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Sec. 7–7 Output Signal-to-Noise Ratio for PCM Systems 527
take on values of +1 or -1. For simplicity, we assume that the PCM code words are related to
the quantized values by † n
Q(x k ) = V a a kj A 1 2B j
(7–71)
j=1
For example, if the PCM word for the kth sample happens to be (+1, +1 , Á , +1), then the
value of the quantized sample will be
Q(x k ) = Va 1 2 + 1 2 + Á + 1 2 n b = V 2 a1 + 1 2 + 1 4 + Á +
From Appendix A, we find that the sum of this finite series is
1
2 n-1 b
Q(x k ) = V A 1 2B n
2 c - 1
1
2 - 1 d = V - V 2 n
where d is the step size of the quantizer (Fig. 7–16). Thus, the PCM word (+1, +1 , .. . , +1)
represents the maximum value of the quantizer, as illustrated in Fig. 7–16. Similarly, the level
of the quantizer corresponding to the other code words can be obtained.
Referring to Fig. 7–15 again, we note that the analog sample output of the PCM system
for the kth sampling time is
y k = x k + n k
where x k is the signal (same as the input sample) and n k is noise. The output peak signal power
to average noise power is then
a S (7–72)
N b = [(x k) max ] 2
= V2
2 2
pk out n k n k
where (x k ) max = V, as is readily seen from Fig. 7–16. As indicated in Chapter 3, it is assumed
that the noise n k consists of two uncorrelated effects:
• Quantizing noise that is due to the quantizing error:
e q = Q(x k ) - x k
• Noise due to bit errors that are caused by the channel noise:
e b = y k - Q (x k )
Thus,
n 2 k = e 2 2
q + e b
(7–73)
(7–74)
(7–75)
First, evaluate the quantizing noise power. For a uniformly distributed signal, the quantizing
noise is uniformly distributed. Furthermore, as indicated by Fig. 3–8c, the interval of
† For mathematical simplicity, natural binary coding levels are used in Eq. (7–71), not the Gray coding used
in Table 3–1.