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From: fisico32 on 20 May 2010 10:41
Hello Forum,

given a signal x(t) with complex spectrum X(w), its power or energy is
given by the integral of x(t)^2.

The same power or energy information is given by the product of
X(w)X*(w)=|X(w)|^2 (it must be a positive quantity, multiplication of a
complex number by its conjugate).

What does the integral of x(t)^3 represent? The kurtosis of the signal?
Is x(t)^3 supposed to be "defined" as positive quantity like energy and
power are? It turns out to be if x(t) is real valued....

Spectrally, how would we get the same information for X(t)^2 using some
particular product of the spectrum X(w) with itself?

The integral of X(w)X(w)X*(w) would not give the same result as the
integral of x(t)^3, right?

In general, when computing inner products, one of the two
vectors(functions) is present in its conjugate form. Ex: Int f(x) g*(x) dx

If we had to calculate a "triple" inner product between 3 functions, would
we need two or just one of the functions to show up in its conjugate form

thanks
fisico32

From: Clay on 20 May 2010 12:15
On May 20, 10:41 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com>
wrote:
> Hello Forum,
>
> given a signal x(t) with complex spectrum X(w), its power or energy is
> given by the integral of x(t)^2.
>
> The same power or energy information is given by the product of
> X(w)X*(w)=|X(w)|^2 (it must be a  positive quantity, multiplication of a
> complex number by its conjugate).
>
> What does the integral of x(t)^3 represent? The kurtosis of the signal?
> Is x(t)^3 supposed to be "defined" as positive quantity like energy and
> power are? It turns out to be if x(t) is real valued....
>
> Spectrally, how would we get the same information for X(t)^2 using some
> particular product of the spectrum X(w) with itself?
>
> The integral of X(w)X(w)X*(w) would not give the same result as the
> integral of x(t)^3, right?
>
> In general, when computing inner products, one of the two
> vectors(functions) is present in its conjugate form. Ex: Int f(x) g*(x) dx
>
> If we had to calculate a "triple" inner product between 3 functions, would
> we need two or just one of the functions to show up in its conjugate form
>
> thanks
> fisico32


I think by doing a little math, you can answer your own question.

First of all realize that most books give the relation between
Int( x(t)^2 ) = scalefactor*Int( |X(w)|^2)

and erroneously call it Parseval's relation(theorem). But actually
this simplified form is due to Bessel. Parseval actually came up with
a much more general relation


Int( g*(t) * h(t) ) = scalefactor * Int( G*(w) * H(w) )


Where the "scalefactor" depends on how you do the Fourier transform -
i.e., where you put the 2pi.

Now looking at Parseval's relation as I gave it above, think about how
you can insert your triple product and apply a product relation for
fourier transforms.

Try working it from here.

Clay





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