Analysis of stochastic gradient tracking of time-varying polynomial Wienersystems

This paper presents analytical and Monte Carlo results for a stochastic gra dient adaptive scheme that tracks a time varying polynomial Wiener system [ i.e., a linear time-invariant (LTI) filter with memory followed by a time-v arying memoryless polynomial nonlinearity], The adaptive scheme consists of two phases: 1) estimation of the LTI memory using the LMS algorithm and 2) tracking the time-varying polynomial-type nonlinearity using a second coup led gradient search for the polynomial coefficients. The time varying polyn omial nonlinearity causes a time-varying scaling; for the optimum Wiener fi lter for Phase 1, These time variations are removed for Phase 2 using a nov el coupling scheme to Phase I. The analysis for Gaussian data includes recu rsions for the mean behavior of the I,RIS algorithm for estimating and trac king the optimum Wiener filter for Phase 1 for several different time-varyi ng polynomial nonlinearities and recursions for the mean behavior of the st ochastic gradient algorithm for Phase 2, The polynomial coefficients are sh own to be accurately tracked, Monte Carlo simulations confirm the theoretic al predictions and support the underlying statistical assumptions.