TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 174

configurations, which we denote as S 0 , ... , S 2 L 1 . These relate directly to the 2 Lþ 1
candidate transitions at each stage of the trellis diagram of the channel. We denote
by H ( S j ) the noise-free channel output induced by the transition captured by S j .

B EXAMPLE 3.9
Noise-Free Channel Outputs. Returning to the trellis diagram of Figure 3.8, the
augmented state vector assumes seven candidate configurations

S 0 ¼ [ þ 1 þ 1 þ 1], S 1 ¼ [ 1 þ 1 þ 1], S 7 ¼ [ 1 1 1] :

The noise-free channel outputs that the channel can produce may be enumerated as
H ( S 0 ) ¼h t j i j i ¼S 0 ¼h 0 þh 1 þh 2
H ( S 1 ) ¼h t j i j i ¼S 1 ¼h 0 þh 1 þh 2
.
H ( S 7 ) ¼h t j i j i ¼S 7 ¼h 0 h 1 h 2

in one-to-one correspondence with the state transitions, and comprise all combi-
nations of sums and differences of the channel coefficients.

Note that knowledge of the channel input sequence ( d i ) implies knowledge of the
state transition sequence ( j i ), and vice-versa. A method of channel identification then
consists of finding the coefficients h which are best compatible with the observed
channel output sequence [18]. To this end, let

u¼ [ H ( S 0 ), ... , H ( S 2 M 1 ), s ]

denote the unknown parameters for the channel, consisting of the noise-free channel
output constellation values fH ( S j ) g and the channel noise standard deviation s .
Consider the likelihood function Pr( yju ) for the given received sequence y ¼ ( y 1 ,
y 2 , ... ). If we maximize this function versus the parameters u
u¼ arg max
u

Pr( yju )

the estimate u is then optimal in the maximum likelihood sense. Direct optimization of
the likelihood function Pr( yju ) versus u is usually computationally difficult, and so
iterative techniques are employed.
Let J¼ ( j 1 , j 2 , ... ) be a valid state transition sequence through the trellis, and
introduce a joint likelihood function Pr( y , Jju ). We can then consider Pr( yju )asa
marginal function obtained by summing over all valid state transition sequences
Pr( yju ) ¼ X
J

Pr(y, Jju ) :