Normal input to a LTI system

asmani

1. The problem statement, all variables and given/known data

The signal s(t) is a deterministic signal with the finite duration (0,T_s) and the energy E_s=∫s²(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.


2. Relevant 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)|²G_n(f)

3. 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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