B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

14.1 1 Turbo Codes 847
following simplifications of the conditional probabilities:
p (1\+1:N I S; = m, Si- 1 = m', ru) = p (ri+l :N 1s1 = m )
p (r; , S; = m jSi- 1 = m', T!:i- 1) = p (r;, S; = m!S;-1 = m' )
(14.49a)
(14.49b)
The MAP detector needs to determine the log-likelihood ratio
"' P[di = Ilr J p (d; = l, r )
A (d; ) = log
➔ = log ➔
P[d; = 0 I r] p (di = 0, r)
(14.50)
We are now ready to explain the operations of the BCJR algorithm. First, let Qi (u) denote
the set of all possible state transitions from S;- 1 = m' to S; = m when d; = u (u = 0, 1).
There are only two such sets for d; = 1 and di = 0. We can see that
p (d; = l, r ) = L p(S;- 1 = m', S; = m, r)
(m 1 , m)Erl;(l)
L p (si-1 = m', ru, S; = m, Ti+l :N )
(m', m) Erl; (1)
L P (si-1 = m', ru- 1 , S; = m, r; )
(m', m)Erl; (1)
. p ( r;+l :N 1 s1 -1 = m', TJ:i, S; = m)
(14.51)
Applying Eqs. ( 14.49a) and ( 14.49b) to the last equality, we have
p (d; = 1, r ) = L p (S1- 1 =m', hi- I , Si = m, r; ) ·P(ri+IN 1si = m ) (14.52)
(m 1 ,m)EQ;(l)
L p (si- 1 = m', T!:i- 1) · p (S; = m, r; jsi-1 = m')
(m', m)Erl;(l)
· p (ri+l :N IS; = m )
Applying the notations used by Bahl et al., 13 we define
a;-1 (m') p (si- 1 = m', ru- 1)
{3;(m) p (rt+1:N ISi = m )
y; (m', m) p (S; = m, ri l si-1 = m')
(14.53a)
(14.53b)
(14.53c)
Given the notations in Eq. (14.53), we can use Eqs. (14.50) to ( 14.52) to write the LLR of each
information bit di as
A (d· ) = log
l
L (
1
-m_, m_E_, ________ _
I
L
(m' , m)Erl;(O)
) n ( l )
a;- 1 (m')Yt (m', m)f3;(m)
a;-1 (m ) y; (m , m)f3;(m)
I
(14.54)