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

14. 1 1 Turbo Codes 853
whereas from encoder 2, the channel outputs are
r;,s = /Ei, (2d; - l) + w;
(2) /Ei, (2)
r . P
= Eb (2p. - 1) + w; 2
I, l '
(14.63a)
(14.63b)
Note that the Gaussian noises w;, w;,1 , and w;, 2 are all independent with identical Gaussian
distribution of zero mean and variance N /2. The first BCJR decoder is given signals r;,s and
rr to decode, whereas the second BCJR decoder is given signals r;. s and rr; to decode.
Let's first denote p;[m', ml as the ith parity bit at a decoder corresponding to message bit
d; . It naturally corresponds to the transition from state m' to state m. For each decoder, the
received channel output signals r; = [r;, s , r;,pl specifies y; (m', m) via
y; (m', m) = p(r; j c; [m', m])P(d;)
= p (r;, s , r;,p ld;, p; [m', ml) P(d;)
= _ l_ exp [- _ l r _ ;,s_ - __E_b ( _ 2 _ d; _ - _l ) _ l 2 _ + _1 r ;,p - - - E_ b _ (2 _ p _ ; [ _ m _' , _ m_l _l_ ) _ l 2 ] P(d;)
nN
N
1
= nN
exp -
[
rf,., + Y(p + 2Eb l 2 ]
{ 2
N
exp
[r;, s (2d; - I)+ r;,p (2p;[m , ml - 1)]
x P(d;) (14.64)
Notice that the first term in Eq. (14.64) is independent of the codeword or the transition from
m' to m. Thus, the LLR at this decoder becomes
A(d; ) = log
L
'°"'
(m', m) EQ;( l)
L..., (m', m) EQ;(O)
P[d·-l)
P[d; =OJ
i:x;- 1 (m 1 )p (r; l c;[m', m]) P[d; = l l/3; (m)
i:x; - 1 (m')p (r; l c;[m', ml) P[d; = Ol/3; (m)
L / {2 [ I 11}
1
L
}
' '
1
}
(14.65)
, . i:x;- 1 (m )exp ----;:-; r;_,+2r;,p p; [m , ml J
/3; (m)
,
{ 2 [
,- (111 , m) EQ,(]) JV
= log --- + log-----------'-------------'---
(111', m) EQ; (O)
i:x;- 1 (m )exp -
N
L-r; s +2r; p p;[m , mJ J
/3;(m)
By defining the gain parameter s = 4 /N, we can simplify the LLR into
A (d; ) = log
P[d· , l j L
- ,
+ S . r;,s + log __
CXi- 1 (m') exp (s · r;,p p;[m', ml) /3;(m)
(_m _ ,_n)_E_Q ; (l) _________ _ __ _
P[d; = 0] .___, '°"' a -_ 1 (m') exp (r . r· p ·[m' m]) R-(m)
'----,.- L..., t
A ( c)
( m , m E><,
A ( a ) A (el
)
r, · (O )
l t,p l , /-'l
(14.66)
In other words, for every information bit d;, the LLR of both decoders can be decomposed
into three parts as in
j = 1, 2