jerri. The Shannon sampling theorem --- its various extensions and applications: a tutorial review. Proc. IEEE, 65(11):1565-- 1596, Nov. 1977.

....which is a result of extremely low number of bits allocated to chrominance. On the other hand, users expect colourful pictures to be available with low trans mission cost. Traditional image processing with regular sampling draws from well established WhittakerShanon s sampling theorem [1] and statistical models based on simplifying assumptions regarding local stationarity. Moreover, uniform sampling and block based algorithms offer the advantage of simplicity in practical implementation due to regular data structures and well defined separable two dimensional algorithms. For ....

A. Jerri, The Shannon sampling theorem-its various extensions and applications: A tutorial review. Proceedings of IEEE, vol. 65, 1977

....A. Erdelyi, Ed. Higher Transcendental Functions, vol. II. Melbourne, FL: Krieger, 1981. [4] D. J. Schmidlin, The Simple Polynomial Set: A Mathematical Tool for Representing and Generating Discrete Time Signals, Univ. of Massachusetts Dartmouth, Tech. Rep. UMD ECE TR98 100, Sept. 1998. Sampling and Generalized Almost Periodic Extension of Functions Paulo J. S. G. Ferreira Abstract ....

....such functions from a knowledge of f(tn) n ) is the subject of sampling theory. For an introduction to the topic, see [1] The historical development of sampling theory is partially discussed in [2] and [3] which also review a number of interesting related results. The review paper by Jerri [4] is an account of the state of the art in sampling as of 1977, whereas Butzer s review [5] concentrates on the results obtained at the Lehrstuhl A fur Mathematik, Aachen, Germany. A number of more recent reviews and books are available, such as [6] or [7] the latter which contains an extensive ....

A. J. Jerri, "The Shannon sampling theorem---Its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, pp. 1565--1596, Nov. 1977.

....demonstrating the potential for practical multifold reductions in sampling rate. Index Terms Error bounds, Landau Nyquist rate, matrix inequalities, multiband, nonuniform periodic sampling, optimal sampling and reconstruction. I. INTRODUCTION T HERE has been a long history of research [1] [4] devoted to sampling theory, with perhaps the most fundamental and importantpieceofworkinthisareabeingtheclassicalsamplingtheorem. AlsoknownastheWhittaker Koteln ikov Shannon(WKS) theorem, it states that a lowpasssignalbandlimited to thefrequencies can be reconstructed perfectly from its ....

A. J. Jerri, "The Shannon sampling theorem---its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, pp. 1565--1596, Nov. 1977.

....some domain needs more attention. Examples include shape reconstruction [12] or landmark interpolation [13] 16] The reconstruction can be done within the class of bandlimited functions [17] 18] or more general wavelet and spline like spaces [19] For an extensive review on sampling, see [20] and [21] B. Related Work The work presented in this paper can be seen as an extension of the theory of radial basis function approximation [22] 23] especially Duchon s thin plate splines [24] 25] to vector functions, nonideal (generalized) sampling, and generating functions that need not ....

A.J. Jerri, "The Shannon sampling theorem--Its various extensions and applications: A tutorial review," Proc. 1EEE, vol. 65, pp. 1565-1596, Nov. 1977.

.... i)b, l (15) 28As a consequence of the discovery of the several independent inTro ductions of the sampling theorem, people starled to refer to the theorem by including the names of the aforementioned authors, resulting in such catchphrases as The Whittaker KoTel nikov Shannon sampling theorem [124] or even The WhiTTaker KoTel nikov Raabe Shannon Someya sampling theorem [121] To avoid confusion, perhaps the best thing to do is to refer to it as the sampling theorem, raTher than Trying to find a title that does justice to all claimants [125] 29The word osculaTory originates from ....

....the kernel is the ,th degree B spline)S In the decades to follow, both Shannon s and Schoenberg s paper would prove most fruitful, but largely in different fields. The former had great impact on communication engineering [144] 147] numerous signal processing and analysis applications [98] [124], 125] 148] 150] and to some degree also numerical analysis [151] 154] Splines, on the other hand and after some two decades of further study by Schoenberg [155] 157] found their way into approximation theory [158] 164] mono and multivariate interpolation [81] 165] 167] numerical ....

A.J. Jerri, "The Shannon sampling theorem its various extensions and applicalions: A tulorial review," Proc. IEEE, vol. 65, pp. 1565 1596, Nov. 1977.

