Path: utzoo!news-server.csri.toronto.edu!cs.utexas.edu!wuarchive!mit-eddie!media-lab!cas
From: cas@media-lab.MEDIA.MIT.EDU (Scud)
Newsgroups: comp.dsp
Subject: Re: Nyquist Rate
Message-ID: <5410@media-lab.MEDIA.MIT.EDU>
Date: 3 Mar 91 17:36:30 GMT
References: <1991Mar2.230829.21497@eedsp.gatech.edu>
Reply-To: cas@media-lab.media.mit.edu (Scud)
Organization: MIT Media Lab, Cambridge MA
Lines: 22

In article <1991Mar2.230829.21497@eedsp.gatech.edu> dar@euler.eedsp.gatech.edu (Doug Reynolds) writes:
>In a pervious post, someone brought up the theoretical problem that you
>sample a sine wave with frequency f at exactly 2*f, but your first sample
>is perfectly lined up at t=0. Thus it would seem that your sample stream would
>then be all zeros makeing it impossible to reconstruct the original sine wave.
>
>I have tried to reconcile this, but to no ava as of yet. Does anyone else
>have any ideas on this? 

It's just a matter of "greater than" versus "greater than or equal to." 
Nyquist's theorem claims that the sampling frequency should be strictly 
greater than twice the frequency of the highest component of your signal.
Usually this distiction isn't a big deal, but there is indeed aliasing if you
sample at exactly the Nyquist rate. In the sine wave example of the previous
posting, you'll get impulses at +/- f with magnitudes +/- 1/2j. When you 
periodically replicate the sine spectrum, the positive impulses cancel the 
negative impulses and you get a DTFT of zero, as predicted. In the case of a
cosine you get lucky and the impulses add, so the aliasing isn't destructive
and you can reconstruct your signal (although you will be off by a constant
factor).

Casimir Wierzynski