Performance of the forgetting factor RLS during the transient phase

The recursive least squares (RLS) algorithm is one of the most well known algorithms used for adaptive filtering and system identification. We consider the convergence properties of the forgetting factor RLS algorithm in a stationary data environment. We study the dependence of the speed of convergence of RLS with respect to the initialization of the input sample covariance matrix and with respect to the observation noise level. By obtaining estimates of the settling time we show that RLS, in a high SNR environment, when initialized with a matrix of small norm, has a very fast convergence. The convergence speed decreases as we increase the norm of the initialization matrix. In a medium SNR environment the optimum convergence speed of the algorithm is reduced, but the RLS becomes more insensitive to initialization. Finally in a low SNR environment it is preferable to start the algorithm with a matrix of large norm.