Tracking analysis results for NLMS and APA

We present the tracking properties of the Normalized LMS and affine projection class of algorithms for a randomly time-varying system under certain simplifying assumptions on the data. An expression is given for the steady-state mean-squared error. The dependence of the steady-state error and of the tracking properties on three user-selectable parameters, namely step size, number of vectors used for adaptation, and delay used to choose input vectors used for adaptation, is discussed. While the lag error depends on all of the above parameters, the fluctuation error depends only on step size. Increasing delay results in a linear increase in the lag error and hence the total steady-state mean-squared error. There is an optimum choice for stepsize and number of input vectors that minimizes the total mean-squared error. Simulation results are provided to corroborate the theoretical conclusions.