A Novel Stochastic Stability Model for the Coefficient Reusing LMS Algorithm

Adaptive filtering algorithms operate recursively on the input data, making the resulting estimation a learning process with highly sophisticated properties. Being closed-loop estimators, such algorithms may experience instability. Therefore, the development of models that provide clear guidelines on the admissible region for adjustable parameters, in order to guarantee the stability of the algorithm, is of extreme practical importance. In this paper, a simplified model for the input vector is adopted that separates its radial distribution from its angular distribution to allow for a new stability analysis of the least-mean-squares algorithm with the coefficient reuse method. This method is responsible for mitigating the algorithm’s sensitivity to asymptotic performance in settings characterized by a low signal-to-noise ratio. The stochastic model is simplified using heuristics to provide insight to the designer. The main conclusion of the advanced stochastic model is that the upper limit of the learning step of the algorithm under analysis capable of avoiding divergence is higher than the respective limit of the traditional least squares algorithm. Theoretical results are supported by computational experiments. Regarding the steady-state mean squared error, in our simulations, the difference between the simulated and theoretical values was less than 0.04 dB.