A Theory of Nonlinear Systems

The Wiener theory of nonlinear system characterization is described and some of its important concepts are discussed. Following these lines, a theory is developed for the experimental determination of optimum time-invariant nonlinear systems. The systems are optimum in a weighted mean-square sense in which the weighting function is at our disposal. The design of nonlinear systems is regarded as the problem of mapping the function space of the past of the input onto a line that corresponds to the amplitude of the filter output. By choosing a series expansion for this mapping operation that partitions the function space into nonoverlapping cells, an orthogonal representation for nonlinear systems is obtained that leads to convenient apparatus for the determination of optimum systems. General methods are described for applying this theory to the determination of systems that have a performance superior to that of given linear or nonlinear systems. A criterion is established relative to which two systems are defined as "nearly equivalent" and the approximation of nonlinear systems by linear and simple nonlinear systems is discussed. The theory is extended to include the problem of multiple nonlinear prediction and apparatus for the determination of optimum predictors is indicated. IY --·I ----I1 · ____*·^II I C II --_I .--L