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61
Chapter 6.
Quantization
To convert an analog signal to a digital signal, the following three procedures are
required.
First, the signal is passed through a lowpass filter to prevent aliasing.
Second,
the signal is sampled by a sampleandhold circuit.
Finally, the samples are quantized by
an analog to digital converter (ADC) in order to be represented in digital form as shown
in Figure 6.1.
x
(
t
)
Antialiasing
x
(
n
)
$
()
x n
Filter
Sampleandhold
A/D Converter
(Analog Signal)
(LPF)
Circuit
(Discretetime
(Digital Signal)
S
i
g
n
a
l
)
Figure 6.1. Typical analog to digital conversion process
There are many different kinds of quantization techniques available.
Quantization
methods such as the linear quantization, the nonlinear quantization, the delta modulation,
and the sigmadelta modulation are described in this chapter.
Also efficient quantization
methods such as the adaptive quantization and the differential quantization are explained.
The Adaptive Differential Pulse Code Modulation (ADPCM) is also described.
6.1
Linear Quantization
In
linear
or
uniform quantization
, the quantization step size is fixed.
The constant
quantization step size is used no matter what the instantaneous signal amplitude is.
Linear quantization with 2
m
quantization levels is shown in Figure 6.2 where
∆
is the
quantization step size and
m
is the number of bits in a quantization word.
Sample value in volts[V]
(2
m
−
1)
∆
/2
(Positive Peak Value = 2
m
∆
/2: loudest)
(2
m
−
3)
∆
/2
3
∆
/2
∆
/2 (softest)
−∆
/2 (softest)
−
3
∆
/2
−
(2
m
−
3)
∆
/2
−
(2
m
−
1)
∆
/2
(Negative Peak Value =
−
2
m
∆
/2: loudest)
Figure 6.2. Constant quantization step size
∆
is used for linear quantization.
(
m
is the number of bits used for quantization)
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For example, if the peak to peak value of a signal is 4 [V] and
m
= 3, then the signal may
be quantized according to the rule shown in Figure 6.3.
Digital value after quantization
(1.75) 111
(1.25) 110
(0.75) 101
(0.25) 100
Sample value before
−
2
−
1.5
−
1
−
0.5
0
0.5
1
1.5
2
quantization [V]
011 (
−
0.25)
010 (
−
0.75)
001 (
−
1.25)
000 (
−
1.75)
Figure 6.3. Threebit linear quantization example.
(signmagnitude binary representation is used)
In this case
∆
= 0.5 [V].
A value between 0 and 0.5 is approximated (or quantized) by
the quantization level
∆
/2 (0.25) and encoded by 100.
A value between 0.5 and 1.0 is
quantized by the quantization level 3
∆
/2 (0.75) and encoded by 101.
Likewise, a value
between
−
2 and
−
1.5 is quantized by the level
−
7
∆
/2 (
−
1.75) and encoded by 000 and so
on.
The process of sampling, quantization and encoding is referred to as the
pulse code
modulation
(PCM).
Keeping a step size fixed is especially essential for highfidelity digital audio.
One
notable example is the compact disc (CD) format.
The CD format uses 16 bit linear
quantization so that the range between the negative peak and the positive peak is divided
by 2
16
(65,536) uniform quantization levels.
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 Spring '09
 B.K.Dey
 Aliasing

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