Fraanje, R., Verhaegen, M. and Elliott, S.J.
Robustness of the filtered-x LMS algorithm: part 1: necessary conditions for convergence and the asymptotic pseudospectrum of Toeplitz matrices
IEEE Transactions on Signal Processing, 55, (8), . (doi:10.1109/TSP.2007.896083).
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Errors in the secondary path model of the filtered-x LMS (FXLMS) algorithm will lead to its divergence when the eigenvalues of the cross-correlation matrix between the estimated filtered reference and the true filtered reference signals are not all located in the right half plane. This cross-correlation matrix has a (block) Toeplitz structure whose dimension is determined by the number of adaptive filter coefficients. Using results on the asymptotic pseudospectrum of Toeplitz matrices, a frequency-domain condition on the model is derived to ensure stability. The condition is sufficient and necessary for a large number of filter coefficients. A transient analysis shows that the sufficient condition given by Wang and Ren  is only necessary to prevent an initial increase of the error in the adaptive filter coefficients (critical behavior).
|Digital Object Identifier (DOI):
||toeplitz matrices, adaptive filter coefficients, asymptotic, pseudospectrum, block Toeplitz structure, cross-correlation matrix eigenvalues, filtered-x LMS algorithm robustness, frequency-domain condition, secondary path model, transient analysis, true filtered, reference signals
||15 Nov 2007
||16 Apr 2017 18:17
|Further Information:||Google Scholar|
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