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

824 ERROR CORRECTING CODES
Based on Eq. (14.27a)
(14.28a)
(14.28b)
For Pbc « 1, the first term in the summation of Eq. (14.28b) dominates all the other terms,
and we are justified in ignoring all but the first term. Hence,
(14.29a)
for Pbc « 1
(14.29b)
For further comparison, we must assume some specific transmission scheme. Let us consider
a coherent PSK scheme. In this case, for an additive white Gaussian noise (AWGN)
channel,
(14.30a)
and because E b for the coded case is k/n times that for the uncoded case,
Pbc = Q ()
nN
(14.30b)
Hence,
Pec -( n ' ) Hf¥J r
P eu = Q ('5)
(14.31a)
(14.31b)
To compare coded and uncoded systems, we could plot P eu and P ec as functions of the raw
E b /N (for the uncoded system). Because Eqs. (14.31) involve parameters t, n, and k, a proper
comparison requires families of plots. For the case of a (7, 4) single-error correcting code
(t = l, n = 7, and k = 4), P e c and P eu in Eqs. (14.31) are plotted in Fig. 14.3 as a function
of E b /N. Observe that the coded scheme is superior to the uncoded scheme at higher E b /N,
but the improvement (about 1 dB) is not too significant. For large n and k, however, the coded
scheme can become significantly superior to the uncoded one. For practical channels plagued
by fading and impulse noise, stronger codes can yield substantial gains, as shown in our next
example.