On the convergence of steepest descent and least mean-square algorithms for MIMO systems

We investigate convergence properties of both the steepest descent (SD) and least mean-square (LMS) algorithms applied to a multiple-input-multiple-output system in a Rayleigh fading environment with correlated fading. For a given value of the adaptation step, we evaluate the probability that the algorithms are stable. Then, we compare two stable strategies to choose the adaptation step of SD and analyze their speed of convergence. Our results are valid for an arbitrary number of transmit and receive antennas for the uncorrelated fading, and for an arbitrary number of receive antennas less or equal to the number of transmit antennas for the case of correlated fading.