- #1
mnb96
- 711
- 5
Hi,
I was studying the derivation of the solution of Wiener filter from the http://en.wikipedia.org/wiki/Wiener_filter#Wiener_filter_problem_setup". There is a step I don't quite understand.
First, we define the square error between the estimated signal \hat{s}(t) and the original true signal s(t):
e^2(t) = s^2(t) - 2s(t)\hat{s}(t) + s^2(t)
then the authors calculate the mean value of e^2, that is E[e^2].
At this point I would note that:
E[e^2] = E[s^2] - 2E[s\hat{s}] + E[s^2]
Unfortunately we don't know the probability density function of the original signal s, so we cannot compute E[s^2]. However, the authors of the article seem to suggest that:
E[s^2]=R_s(0)
where R_s(0) is the autocorrelation function of s evaluated at 0 (though I might have misunderstood this).
Could anyone elaborate this point?
I thought that E[s^2]=\int f_{s}(x) x^2 dx, while R_s(0)=\int s(x)s(x)dx
I don't see either, how they solve this problem without any knowledge on the p.d.f. of s.
Thanks.
I was studying the derivation of the solution of Wiener filter from the http://en.wikipedia.org/wiki/Wiener_filter#Wiener_filter_problem_setup". There is a step I don't quite understand.
First, we define the square error between the estimated signal \hat{s}(t) and the original true signal s(t):
e^2(t) = s^2(t) - 2s(t)\hat{s}(t) + s^2(t)
then the authors calculate the mean value of e^2, that is E[e^2].
At this point I would note that:
E[e^2] = E[s^2] - 2E[s\hat{s}] + E[s^2]
Unfortunately we don't know the probability density function of the original signal s, so we cannot compute E[s^2]. However, the authors of the article seem to suggest that:
E[s^2]=R_s(0)
where R_s(0) is the autocorrelation function of s evaluated at 0 (though I might have misunderstood this).
Could anyone elaborate this point?
I thought that E[s^2]=\int f_{s}(x) x^2 dx, while R_s(0)=\int s(x)s(x)dx
I don't see either, how they solve this problem without any knowledge on the p.d.f. of s.
Thanks.
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