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
[28,53,35,34,68,33].)
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.12
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) [53]:G.13
Thus, the probability that a given roundoff error
assuming of course that
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
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
Note that the variance of