TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 179

Figure 3.13 Showing the channel impulse response coefficient estimates from a single
run of the blind turbo equalizer.

3.8.1 Differential Encoding

A fundamental ambiguity in blind channel estimation is that delay and / or phase shifts
cannot be detected, owing to the absence of an absolute time reference. We review
here how differential encoding can be exploited to remove the phase uncertainty for
the case of antipodal signaling. Differential encoding may be viewed as a rate-one con-
volutional code operation, whose corresponding decoder may be absorbed into the
turbo loop [57]. The resulting scheme gives a special case of nested iterative loops,
introduced earlier in a serial coding context in [3].
The basic sign ambiguity problem of antipodal signaling is that h and 2 h produce
the same output constellation, consisting of sums and differences of the channel coef-
ficients. Thus if a given solution h maximizes the likelihood function of the previous
section, so does 2 h . The channel input values ( d i ) inferred from the inner decoder for
these two choices of the channel coefficients do not agree; rather, one sequence is the
negative of the other. The principle of differential encoding is to render this sign ambi-
guity innocuous, by coding the relevant information into the difference of two succes-
sive channel symbols. In this way, if all channel symbols are negated, the difference
from one to the next is preserved.
A differential encoder is readily absorbed into the communication chain, and
effects the transformation

e i ¼ d i e i 1 , e 0 ¼þ 1

where the seed value e 0 could have equally well been chosen as e 0 ¼ 2 1. In
the absence of noise, the sequence ( d i ) can be recovered from the sequence ( e i )
according to

e i
e i 1 ¼ e i e i 1

d i ¼