Adaptive solution for blind identification/equalization using deterministic maximum likelihood

A deterministic maximum likelihood (DML) approach is presented for the blind channel estimation problem. It is first proposed in a block version, which consists of iterating two steps, each one solving a least-squares problem either in the channel or in the symbols. In the noiseless case and under certain conditions, this algorithm gives the exact channel and the exact symbol vector with a finite number of samples. It is shown that even if the DML method has a single global minimum, the proposed iterative procedure can converge to spurious local minima. This problem can be detected (under some channel diversity conditions) by using a numerical test that is proposed in the paper. Based on these considerations, we extend the maximum likelihood block algorithm (MLBA) to recursive implementations [maximum likelihood recursive algorithm (MLRA)]. The MLRA is able to track variations of the system by the introduction of an exponential forgetting factor in the DML criterion. The link between the adaptive algorithm and a soft decision feedback equalizer (SDFE) is emphasized. Low-complexity versions of the recursive and adaptive algorithm are presented.