# Normal input to a LTI system

1. Apr 7, 2012

### asmani

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

The signal s(t) is a deterministic signal with the finite duration (0,Ts) and the energy Es=∫s2(t)dt. In the following system, n(t) is a normal noise with the mean zero and the power spectral density Gn(f)=η/2. Show that no(Ts) has a normal distribution with the mean zero and the variance (η/2)Es.

2. Relevant equations

E[no(t)]=H(0)E[n(t)]

Rnono(τ)=h(τ)*h(-τ)*Rnn(τ)
(* is the convolution)

var(no(t))=Rnono(0)=∫Gno(f)df
(since E[no(t)]=0)

Gno(f)=|H(f)|2Gn(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?

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Last edited: Apr 7, 2012