Linear and time invariant systems, with random signals at the input

Figure 9 Linear and Time Invariant system with random input

There are very useful relations between the moments of the input and output of the LTI (linear and time invariant) system. The cross-correlation function of the input and output processes can be related to the autocorrelation function of the input, and the impulse response of the LTI system.

Here, only the relations for discrete time systems and signals are presented. There are similar relations for continuous time processes.

ηY = nXH(0) = ηX n=-hn
RXYm = RXXm*h-m SXY(ω) = SXX(ω)H*(ω)
RYXm = RXXm*hm SYX(ω) = SXX(ω)H(ω)
RYYm = RXYm*hm SYY(ω) = SXY(ω)H(ω)
RYYm = RYXm*h-m SYY(ω) = SYX(ω)H*(ω)
RYYm = RXXm*hm *h-mSYY(ω) = SXX(ω)H(ω)2
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