FPGA Central

I have a theorom I am trying to prove, that is part of a proof for a
more important theorom. I hope that someone whose math skills exceed
mine can help!


Start with:


1) Given an infinite-energy signal v(t) (which could be either
discrete or continuous-time)

2) Define the average power of this signal in the usual way;

Power = limit as T approaches infinity of ;

(1/T)* (Integral from t=0 to t=T of v(t)^2 dt)

Theorom:

If the average power of v(t), as defined above, does not converge to
a constant as T approaches infinity, then the Fourier transform F(w) =
FT(v(t)) cannot have any zeros (that is, values of w where F(w) is
zero).

The intuitive reasoning behind this is that any signal that does not
have constant average power over time must have spectral components
that are either decaying or growing, and these changing spectral
components implies spectral smearing which will tend to fill in any
zeros.

Anyone care to tackle this one?


Regards


Bob Adams

[email protected] (Liz) wrote in message news:<[email protected] om>...
> I have a theorom I am trying to prove, that is part of a proof for a
> more important theorom. I hope that someone whose math skills exceed
> mine can help!
>
>
> Start with:
>
>
> 1) Given an infinite-energy signal v(t) (which could be either
> discrete or continuous-time)
>
> 2) Define the average power of this signal in the usual way;
>
> Power = limit as T approaches infinity of ;
>
> (1/T)* (Integral from t=0 to t=T of v(t)^2 dt)
>
> Theorom:
>
> If the average power of v(t), as defined above, does not converge to
> a constant as T approaches infinity, then the Fourier transform F(w) =
> FT(v(t)) cannot have any zeros (that is, values of w where F(w) is
> zero).
>
> The intuitive reasoning behind this is that any signal that does not
> have constant average power over time must have spectral components
> that are either decaying or growing, and these changing spectral
> components implies spectral smearing which will tend to fill in any
> zeros.
>
> Anyone care to tackle this one?
>
>
> Regards
>
>
> Bob Adams

I need some clarification. If the signal has infinite-energy, what do
you mean by the Fourier transform of it?

In any case, I doubt this proposed theorem is true. Given such a v,
dv/dt will have a zero at d.c. and will be a counterexample if it also
has infinite energy and is regular enough that it makes sense to talk
about the zeroes of its Fourier transform.

Sincerely,
Frederick Umminger

Just some sloppy ideas:

since the average power >= 0 for all t, and the average power doesn't
converge to a constant C (C<>0), by Parseval's theorem and induction,
FFT of v(t) can't be zero for any infinite-small increament of w,
otherwise the average power will converge to a constant, which is a
contradition. Setting T to a speical case of t(note the above case is
for all t), then the conclusion is reasonable.

this is not strict proof though.

[email protected] (Liz) wrote in message news:<[email protected] om>...
> I have a theorom I am trying to prove, that is part of a proof for a
> more important theorom. I hope that someone whose math skills exceed
> mine can help!
>
>
> Start with:
>
>
> 1) Given an infinite-energy signal v(t) (which could be either
> discrete or continuous-time)
>
> 2) Define the average power of this signal in the usual way;
>
> Power = limit as T approaches infinity of ;
>
> (1/T)* (Integral from t=0 to t=T of v(t)^2 dt)
>
> Theorom:
>
> If the average power of v(t), as defined above, does not converge to
> a constant as T approaches infinity, then the Fourier transform F(w) =
> FT(v(t)) cannot have any zeros (that is, values of w where F(w) is
> zero).
>
> The intuitive reasoning behind this is that any signal that does not
> have constant average power over time must have spectral components
> that are either decaying or growing, and these changing spectral
> components implies spectral smearing which will tend to fill in any
> zeros.
>
> Anyone care to tackle this one?
>
>
> Regards
>
>
> Bob Adams