H∞ optimality and a posteriori output estimate of the forgetting factor NLMS algorithm

The normalized least mean squares (NLMS) algorithm is widely used for adaptive filtering. The NLMS algorithm may be extended using a variety of weight parameters that improve its performance. One such extension involves appropriately introducing a forgetting factor into the NLMS algorithm using the H∞ framework. The resultant forgetting factor NLMS (FFNLMS) algorithm may be regarded as a special case of the improved proportionate NLMS (IPNLMS) algorithm. This work reveals that the FFNLMS algorithm is H∞-optimal, and the a posteriori output estimate is identical to the observation signal for sufficiently large times.