Tracking Analysis of the \(\ell _0\)-LMS Algorithm

One of the main challenges in using adaptive filtering algorithms is efficiently emulating a system subject to noisy disturbances. This can be facilitated in applications where the system response to impulse is sparse, which allows for acceleration of the convergence rate if appropriate strategies are used. As a result, methods that impose norm constraints on the estimates are widely used. However, in the case of non-stationary plants to be identified, there is a gap in terms of theoretical performance guarantees of these algorithms. This paper proposes a novel stochastic model capable of predicting the performance of the \(\ell _0\) ${\ell }_0$ -LMS algorithm in identifying a plant subjected to a first-order Markovian disturbance. Therefore, a tracking analysis is carried out, including both the average performance of the adaptive coefficients and second-order statistics of these coefficients. The theoretical model offers an analytical equation that predicts the asymptotic mean squared deviation in terms of the variance of the Markovian disturbance. Further, for most simulated scenarios, the theoretical model’s error in mean squared deviation remains below 1 dB, even when the learning step varies across several orders of magnitude. It was possible to observe that the theoretical model can accurately predict the steady-state regime for a wide range of learning step values and calculate an optimal value for this parameter. The findings are confirmed through extensive simulations.