Analytical probability density functions for LMS adaptive filters using the Fokker-Planck equation
Analytical expressions for the theoretical probability density function (pdf) of the LMS (least mean square) adaptive filter weights are obtained for the steady state. The LMS update is formulated as a stochastic differential equation and the weight PDF is derived as the solution to the Fokker-Planck equation. The application of systems identification is examined, and closed-form solutions are obtained for the steady-state PDF for the LMS weights. Only the steady-state PDFs are found for a system identification problem, although the theory developed can also be applied to finding the time-varying PDFs during the adaptation phase.<<ETX>>