Reply by Andor April 6, 20062006-04-06
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 ...
Reply by Rune Allnor April 6, 20062006-04-06
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
Reply by HelpmaBoab April 6, 20062006-04-06
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