  The proportionate normalized least mean square (PNLMS) algorithm and its
variants are by far the most popular adaptive filters that are used to identify
sparse systems. The convergence speed of the PNLMS algorithm, though very high
initially, however, slows down at a later stage, even becoming worse than
sparsity agnostic adaptive filters like the NLMS. In this paper, we address
this problem by introducing a carefully constructed l1 norm (of the
coefficients) penalty in the PNLMS cost function which favors sparsity. This
results in certain zero attractor terms in the PNLMS weight update equation
which help in the shrinkage of the coefficients, especially the inactive taps,
thereby arresting the slowing down of convergence and also producing lesser
steady state excess mean square error (EMSE). We also carry out the convergence
analysis (in mean) of the proposed algorithm.
