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Parsevals Theorem Question

Started by HelpmaBoab April 6, 2006
The basic indea behind Parsevals theroem is that energy is the same whether
it is measured in the time domain or frequency domain ie suppose y(k) is a
random signal k=0,1,2...

E[y^2(k)] = contour integral around the unit circle { Y(z)Y(z^-1)} dz/z

and if y(k)=W(z).noise(k) then we can write this in terms of transfer
functions and a white noise variance term.

I was wondering if there is an equivalent for higher order moments ie

E[y^4(k)] = ? in the freq (z) domain.

After all E[y^2(k)] is thesecond moment if y is random and y^4(k) is the
fourth moment so there should be a relationship if y(k) is non-guassian.


Tam






HelpmaBoab skrev:
> The basic indea behind Parsevals theroem is that energy is the same whether > it is measured in the time domain or frequency domain ie suppose y(k) is a > random signal k=0,1,2... > > E[y^2(k)] = contour integral around the unit circle { Y(z)Y(z^-1)} dz/z > > and if y(k)=W(z).noise(k) then we can write this in terms of transfer > functions and a white noise variance term. > > I was wondering if there is an equivalent for higher order moments ie > > E[y^4(k)] = ? in the freq (z) domain. > > After all E[y^2(k)] is thesecond moment if y is random and y^4(k) is the > fourth moment so there should be a relationship if y(k) is non-guassian.
There is. Check out "higher order statistics" and "cumulant spectra". The peculiar aspect of these techniques is that they are based on the data being non-Gaussian, but require huge amounts of data to lower the variance of the results. I never figured out how one can average lots of data while avoiding the Central Limit Theorem kicking in, driving the avreage towards a Gaussian distribution. Rune
Rune Allnor wrote:
> HelpmaBoab skrev: > > The basic indea behind Parsevals theroem is that energy is the same whether > > it is measured in the time domain or frequency domain ie suppose y(k) is a > > random signal k=0,1,2... > > > > E[y^2(k)] = contour integral around the unit circle { Y(z)Y(z^-1)} dz/z > > > > and if y(k)=W(z).noise(k) then we can write this in terms of transfer > > functions and a white noise variance term. > > > > I was wondering if there is an equivalent for higher order moments ie > > > > E[y^4(k)] = ? in the freq (z) domain. > > > > After all E[y^2(k)] is thesecond moment if y is random and y^4(k) is the > > fourth moment so there should be a relationship if y(k) is non-guassian. > > There is. Check out "higher order statistics" and "cumulant spectra". > The peculiar aspect of these techniques is that they are based on the > data being non-Gaussian, but require huge amounts of data to lower > the variance of the results. I never figured out how one can average > lots of data while avoiding the Central Limit Theorem kicking in, > driving the avreage towards a Gaussian distribution.
Hopefully they didn't use more than 12 data points ...