A note on the convergence analysis of LMS adaptive filters with Gaussian data

Necessary and sufficient conditions for the convergence of LMS (least-mean-squares) adaptive filters with Gaussian data have been established by L.L. Horowitz and K.D. Senne (1981), with the recent support of A. Feuer and E. Weinstein (1985). A feature of both of these studies is the necessity to investigate bounds on the roots of rather unwieldy characteristic equations. The authors show how such an investigation can be avoided through the use of a theorem of F. R. Gantmacher (1959). In formally applying this theorem, similar results to those of the above studies are obtained in a precise and straightforward manner.>