TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 182

initial estimates of the channel coefficients to converge to a correct decoding solution.
This would appear an inherent shortcoming of such blind schemes, that is, local
convergence is often the best that can be achieved [58].

3.9 CONVERGENCE

The feedback behavior of turbo loops induces nonlinear interactions [31], leading to
potentially complicated phenomena such as limit cycles or even chaos in some cases
[59]. Experience shows, however, that iterative decoding usually converges under
reasonable conditions under which the SNR and block length are large enough.
Existing analysis methods to study convergence include those derived from infor-
mation geometry [31, 60-67], numerical analysis [68-70], and density evolution
[71-73]. The information geometry approach views successive iterations of the
turbo loops as projectors in appropriate spaces, but is confounded by the absence of
key invariants in extrinsic information extraction. The numerical analysis approaches
avoid this shortcoming, but give sufficient conditions for convergence that are rather
algebraic and do not readily translate into usable design criteria for engineers. The
density evolution approach treats the extrinsic information probabilities as indepen-
dent, identically distributed random variables whose probability density function
evolves with successive iterations. A popular version of this approach uses extrinsic
information transfer (EXIT) charts, which affords a graphical analysis of the conver-
gence behavior of iterative decoding, and whose results show good agreement with the
experimentally observed convergence behavior of iterative decoding. Its shortcoming
is that a formal justification (e.g. [71, 72]) appeals to asymptotic approximations which
are valid for long block lengths, but which break down for shorter block lengths.
Accordingly, we assume in this section that the block length N is sufficiently long.
Our presentation begins with an adaptation of EXIT analysis to turbo equalization
[39, 74], and then provides a brief overview of some variants of this analysis for
the interference canceler configuration [75], [69].
To begin, we first transform the extrinsic probabilities into the following log extrin-
sic probability ratios

t i 4 log T i ( d i ¼þ 1)
T i ( d i ¼ 1)
Y i 4 log U i ( d i ¼þ 1)

U i ( d i ¼ 1) :

The basic modeling assumption which underlies this analysis is summarized as
follows.

Assumption 1 The variable t i ( resp. Y i ) is modeled as the output of a virtual chan-
nel with input d i, a gain of 0 : 5 6 t ( resp. 0 : 5 6 Y ) , and additive white Gaussian noise
with variance 6 t ( resp. 6 Y ) .