dithering, repudiable steganography

[karn@qualcomm.com (Phil Karn)]

I am not sure that your explanation of "granulation noise" and
dithering are quite correct. You showed how a low-amplitude sine wave
is quantized by the sampling process, but quantization errors are
unavoidable in any digital recording system, with or without
dithering.

Here's another way to look at dithering. Let's say that I want to
represent a very low frequency audio signal ("low frequency" with
respect to the sampling rate). Let's assume further that over a period
of several samples, this signal has a relatively constant value of,
say, .25 microvolts in a system with 1 microvolt quantizing steps. If
I fed this into an ideal A/D converter, it would round it to the
closest representable level, i.e., 0 volts. And it would do so for
every sample, resulting in an constant error of .25 microvolts each
time.

But suppose I add some analog noise to the signal before I sample it.
The analog noise I generate will be a uniformly distributed random
voltage between +.5 and -.5 microvolts. Now the resulting signal will
range uniformly between +.75 and -.25 microvolts. If it's between +.5
and +.75 microvolts, the A/D converter will round it to 1 microvolt;
if it's between -.25 and +.5 microvolts, it will round it to zero. The
first case will happen 25% of the time, and the second will happen 75%
of the time.

Now if you average the resulting samples using enough precision for
the sum, viola -- you get an average of .25 microvolts, instead of
zero. This is exactly what happens at the ear after the D/A
reconstruction process, if you turn the volume up high enough to hear
what's happening.  Sure, there's still noise, but at least the average
signal value is correct.

So one purpose of dithering is to better represent low-frequency, low
amplitude information. The effect on complex signals is to turn the
quantization noise that would otherwise occur in narrow frequency
bands (and be quite audible) and spread it out as constant, low level
white noise over the whole audio band, which is much less objectionable.

Dithering is important only when the original analog source material
is *very* clean. In most cases, the background noise in the room and
the noise generated in the low level microphone preamps is much more
than one LSB, so the signal is "self dithering".

Phil