From: matthews@eecs.ucdavis.edu (Thomas W. Matthews)
Newsgroups: rec.audio.high-end
Subject: Re: Digital processing of sound
Date: 14 Nov 92 19:24:51 GMT
Organization: Division of Electrical Engineering and Computer Science, UC Davis
NNTP-Posting-Host: 129.89.7.4
Originator: tjk@csd4.csd.uwm.edu


I just read this section:

>However, we may be able to show that the process of filtering at 22kHz can
>inject substantial noise and distortion back down into the audio band.
>Consider a tone of 20kHz, being amplitude modulated at 7kHz.  This might be a
>_very_ simple representation of a cymbal.  Ignoring the time domain, the
>spectrum of this signal (a 20kHz tone regularly varying in amplitude) will
>consist of three spikes - one at 20kHz, where we would expect, and one at
>each of 20kHz +/- 7kHz, ie 13kHz and 27kHz.
>
>In the ordinary course of events, playing back these three signals will
>result in the original signal. This happens in the human sensing mechanism,
>because the body is still sensitive to the 27kHz tone, it combines all three
>and we 'hear' the original cymbal.
>
>However, pass this signal through a brick wall filter at 22kHz, and we now
>have only two tones, one at 20kHz and one at 13kHz.  No matter how well these
>are then reproduced by the playing equipment, there is no 27kHz sideband to
>combine with the 13kHz sideband to modulate the 20kHz carrier.  The result is
>a distorted version of the original 20kHz tone, and a new and completely
>unrelated 13kHz tone that wasn't in the original signal!

To me, this implies that the process of filtering ALONE will introduce
aliassing.  It will not.  (if the filter has non-linearities, intermodultion
distortion can be produced, but that is not the issue here).

The writer does make an interesting point:  A 20 kHz. tone cannot be
amplitude modulated at 7 kHz. unless a 27 kHz. component is present.
Traditional thinking says that since one cannot hear 27 kHz. by itself,
its presence or absence will not affect the perception of the amplitude
modulated sound described.  Have any listening tests ever been done
that isolated this one specific issue?  I would like to thank the author
for bringing up a very "sound" argument against deliberate filtering.

The posting goes on:

>Proponents of Nyquist (but who haven't studied his work) will say that his
>theorem states that all signals up to half the sampling frequency can be
>reproduced. This is precisely what's happening in this case, but the
>information content of the signals below half the sampling frequency is being
>eroded, and noise is being added.

Acording to standard definitions used in communications theory,
the "information content of the signals below half the sampling frequency"
never did include the 27 kHz. component.  (As a matter of fact, the 13 kHz.
component had the same information as the 27 kHz. component because the 
modulating 7 kHz. tone was real-valued.  This is why single-sideband AM
transmission works.)  I recognize that I'm being a little picky about
terminology here, and I hope I haven't crossed over the line into being
a jerk over this minor point :-) .

The last paragraph:

>It is this injection of noise back into the audio band that is subjectively
>the most objectionable component of digital sound.  The ear is apparently
>tolerant of noise and/or distortion, as long as it is a harmonic of the
>original signal.  It is extremely sensitive to signals that are not harmonic.
>I was told that this is easy enough to demonstrate in a laboratory using
>signal generators - when a test tone is playing, harmonic signals (several %)
>may be introduced with very little subjective effect, but fractional
>percentages of unrelated signals are extremely easy to detect.  The ear is
>very good (because it practices every day, in the world around us) at picking
>out small sounds from background signals.

makes a strong argument that aliasing is far less tolerable than the same amount
of harmonic distortion.  Once again, though, aliasing arises from the sampling
operation, not the filtering.  And to avoid aliasing when the Nyquist rate
is so close to the desired bandwidth, sharp cutoff filters must be used.  An
earlier posting makes a srrong case in favor of realizing these filters in the
digital domain (but only because they've got to be there somewhere!).

Tom Matthews

Newsgroups: rec.audio