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

834 ERROR CORRECTING CODES
Figure 14. 1 0
Continued
Received bits: 01
0/00
10 11
0/00 2 0/00
10 00 00
1/11
b: [QI]
min (2 + 2, 2 + 0)
c:
[iQJ
QJ min (2 + I, 3 + I )
d:
DIJ
(d)
Received bits: 01
10 1 1
10 00 00 Optimal path
a:
c:
(I)min (2 +1,3 cl)
(c)
In the preceding example, we have illustrated how to progress from one stage to the next
by determining the optimum path (survivor) leading to each of the state. When these survivors
do merge, the merged branches represent the most reliable MLD outputs. For the later stages
that do not exhibit a merged path, we are ready to make a maximum likelihood decision based
on the received data bits up to that stage. This process, known as truncation, is designed to a
force a decision on one path among all the survivors without leading to a long decoding delay.
One way to make a truncated decision is to take the minimum distance path as in Eq. (14.32).
Another alternative is to rely on extra codeword information. In Fig. 14. lOe, if the encoder
always forces the last two data digits to be 00, then we can consider only the survivor ending at
state a.
With the Viterbi algorithm, storage and computational complexity are proportional to 2!'-1
and are very attractive for constraint length N < 10. To achieve very low error probabilities,
longer constraint lengths are required, and sequential decoding (to be discussed next) may
become attractive.
Sequential Decoding
In sequential decoding, a technique proposed by Wozencraft, the complexity of the decoder
increases linearly rather than exponentially. To explain this technique, let us consider an encoder