TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 162

while the backward recursion for the b terms then runs as

b i 1 (0) ¼ g i (0, 0) b i (0) þg i (0, 1) b i (1)
b i 1 (1) ¼ g i (1, 2) b i (2) þg i (1, 3) b i (3)
b i 1 (2) ¼ g i (2, 0) b i (0) þg i (2, 1) b i (1)
b i 1 (3) ¼ g i (3, 2) b i (2) þg i (3, 3) b i (3) :

The changes here concern the channel likelihood evaluations g i ( m 0 , m ), of which
there are eight at each trellis section. These may be tabulated as follows.

g i (0, 0) ¼ Pr( d i ¼þ 1) N i ( H 0 , s 2 ), H 0 ¼ h 0 þh 1 þh 2
g i (0, 1) ¼ Pr( d i ¼ 1) N i ( H 1 , s 2 ), H 1 ¼h 0 þh 1 þh 2
g i (1, 2) ¼ Pr( d i ¼þ 1) N i ( H 2 , s 2 ), H 2 ¼ h 0 h 1 þh 2
g i (1, 3) ¼ Pr( d i ¼ 1) N i ( H 3 , s 2 ), H 3 ¼h 0 h 1 þh 2
g i (2, 0) ¼ Pr( d i ¼þ 1) N i ( H 4 , s 2 ), H 4 ¼ h 0 þh 1 h 2
g i (2, 1) ¼ Pr( d i ¼ 1) N i ( H 5 , s 2 ), H 5 ¼h 0 þh 1 h 2
g i (3, 2) ¼ Pr( d i ¼þ 1) N i ( H 6 , s 2 ), H 6 ¼h 0 h 1 þh 2
g i (3, 3) ¼ Pr( d i ¼ 1) N i ( H 7 , s 2 ), H 7 ¼h 0 h 1 h 2

(3 : 4)

in which each Gaussian term is summarized using the notation

( y i H j ) 2
2 s 2

1

N i ( H j , s 2 ) ¼

2 p s exp

:

The a posteriori probabilities then become

Pr( d i ¼þ 1 jy ) / a i 1 (0) g i (0, 0) b i (0) þa i 1 (1) g i (1, 2) b i (2)
þa i 1 (2) g i (2, 0) b i (0) þa i 1 (3) g i (3, 2) b i (2)
Pr( d i ¼ 1 jy ) / a i 1 (0) g i (0, 1) b i (1) þa i 1 (1) g i (1, 3) b i (3)
þa i 1 (2) g i (2, 1) b i (1) þa i 1 (3) g i (3, 3) b i (3)

in which the former (respectively, latter) probability is obtained by summing terms
a i 2 1 ( m 0 ) g i ( m 0 , m ) b i ( m ) over transitions ( m 0 , m )provokedby d i ¼ þ 1 (resp. d i ¼ 2 1).
Each g i term in the expression for Pr( d i ¼ þ 1 jy ) [resp. Pr( d i ¼ 2 1 jy )] contains a
factor Pr( d i ¼ þ 1) [resp. Pr( d i ¼ 2 1)]. As such, we may rewrite these posterior
probabilities as

Pr( d i ¼þ 1 jy ) / Pr( d i ¼þ 1) T i ( d i ¼þ 1)
Pr( d i ¼ 1 jy ) / Pr( d i ¼ 1) T i ( d i ¼ 1)