This appendix shows how to derive that the noise power of amplitude
quantization error is , where is the quantization step
size. This is an example of a topic in statistical signal
processing, which is beyond the scope of this book. (Some good
textbooks in this area include
[25,50,33,32,63,31].)
However, since the main result is so useful in practice, it is derived
below anyway, with needed definitions given along the way. The
interested reader is encouraged to explore one or more of the
above-cited references in statistical signal processing.^{G.10}

Each round-off error in quantization noise is modeled as a
uniform random variable between and . It therefore
has the following probability density function (pdf) [50]:^{G.11}

Thus, the probability that a given roundoff error lies in the
interval is given by

assuming of course that and lie in the allowed range
. We might loosely refer to as a probability
distribution, but technically it is a probability density function,
and to obtain probabilities, we have to integrate over one or more
intervals, as above. We use probability distributions for variables
which take on discrete values (such as dice), and we use probability
densities for variables which take on continuous values (such
as round-off errors).

The mean of
a random variable is defined as

In our case, the mean is zero because we are assuming the use of
rounding (as opposed to truncation, etc.).

The mean of a signal is the same thing as the
expected value of , which we write as
.
In general, the expected value of any function of a
random variable is given by

Since the quantization-noise signal is modeled as a series of
independent, identically distributed (iid) random variables, we can
estimate the mean by averaging the signal over time.
Such an estimate is called a sample mean.

Probability distributions are often characterized by their
moments.
The th moment of the pdf is defined as

Thus, the mean
is the first moment of the
pdf. The second moment is simply the expected value of the random variable
squared, i.e.,
.

The
variance
of a random variable is defined as
the
second central moment of the pdf:

``Central'' just means that the moment is evaluated after subtracting out
the mean, that is, looking at
instead of . In
the case of round-off errors, the mean is zero, so subtracting out the mean
has no effect. Plugging in the constant pdf for our random variable
which we assume is uniformly distributed on
, we obtain the
variance

Note that the variance of can be estimated by averaging
over time, that is, by computing the mean square. Such an estimate
is called the sample variance. For sampled physical processes, the
sample variance is proportional to the average power in the signal.
Finally, the square root of the sample variance (the rms level) is
sometimes called the standard deviation of the signal, but this term
is only precise when the random variable has a Gaussian pdf.