Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!swrinde!zaphod.mps.ohio-state.edu!sdd.hp.com!hp-pcd!hplsla!bryanh
From: bryanh@hplsla.HP.COM (Bryan Hoog)
Newsgroups: comp.dsp
Subject: Re: Nyquist Rate
Message-ID: <9360019@hplsla.HP.COM>
Date: 6 Mar 91 05:15:40 GMT
References: <1991Mar2.230829.21497@eedsp.gatech.edu>
Organization: HP Lake Stevens, WA
Lines: 34

>
>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? 
>

   Since this is a theoretical discussion, I'll go out on a limb and 
 claim that although a sine wave with frequency Fs/2 could not be
 sampled without ambiguity, a complex exponential [e^j(.5Fst+@)] could
 be, for any phase value @.  This is only possible, though, if there
 is no component at [e^-j(.5Fst+@)].

   In other words, you could accurately sample a "tone" at .5 Hz
 with a 1 Hz A/D converter, provided there was no "tone" at -.5 Hz.
 Of course, you'd need a complex (Re+jIm) A/D converter, since the
 signal to be sampled is not purely real anymore.

   As I recall the "nyquist sampling theorem", the sampling rate 
 dictates the maximum bandwidth that can be sampled unambiguously
 with a given sampling rate, and this bandwidth interval is closed
 on one end and open on the other, (-.5,.5] in the above example.

   It is interesting to note that the nyquist bandwidth in this
 example violates the 2X criteria, with almost a 1 Hz bandwidth
 for a 1 Hz sample rate.

   It's been a long time since my college DSP class, so feel free
 to correct me if I got something wrong!

   Bryan Hoog