Robustness of the Filtered-X LMS Algorithm— Part I: Necessary Conditions for Convergence and the Asymptotic Pseudospectrum of Toeplitz Matrices

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 [1999] is only necessary to prevent an initial increase of the error in the adaptive filter coefficients (critical behavior).