Path: utzoo!attcan!uunet!seismo!sundc!pitstop!sun!decwrl!labrea!polya!rokicki
From: rokicki@polya.STANFORD.EDU (Tomas G. Rokicki)
Newsgroups: comp.sys.amiga
Subject: Sampling at 29KHz
Message-ID: <2845@polya.STANFORD.EDU>
Date: 18 May 88 17:42:18 GMT
Organization: Stanford University
Lines: 81

[ Sample *this*, you soundless hulk! ]

	Why Sampling At 29K Is Usually Sufficient

	First, I'm not sure I know what I'm talking about
here, as I'm not a musician and know very little about sound,
but let's pretend I do know what I'm talking about for a
second or two.

	Some people have been bitching lately about having to
provide waveforms for each octave of a sound sample.  I won't
go into issues here about how the relative frequency
distribution changes as a function of frequency for musical
instruments, other than to note that it does, so actually
sampling the different octaves will give you more realistic
sound, but rather I will explain how using separate samples for
the higher frequencies works better than undersampling higher
resolution samples, by stepping through them at a faster rate.

	First, any infinite periodic waveform of period t with
n samples can be represented as a sum of n sine waves with
periods of t/1, t/2, t/3...t/n.  The discrete fourier transform
can do (and undo) this transformation.

	If you undersample such a waveform, aliasing will cause
some strange effects.  Let us say our sampling rate that we are
undersampling at is m (samples per second.)  According to Nyquist,
our sampling rate should be at least twice the frequency of the
highest component sine wave.  Any components that have a
frequency greater than 2/m will have some portion of their energy
aliased or mirrored down to a lower frequency.

	To demonstrate this, let us look at a 10KHz sine wave.
We'll sample it at 20KHz, exactly the Nyquist rate.  If our
samples happen to fall exactly at the zero crossings of the sine
wave, we will have a zero amplitude output.  If they fall on
the peaks, we will have a maximum amplitude output.  Depending
on the random position of our first sample.

	Now let's sample our 10KHz sine wave at 20.2KHz.  We
are now slightly off our frequency, and we will see a 10KHz
tone modulated by a 200 Hz carrier; this will sound like two
narrowly separated frequencies beating against one another.
Ugly as sin.

	Let's now sample a 16KHz sine wave at 20KHz.  In this
case, we will get as output approximately a 4KHz tone at almost
the full amplitude as the original sine wave.  (Draw it out on
graph paper to see what happens.)

	So, by undersampling our highly accurate waveform, we
map the frequencies which are normally unheard (if they are
really there at all) into frequencies which can be heard and which
will greatly diminish the quality of the output.

	How do we fix this?  Easy.  We generate sound samples at
the higher frequencies that do not contain the unheard higher frequency
components of the original sound, and sample this newer sample at
the correct rate.  This is exactly the way it is recommended you do
it on the Amiga, and for good reason.

	Someone recently complained that they had to go all the way
down to a 32 samples to get the high frequencies they needed.  I claim
that that 32 sample sound, if properly created from the filtered
original, would sound superior to an undersampled larger sample size.

	The final point is, because you know *each* sample will be
used, you can adjust the energy of any of the harmonics essentially
independently, and be assured that the output signal will have that
much energy.  Undersampling does not give you this ability.

	Any further comments?

	Oh, so you want to know how to create the smaller samples from
the larger ones.  I'll leave that to the experts out there.

-- 
    /-- Tomas Rokicki         ///  Box 2081  Stanford, CA  94309
   / o  Radical Eye Software ///                  (415) 326-5312
\ /  |  . . . or I       \\\///   Gig 'em, Aggies! (TAMU EE '85)
 V   |  won't get dressed \XX/ Bay area Amiga Developer's GroupE