Nonlinear systems and exponential eigenfunctions

摘要

It has been shown previously that homogeneous time-invariantnsystems produce exponential outputs in response to similar exponentialninputs, but that the concepts of “impulse response” andn“frequency response” are of little use for their analysis.nIt was also asked whether there exist more general classes of systemsnwith exponential eigenfunctions. In this paper, we recall that thenconcepts of impulse and frequency response can be useless even forncertain linear, time-invariant systems. We discuss the role ofntime-invariance, commuting linear systems, and conditions under whichnthey have common eigenfunctions. Then we exhibit a class of nonlinear,nnonhomogeneous, time-varying systems that still have exponentialneigenfunctions. This class contains homogeneous time-invariant systems,nfinite impulse response (FIR) filters and generalized feedforwardnfilters as special cases, and it shows that the exponentialneigenfunction property does not imply linearity, homogeneity, orntime-invariance