Dither: Making a Lemon Into Lemonade
or
Nature Finally Gets Genenerous
by Bob Masta
Interstellar Research
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Noise is usually considered an "evil" in any measurement
system, but with digitized data a certain amount of noise
is a desireable thing: It actually increases the
resolution of measurements when used in conjunction with
synchronous averaging. If noise is specifically added
for this purpose, it's called dither. Often,
though, you may be able to take advantage of noise that
is already present in the signal to provide a
"selfdither" ... a case of making a lemon into lemonade.
Truelife case: In the early days of CD recording,
engineers labored to produce systems with lower and lower
noise. Unfortunately, when the noise got low enough, the
recording quality actually got worse: A hanging
piano note, for example, which should have decayed
smoothly into silence, instead passed through a range of
ugly "buzz" just before vanishing. At the point where it
was only activating a few bits of the 16bit system, the
output waveform was so rectangular that it had nearly
100% distortion.
But a tiny amount of background hiss, either left in or
deliberately added, completely eliminated the distortion.
The natural integration of the human ear provided the
averaging action, and the listener heard only a pure
tone with a small hiss ... much better! That was
the "miracle of dither", which is now used in one form or
another on all commercial CD music recordings.
How does dither work its magic? Consider a noisefree
continuous signal, such as a ramp waveform, that is
digitized with an ordinary ADC. Normally you would try
to set an input gain such that the waveform used most of
the range of the ADC. However, sometimes that's just not
possible: Perhaps the signal has unwanted transients
that greatly exceed the range of the "normal" portion
that you want to observe. Or maybe the amplitude varies,
like that piano note that decays to silence in a system
that allows crescendos. You have to set the gain low
enough to accomodate the largest input, and simply accept
that lowlevel inputs will use less of the ADC's range.
So, when you magnify the digitized waveform, you may find
that it looks "chunky", more like a staircase than a
ramp. Figure 1 shows an ideal lowlevel ramp (yellow) and
the results of digitizing with inadequate bit resolution
(red.) The waveform can only take on the discrete steps
of the LSB size of the ADC; all information about
intermediate values is lost. If you synchronously
average a number of repetitions of the noisefree
waveform, you get exactly the same staircase. (Note that
this has nothing to do with sample rate, only the ADC
resolution: Sampling faster would just mean more samples
between step transitions.)
Fig. 1: Ideal and Sampled Ramp Waves
If you add noise whose peaktopeak level is equal
to the LSB resolution, you find that each step of the
staircase has "grass" growing out of it, where the noise
plus the signal exceeds the ADC's threshold for the next
bit. Since the noise is random, the grass is different
on each step and on each repetition of the waveform.
Fig. 2: Dithered Ramp, Instantaneous and Averaged
Figure 2 shows a snapshot of the sampled waveform
(red) superimposed with the results of averaging 256
repetitions (yellow.) Notice that the noise causes
frequent transitions in regions that were the edges
of the original stairs in Figure 1, and fewer transitions
near the center of each stair. It turns out that the
center of each original stair is the desired true value
of the ramp, which is unchanged by averaging since there
are no noiseinduced spikes at that point on any
repetition. The noisy transition regions, on the other
hand, average out to values midway between adjacent
steps, forming a nearprefect replica of the original
input.
To see what's going on in detail, consider that the ADC
has transition points midway between each LSB step. For
example, let's say this is an 8bit ADC with an input
range from 0 to 2.55 V, or 10 mV per LSB. If the input
is above 5 mV but less than 15 mV, the output will have
only the LSB set, indicating a 10 mV level for anything
in this entire range. When the input moves above 15 mV,
the ADC output reports 20 mV. Hence, without dither, we
get the staircase of Figure 1.
Now consider the addition of a dither signal whose
maximum excursion is +/0.5 LSB, or +/5 mV. Where the
input ramp is exactly at 10 mV, the added noise will very
rarely be at its extreme limits to bump the total above
the 15 mV or below the 5 mV thresholds. Thus we see no
"grass" at the center points of each bit value.
On the other hand, where the input ramp is just at the
5 mV transition point, any added noise will force the
output either up to the 10 mV step or down to the 0 mV
step, so we see a lot of output transitions here. If the
noise has an average value of 0, we expect about half of
the output values to be 10 mV and half 0, so after a lot
of repetitions we expect the average output value to be
pretty close to 5 mV ... even though the ADC itself can't
resolve this value directly.
Similarly, intermediate values of the input ramp will
produce intermediate averaged values. For example, if
the ramp is at 6 mV, the rampplusnoise will run from
1 mV to 11 mV. Assuming a uniform distribution of noise
values, we would expect about 40% of the noise values to
be lower than 1 mV, such that the total input would be
below 5 mV and thus the output would be 0. About 60%
should be higher, giving an output of 10 mV. The average
value then comes out to 6 mV, just as desired.
