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82
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,
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
, 2009
"... Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, alt ..."
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Cited by 69 (15 self)
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Wideband analog signals push contemporary analogtodigital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.
From theory to practice: SubNyquist sampling of sparse wideband analog signals
 IEEE J. SEL. TOPICS SIGNAL PROCESS
, 2010
"... Conventional subNyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind subNyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. ..."
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Cited by 69 (42 self)
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Conventional subNyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind subNyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then lowpass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, realtime performance for signals with timevarying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of stateoftheart analog conversion technologies such as interleaved converters.
Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals
"... We address the problem of reconstructing a multiband signal from its subNyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multicoset sampling. Prior recovery methods for this sampling strategy either require knowledge of band locations or impose stric ..."
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Cited by 61 (51 self)
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We address the problem of reconstructing a multiband signal from its subNyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multicoset sampling. Prior recovery methods for this sampling strategy either require knowledge of band locations or impose strict limitations on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multicoset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finitedimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of knownspectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.
Blocksparse signals: Uncertainty relations and efficient recovery
 IEEE Trans. Signal Process
, 2010
"... Abstract—We consider efficient methods for the recovery of blocksparse signals—i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, wh ..."
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Cited by 51 (13 self)
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Abstract—We consider efficient methods for the recovery of blocksparse signals—i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we introduce. We then show that a blockversion of the orthogonal matching pursuit algorithm recovers block ksparse signals in no more than k steps if the blockcoherence is sufficiently small. The same condition on blockcoherence is shown to guarantee successful recovery through a mixed `2=`1optimization approach. This complements previous recovery results for the blocksparse case which relied on small blockrestricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of blocksparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem. Index Terms—Basis pursuit, blocksparsity, compressed sensing, matching pursuit. I.
Density theorems for sampling and interpolation in the BargmannFock space
 II, J. Reine Angew. Math
"... Abstract. We give a complete description of sampling and interpolation in the BargmannFock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann latti ..."
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Cited by 41 (1 self)
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Abstract. We give a complete description of sampling and interpolation in the BargmannFock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice. 1. Introduction and
Degrees of freedom in multipleantenna channels: A signal space approach
 IEEE Trans. Inf. Theory
, 2005
"... We consider multipleantenna systems that are limited by the area and geometry of antenna arrays. Given these physical constraints, we determine the limit to the number of spatial degrees of freedom available and find that the commonly used statistical multiinput multioutput model is inadequate. A ..."
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Cited by 34 (4 self)
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We consider multipleantenna systems that are limited by the area and geometry of antenna arrays. Given these physical constraints, we determine the limit to the number of spatial degrees of freedom available and find that the commonly used statistical multiinput multioutput model is inadequate. Antenna theory is applied to take into account the area and geometry constraints, and define the spatial signal space so as to interpret experimental channel measurements in an arrayindependent but manageable description of the physical environment. Based on these modeling strategies, we show that for a spherical array of effective aperture A in a physical environment of angular spread Ω  in solid angle, the number of spatial degrees of freedom is AΩ  for unpolarized antennas and 2AΩ  for polarized antennas. Together with the 2WT degrees of freedom for a system of bandwidth W transmitting in an interval T, the total degrees of freedom of a multipleantenna channel is therefore 4WTAΩ. 1
Incompleteness of Sparse Coherent States
 Appl. Comput. Harmon. Anal
, 1997
"... This paper is concerned with the completeness properties of the set ..."
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Cited by 33 (1 self)
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This paper is concerned with the completeness properties of the set
Spectral Sets and Factorizations of Finite Abelian Groups
, 1997
"... A spectral set is a subset\Omega of R n with Lebesgue measure 0 ! ¯(\Omega\Gamma ! 1 such that there exists a set of exponential functions which form an orthogonal basis of L 2(\Omega\Gamma4 The spectral set conjecture of B. Fuglede states that a set\Omega is a spectral set if and only if\Omega ..."
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Cited by 33 (8 self)
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A spectral set is a subset\Omega of R n with Lebesgue measure 0 ! ¯(\Omega\Gamma ! 1 such that there exists a set of exponential functions which form an orthogonal basis of L 2(\Omega\Gamma4 The spectral set conjecture of B. Fuglede states that a set\Omega is a spectral set if and only if\Omega tiles R n by translation. We study sets\Omega which tile R n using a rational periodic tile set S = Z n + A, where A ` 1 N 1 Z \Theta \Delta \Delta \Delta \Theta 1 Nn Z is finite. We characterize geometrically sets\Omega that tile R n with such a tile set. Certain tile sets S have the property that every bounded measurable set\Omega which tiles R n with S is a spectral set, with a fixed spectrum S . We call such S a universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic set to be a universal spectrum for S, which is expressed in terms of factorizations A \Phi B = G where G = ZN 1 \Theta \Delta \Delta \Delta \Theta ZNn , and A ...
Minimum rate sampling and reconstruction of signals with arbitrary frequency support
 IEEE Trans. Inform. Theory
, 1999
"... Abstract—We examine the question of reconstruction of signals from periodic nonuniform samples. This involves discarding samples from a uniformly sampled signal in some periodic fashion. We give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform ..."
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Cited by 33 (0 self)
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Abstract—We examine the question of reconstruction of signals from periodic nonuniform samples. This involves discarding samples from a uniformly sampled signal in some periodic fashion. We give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed. We give an implicit characterization of the reconstruction system, and a design method by which the ideal reconstruction filters may be approximated. We demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing, as in earlier work. Previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made, or where the minimum rate was approached asymptotically. We show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered. When approaching the minimum rate, this freedom allows us, in certain cases to have a far less complex reconstruction system. Index Terms — Multiband, nonuniform, reconstruction, sampling. I.