Optimum nonlinear filters for quantized inputs

Optimum least-square filters belonging to Zadeh\´s nonlinear class are considered. Attention is restricted to those systems whose present output is influenced only by a portion of the past input. The input signal consists of a message and noise, both of which are stationary random processes. It is assumed that the amplitude of the input time series is bounded and takes on discrete values at all times. This assumption leads to a nonlinear filter which can be realized as a quantizer or amplitude selector followed by a parallel set of linear filters. The system becomes optimum when the impulse responses of the linear filters satisfy a system of integral equations of the Wiener-Hopf type adapted to finite memory filters. By virtue of the assumptions made concerning the joint probabilitY density functions of the message and noise processes, it is found that the Fourier transforms of the kernels of these equations are rational functions. A method is developed for the solution of this set of integral equations. This method is illustrated by an example, and the mean-square error of the nonlinear filter so obtained is compared with the best linear filter.