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

838 ERROR CORRECTING CODES
· There are min(2 k , 2 n -k ) possible states in the trellis.
· The state of a codeword at instant i is determined by the codeword and the parity check
matrix by syndrome from the first codeword bit to the ith codeword bit:
➔ ➔ ➔
z; = c1h; EB c2h2 EB · · · EB c;h;. (14.33)
Note that this syndrome trellis, unlike the state transition trellis of convolutional code, is
typically nonrepeating. In fact, it always starts from the "zero" state and ends in "zero" state.
Indeed, this trellis is a time-varying trellis. We use an example to illustrate the construction
of a syndrome trellis.
Example 14.8 Consider a Hamming (7, 4, 3) code with parity check matrix
1
1 1 6]
1 0 1 0 0 1
(14.34)
Sketch the trellis diagram for this block code.
For this code, there are 3 error syndrome bits defining a total of 2 3 = 8 states. Denote the
eight states as (So, S1 , S2, S3, S 4 , Ss, S6, S1 ). There are 2 k = 2 4 = 16 total codewords
with 7 code bits that are in the null-space of the parity check matrix H. By enumerating
all 16 codewords, we can follow Eq. (14.33) to determine all the paths through the trellis.
The corresponding time-varying trellis diagram is shown in Fig. 14. 14. Notice that
each path corresponds to a codeword. We always start from state So initially and end at
the state So . Unlike the case of convolutional code, it is not necessary to label the trellis
branches in this case. Whenever there is a state transition between different states, the
branch automatically corresponds to a "1" code bit. When a state stays the same, then the
transition branch corresponds to a "O" code bit.
Figure 14. 14
Trellis diagram of
a Hamming
(7, 4, 3) code
with parity check
matrix of
Eq. (14.34).
So
S1
S2
Kl S3
vi S4
S5
s6
S7
Code bits
C 1 C 2 C 3 C4 C 5 C 6 C 7
Once we have a trellis diagram, the Viterbi decoding algorithm can be implemented for the
MLD of the block code at reduced complexity. Maximum likelihood detection of block codes
can perform better than a syndrome-based decoder. Keep in mind that the example we show
is a very short code that does not benefit from Viterbi decoding. Clearly, the Viterbi algorithm
makes more sense when one is decoding a long code.