Asymptotic eigenvalue distribution of block-Toeplitz matrices

Bose, N.K.; Boo, K.J.

doi:10.1109/18.661535

Cited by 21 publications

(14 citation statements)

References 5 publications

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“…Moreover, the sequences of TBT and CBC matrices of increasing size are asymptotically equivalent in the same sense as a Toeplitz matrix and its circulant matrix approximation in [25]. Such equivalence implies that the eigenvalues behave similarly as the size of the matrices increases [27]. Similar to the circulant matrix, the CBC matrix structure also has some very interesting properties, among which is the fact that the eigenvalues of a CBC matrix correspond to the 2-D DFT transform of its first row (or column) [29].…”

Section: Appendixmentioning

confidence: 95%

Minimax Robust MIMO Radar Waveform Design

Yang

Blum

2007

IEEE J. Sel. Top. Signal Process.

146

124

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Abstract-We consider the problem of minimax robust waveform design for multiple-input multiple-output (MIMO) radar based on mutual information (MI) and minimum meansquare error (MMSE) estimation for target identification and classification. Recognizing that a single, precise characterization of target power spectral density (PSD) is rare in practice, we assume the PSD lies in an uncertainty class of spectra bounded by known upper and lower bounds, which markedly relaxes the required target a priori knowledge. Based on this band model, we develop minimax robust waveforms for MIMO radar under both MMSE and MI criteria. These robust waveforms bound the worst-case performance at an acceptable limit, and also could insure performance will be sufficiently good for any PSD in the uncertainty class. Our findings also indicate that the MI and MMSE criteria lead to different minimax robust waveform solutions, which is in contrast to the case of the perfectly known target PSD.Index Terms-Extended radar targets, minimum meansquare error (MMSE), multiple-input multiple-output (MIMO) radar, mutual information (MI), radar waveform design, robust signal processing, target classification and identification.

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Section: Appendixmentioning

confidence: 95%

Minimax Robust MIMO Radar Waveform Design

Yang

Blum

2007

IEEE J. Sel. Top. Signal Process.

146

124

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“…Here, we shall admit that Gray's result extends to TBT matrices, not necessarily Hermitian. which allows to deduce from (18) and (19) that When , the latter inequality drastically simplifies according to which proves the assertion.…”

Section: Proof Of Theoremmentioning

confidence: 71%

“…( is the trace of a square matrix, i.e., the sum of its diagonal elements, which is also the sum of its eigenvalues). Specifically, important results establish the asymptotical equivalence between Toeplitz and circulant matrices [18], and between TBT and CBC matrices [15], [19].…”

Section: Asymptotic Behavior Of the Jls Solutionmentioning

confidence: 99%

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Statistical behavior of joint least-square estimation in the phase diversity context

Idier

Mugnier

Blanc

2005

IEEE Trans. on Image Process.

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Abstract-The images recorded by optical telescopes are often degraded by aberrations that induce phase variations in the pupil plane. Several wavefront sensing techniques have been proposed to estimate aberrated phases. One of them is phase diversity, for which the joint least-square approach introduced by Gonsalves et al. is a reference method to estimate phase coefficients from the recorded images. In this paper, we rely on the asymptotic theory of Toeplitz matrices to show that Gonsalves' technique provides a consistent phase estimator as the size of the images grows. No comparable result is yielded by the classical joint maximum likelihood interpretation (e.g., as found in the work by Paxman et al.). Finally, our theoretical analysis is illustrated through simulated problems.

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“…It is demonstrated that the optimal linear preprocessor for classification can substantially outperform the optimal linear preprocessor for estimation especially at low SNRs. Other signal estimation preprocessors that adaptively estimate the clean signals from the noisy signals are often used but will not be considered here [17].…”

Section: On Preprocessing For Mismatched Classification Of Gaussian Smentioning

confidence: 99%

Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification

Gazzah¹,

Regalia²,

Delmas³

2001

IEEE Trans. Inform. Theory

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Abstract-Szegö's theorem states that the asymptotic behavior of the eigenvalues of a Hermitian Toeplitz matrix is linked to the Fourier transform of its entries. This result was later extended to block Toeplitz matrices, i.e., covariance matrices of multivariate stationary processes. The present work gives a new proof of Szegö's theorem applied to block Toeplitz matrices. We focus on a particular class of Toeplitz matrices, those corresponding to covariance matrices of single-input multiple-output (SIMO) channels. They satisfy some factorization properties that lead to a simpler form of Szegö's theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses. This bound is interpreted in the context of blind channel identification using second-order algorithms, and more particularly in the case of band-limited channels.Index Terms-Asymptotic eigenvalue distribution, band-limited channels, blind identification, block Toeplitz matrices, multivariate processes, second-order statistics algorithms.

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Asymptotic eigenvalue distribution of block-Toeplitz matrices

Cited by 21 publications

References 5 publications

Minimax Robust MIMO Radar Waveform Design

Minimax Robust MIMO Radar Waveform Design

Statistical behavior of joint least-square estimation in the phase diversity context

Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification

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