mohatta2015.pdf

164 ADVANCED TOPICS
Now we can focus on determining joint likelihoods of valid state transitions and
the received samples,
σ(πι', m) = Pr{S t _i = m', S t = m, Y{}. (7.29)
We can expand the set of received samples Υζ into three subsets: Y/ -1 , r t , and
Yt + \- Then (7.29) with some reordering and grouping of terms becomes
By defining
a(m',m) =Pr{(5e = m,r t ,y t ; i ),(5 t _ 1 =m',y l t - 1 )}. (7.30)
and applying (7.22), we obtain
where
A = (S t = m,r t ,Y t
T
+1 ) (7.31)
B = (St-i = m 1 ,Υ/" 1 ) (7.32)
q(m',m) = Pr{5 ( _, = m',Y^l}PT{S
t = m,r t ,Y t
T
+1 \St-i = m', Y/" 1 }
= a t -i(m')Pr{St = m,r t ,Y? +1 \S t -i=m'}, (7.33)
a t {m)è:P I {S t = m,Y 1 t }. (7.34)
Notice that in the second term on the right-hand side, we dropped Y(~ l in the
conditioning term. This has to do with the Markov property of the signal model
in (7.21). Specifically, if we are given the previous state S t _i, then the previous
set of samples Y/ -1 tells us no additional information about S t = m, r t , or Yf +1 .
Because the noise samples are assumed independent, the noise on Y"/ -1 tells us
nothing about the noise on r t ,or Y^+i . While Y{~ 1 could tell us something about
fe ( _i, which affects S t = m and r t , we are already given the value of bt-i by being
given 5(_i. Thus, we can drop Yj t_1 term from the conditioning.
Next, we define
and apply (7.22), giving
A = (Y^St-^m') (7.35)
B = (S t = m,r t \St-i = m') (7.36)
a(m',m) = a t -i{m')PT{S t — m,r t \S t -i = m'yPr{Yf +1 \S t -i = m',S t = m,r t }
where
= a t -i(m')7t(m',m)PT{Y t
T
+1 \S t =m}
= at-i{m')~tt(m',m)ß t (m), (7.37)
lt{m',m) â P T {S t =m,r t \St-i=m'} (7.38)
ß t (m) â Pr{Y; +l \S t = m}. (7.39)
Again, we have used the Markov property by dropping S t -i = m' and r t from
Pr{Y¿!J. 1 \S t -i = m', S t = m, r t }. We now examine how to compute a t (m), 7i(m', m),
and ßt(m).