Time correlation statistics of the LMS adaptive algorithm weights

The transient and steady-state weight correlation statistics of both the real and complex LMS adaptive filters are obtained when the inputs are independent samples from real and circularly Gaussian processes respectively. A matrix relationship is derived between the covariance matrix of the weight vector at two different times and the covariance matrix of the weights at one time. These expressions show that the weight fluctuations have the same time constants as the mean behavior of the LMS algorithm itself (i.e., the weights are correlated over the same number of iterations that it takes for the algorithm to converge to the Wiener weights for stationary inputs).