Sampling question

robert bristow-johnson wrote:

> On Sep 7, 10:28 pm, kronec...@yahoo.co.uk wrote:

>>The sampling theorem tells us that we must sample at least twice the
>>bandwidth of a signal.

> actually, i think it's *more* than twice.  we cannot, from one number,
> represent *both* the phase and the magnitude of the sinusoidal
> component at the Nyquist frequency.

True, but there is usually zero probability of that case.

Or, following another recent post, the Fourier transform
of the two are the same, assuming that the transform is
finite.  (That the spectrum is continuous.)

Only a delta function at Fs/2 will cause problems.

-- glen

On Sep 8, 4:47 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> robert bristow-johnson wrote:
> > On Sep 7, 10:28 pm, kronec...@yahoo.co.uk wrote:
> >>The sampling theorem tells us that we must sample at least twice the
> >>bandwidth of a signal.
> > actually, i think it's *more* than twice.  we cannot, from one number,
> > represent *both* the phase and the magnitude of the sinusoidal
> > component at the Nyquist frequency.
>
> True, but there is usually zero probability of that case.

well, your spinner (which, if it lands utterly precisely at an angle
of -180 degrees gives you a simple nyquist frequency input) doesn't
decide what the input is.  the devil gets to decide.  if you have a
continuous (but discrete-time) stream of ...-A, +A, -A, +A, ..., then
you know there is a non-zero sinusoidal component at nyquist, but you
don't know both the phase and magnitude of that component except that
the magnitude is at least A.

>
> Or, following another recent post, the Fourier transform
> of the two are the same, assuming that the transform is
> finite.  (That the spectrum is continuous.)
>
> Only a delta function at Fs/2 will cause problems.

which, in the time domain, adds a component that looks like ...-A, +A,
-A, +A... to the signal.  it is possible (with vanishing likelihood),
in a finite input sequence (DFT) or infinite input sequence (DTFT), to
have that alternating-sign component added to dota that would actually
cause a form of aliasing by the delta function at -Fs/2 being aliased
with the delta at +Fs/2.  they'll add and you'll lose the phase
information.

r b-j