- #1
JohanL
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This is not homework. I am doing old exams so i have the full solution but need help understanding it.
1. Homework Statement
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 don't understand how you get to lines (i) and (ii) and i can't find any definitions that explains those steps.
1. Homework Statement
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 ##
**************************
I don't understand how you get to lines (i) and (ii) and i can't find any definitions that explains those steps.
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