Klarinet Archive - Posting 000672.txt from 1997/11

From: "Jerry Korten" <jerryk@-----.com>
Subj: Re: Nyquist and Analog
Date: Wed, 19 Nov 1997 18:34:51 -0500

>Here is the disputed definition of Nyquist's theorom
>Nyquist's theorem: A theorem, developed by H. Nyquist, which states that
>an analog signal waveform may be uniquely reconstructed, without error,
>from samples taken at equal time intervals. The sampling rate must be
>equal to, or greater than, twice the highest frequency component in the
>analog signal. Synonym sampling theorem.
>Well, I'm no expert but I can immediately see a case where this is not
>going to be true. Take as an example a 20 KHz sine wave sampled at 40
>KHz. According to the above definition the samples will contain enough
>information to perfectly reconstruct the 20 KHz sine wave.
>Surely you are going to get different results depending on the relative
>phases of the sampling and the sine wave itself. In one extreme case
>you could get a set of samples that are all 0. This occurs because the
>wave crosses the 0 point every 1/2 it's period. From that you aren't
>going to be able to reconstruct very much at all! At the other extreme
>you can get a series of +n, -n, +n, -n ... where n is the amplitude of
>the signal.
>Jerry, Jonathon - am I missing something obvious?
>Ian Dilley
You are absolutely correct, mostly. The Nyquist criterion says > 2X the
highest frequency therefore there will always be two sample points per
waveform period. They can not all lie on zero. One will slightly be above
and the other slightly below providing a low amplitude triangle waveform.

And Mark the citation, no matter where it comes from is in error. Take it
to a DSP engineer in your company and ask them. I have also disproved
mathematical formulas given in biomedical engineering text books as they
too can sometimes be printed in error.

The key element is the assumption of interpolation which is assumed when
trying to do reconsruction. The fact remains that a sine wave sampled at
slightly higher than twice it's frequency looks like a triangle wave.

It is not to hard to grasp the true meaning of a waveform sampled at the
Nyquist rate, Mr. Dilley seems to get it, why is it so hard for anybody
else?

Jerry Korten
NYC