Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 38

A unipolar quantizer deals with analog signals ranging from 0 volt to a positive reference voltage,
and a bipolar quantizer deals with analog signals ranging from a negative reference to a positive
reference. The notations and general rules for quantization are as follows:

D ¼ ðx max x min Þ
L

(2.19)

L ¼ 2 m (2.20)

x x min
D

i ¼ round (2.21)

x q ¼ x min þ iD i ¼ 0 ; 1 ; / ; L 1 (2.22)

where x max and x min are the maximum value and minimum values, respectively, of the analog input
signal x . The symbol L denotes the number of quantization levels, which is determined by Equation
(2.20) , where m is the number of bits used in ADC. The symbol D is the step size of the quantizer or the
ADC resolution. Finally, x q indicates the quantization level, and i is an index corresponding to the

binary code.
Figure 2.29 depicts a 3-bit unipolar quantizer and corresponding binary codes. From Figure 2.29 ,
we see that x min ¼ 0, x max ¼ 8 D , and m ¼ 3. Applying Equation (2.22) gives each quantization
level as follows: x q ¼ 0 þ iD , i ¼ 0 ; 1 ; / ; L 1, where L ¼ 2 3 ¼ 8 and i is the integer corre-

sponding to the 3-bit binary code. Table 2.1 details quantization for each input signal subrange.
Similarly, a 3-bit bipolar quantizer and binary codes are shown in Figure 2.30 , where we have
x min ¼ 4 D , x max ¼ 4 D , and m ¼ 3. The corresponding quantization table is given in Table 2.2 .

Binary code

x q

111

7 Δ

110

6Δ

101

5Δ

100

4 Δ

011

3 Δ

010

2 Δ

001

Δ

000 x

0 Δ 2 Δ 3Δ 4 Δ 5Δ 6Δ 7 Δ 8Δ

e q

Δ /2

−Δ

x

/2

FIGURE 2.29
Characteristics of the unipolar quantizer.