# A signal on a noisy channel is input to a filter

1. Dec 9, 2015

### JohanL

This is not homework. Im doing old exams so i have the full solution but need help understanding it.

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

A WSS random signal {X(t)}t∈R with PSD S_X(ω) is transmitted on a noisy channel where it is disturbed by an additive zero-mean WSS random noise {N(t)}t∈R that is independent of the signal X and has PSD S_N (ω).

The recived signal Y (t) = X(t)+N(t) is input to a linear system (/filter) with output signal Z(t) that has frequency response

$H(ω) = S_X (ω)/(S_X (ω) + S_N (ω)).$

Express the mean-square deviation

$E((Z(t)− X(t))^2)$

in terms of

$S_X , S_N$.

3. The solution

Writing h for the impulse response of the filter and ⋆ for convolution the fact that h ⋆ N is zero-mean and independent of X (as N is) readily gives that

$E((Z(t) − X(t))^2) = E(((h ⋆X)(t) + (h ⋆N)(t) − X(t))^2) =$

$(i) = E((((h−δ)⋆X)(t) + (h ⋆N)(t))^2) =$

$= E(((h−δ)⋆X)(t)^2 + (h⋆N)(t)^2) =$

$(ii) = 1/2π \int^{+\infty}_{-\infty} [|H(ω)−1|^2 S_X(ω) + |H(ω)|^2 S_N(ω)] dω =$

$. . . = 1/2π \int^{+\infty}_{-\infty} S_X(ω)S_N(ω)/(S_X (ω) +S_N(ω)) dω$

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I dont understand how you get to lines (i) and (ii) and i cant find any definitions that explains those steps.

Last edited: Dec 9, 2015
2. Dec 9, 2015

### Ray Vickson

What are WSS and PSD? I can guess the first one, but why should I need to? I cannot even guess about the second one.

3. Dec 9, 2015

### JohanL

Sorry, WSS = Wide sense stationary and PSD = Power spectral density