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

14.1 1 Turbo Codes 849
derived from the basic modulation and channel information:
Yi (m', m) p (S; = m, ri I S;-1 = m 1 )
= p (r; 1si- l = m 1 , Si = m ) . P[S; = m 1si-l = m 1 ]
= p (r; ic;[nl, mJ) • P[d; = u ] (14.57)
where c;[nl, m] is the codeword from the encoder output corresponding to the state transition
from n/ to m, where as d; = u is the corresponding input bit. To determine Yi (m 1 , m) for d i = u
according to Eq. (14.57), P[r; ic;[m t , m]l is determined by the mapping from encoder output
c; [m', m] to the modulated symbol b; and the the channel noise distribution w; .
In the special case of the convolutional code in Fig. 14.5, for every data symbol d i , the
convolutional encoder generates two coded bits {Vi,!, v;,2). The mapping from the coded bits
{v;,1, v;,2} to modulated symbol(s) b; depends on the modulations. In BPSK, then each coded
bit is mapped to ±1 and b; has two entries
b; = [ 2vi, 1 - 1 ]
2v;,2 - 1
If QPSK modulation is applied, then we can use a Gray mapping
b; =<!¢;
where
,
</>; o
= rr:/2,
Tr,
{
-Tr /2,
{v;,1, vi,2} = {O, 0}
{v;,1, v;,2} = {O, 1}
{vu, v;,2} = {l, I}
{ v;, 1, v;,2} = { 1, O}
Hence, in a baseband AWGN channel, the received signal sample under QPSK is
(14.58)
in which w; is the complex, i.i.d. channel noise with probability density function
1
(
lx1 2 )
Pw (x) = rr: N exp - N
As a result, in this case
p (r; j c; [m', m]) = p h Id; = u)
= p (r; lb;= J<t>;)
= Pw (r; - JEsJ<I>;)
1
= - - exp - Ir; - J<t>; 1 2 )
rr:N
N
(
(1 4.59)