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Normal input to a LTI system 
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#1
Apr712, 04:54 AM

P: 84

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}=∫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}. 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)^{2}G_{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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