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RE: Questions for audio gurus

Subject: RE: Questions for audio gurus
From: Tom Duff <>
Date: Thu, 23 Feb 2006 09:56:43 -0800 (PST)
"Don Lloyd" <> says:
> I'm not sure I understand this here Nyquist limit. Wouldn't
> the sampling frequency have to be _much_ greater than
> twice the highest frequency being recorded? F'rinstance,
> say you want to record a 22.05KHz sine wave. The best you
> could do with a 44.1KHz sample rate is a 22.05KHz square
> wave, regardless of bit depth. The quantitative frequency
> would be present, but in a considerably altered shape.

It depends on what your recording and playback systems do with
the samples.  If each sample value denotes setting the signal
level to a particular value and holding it at that level until
the next sample comes along, then, indeed, you can represent
22.05 KHz square waves and not sines.

But playback systems (invariably?) include anti-aliasing
filters (if only to protect your power amplifiers and speakers
from high-frequency signals that they probably weren't designed
to deal with), which round off the corners of the signal and
introduce a certain amount of ringing.  An "ideal" anti-aliasing
filter converts an single-sample impulse (i.e. a sequence of samples
all but one of which are zero) into a signal which looks like
sin(44100 PI t)/(44100 PI t), which is a sinusoid of frequency
22.05 KHz that falls off on either side of the sample position.
This signal is usually called the "sinc function."

If you add up a bunch of sinc functions, each delayed and
scaled according to your recorded samples, you will find,
after slogging through more math than either of us is probably
interested in, that you can indeed represent a 22.05 KHz
sine wave.

In fact, because 22.05 KHz is right on the hairy edge of what
the Nyquist thm says you can represent, its phase had better be
exactly right in order for the reconstruction to work.  But for
any lower frequency, even 22.04999999 Khz, you can, regardless
of phase, exactly reconstruct the signal from its samples.

All of this depends on being able to use an ideal sinc
reconstruction filter.  As you might suspect, real-world
filters can only approximate ideal filters, so real-world
systems, while they can, with enough care, come very close,
must invariably fall short of ideal performance.

Tom Duff.  ``My favorite country musician''  -Eric Clapton


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