Parsevals Theorem Question

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 ...