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From: jbuck@galileo.berkeley.edu (Joe Buck)
Newsgroups: comp.dsp
Subject: Re: Yet another sampling issue
Keywords: stochastic processes
Message-ID: <11895@pasteur.Berkeley.EDU>
Date: 12 Mar 91 02:27:05 GMT
References: <2895@cod.NOSC.MIL>
Sender: news@pasteur.Berkeley.EDU
Reply-To: jbuck@galileo.berkeley.edu (Joe Buck)
Lines: 42

In article <2895@cod.NOSC.MIL>, reuter@cod.NOSC.MIL (Michael Reuter) writes:
|>   The recent deluge of postings regarding sampling and the Nyquist rate led
|> me to think of an issue that I thought about a while back.  The DSP books that
|> I have discuss the sampling rate with regards to a deterministic function
|> whose Fourier transform exists (absolutely integrable etc.).  However in 
|> many applications we don't sample such functions; we sample stochastic
|> processes where the Fourier integral probably doesn't exist.

You need to get a book that discusses so-called "modern" digital signal
processing, where everything is defined in terms of random processes.
This theory was co-invented by Wiener in the US and Kolmogorov in the USSR.
It's where you go to answer questions like yours.

An equivalent form of the Nyquist sampling theorem applies to stochastic
processes.  Let x(t) be a strictly bandlimited process.  Then consider the
sampled process x(nT).  We can reconstruct x(t) from its samples x(nT) using
the sin x/x expansion: this expansion converges in a mean-squares sense to
x(t).  (I believe that Shannon was the first to show this).

|>   Since x(t) is bandlimited the error |r(tau) - r(m)| (or |S(jw) - S(e(jw))|)
|> must be bounded, but by how much?  The error must then be related to the
|> cutoff frequency wc, but how?  My guess is that something interesting must
|> happen as we pass through the Nyquist rate.  The error might plummet just
|> above wc and then monotonically decrease beyond that.  This is probably a 
|> well known problem to mathematicians.

If x(t) is bandlimited there is NO error introduced by sampling at the Nyquist
frequency or above.  If x(t) is not strictly bandlimited then aliasing occurs.
Remember, S is a deterministic function for a WSS random process.

The relation between S(jw) and S(e(jw)) is simple: S(jw) is zero beyond the
nyquist frequency; S(e(jw)) is periodic; the period is 1/(sampling frequency).
The autocorrelation function of the discrete random process is just a
sampled version of the autocorrelation function of the continuous-time
random process.

Basically, the exact same thing happens to the autocorrelation function
(which is a deterministic function) as happens to a deterministic x(t)
signal when you sample it.
--
Joe Buck
jbuck@galileo.berkeley.edu	 {uunet,ucbvax}!galileo.berkeley.edu!jbuck