TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 161

Figure 3.7 Illustrating a simple 3-tap channel.

as sketched in Figure 3.7. This system admits a state-space description as

¼

x 0, i 1
x 1, i 1

d i

x 0 i
x 1 i

00
10

1
0

þ

x 0, i 1
x 1, i 1

a i ¼ [ h 1 h 2 ] þh 0 d i

with y i ¼ a i þ b i , and can be written in the trellis diagram of Figure 3.8. The figure
also lists the constellation values H 0 , ... , H 7 from which each noise-free output
a i is drawn.
The forward-backward algorithm can again be run as in the previous section. The
forward recursion for the a terms proceeds as

a i (0) ¼ a i 1 (0) g i (0, 0) þa i 1 (2) g i (2, 0)
a i (1) ¼ a i 1 (0) g i (0, 1) þa i 1 (2) g i (2, 1)
a i (2) ¼ a i 1 (1) g i (1, 2) þa i 1 (3) g i (3, 1)
a i (3) ¼ a i 1 (1) g i (1, 3) þa i 1 (3) g i (3, 3)

Figure 3.8 Trellis diagram for the channel of Figure 3.7. Solid lines indicate an input of
þ1 to the channel, while dashed lines indicate 1.

2