Path: utzoo!news-server.csri.toronto.edu!cs.utexas.edu!uunet!mtnmath!paul
From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: comp.dsp
Subject: Re: Nyquist Rate
Message-ID: <75@mtnmath.UUCP>
Date: 5 Mar 91 20:33:26 GMT
References: <1991Mar2.230829.21497@eedsp.gatech.edu> <13687@life.ai.mit.edu>
Organization: Mountain Math Software, P. O. Box 2124, Saratoga, CA 95070
Lines: 35

In article <13687@life.ai.mit.edu>, campbell@churchy.ai.mit.edu (Paul Campbell) writes:
> The Nyquist rate is the rate needed to sample a given bandwidth from zero
> up to that frequency but not including that frequency ...

Correct so far.

> This is not as bad
> as the frequencies outside of the band, which will be aliased.

A frequency that is an exact multiple of the sampling rate gets aliased
as a DC signal. The amplitude of this aliased signal depends on the phase
where sampling occurs.

> 
>  ... Realistically, for any decent
> resolution of the wave, you should sample at four times the maximum, but
> the absolute minimum (the limit, which will obviously not work at all)
> is the Nyquist rate, or twice the maximum frequency.

When you are sampling at slightly more than twice the minimum frequency
you can completely reconstruct the original signal *provided your sampling
sequence is infinitely long*. Of course, it never is. You can
oversample to increase frequency resolution, but you can also accomplish
this by increasing your sample length.

One point that may be confusing to some people is that
an FFT allows you to determine the spectrum from a finite sample
series.  However, it assumes that the infinite series from which this
finite series is taken keeps repeating the same sequence of samples out
to infinity. To the degree this assumption is incorrect, the FFT does
not give the exact spectrum of the infinite sequence. Thus
a longer sampling length or increased sample rate can provide better
resolution in an FFT when this assumption is false. 

Paul Budnik