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