Sparse Least Logarithmic Absolute Difference Algorithm with Correntropy-Induced Metric Penalty

Abstract

Sparse adaptive filtering algorithms are utilized to exploit system sparsity as well as to mitigate interferences in many applications such as channel estimation and system identification. In order to improve the robustness of the sparse adaptive filtering, a novel adaptive filter is developed in this work by incorporating a correntropy-induced metric (CIM) constraint into the least logarithmic absolute difference (LLAD) algorithm. The CIM as an $l_{0}$-norm approximation exerts a zero attraction, and hence, the LLAD algorithm performs well with robustness against impulsive noises. Numerical simulation results show that the proposed algorithm may achieve much better performance than other robust and sparse adaptive filtering algorithms such as the least mean p-power algorithm with $l_{1}$-norm or reweighted $l_{1}$-norm constraints.

Appendix

In this appendix, we give the evaluations of the computational complexity of (6). First, we rewrite the algorithm as

$$\begin{aligned} W(n+1)= & {} W(n)+\frac{\mu \tau e(n)}{1+\tau |e(n)|}X(n)-\frac{\rho }{M\sigma ^{3}\sqrt{2\pi }}W(n).^{*}\exp \left( {-\frac{W(n).^{*} W(n)}{2\sigma ^{2}}} \right) \nonumber \\= & {} W(n)+A(n)+B(n), \end{aligned}$$

(14)

where $A(n)=\frac{\mu \tau e(n)}{1+\tau |e(n)|}X(n)$, $B(n)=-\frac{\rho }{M\sigma ^{3}\sqrt{2\pi }}W(n).^{*}\exp \left( {-\frac{W(n).^{*} W(n)}{2\sigma ^{2}}} \right)$.

The detailed evaluations of the computational complexity per iteration are then shown in Table 2.

Table 2 Evaluations of the computational complexity