An estimator for the eigenvalues of the system matrix of a periodic-reference LMS algorithm

The convergence analysis of the Least Mean Square (LMS) algorithm has been conventionally based on stochastic signals and describes thus only the average behavior of the algorithm. It has been shown previously that a periodic-reference LMS system can be regarded as a linear time-periodic system whose stability can be determined from the monodromy matrix. Generally, the monodromy matrix can only be solved numerically and does not thus reveal the actual factors behind the dynamics of the system. This paper derives an estimator for the eigenvalues of the monodromy matrix. The estimator is easy to calculate, and it also reveals the underlying reason for the bad convergence of the LMS algorithm in some special cases. The estimator is confirmed by comparing it to the precise eigenvalues of the monodromy matrix. The estimator is found to be accurate for the eigenvalues close to unity.