Path: utzoo!utgpu!cunews!software.mitel.com!grayt
From: grayt@Software.Mitel.COM (Tom Gray)
Newsgroups: sci.electronics
Subject: Re: A question about the Nyquist theorm
Message-ID: <6648@healey>
Date: 21 Feb 91 14:44:33 GMT
References: <1758@manta.NOSC.MIL> <1759@manta.NOSC.MIL> <2189@umriscc.isc.umr.edu>
Organization: Mitel. Kanata (Ontario). Canada.
Lines: 63

In article <2189@umriscc.isc.umr.edu> robf@mcs213f.cs.umr.edu (Rob Fugina) writes:
>In article <1759@manta.NOSC.MIL> north@manta.NOSC.MIL (Mark H. North) writes:
>>Sorry to answer my own post but I take that last paragraph back. I think
>>you are wrong after all. Look at it this way -- suppose I tell you I'm
>>going to send you one of two signals, either 1 volt 60 Hz or a DC voltage
>>between -1 and 1 volt. You may sample at 120 Hz. You get all identical
>>samples at 0.5 volts. Which signal did I send?
>>Mark
>
>You sent a DC signal of 0.5 volts.  If it were AC, you the samples would
>be alternating positive and negative of the same magnitude.
>
>Rob  robf@cs.umr.edu

This is an accurate statement. The ability to reconstruct signals
at the half-nyquist depends on the sampling method used.
Digital systems use instantaneous (or flat top) sampling and
cannot reconstruct half-nyquist signals. Other systems can
use perfect sampling in which the shape of the sampled wave is
preseved within the finite width sampling period. This type
of sampling can reconstruct half-nyquist signals.

So in general f<= 2B (assuming that you can use perfect
sampling) but for the special case of digital systems
F<2B because of the limitations of the sampling 
method used.
 
I have not seen Nyquist's paper on his theorem so I cannot say for
ceratin what result he derived. However Mischa Schwatz's text
Information Transmission Modulation and Noise gives a derivation
of the Sampling Theorem using finite width pulses. The instantaneous
pulse case is used as a limit.

I was taught the Sampling theorem as the multiplication of
a finite width sampling pulse of unit height with the
sampled signal. The instantaneous pulse case was then
derived as a limiting case. In those days (early 70's)
PAM systems (Pulse Amplitude Modulation) were used
in telephoney. These systems definitely used one
of two systems - Perfect Sampling for smaller
systems and Flat Top Sampling (by a resonanat transfer)
method for larger systems. The distinctions in
sampling method were important since the flat top
method produce another sinc effect in the transfer
function.

Nowadays digital systems prevail and the old
perfect sampling systems are no longer important.
The derivation with instantaneous pulses is suitable
for digital systems since the flat top sampling
is assumed by it. However the math has not changed
and as Mischa Schwartz says f >= 2B is the
Nyquist criterion.

Hope this is coherent.



 


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