  In this paper, we propose two novel p-norm penalty least mean square (Lp-LMS)
algorithms as supplements of the conventional Lp-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 (LpGC-LMS). We explain that the LpGC-LMS can achieve
lower mean square error than the standard Lp-LMS algorithm theoretically and
experimentally. To further improve the performance of the filter, the LpNGC-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 Lp-LMS algorithm in terms of
convergence rate and steady-state behavior in sparse system identification
settings.