The improvement in resolution is proportional to the
number of repetitions that are averaged together. If you
average only 2 observations that are each either 0 or 10
mV, the only possible average values are 0, 5, or 10 mV.
If you average 4 observations, you can get 0, 2.5, 5,
7.5, or 10 mV. Essentially, each doubling of the number
of repetitions adds one bit of resolution by cutting the
effective output step size in half.
There is no particular limit to this ability to increase
resolution, except the practicality of how long you want
to wait for the next bit. Adding an additional 8 bits of
resolution only takes 256 repetitions; adding 16 bits
takes 65536.
However, there is still the issue of the dither
noise itself, which typically subsides as the square
root of the
number of repetitions averaged.
Doubling the repetitions will double the resolution, but it
will only cut the noise by 3 dB. Thus, the residual noise will
be the limiting factor on true resolution, which improves much
more slowly; you'll quickly get high apparent
resolution, but the fine detail you see will just be noise
until you really build up the repetitions. While the dither
phenomenon may be a case where Nature finally gets generous,
She's not that generous.
But that's with dither noise that is uncorrelated with
itself and/or the repetition length. If you are adding
dither just for the purpose of improving resolution, you
can indeed get an improvement proportional to repetitions,
by careful control of the dither. More on this next
time.
The previous discussion also assumes noise with an
amplitude of exactly +/ 0.5 LSB and a uniform
distribution. If the noise level is higher, the dither
process still works, but there will be a higher noise
level in the averaged output. Since many realworld
signals are contaminated with enough noise to require
averaging simply for noise reduction, the side effect of
improved resolution may be considered a pure bonus.
What about the fact that your contaminating noise may
have an unknown distribution? That's not likely to be
a problem unless the noise peaks are just at the +/ 0.5
LSB level. In the case where the distribution is closer
to Gaussian, there might not be enough nearpeak values
to smooth out the transfer function as it approaches
the center of each stair; effectively, this is similar to
having noise with a uniform distribution but with a level
that is too low. The staircase would average into a
scalloped function instead of a straight line. The cure
is often as simple as increasing the input gain or the
noise level.
If the noise level in your signal is fairly well known,
you can use that level to determine the optimum input
gain for best resolution and dynamic range with a given
ADC, or to determine if you need an ADC with more bits of
resolution:
 If the required dynamic range is not known ahead of
time, you can maximize it by setting the system gain
so that the peaktopeak noise level is the same as
the LSB span. Let's say you have 1 mV of noise and
an 8bit ADC; you should set the input gain so that
the ADC covers no more than a 255 mV range. If you
use less gain, the noise won't be enough to provide
the selfdither you need to fully resolve the true
signal by averaging. More gain, on the other hand,
would reduce the input range you could handle before
the ADC is overdriven. To get a larger input range,
you'll need an ADC with more bits: 12 bits will
allow a 4095 mV range, while 16 would allow a
whopping 65,535 mV.
 Alternatively, if you know the peaktopeak range
that the system must handle, divide it by the
peaktopeak noise level. That gives the approximate
number of steps you need from your ADC. Round that
up to the next power of 2, take the log, and divide
by the log of 2. That's the number of useful ADC
bits; buying an ADC with substantially more bits is
overkill for this particular task.
Of course, it never hurts to have an ADC with more
resolution than you absolutely need. That can provide you
with a reserve of dynamic range for unexpected transients,
or conversely a reserve of resolution for lowernoise
signals. But in the real world there are often
compromises. Whether you are selecting a singlechip
processor with a builtin ADC for a highvolume embedded
application, or just buying one data acquisition board
for a test setup, you will need to balance other features
as well: An ADC with more bits may be slower, or cost
substantially more. The particular singlechip processor
or data acquisition board that supports the ADC may lack
some other needed feature like enough digital I/O lines,
or may not be compatible with the intended software.
Thus, it's always a good idea to begin with a clear
vision of how the system will operate. Uninformed users
have been known to specify 16bit systems for
applications where the noise spanned more than 12 of
those bits, thinking that this somehow gave superior
performance compared to a 12bit or even 8bit system.
In fact, so many repetitions had to be averaged just for
noise reduction, that even a 4bit system would have
given the same ultimate resolution!
Next time, we'll look at some unusual dither
applications, including a singlebit ADC, DC dither,
correlated dither, and puretone dither. In the
meantime, you can experiment with dither plus averaging
using the builtin signal generator of the author's
Daqarta for Windows software.
It turns your Windows sound card into a
data acquisition system, including extensive builtin
signal generation options.
All Daqarta features are free to use for 30 days or
30 sessions, after which it becomes a freeware
signal generator... with full analysis capabilities.
(Only the sound card inputs are ignored.)
You don't even need any external connections to the sound card
to see the results of dither plus averaging visually, since you
can average the raw signal before it gets to the outputs.
But for a dramatic demonstration of audible dither (where
your auditory system provides the averaging), you should
listen to the output as you restrict the effective bits of
the signalplusnoise combination.
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