A Generalized Correlation Theorem for LTI Systems

In this paper we derive a generalized correlation theorem for linear time-invariant (LTI) system with wide-sense stationary (WSS) input. This theorem shows that if z(n) and x(n) are jointly WSS random signals, then z(n) and y(n), the output of x(n) passing an LIT system h(n), are also jointly WSS. In addition, the cross-correlation function (CCF) between y(n) and z(n) can be computed by the convolution of h(n) and the CCF between x(n) and z(n). Many results of the correlations between the input and output of an LTI system are special cases of this theorem. Moreover, the correlations between the inputs and outputs of multiple LTI systems can be analysed by this theorem.