A unifying framework for the analysis of proportionate NLMS algorithms

Summary Despite being a de facto standard in sparse adaptive filtering, the two most important members of the class of proportionate normalised least mean square (PNLMS) algorithms are introduced empirically. Our aim is to provide a unifying framework for the derivation of PNLMS algorithms and their variants with an adaptive step-size. These include algorithms with gradient adaptive learning rates and algorithms with adaptive regularisation parameters. Convergence analysis is provided for the proportionate least mean square (PLMS) algorithm in both the mean and mean square sense and bounds on its parameters are derived. An alternative, more insightful approach to the convergence analysis is also presented and is shown to provide an estimate of the optimal step-size of the PLMS. Incorporating the so obtained step-size into the PLMS gives the standard PNLMS together with a unified framework for introducing other adaptive learning rates. Simulations on benchmark sparse impulse responses support the approach.