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

# Parsevals Theorem Question

Started by ●April 6, 2006

Reply by ●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 ●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 ...