Maarten Hofman wrote:

> However, pretty good approximations are possible (as you mentioned later
> in your post: especially the function becomes 0 to closer it gets to
> infinity, as it includes the 1/x factor: together with a buffer for a
> certain amount of time, so that previous values can be modified
> accordingly, this solves most problems), and I don't see why the signal
> has to be periodic: a CD player uses 44100 samples/second to recreate a
> 22000 Hz waveform with rather high accuracy (the main problem there is
> usually the filter to limit the signal to its band).

The CD player situation is not exactly a real-time system. A CD player does
have access to the complete set of samples, so to speak from -infinity to
+infinity. They don't do that in practice, but they could read a full track
and add up the sinc functions for all samples. In any case, they can read
any relevant number of samples ahead and use them to recreate the function
in the time domain (or use the appropriate filters, which then have the
delay necessary for doing exactly that).

It's also not as close as you said. HiFi audio is defined to go up to
20kHz, not 22kHz. There is also some equipment that samples with 48kHz. So
both common sampling frequencies are an indication that while in theory it
is possible to approach half the sampling frequency, in practice normally a
certain distance from that point is necessary. This is because the closer
you get, the more into the future you need to look to be able to recreate
the signal (or the longer is the delay in the output).

> Note that as far as I can tell we're not really disagreeing: 

I know... we're just elaborating (and refreshing the Nyquist-Shannon
theorem in our memories :)

> I myself don't think that (except maybe for dsPICs) there are many cases
> where people would use a PICmicro towards the edges of the
> Nyquist-Shannon theorem (and I would enjoy hearing from people that do
> use it that way), mostly because the fact that they would never be able
> to do anything sensible with the sampled signal, as the reconstruction
> would be way too costly compared to the acquisition.

That's the one thing, and the other thing is the theoretic limit. The
closer you get, the more influence have the far away (in the past /and/ in
the future) samples. This introduces either the need to have the complete
sample set available or creates a long delay that's for many practical
purposes not desired. (The closer you get, the longer that delay.)

There's also the limited Q of the anti-aliasing filter. For the theorem to
work, you may have no energy at or above half the sampling frequency. This
makes it necessary to stay away from the theoretical limit by a distance
that is enough for the filter curve to fall below a certain threshold.
Which again is one more reason why you can't run an ADC at slightly above
twice the frequency you want to capture and think you get something useful.
(This of course doesn't apply if for some reason you know that the signal
won't contain content above the threshold frequency. But in reality, every
signal is only filtered, whether by the process or by electronics, with a
real Q, and therefore will contain frequency components above the signal
frequency.)

I suspect that's one of the reasons why they chose 44.1kHz and 48kHz as
sampling frequencies, and not 40.5kHz -- which would theoretically be
enough to sample the common 20kHz bandwidth audio signals.


Anyway, to get back to the initial question, it seems that nobody had ever
a problem with using an acquisition time of 0 when sampling only one
channel. In any case, while possible, it does seem to be rare :)

Gerhard

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