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67
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 55 (44 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.
Formally Biorthogonal Polynomials and a LookAhead Levinson Algorithm for General Toeplitz Systems
 Linear Algebra Appl
, 1993
"... Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear syst ..."
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Cited by 25 (2 self)
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Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. In this paper, an extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses lookahead to skip over exactly singular, as well as illconditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials. In particular, the occurrence of singular or illconditioned submatrices corresponds to The research of this author was performed at the Research Institute for A...
Theory And Design Of Optimum FIR Compaction Filters
 IEEE TRANS. SIGNAL PROCESSING
, 1998
"... The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N ! M and N = 1 have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respec ..."
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Cited by 22 (9 self)
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The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N ! M and N = 1 have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively. In this paper, we deal with the more difficult case of M ! N ! 1. For the twochannel case and for a restricted but important class of random processes, we give an analytical solution for the compaction filter which is characterized by its zeros on the unitcircle. This also corresponds to the optimal twochannel FIR filter bank that maximizes the coding gain under the traditional quantization noise assumptions. This can also be used to generate optimal wavelets. For the arbitrary M \Gammachannel case, we provide a very efficient suboptimal design method called the window method. The method involves two stages that are associated with the above two special cases. As the order incre...
On A Sturm Sequence Of Polynomials For Unitary Hessenberg Matrices
 SIAM J. Matrix Anal. Appl
, 1993
"... Unitary matrices have a rich mathematical structure which is closely analogous to real symmetric matrices. For real symmetric matrices this structure can be exploited to develop very efficient numerical algorithms and for some of these algorithms unitary analogues are known. Here we present a unitar ..."
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Cited by 15 (3 self)
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Unitary matrices have a rich mathematical structure which is closely analogous to real symmetric matrices. For real symmetric matrices this structure can be exploited to develop very efficient numerical algorithms and for some of these algorithms unitary analogues are known. Here we present a unitary analogue of the bisection method for symmetric tridiagonal matrices. Recently Delsarte and Genin introduced a sequence of socalled fl n symmetric polynomials which can be used to replace the classical Szego polynomials in several signal processing problems. These polynomials satisfy a three term recurrence relation and their roots interlace on the unit circle. Here we explain this sequence of polynomials in matrix terms. For an n \Theta n unitary Hessenberg matrix, we introduce, motivated by the Cayley transformation, a sequence of modified unitary submatrices. The characteristic polynomials of the modified unitary submatrices p k (z); k = 1; 2; : : : ; n are exactly the fl n symmetric ...
Spectral Estimation via Selective Harmonic Amplification
, 2001
"... The statecovariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter, and also imposes a generalized interpolation constraint on the power spectrum of the input process. This algebraic property and the relationship between statec ..."
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Cited by 11 (7 self)
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The statecovariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter, and also imposes a generalized interpolation constraint on the power spectrum of the input process. This algebraic property and the relationship between statecovariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis. Several algorithms for spectral estimation will be developed with resolution higher than state of the art. Index TermsAnalytic interpolation, nonlinear spectral estimation, spectral analysis, state covariances. I.
Regularized Estimation of Mixed Spectra Using a Circular GibbsMarkov Model
 IEEE Trans. Signal Processing
, 2001
"... Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such illposed problems. Following Sacchi et al., we first address line spectra reco ..."
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Cited by 9 (3 self)
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Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such illposed problems. Following Sacchi et al., we first address line spectra recovering in this framework. Then, we extend the methodology to situations of increasing difficulty: the case of smooth spectra and the case of mixed spectra, i.e., peaks embedded in smooth spectral contributions.
Sparsity and uniqueness for some specific underdetermined linear systems
 in Proc. of IEEE ICASSP ’05
, 2005
"... The purpose of this contribution is to extend some results on sparse representations of signals in redundant bases developed for arbitrary bases to two frequently encountered bases. The general problem is the following: given a matrix with and a vector with having nonzero components, find sufficient ..."
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Cited by 9 (0 self)
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The purpose of this contribution is to extend some results on sparse representations of signals in redundant bases developed for arbitrary bases to two frequently encountered bases. The general problem is the following: given a matrix with and a vector with having nonzero components, find sufficient conditions for to be the unique sparsest solution to. The answer gives an upperbound on depending upon. We consider the cases where is a Vandermonde matrix or a real Fourier matrix and the components of are known to be greater than or equal to zero. The sufficient conditions we get are much weaker than those valid for arbitrary matrices and guarantee further that can be recovered by solving a linear program. 1.