Digital Signal Processing Reference
In-Depth Information
Fig. 9.12. Input-output
relationship of an
L
-level
quantizer used to discretize the
sample values
x
[
kT
s
]ofaDT
sequence
x
[
k
]. (a) Uniform
quantizer; (b) non-uniform
quantizer.
output
output
r
L
−1
r
L
−1
r
L
−2
r
L
−2
r
k
+ 4
r
k
+ 4
d
k
input
input
d
L
d
L
d
0
d
1
d
2
d
k
d
k
+ 4
d
L
−1
d
0
d
1
d
2
d
k
+ 4
d
L
−1
r
k
r
k
r
2
r
2
r
1
r
1
r
0
r
0
(a)
(b)
is used. The above filter is referred to as the compensation, or anti-imaging,
filter. Filtering
X
s
(
ω
) with the anti-imaging filter introduces a linear phase
−ω
T
s
corresponding to the exponential term exp(
−
j
ω
T
s
). Inclusion of a linear phase
in the frequency domain is equivalent to a delay in the time domain and is
therefore harmless and not considered as a distortion.
9.3 Quantization
The process of sampling, discussed in Sections 9.1 and 9.2, converts a CT signal
x
(
t
) into a DT sequence
x
[
k
], with each sample representing the amplitude of
the CT signal
x
(
t
) at a particular instant
t
=
kT
s
. The amplitude
x
[
kT
s
]ofa
sample in
x
[
k
] can still have an infinite number of possible values. To produce
a true digital sequence, each sample in
x
[
k
] is approximated to a finite set
of values. The last step is referred to as
quantization
and is the focus of our
discussion in this section.
9.3.1 Uniform and non-uniform quantization
Figure 9.12(a) illustrates the input-output relationship for an
L
-level uniform
quantizer. The peak-to-peak range of the input sequence
x
[
k
] is divided uni-
formly into (
L
+
1) quantization levels
{
d
0
,
d
1
, ...,
d
L
}
such that the sepa-
ration
=
(
d
m
+
1
-
d
m
) is the same between any two consecutive levels. The
separation
between two quantization levels is referred to as the
quantile inter-
val
or quantization step size. For a given input, the output of the quantizer is
calculated from the following relationship:
=
1
y
[
k
]
=
r
m
2
[
d
m
+
d
m
+
1
]
for
d
m
≤
x
[
k
]
<
d
m
+
1
and
0
≤
m
<
L
.
(9.27)
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