Re: Frequency/Sample rate

And of course sampling theory states that you need MORE THAN
two samples per wave, not at least two samples per wave.

...so, if i require "more than two per wave", then
are you saying that the highest frequency distinguishable
from 44.1k is something less than 15k (44.1 divided by 3)?

Any decent CD player has a frequency response which is flat to within
a small fraction of one dB, up to 20 kHz, and accurately resolves
frequencies within that bandwidth.

As long as you have more than two, the wave is uniquely
and totally described - any further samples are unneeded
and superfluous.

only if you assume a perfect sine wave with no
overtones.

If it has overtones (i.e. harmonics), then by definition these
harmonics have to be taken into account as part of the "highest
frequency distinguishable". For example, a non-sinusoidal 20 kHz
signal contains overtones (e.g. at 40 and 60 kHz, the second and third
harmonics) and is actually a composite signal which has 60 kHz of
bandwidth. This cannot be accurately sampled and reconstructed at a
sampling rate of 44.1 ksamples/second.

That's why signals must be low-pass-filtered before being sampled.
It's an essential part of the process.

To achieve faithful replication of up to 20k,
then at least 3 samples per wave at 20k would be necessary,
true?

No, not true. You don't need an _integral_ number of samples per
cycle (e.g. 3 or 4) to accurately distinguish, and reproduce, the
signal. All that's required is that you have somewhat more than two.

With CDs, a 20 kHz audio bandwidth, and a 44.1 kHz sampling rate,
gives you a minimum of 2.2 samples per cycle (at the 20 kHz bandwidth
limit) and more than that at lower frequencies. This is sufficient
to accurately reproduce the signal.

This may seem counter-intuitive, but it actually does work (both in
practice, and in the underlying mathematics).

I mean, 44.1 and 48k wouldn't cut it.
only something greater than 60k. e.g. 96k.

This turns out not to be the case. 44.1 ksamples/second *does*
allow the accurate sampling, and reconstruction, of an audio signal
with 20 kHz of bandwidth.

There are two *essential* steps in this process. You *must* filter
the incoming continuous signal before you sample it, to ensure that it
actually has no more than 20 kHz of bandwidth (i.e. you must filter
out any individual signals, or harmonics/overtones which lie above 20
kHz). This is usually known as the "anti-aliasing filter" step.

Then, when you convert the samples back to continuous form, you *must*
run the samples through another bandwidth-limiting filter (again, DC
to 20 kHz in the case of CDs) to eliminate the image frequencies lying
above 20 kHz. This is usually referred to as the "reconstruction
filter".

of course, such a situation is not likely in the 'real world',
because no natural sound source would be that perfectly
in lock step with the sampling frequency. Nevertheless,
'just two' samples per wave is insufficient to faithfully
replicate a waveform, and invites distortion.

i would agree that 'more than two' are thus necessary.
which, in my klunker way of putting it, was what i
was trying to say in the first place.

You're correct. You need more than two.

You just don't need all that much *more* than two. You don't need
three. 2.2 turns out to be sufficient, if you do a proper job
implementing the anti-aliasing and reconstruction filters.

--
Dave Platt <dplatt@xxxxxxxxxxxx> AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!
.

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