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From: jmuller@Stardent.COM (Jim Muller)
Newsgroups: rec.music.synth,comp.music,rec.music.makers
Subject: Re: White Noise???????
Message-ID: <1990Dec18.180227.983@Stardent.COM>
Date: 18 Dec 90 18:02:27 GMT
References: <12263@bsu-cs.bsu.edu>
Organization: Stardent Computer, Newton MA
Lines: 66

In <12263@bsu-cs.bsu.edu> travis@bsu-cs.bsu.edu
  (Travis Michael Banks) writes:

>Can someone please tell me what WHITE NOISE is?

I see that metlay wrote a description that also mentioned pink noise.  His
explanation is okay but contains a few "imprecisions", well, okay, technical
inaccuracies.  I like the green noise description though... :-)  (Neither
white nor pink noise definitions have anything to do with the human ear.)

White noise is noise, actually *any* time signal, in which the power per
frequency is constant.  In continuous space (i.e. calculus), this means
dP/dF = k.  In discrete space, such as time measurements of voltage (you
can use either amplitude, e.g. voltage, or power), analyzed via FFT, this
means a constant power (or constant amplitude) for all frequencies, given
a fixed delta-F.

More properly, we should speak of the expected value, with a given
distribution of sampling variation, since we usually mean random,
aperiodic, non-deteministic noise.  At any given time, the amplitude
has only one value, of course.  A "spectrum", be it discrete FFT or
continuous via analytical Fourier Transform, is a time-integral, and
thus assumes a time-window over which that "spectrum" applies.  We
rarely have a complete time history over which to calculate a spectrum, 
and in music, we never do.  However, we can calculate a spectrum over a
time interval, then do it again and again, forming a time-dependent
spectrum.  This is valid provided there is enough time in the interval
to cover the frequencies desired, and that there be enough overlap
(in time) of successive calculations, typically a factor of 2.  With
less overlap, there is a possibility of aliasing of the high frequencies
to appear to have pulsating amplitudes.  This is generally not an issue
with musical applications, and probably is rarely considered.  The net
result is that we generally speak of the power per frequency being
constant, but in reality we mean:

If we measure it over a number of time intervals, and if the noise is
more or less "constant", we will see variation with time but the mean
value will be constant over time and over frequency.  The constant-over-
time part is assumed anyway, with "mean value" being explicit when we say
"is constant".  What makes it "white" is the constant-over-frequency part.
So it is a simplification, but a reasonable one, to say dP/dF = k.

So what then is "pink" noise?  How does it differ from white and what
is it used for?  Pink noise is a similer aperiodic noise, but with the
frequency distribution different.  It has a decrease in power of 3dB
per octave; more precisely, the power at 2f is 1/2 of that at f.
Why use such a power distribution?  In musical applications, we are
usually concerned with octaves, i.e. frequency bands which get wider
as you go up.  We speak of 30-60Hz, 60-125Hz, 125-250Hz, 250-500Hz, etc.
If we used equal-width frequency intervals like those produced by an
FFT, or as appropriate for white noise, we would need twice as many
intervals for 60-125Hz as for 30-60Hz, and twice as many again for
125-250Hz, etc.  To avoid this overload, we use a logarithmic frequency
scale.  But if you analyze and display the time-dependent spectrum of white
noise on a scale in which the frequency intervals get wider with increasing
frequency, the apparant power in each band will go up by a factor of two
for each band.  What is "flat" on a linear frequency scale is ramped up
with frequency on this logarithmic scale.  So we just start with a noise
that has a power reduction of a factor of two for every increase of that
much in frequency, i.e. P = k/f.  This way, it looks "flat" when analyzed
and displayed on a typical music-oriented octave-based frequency scale.
It has nothing to do with the human ear.  It is simple math...

Just what you didn't really wnat to know...
-- 
 - Jim Muller