Sparse Least Mean Square Algorithm for Estimation of Truncated Volterra Kernels

The Volterra series model, though a popular tool in modeling many practical nonlinear systems, suffers from the problem of over-parameterization, as too many coefficients need to be identified, requiring very long data records. On the other hand, often it is observed that of all the model coefficients, only a few are prominent while the others are relatively insignificant. The sparsity inherent in such systems is, however, not exploited by standard estimators which are based on minimization of some L2 norm like mean square error or sum of error squared. This paper draws inspiration from the domain of compressive sampling and proposes an adaptive algorithm for estimating sparse Volterra Kernels, by embedding a L1 norm penalty on the coefficients into the quadratic least mean squares (LMS) cost function. It is shown that the proposed algorithm can achieve a lower steadystate mean square error than that of a standard LMS based algorithm for identifying the Volterra model. Index terms : Volterra Series, L1 norm, Sparse Systems, LMS adaptation