Abstract
Some time invariant non-linear filters of Zadeh's class ? are optimized. A method is proposed for the physical realization of these filters as multipath structures, which consist of, in general, an orthonormal set of non-linear zero memory polynomial functions, each of which is followed by a linear memory network in cascade. The use of ortho-normal polynomial functions instead of, for instance, power law functions simplifies the optimization. An almost routine optimization procedure is proposed and found to work in most practical cases. The theory is extended where necessary to enable its application to the following problems: Noise filtering; prediction; systems analysis; non-linear compensation of a feedback control system. Examples are given. An expansion of second probability density functions in terms of the same orthonormal polynomials which are chosen for the non-linear filters is discussed. Methods of obtaining the form of the polynomials and the expansion coefficients both analytically and experimentally are proposed. The information, in the form required for the optimization procedure, is shown to consist of certain of the expansion coefficients of both the second probability density of the input and the joint probability density of the input and desired output. Some special classes of random processes are defined and their properties are derived. A theorem is proved, showing that an optimum filter of class eta becomes linear when the input and desired output processes belong to a broad class, which includes Gaussian and many other special processes. Examples of processes of each classification are given and used in the optimization problems; expansion coefficients are worked out for these cases.