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