B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

6.2 Pulse Code Modulation {PCM) 273
Assuming that the error is equally likely to lie anywhere in the range (- 1'1v/2, 1'1v/2), the
mean square quantizing error q 2 is given by*
q 2 = _ 1 1 1:lv/2
q 2 dq
1'1v - 1:lv/2
(1'1v) 2
(6.32)
12
m 2
(6.33)
3L 2
Because q 2 (t) is the mean square value or power of the quantization noise, we shall denote it
by Nq ,
Assuming that the pulse detection error at the receiver is negligible, the reconstructed signal
m(t) at the receiver output is
m(t) = m(t) + q(t)
The desired signal at the output is m(t), and the (quantization) noise is q(t). Since the power
of the message signal m(t) is m 2 (t), then
and
(6.34)
In this equation, mp is the peak amplitude value that a quantizer can accept, and is therefore
a parameter of the quantizer. This means S0/N 0 , the SNR, is a linear function of the message
signal power m 2 (t) (see Fig. 6.18 with µ, = 0).
* Those who are familiar with the theory of probability can derive this result directly by noting that the probability
density of the quantization error q is 1/(2m p
/ L) = L/2m p over the range lql ::= mp/L and is zero elsewhere. Hence,