Re: sampling frequency

[Tom Combs <tncombs@gmail.com>]

> sample a 22kHz signal at 44.1kHz

> the result is identical to sampling a 22.05kHz tone
> which is amplitude modulated  at 500Hz

> agree? (if no, then try working it out on paper)
 

Unless my math is wrong, I don't agree.  A 22khz tone sampled at 44.1khz produces a signal containing the fundamental (22k) plus additional components at 22.1k, 66.1k, 66.2k, 110.2k, 110.3k, 154.3k, ad infinitum.  Do you agree?  I may be misinterpreting "the result is identical to sampling a 22.05kHz tone
which is amplitude modulated  at 500Hz".  I'm assuming you're first modulating a 22.05k tone at 500hz, producing the original 22.05k tone plus sidebands at 22.55k and 21.55k.  Sample that signal and you don't get the same result as the 22k tone sampled at 44.1k.

 

Which is wrong? My math, my interpretation, or both?

 

 

The ideal Nyquist sampling theorem is based on the ideal Fourier transform - ideal Fourier transforms are lossless aren't they?  Granted it all falls apart when you try to put it into practice.

 

 

On 12/21/05, a k butler <akbutler@tiscali.co.uk> wrote:

>Although you might be correct for a frequency of f
>when the sampling frequency is 2f, the theorem correctly stated says
>that it will be good for frquencies UP TO f Hz, i.e. not including f.

as its not an integer measurement then
"up to" and "up to and including" mean the same in real terms

>  So while you're correct for one frequency, f, the theorem holds
> 100% true for all frequencies below f and no information is lost.

Ok, let's take it through slowly,

sample a 22kHz signal at 44.1kHz

the result is identical to sampling a 22.05kHz tone
which is amplitude modulated  at 500Hz

agree? (if no, then try working it out on paper)

and there are other signals (with FM and AM) which would produce the
same result
when sampled

so how can we be sure and re-create 22kHz?

We can't, that's a classic case of loss of information.

>  The mathematics bear out.

The mathematics is based on an assumption zero information loss in a
fourier transform,
where the signal is periodically repeating at a known frequency, and
the sampling
frequency is picked to be a whole multiple of that periodic frequency.

...but audio is non-periodic and of unknown frequency.

so the maths doesn't bear out your claim.


i.e. information loss increases as you approach the Nyquist frequency.

andy butler