Convergence and fixed-point roundoff error analysis of the RLS algorithm and an application to mobile communications

New expressions are derived for the mean weight misadjustment in the fixed-point recursive least squares (RLS) algorithm. These expressions are general, in that they take into account the correlation in the input. Input correlation is shown to be an important factor affecting the performance of the algorithm, and its interaction with the noise sources in the algorithm is analyzed in detail. It is shown that the additive system noise is amplified by a correlation amplification factor which is defined as a function of the input autocorrelation matrix eigenvalues. Correlation also amplifies the effect of roundoff error due to the desired signal estimate computation. It is also demonstrated that tracking noise and weight update roundoff error behave similarly in that their contributions are proportional to the time constant of the algorithm. The roundoff error due to weight update increases when the forgetting factor approaches 1, and leads to the divergence of the algorithm when the forgetting factor is equal to 1. A new stabilization technique is proposed to eliminate this random walk phenomenon. This method is analyzed for the conventional RLS algorithm showing that the divergence is prevented. Conditions are determined for the termination of updating which is an important problem in the finite precision implementation of adaptive filters. Since additive system noise and tracking noise can be present even if the algorithm is implemented in infinite precision, these results are applicable to the infinite precision RLS problem, and yield new expressions for the weight misadjustment also in this case. The original results of the analyses are checked by simulations and perfect agreement with the theory is observed. An important application of the RLS algorithm in adaptive filtering is equalization of mobile communication channels where fast convergence is a strong requirement. A stable multichannel general order fast RLS algorithm is derived and applied to the equalization of mobile communication channels with decision feedback. It is shown that the algorithm successfully equalizes the channel.