Crosscorrelation between linearly and nonlinearly distorted versions of a given signal

Abstract

A given deterministic signal x(.) is distorted by passing it through a linear time-invariant filter and also by subjecting it to the action of an instantaneous nonlinearity. The resulting time crosscorrelation of the two distorted versions of the original signal is expressed by the function R 2(s)≜∫ −∞ ∞≜∫ −∞ ∞g[x(t)]k(t−t′)x(t−s)dt dt′ , where x(.) is the given signal, k(.) is the nonnegative definite impulse response of the linear filter, and g(.) is the output-input characteristic of the zero-memory nonlinear device. The problem considered is that of determining conditions on the pair ( x, g) such that R 2( s) ⩽ R 2(0) for all s and any choice of nonnegative definite filter function k; the principal result is the formulation of a necessary and sufficient condition for R 2 to have a global maximum at the origin. In particular, the peak value occurs at the origin if and only if G x ∗ (ω)X(ω) is real and nonnegative for all ω ⩾ 0, where G x (.) and X(.) are the Fourier transforms of g[ x(.)] and x(.), respectively. An equivalent condition is that the correlation function R 2(s)≜∫ −∞ ∞g[x(t)]x(t−s)dt , previously studied by Richardson, be nonnegative definite. Several examples are given, and it is shown that, unlike the case for R 1(.), monotonicity of g(.) is not a sufficient condition for R 2(.) to have a global maximum at s = 0 independently of the choice of filter characteristic k. Certain extensions of these results are given for the case when x(.) is a Gaussian random input.