Sparse least mean fourth filter with zero-attracting ℓ₁-norm constraint

Traditional stable adaptive filter was used normalized least-mean square (NLMS) algorithm. However, identification performance of the traditional filter was especially vulnerable to degradation in low signal-noise-ratio (SRN) regime. Recently, adaptive filter using normalized least-mean fourth (NLMF) is attracting attention in adaptive system identifications (ASI) due to its high identification performance and stability. In the case of sparse system, however, the NLMF filter cannot identify effectively due to the fact that its algorithm neglects the inherent sparse structure. In this paper, we proposed a sparse NLMF filter using zero-attracting ℓ₁-norm constraint to exploit the sparsity and to improve the identification performance. Effectiveness of the proposed filter is confirmed from two aspects: 1) stability is derived equivalent to well-known stable NLMS filter; 2) identification performance of the proposed is verified by mean square deviation (MSD) standard in computer simulations. When comparing with conventional adaptive filter, the proposed one can achieve much better identification performance especially in low SNR regime.