kb:wss_processes_lti_systems

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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)=hx(t). Then, the following relations are true:

E[y(t)]=H(j0)μx

Ryxτ=hRxx(τ)

Cyx(τ)=hCxx(τ)

Rxy(τ)=hRxx(τ)

Cxy(τ)=hCxx(τ)

Ryy(τ)=hhRxx(τ)

Ryy(τ)=hhCxx(τ)

  • y(t) is also wide-sense stationary.
  • y(t) is jointly wide-sense stationary with its input.

Given y=hx and z=gw:

Ryz(τ)=hgRxw(τ)

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)

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  • Last modified: 2024-04-30 04:03
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