|
On Dither
by Mithat Konar
Of the all the things that are commonly misunderstood
about digital audio, one of the most common is dither. Dither is the
deliberate introduction of a random signal (i.e., noise) prior to quantization
in an analog-to-digital converter or prior to reducing the wordlength
(e.g., 24 to 16 bits) in a DSP algorithm. Dither comes in various forms
and in varying levels levels of complexity, but all forms have in common
the introduction of a random signal into the signal chain.
It might seem like a stupid idea to add noise to a system, but
when dither is done correctly, it completely eliminates the low-level
distortion found in quantized systems while increasing the resolution
of the system to a level well below the noise floor. (In theory, the
resolution becomes infinite). In these respects, dither makes digital
systems behave exactly like analog ones.
The primary consequence of adding dither is a slight decrease in signal-to-noise
ratio -- typically a few decibels. However, advanced schemes add the
noise in such a way that its audibility is minimized. The end result
with 16-bit, 44.1 kHz systems is a digital signal with all the benefits
of dither (elimination low-level distortion and an increase in resolution)
but virtually no perceived increase in noise.
It is often stated that dither simply masks, not eliminates, the low-level
errors found in digital systems, that the errors are still present,
but they are simply buried in hiss. This is very much not the
case. The following thought experiments may help to put these issues
into intuitive perspective.
(1) Imagine you have a bipolar, DC coupled analog-to-digital converter
with a least significant bit (LSB) threshold of 1 volt. (Just go with
it -- this is a thought experiment!) Furthermore, assume the ADC stairstep
function is aligned such that it is symmetric about zero. In other words,
an ADC input over the 1-volt interval [-0.5, 0.5] produces an ADC output
of 0, an input over the range [0.5, 1.5] produces an output of 1, and
so on.
(2) Without applying any dither, apply a 0.25 volt signal to the input
of the ADC. The output of the ADC will be a string of zeros. In fact,
any signal between -0.5 and 0.5 volt will result in an ADC output of
zero. Any information below the LSB threshold is completely lost.
(3) Remove the 0.25 volt signal and apply dither to the input of the
ADC in the form of a completely random signal (i.e., noise) centered
around 0 volts and large enough to just barely twiddle the LSBs of the
ADC. The output of the ADC will be a stream of very small random numbers.
However, the AVERAGE of all these values will be zero.
(4) Now let's apply our 0.25 volt signal again (with the dither on).
The output stream will again look like a stream of very small random
numbers, but guess what? The AVERAGE of all those numbers will now be...you
guessed it, 0.25. We have thus retained the information that was previously
lost (even though it's buried in "noise"). In other words,
our system's resolution has improved. The conversion is still essentially
random, but the presence of the 0.25 volt signal biases the randomness.
The mathematician would say that the characterization of the system
with dither is transformed from a completely deterministic one to a
statistical one.
(5) With the dither on, we can now change the input signal over a continuous
range and the average of the ADC output will track it perfectly. An
input signal of 0.373476 volts will have an average ADC output of 0.373476.
The same will hold true of inputs going over the +/- 0.5 volt LSB threshold:
e.g., an input of 3.22278 will have an average ADC output of 3.22278.
So not only has the dither enhanced the resolution of the system, but
it has also eliminated quantization effects!
(6) The results in (4) and (5) will not happen by adding noise
after the A/D conversion. Go back to our first experiment and
add the random signal to the output of the ADC. As long as the input
signal is between -0.5 and 0.5 volt, the ADC's average output will still
be zero. Between 0.5 and 1.5 volts the average will jump to 1. There
is no resolution enhancement and the quantization effects remain.
So, you can see that dither's resolution enhancement and error elimination
are truly physical/mathematical in nature and not a masking trick. You
should also keep in mind that human beings are able to hear signals
in the presence of noise of greater energy than the signal, i.e., with
negative signal-to-noise ratios. Therefore, even though a given signal
is below the noise-floor of the system, and therefore you might think
it irrelevant, it in fact may not be. Depending on what else is going
on around that signal, it can be audible at several decibels below the
noise-floor. Therefore, both benefits of dither -- eliminating low-level
quantization errors and increasing the system's resolution -- are truly
beneficial, perceivable effects.
r 0.00 8/27/98
copyright © 1998 Mithat Konar--all rights reserved
|
