TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 201

Show that

2 y i
s 2 ¼

2 d i
s 2 þ

2 b i
s 2 :

h¼

(b) Introduce the conditional means m þ and m 2 as

m þ ¼ E ( hjd i ¼þ 1), m ¼ E ( hjd i ¼ 1)

where the expectation is with respect to the probability density function of
b i . Show that

2
s 2 :

m þ ¼m ¼

(c) Introduce the conditional variances 6 2
þ

and 6 2 as

6 2

þ ¼ E [( hm þ ) 2

jd i ¼þ 1], 6 1 ¼ E [( hm ) 2

jd i ¼ 1] :

Show that

4
s 2

6 2

þ ¼ 6 2 ¼

so that the conditional mean is half the conditional variance (give or
take a sign factor). What happens to the conditional means and variances
as s 2 ! 0?

3.3 Consider the log likelihood ratio as in Problem 3.2 (dropping the subscript i for
convenience)

h¼ log Pr( y j d ¼þ 1)

2 y
s 2 :

Pr( yjd ¼ 1) ¼

The variable h is then characterized by a conditional variance parameter 6 h ¼
4 =s 2 . Assuming d is binary and equiprobable [Pr( d¼þ 1) ¼ Pr( d¼ 1) ¼

2 ],
the mutual information between d and y is expressed in terms of the conditional
distribution Pr( yjd )as

1

0
@

ð

1
2

2Pr( yjþ 1)
Pr( yjþ 1) þ Pr( yj 1) dy

I ( d , y ) ¼ Pr( yjþ 1) log

y

Pr( yjþ 1) þ Pr( yj 1) dy 1

ð

2Pr( yj 1)

þ Pr( yj 1) log

A :

y