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26
Structured Inverse Eigenvalue Problems
 ACTA NUMERICA
, 2002
"... this paper. More should be said about these constraints in order to define an IEP. First we recall one condition under which two geometric entities intersect transversally. Loosely speaking, we may assume that the structural constraint and the spectral constraint define, respectively, smooth manifo ..."
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Cited by 42 (14 self)
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this paper. More should be said about these constraints in order to define an IEP. First we recall one condition under which two geometric entities intersect transversally. Loosely speaking, we may assume that the structural constraint and the spectral constraint define, respectively, smooth manifolds in the space of matrices of a fixed size. If the sum of the dimensions of these two manifolds exceeds the dimension of the ambient space, then under some mild conditions one can argue that the two manifolds must intersect and the IEP must have a solution. A more challenging situation is when the sum of dimensions emerging from both structural and spectral constraints does not add up to the transversal property. In that case, it is much harder to tell whether or not an IEP is solvable. Secondly we note that in a complicated physical system it is not always possible to know the entire spectrum. On the other hand, especially in structural design, it is often demanded that certain eigenvectors should also satisfy some specific conditions. The spectral constraints involved in an IEP, therefore, may consist of complete or only partial information on eigenvalues or eigenvectors. We further observe that in practice it may occur that one of the two constraints in an IEP should be enforced more critically than the other due, say, to the physical realizability. Without the realizability, the physical system simply cannot be built. There are also situations when one constraint could be more relaxed than the other due, say, to the physical uncertainty. The uncertainty arises when there is simply no accurate way to measure the spectrum or there is no reasonable means to obtain the entire information. When the two constraints cannot be satisfied simultaneously, the IEP could be formulat...
Inverse eigenvalue problems
 SIAM Rev
, 1998
"... Abstract. A collection of inverse eigenvalue problems are identi ed and classi ed according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserves furth ..."
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Cited by 41 (6 self)
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Abstract. A collection of inverse eigenvalue problems are identi ed and classi ed according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserves further study. An extensive bibliography of pertinent literature is attached.
GSVDBased Optimal Filtering for Single and MultiMicrophone Speech Enhancement
, 2002
"... In this paper a Generalised Singular Value Decomposition (GSVD) based algorithm is proposed for enhancing multimicrophone speech signals degraded by additive coloured noise. This GSVDbased multimicrophone speech signal... ..."
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Cited by 18 (7 self)
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In this paper a Generalised Singular Value Decomposition (GSVD) based algorithm is proposed for enhancing multimicrophone speech signals degraded by additive coloured noise. This GSVDbased multimicrophone speech signal...
Efficient Channel Shortening Equalizer Design
, 2004
"... Timedomain equalization is crucial in reducing channel state dimension in maximum likelihood sequence estimation, and intercarrier and intersymbol interference in multicarrier systems. A timedomain equalizer (TEQ) placed in cascade with the channel produces an effective impulse response that is ..."
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Cited by 14 (8 self)
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Timedomain equalization is crucial in reducing channel state dimension in maximum likelihood sequence estimation, and intercarrier and intersymbol interference in multicarrier systems. A timedomain equalizer (TEQ) placed in cascade with the channel produces an effective impulse response that is shorter than the channel impulse response. This paper analyzes two TEQ design methods amenable to costeffective realtime implementation: minimum mean squared error (MMSE) and maximum shortening SNR (MSSNR) methods. We reduce the complexity of computing the matrices in the MSSNR and MMSE designs by a factor of 140 and a factor of 16 (respectively) relative to existing approaches, without degrading performance. We prove that an infinite length MSSNR TEQ with unit norm TEQ constraint is symmetric. A symmetric TEQ halves FIR implementation complexity, enables parallel training of the frequencydomain equalizer and TEQ, reduces TEQ training complexity by a factor of 4 and doubles the length of the TEQ that can be designed using fixedpoint arithmetic, with only a small loss in bit rate. Simulations are presented for designs with a symmetric TEQ or target impulse response.
Extreme eigenvalues of real symmetric Toeplitz matrices
 Math. Comp
, 2000
"... Abstract. We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, ..."
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Cited by 7 (1 self)
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Abstract. We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results. 1.
