A novel combination scheme of proportionate filter

The conventional convex combination schemes with two proportionate filters have high computational burdens due to the two proportionate filters running at the same time. Meanwhile, they also have poor convergence behavior at the intersection of the large step size filter and the small step size filter. To overcome these drawbacks, this paper proposes a family of combined-step-size (CSS) proportionate filters. The proposed CSS proportionate filters use a variable mixing factor to adaptively combine two different step sizes of one proportionate filter, where the large step size affords a fast convergence rate and the small one offers a small steady-state error. The variable mixing factor is defined as the output of a modified sigmoidal activation function and it updates indirectly by using the stochastic gradient descent method to minimize the L1-norm of the system output error. The proposed CSS proportionate filters have lower computational complexities than the corresponding convex combination proportionate filters because they only require one filter run at every moment. Simulations in three echo channels with different degrees of sparsity have demonstrated the superior convergence performance of this family of CSS proportionate filters.