TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 202

An analogous expression gives I ( d , h ) in terms of Pr( hjd ). Show that

I ( d , h ) ¼ I ( d , y ) :

3.4 Let t be a log extrinsic ratio whose probability distribution, conditioned on
d ¼ 2 1, is:

( t þ 0 : 5 6 2 ) 2
2 6 2

1

2 p 6 exp

Pr( tjd ¼ 1) ¼ :

Let t be the corresponding extrinsic probability

e t
1 þe t :

t ¼

(a) Show that t is a bounded variable

0 t 1 :

(b) Show that t increases monotonically with t

dt
dt . 0 :

(c) Show that the probability distribution function for t (conditioned on
d ¼ 2 1) is given by

" #

2

. 2 6 2

6 2
2

1 t

1 t þ

2 p 6t (1 t ) exp log

Pr( tjd ¼ 1) ¼ :

Hint : Use the fact that if t is a unique function of t , that is, t ¼ f ( t ) where
f ( . ) is monotonic, then Pr( tjd ) ¼ Pr( tjd ) =jdf =dtj .

3.5 Consider the simplified schematic of Figure 3.30 in which the sequence ( y i )is
obtained from an additive white Gaussian noise channel

y i ¼ d i þb i , i ¼ 1, 2, ... , N:

Here b i has zero mean and variance s 2 , the bits d 1 , ... , d K are the antipodal
forms of the information bits, and the remaining terms d Kþ 1 , ... , d N are the anti-
podal forms of the parity-check bits, and are thus uniquely determined from
d 1 , ... , d K . The log extrinsic ratios ( t i ) are produced by the turbo decoder,
whose input is the sequence ( y i ) alone. The variables ( y i ) and ( t i ) are assumed