MMSE precoder design for diversity systems with temporal correlation

This paper describes a numerical procedure for the design of precoders for diversity systems, where temporal correlation in the channel is present, and can be well modelled by a multichannel autoregressive process. The proposed procedure yields the class of diversity codes which minimise the greatest lower bound on the achievable channel estimation error (in the mean square sense) for any receiver. The paper describes the use matrix differential calculus to derive the Jacobian (first derivative) of the trace of the optimal (MMSE) estimation error covariance matrix with respect to the channel precoder matrix. This is used to define a gradient descent algorithm to minimise this error over all unit norm precoders. Simulation results using a previously derived per-survivor processing (PSP) based receiver illustrate the improvement gained over unitary codes. The paper shows by example that the PSP receiver attains the designed lower bound even at relatively high symbol error rates, although this design does not necessarily lead on its own to optimal symbol detection error probability performance.