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Wide-sense stationary processes and LTI systems
Let x(˙) be a wide-sense stationary process with:
- Mean μx
- Autocorrelation Rxx(τ)
- Autocovariance Cxx(τ)
- E[x2(t)]<∞
Let y(t)=h∗x(t). Then, the following relations are true:
E[y(t)]=H(j0)μx
Ryxτ=h∗Rxx(τ)
Cyx(τ)=h∗Cxx(τ)
Rxy(τ)=←h∗Rxx(τ)
Cxy(τ)=←h∗Cxx(τ)
Ryy(τ)=h∗←h∗Rxx(τ)
Ryy(τ)=h∗←h∗Cxx(τ)
- y(t) is also wide-sense stationary.
- y(t) is jointly wide-sense stationary with its input.
General form:
Given y=h∗x and z=g∗w:
Ryz(τ)=h∗←g∗Rxw(τ)
Power spectral density
Main article: Power spectral density
CT case:
Rxx(τ)↔Sxx(jω) Cxx(τ)↔Dxx(jω)
DT case:
Rxx[m]↔Sxx(ejΩ) Cxx[m]↔Dxx(ejΩ)