Gradient compared ℓp-LMS algorithms for sparse system identification

In this paper, we propose two novel p-norm penalty least mean square (ℓ_p-LMS) algorithms as supplements of the conventional ℓ_p-LMS algorithm established for sparse adaptive filtering recently. A gradient comparator is employed to selectively apply the zero attractor of p-norm constraint for only those taps that have the same polarity as that of the gradient of the squared instantaneous error, which leads to the new proposed gradient compared p-norm constraint LMS algorithm (ℓ_pGC-LMS). We explain that the ℓ_pGC-LMS can achieve lower mean square error than the standard ℓ_p-LMS algorithm theoretically and experimentally. To further improve the performance of the filter, the ℓ_pNGC-LMS algorithm is derived using a new gradient comparator which takes the sign-smoothed version of the previous one. The performance of the ℓpNGC-LMS is superior to that of the ℓpGC-LMS in theory and in simulations. Moreover, these two comparators can be easily applied to other norm constraint LMS algorithms to derive some new approaches for sparse adaptive filtering. The numerical simulation results show that the two proposed algorithms achieve better performance than the standard LMS algorithm and ℓ_p-LMS algorithm in terms of convergence rate and steady-state behavior in sparse system identification settings.