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Performance of Communication Systems Corrupted by Noise Chap. 7
the uniform distribution is (-d/2, d/2), where d is the step size (d = 2 in Fig. 3–8c). Thus,
with M = 2 n = 2V/d (from Fig. 7–16),
q
d/2
e 2 2
2
q = e q f(e q ) de q = e 1 q (7–76)
L - q
L -d/ 2 d de q = d2
12 = V2
3M 2
The noise power due to bit errors is evaluated by the use of Eq. (7–74), yielding
e 2 b = [y k - Q(x k )] 2
(7–77)
where Q (x k ) is given by Eq. (7–71). The recovered analog sample y k is reconstructed from the
received PCM code word using the same algorithm as that of Eq. (7–71). Assuming that the
received PCM word for the kth sample is (b k1 , b k2 , .. . , b kn ), we see that
y k = V a
n
j=1
b kj A 1 2B j
(7–78)
The b’s will be different from the a’s whenever there is a bit error in the recovered digital
(PCM) waveform. By using Eqs. (7–78) and (7–71), Eq. (7–77) becomes
e 2 b = V 2 n
c a (b kj - a kj )A 1 2B j 2
d
j=1
= V 2 n n
a a ab kj b k/ - a kj b k/ - b kj a k/ + a kj a k/ b2 -j-/ (7–79)
j=1 /=1
where b kj and b k are two bits in the received PCM word that occur at different bit positions
when j Z . Similarly, a kj and b k are the transmitted (Tx) and received (Rx) bits in two different
bit positions, where j Z (for the PCM word corresponding to the k th sample). The
encoding process produces bits that are independent if j Z . Furthermore, the bits have a
zero-mean value. Thus, b kj b k/ = b kj b k/ = 0 for j Z . Similar results are obtained for the
other averages on the right-hand side of Eq. (7–79). Equation (7–79) becomes
e 2 b = V 2 n
2
a ab kj - 2a kj b kj + a 2 kj b2 -2j
(7–80)
j=1
Evaluating the averages in this equation, we obtain †
b 2 kj = (+1) 2 P(+1Rx) + (-1) 2 P(-1Rx) = 1
2
a kj = (+1) 2 P(+1Tx) + (-1) 2 P(-1Tx) = 1
a kj b kj = (+1) (+1)P(+1Tx, +1Rx) + (-1)(-1)P(-1Tx, -1Rx)
+ (-1)(+1)P(-1Tx, +1Rx) + (+1)(-1)P(+1Tx, -1Rx)
= [P(+1Tx, +1Rx) + P(-1Tx, -1Rx)]
- [P(-1Tx, +1Rx) + P(+1Tx, -1Rx)]
= [1 - P e ] - [P e ] = 1 - 2P e
† The notation +1Tx denotes a binary 1 transmitted, -1Tx denotes a binary 0 transmitted, +1Rx denotes a
binary 1 received, and -1Rx denotes a binary 0 received.