How Does Normality Arise in the Output of an LTI System with Normal Noise Input?

[asmani]

Homework Statement

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.

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)|²G_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.

 

[KataKoniK]

Dear poster,

Thank you for your question. To show that no(Ts) has a normal distribution with the mean zero and the variance (η/2)Es, we can use the central limit theorem. This theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the distribution of the individual variables.

In this case, we can consider the signal s(t) to be composed of an infinite number of small segments, each with duration Ts. Each of these segments can be seen as a random variable with a normal distribution, since it is a deterministic signal with finite duration. Therefore, by the central limit theorem, the sum of these segments, which is no(Ts), will also have a normal distribution.

Furthermore, since n(t) is a normal noise with mean zero and power spectral density Gn(f)=η/2, we can use the properties of LTI systems to show that the output of the system, no(Ts), will also have a normal distribution with mean zero and variance (η/2)Es. This is because the output of an LTI system is the convolution of the input with the impulse response, which in this case is n(t). Since n(t) is a normal noise, the output will also be normal.

I hope this helps to answer your question. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your studies!