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

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ERROR CORRECTING CODES
14. 10 SOFT-OUTPUT VITERBI ALGORITHM (SOVA)
Chase algorithms can generate multiple candidate codewords and the associated reliability
metrics. The metric information can be exploited by other receiver processing units to determine
the final decoded codeword. If the decoder can produce soft reliability information on
every decoded bit, then it can be much better utilized jointly with other soft-output decoders and
processors. Unlike Chase algorithms, soft-output Viterbi algorithms (SOVA) 10 and the maximum
a posterior; (MAP) algorithms are two most general soft decoding methods to produce
bit reliability information. We first describe the principles of SOVA here.
The most reliable and informative soft bit information is the log-likelihood ratio (LLR)
of a particular code bit c; based on the received signal vector
r = (r1, r2, ... , r n )
In other words, the LLR 11 as defined by
P [c; = l l r = r]
A(c;) = log ----­
P [c; = Oi r = r]
(14.40)
indicates the degree of certainty by the decoder on the decision of c; = 1. The degree of
certainty various from -oo when P [c; = Olr] = I to +oo when P [c; = Olr] = 0
Once again, we consider the BPSK case in which (2c; - 1) = ±1 is the transmitted
data and
r; = (2c; - 1) + W; i = 1, 2, ... , n (14.41)
where w1 is the AWGN sample. Similar to the Chase algorithms, the path metric is computed
by the correlation between {r; } and the BPSK signal { c;}. In other words, based on the received
data samples {rd, we can estimate
n2
path metric between stages 11 1 and n2 = L ri • (2cj - 1)
j=n1+l
(14.42)
Like the traditional Viterbi algorithm, the SOVA decoder operates on the corresponding trellis
of the (convolutional or block) code. SOVA consists of a forward step and a backward step.
During the forward step, as in the conventional Viterbi algorithm, SOVA first finds the most
likely sequence (survivor path). Unlike conventional VA, which stores only the surviving path
metrics at the states in the current stage, SOVA stores the metric of every surviving path leading
to a state for all stages.
To formulate the idea formally, denote
Se(i) = state e at stage (time) i
For each survivor at state Se in stage i, we will determine the forward path metric leading to this
state. These forward metrics ending in state £ at time i are denoted as M{ (i). The maximum
total path metric at the final state of the forward VA, denoted M max, corresponds to the optimum
forward path. During the backward step, SOVA then applies VA backward from the terminal