The complex LMS adaptive algorithm--Transient weight mean and covariance with applications to the ALE

The transient and steady-state mean and covariance of the complex-valued LMS adaptive element weights are investigated when the inputs are samples from circularly normal processes. It is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix. This result permits describing the complete transient behavior of the adaptive line enhancer (ALE) weight covariance matrix in closed form for the case of multiple, equal power, narrow-band, statistically independent, orthogonal, Rayleigh fading sinusoids in broad-band noise. Based on these transient covariance results, it is shown that for any stage of adaptation, there exists an optimum ALE gain constant that minimizes the excess mean squared prediction error. This result is of particular significance when processing time-limited data.