Frequency domain computations for nonlinear steady-state solutions

Steady-state analysis and Fourier analysis play a major role in linear signal processing. In response to a bounded input, a steady-state solution exists if all the poles of the discrete-time linear system are inside the unit circle. Despite the fact that there is no principle of superposition for nonlinear systems, under appropriate sufficient conditions (including all poles inside the unit circle for the linear part of the nonlinear system), there is a bounded solution for all time in response to a bounded input for all time for a time-varying nonlinear difference equation. All solutions that start sufficiently close to this unique solution converge to it as time goes to infinity. This steady-state solution can be computed by applying Fourier and inverse Fourier transforms to each step in a Picard process. In this paper, we develop an algorithm to compute (approximate) steady-state solutions for discrete-time, nonlinear difference equations by employing fast Fourier transforms and inverse fast Fourier transforms at each step of the iterative process. Simulations are provided to illustrate our algorithm.