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19
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 207 (22 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,
Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in splinelike spaces: the Lp theory
 Proc. Amer. Math. Soc
"... Abstract. We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense ” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of splinelike spaces in Lp (Rn). Moreover, the reconstruction can be implemented using fast alg ..."
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Cited by 42 (5 self)
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Abstract. We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense ” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of splinelike spaces in Lp (Rn). Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical ShannonWhittaker sampling theorem on regular sampling and the PaleyWiener theorem on nonuniform sampling. 1.
Optimal tight frames and quantum measurement
 IEEE Trans. Inform. Theory
, 2002
"... Tight frames and rankone quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rankone generalized quantum measurements (POVMs) on that space. Using this relationshi ..."
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Cited by 41 (10 self)
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Tight frames and rankone quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rankone generalized quantum measurements (POVMs) on that space. Using this relationship, frametheoretical analogues of various quantummechanical concepts and results are developed. The analogue of a leastsquares quantum measurement is a tight frame that is closest in a leastsquares sense to a given set of vectors. The leastsquares tight frame is found for both the case in which the scaling of the frame is specified (constrained leastsquares frame (CLSF)) and the case in which the scaling is free (unconstrained leastsquares frame (ULSF)). The wellknown canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling. Finally, the canonical frame vectors corresponding to a geometrically uniform vector set are shown to be geometrically uniform and to have the same symmetries as the original vector set.
Geometrically uniform frames
 IEEE Trans. Inform. Theory
, 2003
"... Abstract—We introduce a new class of finitedimensional frames with strong symmetry properties, called geometrically uniform (GU) frames, that are defined over a finite Abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to comp ..."
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Cited by 37 (12 self)
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Abstract—We introduce a new class of finitedimensional frames with strong symmetry properties, called geometrically uniform (GU) frames, that are defined over a finite Abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by a finite Abelian group of unitary matrices using multiple generating vectors. The dual frame vectors and canonical tight frame vectors associated with GU frames are shown to be GU and, therefore, also generated by a single generating vector, which can be computed very efficiently using a Fourier transform (FT) defined over the generating group of the frame. Similarly, the dual frame vectors and canonical tight frame vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the Abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the generating vector. Index Terms—Compound geometrically uniform (CGU) frames, generalized Fourier transform (FT), geometrically uniform (GU) frames, least squares. I.
Orthogonal Sampling Formulas: A Unified Approach
 SIAM Rev
, 2000
"... Abstract. This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values. In orde ..."
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Cited by 11 (0 self)
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Abstract. This paper intends to serve as an educational introduction to sampling theory. Basically, sampling theory deals with the reconstruction of functions (signals) through their values (samples) on an appropriate sequence of points by means of sampling expansions involving these values. In order to obtain such sampling expansions in a unified way, we propose an inductive procedure leading to various orthogonal formulas. This procedure, which we illustrate with a number of examples, closely parallels the theory of orthonormal bases in a Hilbert space. All intermediate steps will be described in detail, so that the presentation is selfcontained. The required mathematical background is a basic knowledge of Hilbert space theory. Finally, despite the introductory level, some hints are given on more advanced problems in sampling theory, which we motivate through the examples. Key words. orthonormal bases, sampling expansions, reproducing kernel Hilbert spaces
Optimal SubNyquist Nonuniform Sampling and Reconstruction for Multiband Signals
, 2001
"... We study the problem of optimal subNyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau ..."
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Cited by 11 (2 self)
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We study the problem of optimal subNyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's lower bound equal to the measure of . For signals with sparse , this rate can be much smaller than the Nyquist rate. Unfortunately, the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a recent study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with that tiles under translation. For signals with nontiling , which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with subNyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate.
Numerical analysis of the nonuniform sampling problem,”J
 Comput. Appl. Math
"... We give an overview of recent developments in the problem of reconstructing a bandlimited signal from nonuniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finitedimensional model plays a key role in the numerical solution of the nonuniform samp ..."
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Cited by 10 (0 self)
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We give an overview of recent developments in the problem of reconstructing a bandlimited signal from nonuniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finitedimensional model plays a key role in the numerical solution of the nonuniform sampling problem. In the one approach (often proposed in the literature) the finitedimensional model leads to an illposed problem even in very simple situations. The other approach that we consider leads to a wellposed problem that preserves important structural properties of the original infinitedimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples. We also discuss the design of efficient regularization methods for illconditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods.
Sampling of Bandlimited Functions on Unions of Shifted Lattices
 J. Fourier Anal. Appl
, 2000
"... We consider Shannon sampling theory for sampling sets which are unions of shifted lattices. These sets are not necessarily periodic. A function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. Wh ..."
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Cited by 9 (1 self)
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We consider Shannon sampling theory for sampling sets which are unions of shifted lattices. These sets are not necessarily periodic. A function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. While explicit reconstruction formulas are possible, it is most convenient to use a recursive algorithm. The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given, including a numerical example implemented in MATLAB. 2000 Mathematics Subject Classification: 94A20, 94A12, 43A25, 42B99 Key words: Shannon sampling, multidimensional sampling, nonuniform sampling, periodic sampling, nonperiodic sampling, irregular sampling, locally compact abelian groups. # Mathematics Department, Western Oregon University, Monmouth, Oregon 97361 + Department of Mathematics, Oregon State University, Corvallis, OR 97331. This work was supported by ...