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

14.7 Trellis Diagram of Block Codes 837
Fi g ure 14. 13
Setting the
threshold in
sequential
decoding.
d--
digits d is a straight line (n e = P e d) with slope P e , as shown in Fig. 14.13. The actual number
of errors along the path is also plotted. If the errors remain within a limit (the discard level),
the decoding continues. If at some point the errors exceed the discard level, we go back to the
nearest decision node and try an alternate path. If errors still increase beyond the discard level,
we then go back one more node along the path and try an alternate path. The process continues
until the errors are within the set limit. By making the discard level very stringent (close to the
expected error curve), we reduce the average number of computations. On the other hand, if the
discard level is made too stringent, the decoder will discard all possible paths in some extremely
rare cases of an unusually large number of errors due to noise. This difficulty is usually resolved
by starting with a stringent discard level. If on rare occasions the decoder rejects all paths, the
discard level can be relaxed little by little until one of the paths is acceptable.
It can be shown that the error probability in this scheme decreases exponentially as N,
whereas the system complexity grows only linearly with k. The code rate is rJ ::::: 1 / e. It can be
shown that for rJ < 17 0 , the average number of incorrect branches searched per decoded digit
is bounded, whereas for 7/ > r, 0 it is not; hence 71 0 is called the computational cutoff rate.
There are several disadvantages to sequential decoding:
1. The number of incorrect path branches, and consequently the computation complexity, is a
random variable depending on the channel noise.
2. To make storage requirements easier, the decoding speed has to be maintained at 10 to 20
times faster than the incoming data rate. This limits the maximum data rate capability.
3. The average number of branches can occasionally become very large and may result in a
storage overflow, causing relatively long sequences to be erased.
A third technique for decoding convolutional codes is feedback decoding, with threshold
decoding 6 as a subclass. Threshold decoders are easily implemented. Their performance,
however, does not compare favorably with the previous two methods.
14.7 TRELLIS DIAGRAM OF BLOCK CODES
Whereas a trellis diagram is connected with convolutional code in a direct and simple way, a
syndrome trellis can also be constructed for a binary linear (n, k) block code according to its
parity check matrix 7 H. The construction can be stated as follows:
Let (c1, c2 , ... , e n ) be a codeword of the block code.
Let H = [ h1 h 2 • • • h n ]
be the (n - k) x n parity check matrix with columns {hi}.