Re: Loopers-Delight-d Digest V05 #834

[Tom Combs <tncombs@gmail.com>]

Yes, fourier falls apart (even in theory) when applied to non-stationary signals.  I agree with that.

Maybe we're comparing apples and oranges here - I'm claiming that the
Nyquist theorem is a mathematical proof that you can perfectly
reconstruct a stationary, periodic signal given perfect components to
work with.

>>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?

>no,
>for one thing this ignores aliasing.

What are the alias frequencies in this example?  I'm not seeing them.

I don't see how the two sampled signals you described are mathematically equivalent.

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

At 22:24 21/12/05, you wrote:
> > 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)

just draw the waveform,
and put the sample points on
...easy

then you'll see it

sorry
for being unclear, I didn't mean to do the arithmetic
>
>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?

no,
for one thing this ignores aliasing.

secondly, only 22.1k is within the limits imposed by the sampling frequency.

thirdly, ;-) it's getting late here and I'm a bit tired to crunch the numbers,

I will do later, but once you see the samples plotted the maths won't matter

>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.

um, I already dealt with that.
Ideal fourier presumes a periodic signal of known frequency.
So it falls apart in theory, even before the practice

andy