Fading Memory and the Problem of Approximating Nonlinear Operators with Volterra Series

S. Boyd and L. O. Chua

IEEE Transactions on Circuits and Systems, CAS-32(11):1150-1171, November 1985.

Using the notion of fading memory, very strong versions of two theorems are proved. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented hold for all time and for signals in useful (noncompact) sets. The discrete-time analog of the second theorem asserts that any TI operator with fading memory can be approximated (in the strong sense) by a nonlinear moving-average operator. Some further discussion of the notion of fading memory and practical applications for the material presented are given.