mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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162 ADVANCED TOPICS<br />
If we assume no prior information (all sequences equi-likely), then (7.19) simplifies<br />
to<br />
P{s = h(m)\r{t) Vi} = ^ Pr{r(t) \/t\s(m) = h(m),s m = h m }, (7.20)<br />
where the common sequence likelihood has been dropped. Thus, when symbols are<br />
all equi-likely, MAPSD becomes ML symbol detection.<br />
Recall that MLSD had an efficient implementation, the Viterbi algorithm. MAPSD<br />
has a similar, efficient form, the BCJR algorithm. After introducing the BCJR algorithm,<br />
certain approximate MAPSD forms are introduced.<br />
7.3.1.1 The BCJR algorithm To explain the BCJR algorithm, it helps to consider<br />
the three-path example used to explain the Viterbi algorithm. However, to make<br />
the notation closer to that used in the original explanation of the BCJR algorithm<br />
[Bah74], we rewrite (6.30) as<br />
r t μ cb t + dbt-i + efe t _ 2 + n t , (7.21)<br />
where b t is a binary symbol (+1 or —1) and t is a discrete symbol period index.<br />
The trellis diagram is shown in Fig. 6.8, which is reproduced in Fig. 7.1 using the<br />
new notation.<br />
We assume we have a set of received samples from ί = 1 through t = r, denoted<br />
Y{. Consider detecting bit b t -i- MAPSD involves determining the larger<br />
of two conditional bit likelihoods: Pr{b É _i = +1|V7} and Pr{b s _i = -1|Y7}. As<br />
computing these two quantities is similar, we will focus on the first.<br />
First, it is convenient to consider joint probabilities instead of conditional probabilities.<br />
Here, and elsewhere, we will use the fact that for events A and B,<br />
PT(A, B) = Ρτ(Α\Β)Ρτ(Β) = Ρτ{Β)Ρτ{Α\Β), (7.22)<br />
where (A, B) denotes A and B. We can rewrite (7.22) as<br />
Using (7.23), we can write<br />
Ρτ{Α\Β) = Ρτ(Α,Β)/Ρτ{Β). (7.23)<br />
Pr{6 t _! = +1|*7} = Pri&t-j = +l,y 1 T }/Pr{y 1 r } (7.24)<br />
The term in the denominator will be the same for both bit values and will not<br />
impact which one is larger, so we can drop it. This leaves us with computing the<br />
symbol metric<br />
P(bt-i = +1) = Pr{& ( -i = +1, Y{}- (7-25)<br />
Second, instead of writing the symbol likelihood directly in terms of the set of<br />
received samples Y{, we can write it in terms of intermediate quantities, state<br />
transition likelihoods. To do this, we need to set up some notation. As shown in<br />
Fig. 7.1, we use m! to identify a state value for the state at time t — 1, denoted<br />
S t _i. Similarly, we use m to identify a state value for the state at time t, denoted<br />
St. As there are four possible states at each time, the possible values for m' and m