A new approach to introducing a forgetting factor into the normalized least mean squares algorithm

• A new approach was developed for introducing the forgetting factor into the theoretical framework of the NLMS algorithm.
• The resultant forgetting factor NLMS (FFNLMS) algorithm can be performed at the expense of only 3N multiplications per sample.
• The FFNLMS algorithm is H∞-optimalH∞-optimal.

The performances of adaptive filtering algorithms are critically controlled by specific tunable parameters. The convergence rate of the normalized least mean squares (NLMS) algorithm may be accelerated by adjusting the step size parameter. The tracking speed of the recursive least squares (RLS) algorithm may be improved by using the forgetting factor, which has not yet been appropriately introduced into the NLMS algorithm. This work aims to successfully introduce the forgetting factor into the NLMS algorithm using an H∞H∞ theoretical framework developed to create a unified view of adaptive algorithms for recursively identifying the finite impulse response (FIR) filter coefficients. The performances of the forgetting factor NLMS (FFNLMS) algorithm developed here, in the context of several adaptive filtering applications, are evaluated using computer simulations.