TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 192

3.9.3 EXIT Chart for Interference Canceler

The EXIT analysis can also be extended to the interference canceler configuration [38,
39, 74], with an additional complication: The output of the interference canceler, once
converted to log likelihood ratios ( t i ), does not adhere to the modeling Assumption 1,
except when perfect symbol estimates are fed back to cancel the intersymbol inter-
ference. In effect, the residual intersymbol interference in the presence of imperfect
feedback is decidedly non-Gaussian, such that the conditional probabilities Pr( t i jd i )
assume a more complicated form. Accurate extrinsic transfer function estimation
then requires delicate histogram estimation [39], which can be computationally inten-
sive. An alternative is to observe that the transfer function relating mutual information
through the interference canceler appears as a straight line in the examples from [38,
39, 74]. (This character is also apparent in Figure 3.19, even though that plot corre-
sponds to a forward-backward equalizer, not an interference canceler.) A simplified
approach thus consists in evaluating the extrinsic information transfer function of
the interference canceler at its extreme points of I ( d i , Y i ) ¼ 0 (no feedback) and
I ( d i , Y i ) ¼ 1 (perfect feedback), and then connecting the resulting output mutual infor-
mation values with a straight line [39].
To illustrate, consider first the case of perfect feedback to the interference canceler.
The intersymbol interference is perfectly canceled, and the equalizer output consists of
a delayed input symbol plus filtered noise

v i ¼ r 0 d iL þ X

L

p l b il , i . L

l¼ 0

where r 0 ¼ P h l ¼ P p l . For simplicity, we assume that r 0 ¼ 1; otherwise we need
only replace v i by v i / r 0 in what follows, to reach the same conclusions. Thus with

r 0 ¼ 1, the filtered noise term P l p l b il remains Gaussian, with variance s 2 . Thus,
the conditional distribution of v i remains Gaussian

( v i + 1) 2
2 s 2

1

2 p s exp

Pr( v i jd iL ¼+ 1) ¼ :

Accordingly, the log likelihood ratios t j (obtained after an L -sample offset and inter-
leaving) become

t j ¼ log Pr( v i þ L j d i ¼þ 1)

2 v i þ L
s 2 Pr( v iþL jd i ¼ 1) ¼

j ¼P ( i ) :

The variable t j thus remains conditionally Gaussian given d i [with j ¼ P ( i )], with
conditional mean 2 2 d i / s 2 and conditional variance 6 t ¼ 4 =s 2 . As such, the
output mutual information I ( d j , t j ) at perfect feedback may be found from the graphi-
cal relation of Figure 3.17, using 6 t ¼ 2 / s .