Signals and Systems - Electrical Engineering

442 CHAPTER 7: Sampling Theory

FIGURE 7.12

Four-level quantizer and coder.

Δ

Δ

10

11

01

00

− 2 Δ −Δ x(nTs)

−Δ

− 2 Δ

2 Δ

x∧(nTs)

That is, 1 is assigned so as to cover the possible peak-to-peak range of values of the signal, or its

dynamic range. To each of the levels a binary code is assigned. The code assigned to each of the

levels uniquely represents the different levels [− 21 ,− 1 , 0, 1 ]. As to the way to approximate the

given sample to one of these levels, it can be done byroundingor bytruncating. The quantizer shown

in Figure 7.12 approximates by truncation—that is, if the samplek 1 ≤x(nTs) < (k+ 1 )1, fork=

−2,−1, 0, 1, then it is approximated by the levelk 1.

To see the quantization, coding, and quantization error, let the sampled signal be

x(nTs)=x(t)|t=nTS

The given four-level quantizer is such that

k 1 ≤x(nTs) < (k+ 1 )1 ⇒ ˆx(nTs)=k 1 k=−2,−1, 0, 1 (7.24)

where the sampled signalx(nTs)is the input and the quantized signalxˆ(nTs)is the output. Therefore,

− 21 ≤x(nTs) <− 1 ⇒ ˆx(nTs)=− 21

− 1 ≤x(nTs) < 0 ⇒ ˆx(nTs)=− 1

0 ≤x(nTs) < 1 ⇒ ˆx(nTs)= 0

1 ≤x(nTs) < 21 ⇒ ˆx(nTs)= 1

To transform the quantized values into unique binary 2-bit values, one could use a code such as

xˆ(nTs) = − 21 ⇒ 10

xˆ(nTs) = − 1 ⇒ 11

xˆ(nTs) = 01 ⇒ 00

xˆ(nTs) = 1 ⇒ 01

which assigns a unique 2-bit binary number to each of the four quantization levels.

If we define the quantization error as

ε(nTs)=x(nTs)−ˆx(nTs)