15.3 DIGITAL COMMUNICATION SYSTEMS 713Quantization Error
Sampling followed by quantization is equivalent to quantization followed by sampling. Figure
15.3.3 illustrates a message signalf(t) and its quantized version denoted byfq(t). The difference
betweenfq(t) andf(t) is known as thequantization errorεq(t),
εq(t)=fq(t)−f(t) (15.3.3)
Theoretically,fq(t) can be recovered in the receiver without error. The recovery offq(t) can
be viewed as the recovery off(t) with an error (or noise)εq(t) present. For a smallδvwith a large
number of levels, it can be shown that the mean-squared value ofεq(t) is given by
εq^2 (t)=(δv)^2
12(15.3.4)When a digital communication system transmits an analog signal processed by a uniform
quantizer, the best SNR that can be attained is given by
(
So
Nq
)
=f^2 (t)
εq^2 (t)=12 f^2 (t)
(δv)^2(15.3.5)whereS 0 andNqrepresent the average powers inf(t) andεq(t), respectively. Whenf(t) fluctu-
ates symmetrically between equal-magnitude extremes, i.e.,−|f(t)|max≤f(t)≤|f(t)|max,
choosing a sufficiently large number of levelsL, the step sizeδvcomes out as
δv=2 |f(t)|max
L(15.3.6)and the SNR works out as
(
S 0
Nq
)
=
3 L^2 f^2 (t)
|f(t)|^2 max(15.3.7)By defining the messagecrest factor KCRas the ratio of peak amplitude to rms value,
KCR^2 =|f(t)|^2 max
f^2 (t)(15.3.8)Equation (15.3.7) can be rewritten as
Quantized
message fq(t)Quantization
error εq(t)Message f(t)δvδvttFigure 15.3.3Message signalf(t), its quantized
versionfq(t), and quantization errorεq(t).