Robustness of SVDbased Optimal Filtering for Noise Reduction in MultiMicrophone Speech Signals
 in Proc. of the 1999 IEEE International Workshop on Acoustic Echo and Noise Control (IWAENC'99), Pocono
, 1999
"... This paper discusses an SVDbased signal enhancement procedure, applied to noise reduction in multimicrophone speech signals. The SVDbased signal enhancement procedure amounts to a specific optimal filtering problem when the socalled `desired response' signal cannot be observed. The optimal filte ..."
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Cited by 6 (6 self)
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This paper discusses an SVDbased signal enhancement procedure, applied to noise reduction in multimicrophone speech signals. The SVDbased signal enhancement procedure amounts to a specific optimal filtering problem when the socalled `desired response' signal cannot be observed. The optimal filter can then be written as a function of the generalized singular vectors and singular values of a speech and noise data matrix. It is shown that the SNR improvement provided by the SVDbased optimal filtering technique is better than the improvement obtained with standard beamforming techniques. Moreover most beamforming techniques assume the position of the speech source and the microphone array configuration to be known. Therefore the performance of these techniques is rather sensitive to deviations from these assumptions, i.e. incorrect estimation of the position of the speech source and uncalibrated microphone arrays. It is shown that the SVDbased optimal filtering technique is more robu...
A Symmetry Exploiting Lanczos Method for Symmetric Toeplitz Matrices
 Numerical Algorithms, 25:377 – 385
, 1999
"... Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. The most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos ..."
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Cited by 6 (3 self)
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Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. The most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach. Keywords: Toeplitz matrix, eigenvalue problem, Lanczos method, symmetry AMSclassification: 65F15 1 Introduction Several approaches have been reported in the literature for computing the smallest eigenvalue of a real symmetric, positive definite Toeplitz matrix. This problem is of considerable interest in signal processing. Given the covariance sequence of the observed data, Pisarenko [14] suggested a method which determines the sinusoidal frequencies from the eigenvector of the covariance matrix associated with its minimum eigenvalue. Cybenko and Van Loan [3] pre...
SVDbased Optimal Filtering with Applications to Noise Reduction in Speech Signals
 In Proc. of the 1999 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA'99), New Paltz
, 1999
"... In this report, a compact review is given of a class of SVDbased signal enhancement procedures, which amount to a specific optimal filtering technique for the case where the socalled `desired response' signal cannot be observed. A number of simple properties (e.g. symmetry properties) of the obtai ..."
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Cited by 6 (6 self)
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In this report, a compact review is given of a class of SVDbased signal enhancement procedures, which amount to a specific optimal filtering technique for the case where the socalled `desired response' signal cannot be observed. A number of simple properties (e.g. symmetry properties) of the obtained estimators are derived, which to our knowledge have not been published before and which are valid for the white noise case as well as for the coloured noise case. Also a standard procedure based on averaging is investigated, leading to serious doubts about the necessity of the averaging step. When applying this technique to multimicrophone noise reduction, the optimal filter exhibits a kind of beamforming behaviour for highly correlated noise sources. When comparing this technique to standard beamforming algorithms, its performance is equally good for highly correlated noise sources. For less correlated noise sources  a situation where standard beamforming typically fails  it is sho...
Structured Low Rank Approximation
 LINEAR ALGEBRA APPL
, 2002
"... This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matr ..."
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Cited by 6 (1 self)
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This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.
Infinite Length Results and Design Implications for TimeDomain Equalizers
 IEEE Trans. on Signal Processing
, 2004
"... We show that maximum shortening SNR TEQs are often nearly symmetric. Constraining the TEQ to be symmetric causes only a 3% loss in bit rate (averaged over 8 standard ADSL channels). Symmetric TEQs have greatly reduced design and implementation complexity. We also show that for infinite length TEQs, ..."
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Cited by 5 (5 self)
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We show that maximum shortening SNR TEQs are often nearly symmetric. Constraining the TEQ to be symmetric causes only a 3% loss in bit rate (averaged over 8 standard ADSL channels). Symmetric TEQs have greatly reduced design and implementation complexity. We also show that for infinite length TEQs, minimum mean squared error target impulse responses have all zeros on the unit circle, which can lead to poor bit rate performance.