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From: reuter@cod.NOSC.MIL (Michael Reuter)
Newsgroups: comp.dsp
Subject: Yet another sampling issue
Keywords: stochastic processes
Message-ID: <2895@cod.NOSC.MIL>
Date: 8 Mar 91 22:41:42 GMT
Organization: Naval Ocean Systems Center, San Diego
Lines: 38


  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.
  Let's assume we have a classical spectral estimation problem where we
sample a second-order wide sense stationary, ergodic, bandlimited process x(t).
Let's also assume we can analytically compute the autocorrelation function 
r(tau) of the continuous process via the integral equation defining the time 
average, i.e.,
	                     /L
	r(tau) = lim   1/2L | x(t + tau)x(t) dt.                         (1)
              L -> inf     /-L

Its PSD is S(jw). Now the sampling problem applied to r(tau) directly relates
to the traditional Nyquist problem.  However we don't sample r(tau) we sample
x(t).  The problem is: how often must we temporally sample x(t) (getting x(n))
so that we can get a "good" estimate of r(m) (m is a discrete time lag) and
thus a "good" estimate of S(e(jw)).  Let's say S(jw) is bandpass with cutoff
frequency of wc.  Do we just have to sample > wc?  In practice we do, but 
S(e(jw)) is not equal to S(jw) even at this rate.  To see this just interpret
(1) in the Riemann sense, sample x(t), and compare the discrete time average
summation equation 
	                         _N
         r(m) = lim    1/(2N+1) >_  x(n+m)x(n)                           (2)
               N -> inf          -N

to (1).  You will see that the discrete time average (2) is an approximation of 
(1).  However we can get r(m) arbitrarily close to r(tau) as we sample at 
higher and higher rates.  
  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.