Modified Champernowne Function Based Robust and Sparsity-Aware Adaptive Filters

A robust adaptive filter is usually unaffected by spurious disturbances at the error sensor. In an endeavour to improve robustness of the adaptive filter, a novel modified Champernowne function (MCF) is proposed as a robust norm and the corresponding robust Champernowne adaptive filter (CMAF) is derived. To improve modelling accuracy and convergence performance for sparse systems along with being robust, a reweighted zero attraction (RZA) norm is incorporated in the cost function along with MCF and the corresponding RZA-CMAF algorithm is proposed. To further improve filter performance, the CMAF- $l_{0}$ algorithm is proposed where the $l_{0}$ -norm is approximated using the multivariate Geman-McClure function (GMF). Bound on learning rate for the proposed algorithms is also derived. Extensive simulation study shows the improved robustness achieved by the CMAF algorithm, especially when impulsive noises are present for a longer duration. On the other hand, RZA-CMAF and CMAF- $l_{0}$ can provide improved convergence performance under sparse and impulsive noise conditions, with CMAF- $l_{0}$ providing the best performance.