A frequency-independent and parallel algorithm for computing the zeros of strictly proper rational transfer functions

We develop an algorithm for computation of the zeros of a strictly proper rational transfer function in partial fraction form, by transforming the problem of finding the roots of the determinant of a frequency-dependent matrix into one of finding the eigenvalues of a companion matrix comprised of th...

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Veröffentlicht in:Applied mathematics and computation Jg. 274; S. 229 - 236
1. Verfasser: Zadehgol, Ata
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.02.2016
Schlagworte:
Linear time invariant system
Partial fraction
Pole/residue
Pole/zero
Rational function
Signal integrity
Pole/residue
Pole/zero
Linear time invariant system
Rational function
Partial fraction
Signal integrity
ISSN:0096-3003, 1873-5649
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Zusammenfassung:We develop an algorithm for computation of the zeros of a strictly proper rational transfer function in partial fraction form, by transforming the problem of finding the roots of the determinant of a frequency-dependent matrix into one of finding the eigenvalues of a companion matrix comprised of the determinants of a binomial-based set of frequency-independent matrices. The proposed algorithm offers a fundamentally new approach that avoids solving severely ill-conditioned system of linear equations, where condition numbers increase rapidly with frequency. The developed algorithm is straightforward, and enables parallel computation of the characteristic polynomial coefficients an′ that comprise the companion matrix to the characteristic polynomial ∑nansn of the frequency-dependent matrix. Additionally, the algorithm allows for relatively inexpensive computation of asymptotically accurate approximants of the transfer function, such that an′ need be computed only for selected powers of s=jω, where the number of required determinant operations are shown to be relatively small. Additionally, limitations of the developed algorithm are highlighted, where the computational cost is shown to be on the order O(2Np) determinant operations on matrices of dimensions (Np+1)×(Np+1), and Np is the number of poles. Illustrative numerical examples are selected and discussed to provide further insight about pros/cons of the proposed method, and to identify potential areas for further research.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2015.11.022