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16
Interpolation and the Discrete PapoulisGerchberg Algorithm
 IEEE Trans. Signal Processing
, 1994
"... In this paper we analyze the performance of an iterative algorithm, similar to the discrete PaponiisGerchberg algorithm, and which can be used to recover missing samples in finitelength records of bandlimited data. No assumptions are made regarding the distribution of the missing samples, in cont ..."
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Cited by 38 (21 self)
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In this paper we analyze the performance of an iterative algorithm, similar to the discrete PaponiisGerchberg algorithm, and which can be used to recover missing samples in finitelength records of bandlimited data. No assumptions are made regarding the distribution of the missing samples, in contrast with the often studied extrapolation problem, in which the known samples are grouped together. Indeed, it is possible to regard the observed signal as a sampled version of the original one, and to interpret the reconstruction result studied herein as a sampling result. We show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data. We give upper and lower bounds for the error as a function of the number of iterations, together with the signals for which the bounds are attained. Also, we analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix. From this analysis we infer the optimum value of the relaxation constant. We also point out, among all sampling sets with the same density, those for which the convergence rate of the recovery algorithm is maximum or minimum. For lowpass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer. The worst convergence rates generally occur when the missing samples are contiguous.
Dualization of signal recovery problems
, 2009
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenab ..."
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Cited by 18 (7 self)
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In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved by forwardbackward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.
Analysis And Design Of MinimaxOptimal Interpolators
 IEEE Trans. Signal Proc
, 1998
"... We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sa ..."
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Cited by 17 (3 self)
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We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix which is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical illconditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution, consisting of a sinckernel interpolator with specially chosen weighting coefficients. The newly designed sinckernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting, through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinckernel interpolator is shown to perform better than ...
MaximumEntropy Spatial Processing of Array Data
 Geophysics
, 1974
"... The procedure of maximumentropy spectral analysis (MESA), used in the processing of time series data, also applies to wavenumber (bearing) analysis of signals received from a spatially distributed linear array of sensors. The method is precisely the use of autoregressive spectral analysis in the ..."
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Cited by 7 (0 self)
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The procedure of maximumentropy spectral analysis (MESA), used in the processing of time series data, also applies to wavenumber (bearing) analysis of signals received from a spatially distributed linear array of sensors. The method is precisely the use of autoregressive spectral analysis in the space dimension rather than in 1 Ime. There are also close links to the predictive deconvolution method used in geophysical work, and to the process of constructing noisewhitening filters in communication theory, as well as to leastsquares model building. In this note, we review the maximumentropy procedure pointing out all these links. The specific algorithm appropriate to a uniformly spaced line array of sensors is given, as well as one possible algorithm for use in the case of nonuniform sensor spacing.
Duality and Convex Programming
, 2010
"... We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging. ..."
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Cited by 7 (3 self)
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We survey some key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.
Superoscillations: Faster Than the Nyquist Rate
 IEEE Trans. on Signal Processing
, 2006
"... Abstract—It is commonly assumed that a signal bandlimited to 2 Hz cannot oscillate at frequencies higher than Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called ..."
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Cited by 3 (1 self)
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Abstract—It is commonly assumed that a signal bandlimited to 2 Hz cannot oscillate at frequencies higher than Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called superoscillations, can only occur in signals that possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared 2 norm) as a function of the superoscillation’s frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and timefrequency analysis, for example. It also shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations ’ period. Index Terms—Bandlimited signals, information rates, quantum theory, signal sampling, superoscillations, time–frequency analysis. I.
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
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unknown title
, 907
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution ..."
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In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved reliably by forwardbackward splitting and allow for a simple construction of primal solutions from dual solutions. The proposed framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.
unknown title
, 907
"... In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution ..."
Abstract
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In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in signal recovery. These problems are not easily amenable to solution by current methods but they feature FenchelMoreauRockafellar dual problems that can be solved reliably by forwardbackward splitting and allow for a simple construction of primal solutions from dual solutions. The proposed framework is shown to capture and extend several existing dualitybased signal recovery methods and to be applicable to a variety of new problems beyond their scope.
Multiband signal reconstruction from finite samples*
, 1994
"... The minimum meansquared error (MMSE) estimator has been used to reconstruct a bandlimited signal from its finite samples in a bounded interval and shown to have many nice properties. In this research, we consider a special class of bandlimited 1D and 2D signals which have a multiband structure ..."
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The minimum meansquared error (MMSE) estimator has been used to reconstruct a bandlimited signal from its finite samples in a bounded interval and shown to have many nice properties. In this research, we consider a special class of bandlimited 1D and 2D signals which have a multiband structure in the frequency domain, and propose a new reconstruction algorithm to exploit the multiband feature of the underlying signals. The concept of the critical value and region is introduced to measure the performance of a reconstruction algorithm. We show analytically that the new algorithm performs #better than the MMSE estimator for bandlimited/multiband signals in terms of the critical value and region measure. Finally, numerical examples of 1D and 2D signal reconstruction are given for performance comparison of various methods.