Products of independent wss stochastic processes Let {X(t)} t ? R and {Y (t)} t ? R be two centered.

Products of independent wss stochastic processes

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Products of independent wss stochastic processes Let {X(t)} t ? R and {Y (t)} t ? R be two centered.
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Let {X(t)}tR
and {Y (t)}tR
be two centered wss stochastic processes of respective covariance functions CX
(τ ) and CY (τ ).

1. Assume the two signals to be independent. Show that Z (t)
:= X(t)Y (t) (t ∈
R) is a wss stochastic process. Give its mean and covariance function.

2. Assume the same hypothesis as in the previous question,
but now {X(t)}t∈R
is the harmonic process of Example 5.1.20. Suppose that {Y (t)}tR
admits a power spectral density fY (ν). Give the power spectral
density fZ (ν) of {Z (t)}t∈R .