Stochastic convergence properties of the adaptive gradient lattice

A stochastic fixed-point theorem is used as a basis for the study of stochastic convergence properties (in mean-squares sense) of the adaptive gradient lattice filter. Such properties include conditions on the stepsize in the adaptive algorithm and analytic expressions for the misadjustment and convergence rate. Our results indicate that the limits on the stepsize are stricter than the ones obtained by considering convergence of the mean of the reflection coefficients and, therefore, only a slower convergence of the mean-square error can be obtained. It is shown that faster convergence is achieved for highly uncorrelated sequences than for almost deterministic sequences. The misadjustment is shown to be exponentially dependent on the number of stages in the lattice and is higher for uncorrelated sequences than for almost deterministic sequences.