TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 204

Note that if the variables ( t i ) are indeed conditionally independent given the
values ( d i ), then

I ( d ; t ) ¼ X

K

I ( d i ; t i ) :

i¼ 1

(c) The chain rule for mutual information [44] can be invoked to show that

I ( d ; y ) ¼ I ( d ; y 1 , ... , y K ) þI ( d ; y Kþ 1 , ... , y N jy 1 , ... , y K )
¼ X

K

I ( d i ; y i ) þI ( d ; y Kþ 1 , ... , y N jy 1 , ... , y K ) :

i¼ 1

In the absence of coding, the variables y Kþ 1 , ... , y N would be absent. Show
that, in this case, one would have I ( d i ; t i ) I ( d i ; y i ) and thus

4
s 2

6 t

(without coding) :

3.6 Consider a single-input / multiple-output channel,whose received (vector-valued)
sequence is

y i ¼ X

L

h k d ik þb i

k¼ 0

where y i , h k and b i are all column vectors having N elements, and the sequence
fb k g is spatially and temporally white

s 2 I , j ¼ k ;

E ( b j b k ) ¼

0, otherwise :

The output of the interference canceler is

v i ¼ X

L

p k y ik X

2 L

d ik

q k

k¼ 0 k¼ 0

¼ X r k d ik X

d ik þ X

2 L 2 L L

q k p k b ik

k¼ 0 k¼ 0 k¼ 0

where each impulse response term p k has dimensions 1 N , and where ( r k )is
the combined impulse response, obtained by convolving ( p k ) and ( h k )

r k ¼ X
l

p l h kl , k ¼ 0, 1, ... ,2 L: