mohatta2015.pdf

THE MATH 163
Figure 7.1
MAPSD trellis diagram, three-path channel.
are 0, 1,2, and 3. One possible transition is S t -i = 0, S t = 0. This corresponds to
the sequence 6t-2 = +1> öt—i = +1, bt = +1. Notice from the trellis diagram that
not all state transitions are possible, as the values for b t -i for states at times ί — 1
and t must be consistent.
Now we are ready to write the symbol metric in terms of state transition likelihoods.
Recall the fact from probability theory that
Ρτ{Α} = Σ PI{A\BÍ}PI{BÍ} (7.26)
i
where the ß; are independent and span the probability space. By defining the B¡
to be the state transitions, the symbol metric in (7.25) becomes
3 3
P(bt-i = +1) = Σ Σ ϊΜ 5 «-! = m '> S * = m < 6 *-i = +1 > y i T }· ( 7·27 )
Prom the trellis diagram, we see that half of the terms on the right-hand side will
be zero, as the state transition and bt-i value are inconsistent. When they are
consistent, we have the added requirement that bt-i = +1. Thus, we can write
(7.27) as
3 3
P(6 t _! =+1) = J2 ^ Pr{5 t _i =m',S t = m,F T 1 }Tr(m',m)/(m',m,+l),
τη'=0 m=0
(7.28)
where Tr(m', m) denotes a function based on the trellis that is 1 if the transition
is valid and zero otherwise. The term I(m',m, +1) is 1 if the transition is valid
and 6 t _i = +1. While the term Tr(m',m) is redundant in this case, we will find it
useful later.