reuter@cod.NOSC.MIL (Michael Reuter) (03/09/91)
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.jbuck@galileo.berkeley.edu (Joe Buck) (03/12/91)
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