mohatta2015.pdf

162 ADVANCED TOPICS
If we assume no prior information (all sequences equi-likely), then (7.19) simplifies
to
P{s = h(m)\r{t) Vi} = ^ Pr{r(t) \/t\s(m) = h(m),s m = h m }, (7.20)
where the common sequence likelihood has been dropped. Thus, when symbols are
all equi-likely, MAPSD becomes ML symbol detection.
Recall that MLSD had an efficient implementation, the Viterbi algorithm. MAPSD
has a similar, efficient form, the BCJR algorithm. After introducing the BCJR algorithm,
certain approximate MAPSD forms are introduced.
7.3.1.1 The BCJR algorithm To explain the BCJR algorithm, it helps to consider
the three-path example used to explain the Viterbi algorithm. However, to make
the notation closer to that used in the original explanation of the BCJR algorithm
[Bah74], we rewrite (6.30) as
r t μ cb t + dbt-i + efe t _ 2 + n t , (7.21)
where b t is a binary symbol (+1 or —1) and t is a discrete symbol period index.
The trellis diagram is shown in Fig. 6.8, which is reproduced in Fig. 7.1 using the
new notation.
We assume we have a set of received samples from ί = 1 through t = r, denoted
Y{. Consider detecting bit b t -i- MAPSD involves determining the larger
of two conditional bit likelihoods: Pr{b É _i = +1|V7} and Pr{b s _i = -1|Y7}. As
computing these two quantities is similar, we will focus on the first.
First, it is convenient to consider joint probabilities instead of conditional probabilities.
Here, and elsewhere, we will use the fact that for events A and B,
PT(A, B) = Ρτ(Α\Β)Ρτ(Β) = Ρτ{Β)Ρτ{Α\Β), (7.22)
where (A, B) denotes A and B. We can rewrite (7.22) as
Using (7.23), we can write
Ρτ{Α\Β) = Ρτ(Α,Β)/Ρτ{Β). (7.23)
Pr{6 t _! = +1|*7} = Pri&t-j = +l,y 1 T }/Pr{y 1 r } (7.24)
The term in the denominator will be the same for both bit values and will not
impact which one is larger, so we can drop it. This leaves us with computing the
symbol metric
P(bt-i = +1) = Pr{& ( -i = +1, Y{}- (7-25)
Second, instead of writing the symbol likelihood directly in terms of the set of
received samples Y{, we can write it in terms of intermediate quantities, state
transition likelihoods. To do this, we need to set up some notation. As shown in
Fig. 7.1, we use m! to identify a state value for the state at time t — 1, denoted
S t _i. Similarly, we use m to identify a state value for the state at time t, denoted
St. As there are four possible states at each time, the possible values for m' and m