A Zero Attracting Proportionate Normalized Least Mean Square Algorithm

The proportionate normalized least mean square (PNLMS) algorithm, a popular tool for sparse system identification, achieves fast initial convergence by assigning independent step sizes to the different taps, each being proportional to the magnitude of the respective tap weight. However, once the active (i.e., non-zero) taps converge, the speed of convergence slows down as the effective step sizes for the inactive (i.e., zero or near zero) taps become progressively less. In this paper, we try to improve upon both the convergence speed and the steady state excess mean square error (EMSE) of the PNLMS algorithm, by introducing a l(1) norm (of the coefficients) penalty in the cost function which introduces a so-called zero-attractor term in the PNLMS weight update recursion. The zero attractor induces further shrinkage of the coefficients, especially of those which correspond to the inactive taps and thus arrests the slowing down of the convergence of the PNLMS algorithm, apart from bringing down the steady state EMSE. We have also modified the cost function further generating a reweighted zero attractor which helps in confining the "Zero Attraction" to the inactive taps only.