....of discretization and reconstruction. 10.4.1 Aliasing Errors Sampling Theorems What we normally think of as the sampling theorem is the theorem named after E. T. Whittaker, J. M. Whittaker, V. A. Kotelnikov or C. E. Shannon [Whi15, Whi29, Whi35] Kot33] Sha49] cited here after [Jer77]. It states, that a band limited function can be reconstructed exactly from regular samples of the function, if these samples are close enough (at least two samples per smallest involved wavelength) It is less known however, that generalizations of this WKS sampling theorem exist for more ....

.... samples of the function, if these samples are close enough (at least two samples per smallest involved wavelength) It is less known however, that generalizations of this WKS sampling theorem exist for more general, finite limit (truncated) integral transforms besides the usual Fourier transform [Jer77]. In addition to those sampling theorems there exist theorems that make statements about reconstruction from irregular samples [FG92] The sampling theorems contain statements about the conditions under which functions can be reconstructed from samples, some also about the kind of errors that ....

Abdul J. Jerri. The shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE, 65(11):1565--1596, November 1977.

.... (important points in the image) whose location is specified manually or automatically detected [14, 38, 42, 86] The reconstruction can be done within the class of bandlimited functions [118, 119] or more general wavelet and spline like spaces [120] For an extensive review on sampling, see [121, 122]. 113 0 5 10 15 20 25 0 1 2 3 4 5 6 7 Given points Rugged interpolation Smooth interpolation Figure 6.2: When interpolating a function from its values (circles) many solutions are possible. However, smooth interpolation (solid line) is usually preferable to a rugged one (dashed ....

A. J. Jerri, "The Shannon sampling theorem---its various extensions and applications: A tutorial review," Proceedings of IEEE, vol. 65, no. 11, pp. 1565--1596, 1977.

....find the relationship between the values 9 , 9 , and 9Q . Here, 9 denotes the estimate of the time derivative, e.g. 9 9G 9 . This particular estimate is of only low order in accuracy; for higher order finite difference schemes see e.g. 22, 23] for error estimates see [24, 25], and for an application with estimation in the frequency domain e.g. 26] The analysis of delay differential systems based on these insights was first performed by B unner et al. 27, 28] The triple Q can be seen as a three dimensional embedding vector where the first component comes from ....

A.J. Jerri, The Shannon sampling theorem---Its various extensions and applications: A tutorial review, Proceedings of the IEEE 65, 1565--1596 (1977).

....from di erent but equivalent Fourier analytic tools [9] Generalizing the Fourier transform extends the sampling 23 theorem to a larger class of Banach spaces over R or locally compact abelian groups. In what follows we are dealing with real valued L 2 signals and we refer to [48] [50], 54] and [82] for general results. Recalling the L 2 space of bandlimited functions B 2 0 (R) ff 2 L 2 (R) supp( f) 0 ; 0 ]g; the classical sampling theorem is as follows. Theorem 4.1 Let f be in B 2 0 (R) for some 0 0. Then, for every 0 with 0 2 0 ....

A. J. Jerri, The Shannon sampling theorem - its various extensions and applications: A tutorial review, Proc. IEEE, 65 (1977), pp. 1565{ 1596.

....while uniform sampling theorems work well for low pass signals, they are quite inefficient for representing certain bandpass signals and, more generally, for multiband signals, i.e. signals containing several bands in the frequency domain. We refer the reader to Papoulis [5] and Jerri s tutorial [6] for some generalizations of the WKS sampling theorem. To quantify the sampling efficiency for signals with a given spectral support , we define its spectral span, as the smallest interval containing , and its spectral occupancy as , where denotes the Lebesgue measure. The Nyquist rate for ....

A. J. Jerri, "The Shannon sampling theorem---Its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, pp. 1565--1596, Nov. 1977.

....of our noise analysis of oversampled FBs (to be presented in Section 2.3) this subsection provides a frame theoretic, subspace based interpretation of noise reduction in oversampled A D conversion. We shall rst interpret A D conversion as a frame expansion [19] 21] From the sampling theorem [22, 23], we know that a band limited continuous time signal x(t) with bandwidth B 0 can be perfectly 1 Here L 2 (IR) denotes the space of square integrable functions x(t) Furthermore, hx; yi = R 1 1 x(t) y (t) dt (where the superscript stands for complex conjugation) denotes the inner ....

A. J. Jerri, \The Shannon sampling theorem { Its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, pp. 1565-1596, 1977.

....Photoshop TM . m Taken from [16] 90 3.5. Issues 3.5.1. Aliasing Errors As with the standard sampling theorem, there is the usual problem with aliasing when spectra are not band limited or when signals are undersampled. These issues are discussed in any standard reference on sampling theory [40,87]. Typically bounds can be determined by the amount of a spectrum that is above the Nyquist sampling rate. Since standard aliasing analysis applies, the topic is deferred to the references. Of more interest here is the e#ect of unbounded frequencies images have on the isometry theorem. Though no ....

