Systems, Sampling and Quantization
15
signal quantization
linear quantization
analog signal
digital signal
quantization levels
quantum interval
quantization error
quantization noise
white noise
appendix A.9. So far, in this chapter we have described discrete-time systems by
means of signals that are functions of a discrete variable and having a codomain
described by a continuous variable. Actually, the internal arithmetic of comput-
ing systems imposes a signal quantization, which can produce various kinds of
effects on the output sounds.
For the scope of this book the most interesting quantization is the linear
quantization introduced, for instance, in the process of conversion of an analog
signal into a digital signal. If the word representing numerical data is b bits long,
the range of variation of the analog signal can be divided into 2
b
quantization
levels. Any signal amplitude between two quantization levels can be quantized
to the closest level. The processes of sampling and quantization are illustrated
in fig. 4 for a wordlength of 3 bits. The minimal amplitude difference that can
be represented is called the quantum interval and we indicate it with the symbol
q. We can notice from fig. 4 that, due to two's complement representation, the
representation levels for negative amplitude exceed by one the levels used for
positive amplitude. It is also evident from fig. 4 how quantization introduces an
y(t)
2q
0
-2q
-4q
t
T
5T
10T
Figure 4: Sampling and 3-bit quantization of a continuous-time signal
approximation in the representation of a discrete-time signal. This approxima-
tion is called quantization error and can be expressed as
(n) = y
q
(n) - y(n) ,
(41)
where the symbol y
q
(n) indicates the value y(n) quantized by rounding it to
the nearest discrete level. From the viewpoint of the designer, the quantization
noise can be considered as a noise superimposed to the unquantized signal. This
noise takes values in the range
-
q
2
q
2
,
(42)
and it is spectrally colored according to the nature and form of the unquantized
signal.
What follows is a superficial analysis of quantization noises. In order to do a
rigorous analysis we should assume that the reader has a background in random
variables and processes. We rather refer to signal processing books [58, 67, 65]
for a more accurate exposition.
In order to study the effects of quantization noise analytically, it is often
assumed that it is a white noise (i.e., a noise with a constant-magnitude spec-
trum) with values uniformly distributed in the interval (42), and that there is
no correlation between the noise and the unquantized signal. This assumption is
false in general but, nevertheless, it leads to results which are good estimates of