Steady-state analysis of the long LMS adaptive filter

For the well-known LMS adaptive algorithm no general analytic solutions are available for the steady-state weight-error statistics under stationary stochastic excitation. Only approximate tools have been developed using certain assumptions (like the “independence assumption”) and producing more or less reliable results in practical situations. It is only for the case of a vanishingly small stepsize that such assumptions are not required. There a closed-form solution can be determined for any colouring of the input signal and the additive noise.

Here another particular problem is analyzed: the long filter, i.e. a tapped-delay line structure with a large number of taps. For the limiting case of an infinitely long filter, exact closed-form solutions are derived for the steady-state weight-error correlations and the associated “misadjustment”, again valid for any colouring of the input signal and the additive noise, and now also for any stepsize guaranteeing stability.

The analysis is based upon a feedback approach, with a forward branch generating the above-mentioned solution for vanishing stepsize and a peculiar feedback branch responsible for higher-order corrections. As in any feedback structure, instability can occur beyond a critical value of the feedback parameter. In our case an experimentally supported maximum stepsize is found, beyond which spontaneous oscillations might occur.