A. J. Jerri, "The Shannon Sampling Theorem- Its Various Extensions and Applications: A Tutorial Review," Proceedings of the IEEE, 65 (11), pp. 15651596, 1977.

....of such functions from a knowledge of f (t n ) n 2 Z) is the subject of sampling theory. For an introduction to the topic see [1] The historical development of sampling theory is partially discussed in [2, 3] which also review a number of interesting related results. The review paper by Jerri [4] is an account of the state of the art in sampling as of 1977, whereas Butzer s review [5] concentrates on the results obtained at the Lehrstuhl A fur Mathematik, Aachen, Germany. A number of more recent reviews and books are available, such as [6] or [7] which contains an extensive ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review," Proc. IEEE, vol. 65, pp. 1565--1596, Nov. 1977.

....mathematicians (such as G. H. Hardy) quite independently of its important engineering applications, unforeseen at the time. The historical development of sampling theory is partially discussed in [3, 11] which also review a number of interesting related results. The review paper by Jerri [12] contains an account of the state of the art in sampling theory as of 1977, whereas Butzer s review [2] concentrates on the results obtained by his group at the Lehrsthul A fur Mathematik, Aachen. A number of advanced topics and an extensive bibliography, with more than a thousand entries, may be ....

A. J. Jerri. The Shannon sampling theorem --- its various extensions and applications: a tutorial review. Proc. IEEE, 65(11):1565--1596, Nov. 1977.

....The WKS theorem asserts that any function f band limited to w satisfies 1 f(t) 1 X k= Gamma1 f k w sin[w(t Gamma k w ) w(t Gamma k w ) 1) the convergence being absolute and uniform. An elementary introduction to sampling theory may be found in [1] The survey papers [2 5] and the books [6, 7] give an account of the field, its history, and contain detailed bibliographies. There are many important results concerning the approximation of not necessarily band limited signals by 1 A function f 2 L2 (R) is band limited to w if its Fourier transform vanishes outside ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review", Proc. IEEE, vol. 65, no. 11, pp. 1565--1596, Nov. 1977.

....of our new approach can be found in [13] The objective of this paper is the investigation of the error analysis and the stability of the new algorithms. Whereas there exists a vast literature on the error analysis of regular sampling, the cardinal series and generalized regular sampling, e.g. [24, 2, 15, 16, 29] and the references mentioned there, an error analysis for the irregular sampling is virtually non existent. Error analysis for the iterative methods in [10, 12, 26, 21] would require to chase the propagation of an error from one iteration to the next, which seemed quite unfeasible to us. The ....

A.J. Jerri. The Shannon sampling theorem --- its various extensions and applications. A tutorial review. Proc. IEEE 65 (1977), 1565--1596.

....give a nonuniform sampling series for non band limited functions, together with error bounds. I. Introduction S AMPLING theory is a topic with important applications in several fields, which has attracted the attention of many authors. For an introduction to sampling theory see [1] The papers [2] [4] give a rather complete survey of the field and of its history. For more recent developments and up to date bibliographies see [5,6] Briefly, sampling theory is the study of series of the form f(t) 1 X k= Gamma1 f(t k )OE k (t) also called sampling expansions, or sampling series. In ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review", Proc. IEEE, vol. 65, no. 11, pp. 1565--1596, Nov. 1977. 3

.... These are well documented subjects: Shannon sampling by itself is the subject of the four books [15, 19, 20, 27] and of a forthcoming volume by Higgins and Stens; some of the chapters of [1] are concerned with sampling and related issues for example, 5] There are several review papers [2 4, 14, 18]; the conference proceedings [16, 17] also contain many relevant works. The band limited interpolation and extrapolation problems have also been exhaustively studied. The Papoulis Gerchberg iteration [12,23] can be readily adapted to the discrete context. This was studied in [6] a work that ....

A. J. Jerri. The Shannon sampling theorem --- its various extensions and applications: a tutorial review. Proc. IEEE, 65(11):1565-- 1596, Nov. 1977.

.... of square integrable functions and which has a Fourier transform f f( 1 p 2 Z 1 Gamma1 f(t)e Gammaj t dt such that f(t) 1 p 2 Z oe Gammaoe f( e j t d : The set of all such f 2 L 2 will be denoted by B 2 (oe) By the Whittaker Kotelnikov Shannon sampling theorem [1, 2], any f 2 B 2 (oe) can be reconstructed from samples f(nT ) Gamma1 n 1) taken T seconds apart if T =oe. When the oversampling parameter r = oeT= is less than unity, the samples ff(nT )g are redundant, that is, any finite number of them can be obtained from the remaining ones by solving a ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review", Proceedings of the IEEE, vol. 65, no. 11, pp. 1565--1596, Nov. 1977.

