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Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 233 (24 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator
"... We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically t ..."
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Cited by 61 (9 self)
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We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about nonuniform sampling in shiftinvariant spaces. 1.
BeurlingLandauType Theorems For NonUniform Sampling In Shift Invariant Spline Spaces
, 1999
"... . Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its nonuniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invari ..."
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Cited by 31 (8 self)
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. Under the appropriate definition of sampling density D # , a function f that belongs to a shift invariant space can be reconstructed in a stable way from its nonuniform samples only if D # # 1. This result is similar to Landau's result for the PaleyWiener space B 1/2 . If the shift invariant space consists of polynomial splines, then we show that D # < 1 is su#cient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B 1/2 . In the sampling problem one seeks to recover a function f from its samples {f(x j ) : j # ZZ}. Clearly, this problem is illposed, and becomes meaningful only after some a priori conditions on f . Until recently, the standard assumption is that f is bandlimited, i.e., supp f # [#, #] [7, 19, 29]. Although this is a reasonable assumption, in many applications di#erent hypothesis on the functions are desirable, e.g., for taking into account real acquisition and reconstruction devices, for obtaini...
pframes and Shift Invariant Subspaces of L p
, 1999
"... We investigate the frame properties and closedness for the shift invariant space V p (#) = n r X i=1 X j#Z d d i (j)# i (  j) : (d i (j)) j#Z d # l p o , 1 # p # #. We derive necessary and su#cient conditions for a set {# i : 1 # i # r} to constitute a pframe for V p (#), ..."
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Cited by 29 (13 self)
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We investigate the frame properties and closedness for the shift invariant space V p (#) = n r X i=1 X j#Z d d i (j)# i (  j) : (d i (j)) j#Z d # l p o , 1 # p # #. We derive necessary and su#cient conditions for a set {# i : 1 # i # r} to constitute a pframe for V p (#), and to generate a closed, shift invariant subspace of L p . A function in the L p closure of V p (#) is not necessarily generated by l p coe#cients. Hence we often want V p (#) itself to be closed, i.e., a Banach space. For p #= 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of pframes, and the closedness of the space they generate. Since the frame vectors may be dependent, the relation between a function f # V p (#) and the coe#cients of its representations is neither obvious, nor unique, in general. In Hilbert spaces, it is well known that the coe#cients sequence obtained by using...
A generalized sampling theorem for stable reconstructions in arbitrary bases
 J. Fourier Anal. Appl
"... We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be vie ..."
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Cited by 15 (14 self)
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We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the wellknown consistent reconstruction technique (Eldar et al). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable. Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.
Generalized Sampling: A Variational Approach. Part I: Theory
 IEEE Transactions on Signal Processing, 2001. In preparation
, 2002
"... We consider the problem of lconstructing a multidimensional vector function fln: "* from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions ..."
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Cited by 13 (5 self)
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We consider the problem of lconstructing a multidimensional vector function fln: "* from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions. Instead, we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion J. First, we plsent the solution of the generalized interpolation problem. Latel; we also consider the approximation plblem, and we show that both lead to the same class of solutions.
On simple oversampled A/D conversion in shift invariant spaces
 IEEE Trans. Information Theory
, 2005
"... In this paper we study a simple oversampled analogtodigital (A/D) conversion in shift invariant spaces. The BeurlingLandau type theorem for bandlimited signal spaces is extended to shift invariant spaces, and then a nonuniform sampling theorem for shift invariant spaces is established, which say ..."
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Cited by 10 (3 self)
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In this paper we study a simple oversampled analogtodigital (A/D) conversion in shift invariant spaces. The BeurlingLandau type theorem for bandlimited signal spaces is extended to shift invariant spaces, and then a nonuniform sampling theorem for shift invariant spaces is established, which says, a uniformly discrete set is a stable sampling set for a shift invariant space if its Beurling lower density is larger than a fixed density determined by the generator of the shift invariant space. Consequently, an oversampling theorem for shift invariant spaces is attained. These sampling theorems together with a theorem concerning the stability of stable sampling in shift invariant spaces shown by us, are used to build a simple oversampled A/D conversion scheme in shift invariant spaces. In such a scheme, the quantization error e is found to behave as �e � 2 = O(τ 2) with respect to the sampling interval τ, which is the same as that for bandlimited signal spaces derived very recently. Moreover, we demonstrate that the bitrate required to encode the converted digital signal only increases as logarithm of sampling ratio. Keywords: sampling, oversampling, A/D conversion, shift invariant space, generator, BeurlingLandau theorem, quantization error, bitrate.
SAMPLING AND RECONSTRUCTION OF SURFACES AND HIGHER DIMENSIONAL MANIFOLDS
"... Abstract. We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2dimensional manif ..."
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Cited by 10 (7 self)
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Abstract. We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis. 1.
NonUniform Sampling In ShiftInvariant Spaces
, 2000
"... . This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. The reconstruction of a function or signal or image f from its nonuniform samples f(x j ) is a common task in many applications in data or signal or image processing. The n ..."
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Cited by 9 (4 self)
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. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. The reconstruction of a function or signal or image f from its nonuniform samples f(x j ) is a common task in many applications in data or signal or image processing. The nonuniformity of the sampling set is often a fact of life and prevents the use of the standard methods from Fourier analysis. The nonuniform sampling problem is usually treated in the context of bandlimited functions and with tools from complex analysis. However, many applied problems impose di#erent a priori constraints on the type of function. These constraints are taken into consideration by investigating the problem in general shiftinvariant spaces rather than for bandlimited functions only. This generalization requires a new set of techniques and ideas. While some of the tools are implicitly used in certain questions in approximation theory, wavelet theory and frame theory, they hav...
RequestDriven ECG Interpretation Based on Individual Data Validity Periods
"... Abstract—Traditional approach to the automated ECG analysis assumes the calculation is triggered for each data point acquired in a uniform time interval. This approach reveals the variability of selected diagnostic parameters differing by more than three orders. Distributed systems using global wire ..."
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Cited by 6 (5 self)
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Abstract—Traditional approach to the automated ECG analysis assumes the calculation is triggered for each data point acquired in a uniform time interval. This approach reveals the variability of selected diagnostic parameters differing by more than three orders. Distributed systems using global wireless digital communication may benefit from this difference using variable reporting interval or dataadaptive report content. More natural consequence is a requestdriven interpretation described in this paper. Our approach assumes the processing of the acquired ECG is triggered by the data validity period expiry. Such solution significantly reduces unnecessary computation and is particularly interesting in aspect of wearable devices autonomy. Correct definition of data validity for each particular diagnostic parameter guarantees the completeness of the patient status and its convergence to the result of traditional approaches. I.