Greatest common divisor via generalized Sylvester and Bezout matrices

Bitmead, Robert R.; Kung, Sun‐Yuan; Anderson, Brian D. O.; Kailath, T.

doi:10.1109/tac.1978.1101890

Cited by 173 publications

(85 citation statements)

References 16 publications

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“…Sylvester matrices arise in several areas [2,23,31,32], and the idea is certainly not new. In this case, the approach allows to obtain z(s) by computing the constant null-space of T δ+1 .…”

Section: Lemma 1 Let N F Be the Number Of Finite Zeros (Including Mulmentioning

confidence: 99%

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An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

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In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

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Section: Lemma 1 Let N F Be the Number Of Finite Zeros (Including Mulmentioning

confidence: 99%

“…An algorithm to obtain a full-rank minimal degree polynomial matrix Z(s) such that A(s)Z(s) = 0 was sketched in the above Proposition and its proof, see also Theorem 1 and its proof in [31]. Some remarks on this algorithm are as follows: Remark 1.…”

Section: Proposition 1 the Following Structural Information Of A(s) Imentioning

confidence: 99%

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

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“…The matrix E is a generalized Sylvester matrix with L basic building blocks [13]. By defining P i (z) = ni j=0 e i,j z −j (i = 0, 1, .…”

Section: Construction Ofmentioning

confidence: 99%

Generalized periodic non-uniform sampling of non-bandlimited signals

Meng¹,

Tuqan

2008

2008 Information Theory and Applications Workshop

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A new paradigm for the periodic non uniform sampling of a class of non bandlimited signals was recently proposed. The main idea is to generate the periodic non uniform samples by retaining a select group of samples from a larger set, obtained by oversampling the continuous time signal by an integer factor L. In this paper, we revisit the problem and consider the more challenging case where the sampler operates at L/M time the Nyquist rate with L and M being coprime integers. We derive a new multi-input multi-output (MIMO) discretetime model, and then use this model to obtain a set of necessary conditions for perfect signal reconstruction with FIR filters. We also show that unlike the case of M = 1, the newly derived conditions are not sufficient for FIR perfect recovery.

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“…The H above is a generalized Sylvester matrix that is known to have full column rank equal to N, if the polynomials are coprime, i.e., there is no complex number z that is a common zero for all these polynomials Qiu, Hua, & Abed-Meraim 1997 ;Bitmead et al 1978). Hence, under the previous weak (sufficient) coprimeness condition the inverse in equation (32) exists.…”

Section: Appendix a Sufficient Condition For Equation (32) To Existmentioning

confidence: 99%

Adaptive Filter-bank Approach to Restoration and Spectral Analysis of Gapped Data

Stoica¹,

Larsson²,

Li³

2000

The Astronomical Journal

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The main topic of this paper is the nonparametric estimation of complex (both amplitude and phase) spectra from gapped data, as well as the restoration of such data. The focus is on the extension of the APES (amplitude and phase estimation) approach to data sequences with gaps. APES, which is one of the most successful existing nonparametric approaches to the spectral analysis of full data sequences, uses a bank of narrowband adaptive (both frequency and data dependent) Ðlters to estimate the spectrum. A recent interpretation of this approach showed that the Ðlterbank used by APES and the resulting spectrum minimize a least-squares (LS) Ðtting criterion between the Ðltered sequence and its spectral decomposition. The extended approach, which is called GAPES for somewhat obvious reasons, capitalizes on the aforementioned interpretation : it minimizes the APES-LS Ðtting criterion with respect to the missing data as well. This should be a sensible thing to do whenever the full data sequence is stationary, and hence the missing data have the same spectral content as the available data. We use both simulated and real data examples to show that GAPES estimated spectra and interpolated data sequences have excellent accuracy. We also show the performance gain achieved by GAPES over two of the most commonly used approaches for gapped-data spectral analysis, viz., the periodogram and the parametric CLEAN method.

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Greatest common divisor via generalized Sylvester and Bezout matrices

Cited by 173 publications

References 16 publications

An improved Toeplitz algorithm for polynomial matrix null-space computation

An improved Toeplitz algorithm for polynomial matrix null-space computation

Generalized periodic non-uniform sampling of non-bandlimited signals

Adaptive Filter-bank Approach to Restoration and Spectral Analysis of Gapped Data

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