....advance, as well as the convergence rate of some of the iterations. 1. INTRODUCTION The reconstruction of signals and images from sets of nonuniform samples is one of the most often studied problems in signal theory. Some recent general references on sampling are [35, 36, 78] The review papers [2,3,23,28] cover a wide range of topics connected with sampling theory and its history. In this work we focus on finite dimensional problems, meaning that we will work, throughout most of the paper, in the finitedimensional space C N , with the usual norm and inner product. Although we will consider a ....

A. J. Jerri. The Shannon sampling theorem --- its various extensions and applications: a tutorial review. Proc. IEEE, 65(11):1565--1596, November 1977.

....w ) w(t Gamma k w ) 1) absolutely and uniformly. Given the many important applications that sampling theory has in several fields, it is not surprising to find that it has attracted the attention of many authors. An elementary introduction to the subject may be found in [1] The papers [2 4] give a rather complete survey of the field and its history. An account of some subsequent developments and a more complete bibliography may be found in [5, 6] The WKS theorem has been generalized along several broad lines of research, including 1. Extension to other spaces (for example, spaces ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review", Proc. IEEE, vol. 65, no. 11, pp. 1565-- 1596, Nov. 1977.

....f(t n ) n 2 Z) is the subject of sampling theory. The reconstruction method is often of the form f(t) 1 X n= Gamma1 f(t n )s n (t) which is called a sampling expansion of f , or a sampling series. For an introduction to the topic see [1] A number of excellent review papers is available [2 4], which together cover an impressive number of topics, including rather complete bibliographies. A more upto date and extensive bibliography, with more than a thousand entries, may be found in [5] Several recent developments in sampling theory are discussed in [6] Sampling expansions have been ....

A. J. Jerri, "The Shannon sampling theorem --- its various extensions and applications: a tutorial review", Proc. IEEE, vol. 65, no. 11, pp. 1565-- 1596, Nov. 1977.

....was supported by NSF grant DMS 9803352. 1 1 Introduction The classical Whittaker and Shannon sampling theorem permits reconstruction of a bandlimited function from its values on a set of equidistant points on the real line IR [19, 23] It has been extended in many directions; see the reviews [3, 10, 13] as well as the volumes [11, 12, 16, 17, 25] Kluvanek s important generalization results from replacing IR by an arbitrary locally compact abelian (LCA) group G [14] The sampling set is then a coset of a closed subgroup of G. Periodic sampling, introduced by Kohlenberg [15] is well established ....

A. J. Jerri, The Shannon sampling theorem - its various extensions and applications: a tutorial review, Proc. IEEE, 65(1977), pp. 1565-1596.

....and becomes meaningful only after imposing some a priori conditions on f . A classical and common assumption in dimension d = 1 is that f belongs to the space of band limited functions B# , i.e. supp f # [ #, #] see for example [13, 35, 37, 48, 57, 64, 74, 77, 81, 95] and the review papers [21, 47, 51]) The reason for this assumption is a classical result of Whittaker in complex analysis which states that a function f # L 2 # B [ 1 2,1 2] can be recovered exactly from its samples f(j) j # ZZ by the interpolation formula f(x) X k#ZZ f(k) sinc(x k) 1.1) where sinc(x) ....

A. Jerri. The Shannon sampling theorem---its various extensions and applications: A tutorial review. Proc. IEEE, 65:1565--1596, 1977.

....or analogue signals and discrete signals are equivalent. This is the content of the Sampling Theorem, associated with C. E. Shannon [22, 23] and fundamental in information theory and communication, particularly since the advent of modern digital computers. This and its numerous applications [17] have generated a great deal of interest in the mathematics surrounding the Sampling Theorem (excellent accounts are given in the survey articles [5, 15] Naturally there are particularly close connections with interpolation theory and Fourier analysis (see for example [16, 18, 19] but in ....

A. J. Jerri. The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE, 65 (1977), 1565-1596.

....window size. We thus expose the sparse structure of speech and use the residual correlations within the subbands as predictors to perform non uniform subsampling, or sparsification, which now allows a significant reduction over the Nyquist rate. Non uniform sampling has been considered previously [9, 15], albeit with unsatisfactory results. Time Code Modulation is a related method, in which the amplitude of the waveform is sampled uniformly, hence the sampling time is non uniform [4, 5] Better performance has been achieved by sample and hold (zeroth order) and first order hold predictive models ....

