Error analysis of signal zeros: A projected companion matrix approach. (English) Zbl 1024.93055
Regarding the signal zeros as eigenvalues of a small \(n\times n\) matrix, obtained by a projection of the companion matrix, a new error bound for the signal zeros, concentrating on bounds that relate the signal zeros to a small eigenvalue problem, is obtained. The theoretical contribution of the paper is confirmed by the numerical simulation presented. Also we can see that the eigenvalues computed through the projected companion matrix approach are more accurate than the eigenvalues computed through the polynomial approach.
Reviewer: I.Valuşescu (Bucureşti)
MSC:
| 93E10 | Estimation and detection in stochastic control theory |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 93B60 | Eigenvalue problems |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
References:
| [1] | Bazán, F. S.V., Conditioning of rectangular Vandermonde matrices with nodes in the unit disk, SIAM J. Matrix Anal. Appl., 21, 2, 679-693 (2000) · Zbl 0952.15006 |
| [2] | Bazán, F. S.V.; Bezerra, L. H., On zero locations of predictor polynomials, Numer. Linear Algebra Appl., 4, 459-468 (1997) · Zbl 0887.93026 |
| [3] | Bazán, F. S.V.; Toint, Ph. L., Conditioning of Infinite Hankel Matrices of finite rank, Syst. Control Lett., 41, 347-359 (2000) · Zbl 0980.93028 |
| [4] | Bazán, F. S.V.; Toint, Ph. L., Error analysis of signal zeros from a related companion matrix eigenvalue problem, Appl. Math. Lett., 14, 859-866 (2001) · Zbl 1037.65038 |
| [5] | Björk, Å.; Golub, G. H., Numerical methods for computing angles between linear subspaces, Math. Comput., 27, 579-594 (1973) · Zbl 0282.65031 |
| [6] | Edelman, A.; Murakami, H., Polynomial roots from companion matrix eigenvalues, Math. Comput., 64, 210, 763-776 (1995) · Zbl 0833.65041 |
| [7] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0865.65009 |
| [8] | Hansen, P. C., Rank-Deficient and Discrete Ill-Posed Problems-Numerical Aspects of Linear Inversion (1998), SIAM: SIAM Philadelphia, PA |
| [9] | Kumaresan, R., On the zeros of the linear prediction-error filters for deterministic signals, IEEE Trans. Acoust. Speech Signal Process., ASSP, 31, 1, 217-221 (1983) |
| [10] | Kung, S.; Arun, K. S.; Bhaskar Rao, D. V., State-space and singular value decomposition-based approximation methods for the harmonic retrieval problem, Opt. Soc. Amer., 73, December, 1799-1811 (1983) |
| [11] | Li, F.; Vaccaro, R. J., Unified analysis for DOA estimation algorithms in array signal processing, Signal Process., 25, 147-169 (1991) · Zbl 0738.93070 |
| [12] | Makhoul, J., Linear prediction: a tutorial review, Proc. IEEE, 63, 4, 561-580 (1975) |
| [13] | Mayrargue, S.; Jouveau, J. P., A new application of SVD to harmonic retrieval, (Deprettere, E. F., SVD and Signal Processing (1988), North-Holland: North-Holland Amsterdam), 467-472 |
| [14] | Pisarenko, V., The retrieval of harmonics from a covariance function, Geophys. J. Roy. Astron. Soc., 33, 347-366 (1973) · Zbl 0287.62048 |
| [15] | Smith, R. A., The condition numbers of the matrix eigenvalue problem, Numer. Math., 10, 232-240 (1967) · Zbl 0189.47801 |
| [16] | Stewart, G. W., On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev., 19, 4, 634-662 (1977) · Zbl 0379.65021 |
| [17] | Stewart, G. W.; Sun, J.-G., Matrix Perturbation Theory (1990), Academic Press: Academic Press San Diego · Zbl 0706.65013 |
| [18] | Toh, K. C.; Trefethen, L. N., Pseudozeros of polynomials and pseudospectra of companion matrices, Numer. Math., 68, 403-425 (1994) · Zbl 0808.65053 |
| [19] | Van Der Veen, A.; Deprettere, E. F.; Lee Swindlehurst, A., Subspace-based signal analysis using singular value decomposition, Proc. IEEE, 81, 9, 1277-1309 (1993) |
| [20] | Wei, M.; Majda, G., A new theoretical approach for Prony’s method, Linear Algebra Appl., 136, 119-132 (1990) · Zbl 0711.62082 |
| [22] | Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Oxford University Press: Oxford University Press Oxford, UK · Zbl 0258.65037 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
