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Homework Statement
The signal s(t) is a deterministic signal with the finite duration (0,T_{s}) and the energy E_{s}=∫s^{2}(t)dt. In the following system, n(t) is a normal noise with the mean zero and the power spectral density G_{n}(f)=η/2. Show that n_{o}(T_{s}) has a normal distribution with the mean zero and the variance (η/2)E_{s}.
Homework Equations
E[n_{o}(t)]=H(0)E[n(t)]
R_{nono}(τ)=h(τ)*h(τ)*R_{nn}(τ)
(* is the convolution)
var(n_{o}(t))=R_{nono}(0)=∫G_{no}(f)df
(since E[n_{o}(t)]=0)
G_{no}(f)=H(f)^{2}G_{n}(f)
The Attempt at a Solution
I know how to derive mean and variance , but don't know how to show normality. The prof just mentioned in the class that if the input to a LTI system is normal, then the output is so. How to prove this?
Thanks in advance.
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