A. J. Jerri. The Shannon sampling theorem---its various extensions and applications: A tutorial review. Proc. IEEE, 65(11):1565--1598, November 1977.

....and approximation theory in a completely discrete setting. I. Introduction The well known Shannon s sampling theorem for band limited signal with finite energy (i.e. L 2 functions with compactly supported Fourier transform) has been extended to other classes of signals. See the surveys [1], 2] and [3] and the books [4] and [5] for references. One of the most general version of the sampling theorem is in our work in [6] where we have shown that band limited signals in Besov spaces can be recovered from their samples. Moreover, along the lines of the classical Plancherel Polya ....

A. Jerri, "The Shannon sampling theorem-- Its various extensions and applications", Proc. IEEE 65 (1977), pp. 1565--1596.

....of computer numbers requires discretization both in time and amplitude: sampling and quantization must be performed (see Fig. 1) 0 1 2 3 4 5 6 7 8 9 10 Gamma3 Gamma2 Gamma1 0 1 2 3 t x Fig. 1. Sampling and quantization Sampling theory has been elaborately described in the literature [1] [2], 3] and is well understood. Sampling is a linear operation, therefore linear system theory can be applied to the analysis of it. The sampled signal can be obtained from the continuous time signal by multiplying it (modulation) with a certain impulse carrier (Fig. 2a) 4] The spectrum of the ....

Jerri, A. J., "The Shannon Sampling Theorem --- Its Various Extensions and Applications: A Tutorial Review," Proc. IEEE, Vol. 65, No. 11, Nov. 1977, pp. 1565--1596.

....be carried out when interlaced images are processed under the assumption of a translational motion: the information contained in two fields is used to recover exactly the analog signal hidden behind the sampling . The philosophy of the method is based on the periodic nonuniform sampling theory [8, 9]. Based on the same approach, the authors have proposed a deinterlacing algorithm [10] up and downsampling methods [11] and a 50 to 60 Hz converter [12] For all these problems, the interpolation filters are of the infinite impulse response type. In [11] it has been shown how finite impulse ....

....pictures. 3 Generalized sampling and interpolation In this section, we remind the reader of results presented in [7] and generalize them for the twodimensional case. The theory presented here is only valid for interlaced images. 3. 1 One dimensional interpolation formulas From the Nyquist theorem [9], it is well known that any analog signal whose spectrum is limited to f m can theoretically be recovered exactly as long as it is sampled at a rate greater than 2f m . Actually, this condition is sufficient but not necessary. In 1956, Yen [8] published a generalized form of the Nyquist theorem ....

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A. J. Jerri, "The Shannon sampling theorem - its various extensions and applications: a tutorial review", Proceedings IEEE, Vol. 65, No. 11, November 1977, pp. 1565-1596.

.... of Whittaker, Kotel nikov, and Shannon (WKS) establishes that a bandlimited signal f of the form (1) can be reconstructed from uniformly spaced samples by the WKS formula f(t) X n2Z f(nT ) sinc t Gamma nT ) 4) where T = Omega (as above) and the convergence is absolute [7]. To set the stage for the results in the following sections of this paper, this section summarizes Clark s and Kramer s extensions of this theorem. Since no generality is sacrificed, the remainder of the paper will assume Omega = to make T = 1 and simplify the formulae presented. 2.1. Clark s ....

A.J. Jerri, "The Shannon sampling theorem -- its various extensions and applications: A tutorial review, " Proceedings of the IEEE, vol. 65, pp. 1565-- 1594, November 1977.

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Jerri, A.J. (1977). The Shannon sampling theorem -- Its various extensions and applications: A tutorial review. Proceedings of the IEEE, 65, 1565-1596.

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A. Jerri, "The Shannon sampling theorem-- Its various extensions and applications", Proc. IEEE 65 (1977), pp. 1565--1596.

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A. J. Jerri. The Shannon sampling theorem---its various extensions and applications: A tutorial review. Proc. IEEE, 65(11):1565--1598, November 1977.

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A. J. Jerri, The Shannon sampling theorem - its various extensions and applications: a tutorial review, Proc. IEEE, 65(1977), pp. 1565-1596.

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Abdul J. Jerri, "The Shannon Sampling Theorem -- Its Various Extensions and Applications: A Tutorial Review", in Proceedings of the IEEE, vol.65, no.11, 1565-1596, 1977.

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J. A. Jerri, "The Shannon sampling theorem -- its various extensions and applications: a tutorial review," Proceedings of the IEEE, vol. 65, pp. 1565--1596, 1977.

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A. J. Jerri, "The Shannon sampling theorem-its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, no. 11, pp. 1565-1596, 1977.

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