mohatta2015.pdf

THE MATH 165
Let's start with the "gamma" term, 7t(m', m). By defining
A = (r t |5 t _ 1 =m') (7.40)
B = {S t = m|St_i = m') (7.41)
and applying (7.22), we obtain
7t(m', m) = Pr{r t |5 f _! = m', S t = m}Pr{5 t _i = m', 5 ( = m} (7.42)
From the model in (7.21), this becomes
, 1 Í \r t -cq t {m',m) - dq t -i{m',m) - eq t - 2 (m', m)\ 2 \
xPr{S t _ 1 = m',S t = m}, (7.43)
where qt{m', m) is the hypothesized value for b t corresponding to the state transition
from St-i = m' to S¡ = m.
Thus, the gamma term consists of two parts. Notice that the first part is a
likelihood of r t , conditioned on hypothesized values for the bit sequence. The log
of this part, dropping the log of 1/(πΝη), gives the Euclidean distance metric used
in MLSD. The second part is the a priori or prior likelihood of the state transition.
Here is where prior information can be introduced, if available. Otherwise, this part
will be the same for all valid state transitions (it is zero for invalid transitions). For
the case of no prior information, we can redefine
, , Λ_ πν/ / N / \n -cq t {m',m) - dq t -i(rn',m) - eg t _ 2 (m',m)| 2 Ί
7 t (m , m) = lr(m , m) exp i ^ '- ^ i '- ^ .
(7.44)
Notice we have ignored the common scaling by the valid state transition probabilities
and represented the fact that some are invalid by using the previously defined
Tr(m', m) function.
Next consider the "alpha" term, a t {m). Splitting Y{ into F/ -1 and r t , using
(7.26), and defining the B¡ to be all possible past state values, the expression in
(7.34) becomes
a t (m) = Σ Pr{S t = m,S t -i = m',Y*-\r t }. (7.45)
m'=()
Now we can use (7.22) with
A = (S t = m,r t ) (7.46)
B = (St-^m'tf- 1 ), (7.47)
so that (7.45) becomes
:¡
a t (m) = Σ Pr{S t _i = m'rf-^PriSt = m,r t \S t -i = τη',Κ/" 1 }
m'=()
3
= Σ a t -i(m')Pr{S t = m,r t \S t -i = m'}
m'=i)
:¡
= Σ at-i(m')7t(m',m). (7.48)
m'=(l