Improving Steady-State Performance of the UT-ZA-PNLMS Algorithm for Sparse Systems

For identifying sparse systems, a recently proposed algorithm called upper threshold based zero attracting proportionate normalized least mean square (UT-ZA-PNLMS) algorithm has shown improved performance in terms of both the convergence rate and steady-state error in comparison to the ZAPNLMS algorithm. The UT-ZA-PNLMS algorithm employs adaptive threshold based gain function in order to improve convergence rate of the active taps, especially the taps with low magnitude, and appends zero attracting term in the update equation in order to bring the inactive taps to their optimum zero level. However, as the UT-ZA-PNLMS algorithm uses uniform shrinkage for that zero attraction, the active taps experience significant bias which limits overall steady-state performance. In this paper, we introduce selective shrinkage for the zero attracting term so that the inactive taps get strong attractive force whereas the active taps would experience negligibly small attractive force, and thus the bias in the active tap is reduced. In particular, we propose three different algorithms incorporating log-sum, $\ell_{p^{-}}$ norm and $\ell_{0}$-norm penalties to the cost function of the upper threshold based PNLMS algorithm. The resulting algorithms are studied extensively and the simulation results show their improved steady-state performances.