Cryptography Reference
In-Depth Information
to the soft estimate
x
i−
Δ
in
x
i
is set to zero in order not to cancel the signal
of interest. Vectors
f
i
=(
f
i,F
,...,f
i,−F
)
T
and
g
i
=(
g
i,G
,...,g
i,−G
)
T
represent
the coecients of filters
f
and
g
, respectively. Both vectors are a function of
time since they are updated at each new received symbol.
The relations used to update the vectors of the coecients can be obtained
from a least-mean square (LMS) gradient algorithm:
x
i−
Δ
)
y
i
f
i
+1
=
f
i
−
μ
(
z
i
−
(11.56)
x
i−
Δ
)
x
i
g
i
+1
=
g
i
−
μ
(
z
i
−
where
μ
is a small, positive, step-size that controls the convergence properties
of the algorithm.
During the first iteration of the turbo equalizer,
x
i
is a vector all the compo-
nents of which are null; the result is that the coe
cients vector
g
i
is also null.
The MMSE equalizer then converges adaptively towards an MMSE transversal
equalizer. When the estimated data are very reliable and close to the transmit-
ted data, the MMSE equalizer converges towards an ideal (genie) interference
canceller, then having the performance of a transmission without intersymbol
interference. The limiting forms of the adaptive equalizer are therefore totally
identical to those obtained in (11.43) and (11.44), on condition of course that the
adaptive algorithm can converge towards a local minimum close to the optimal
solution.
Note, however, that for intermediate iterations where the estimated infor-
mation symbols
x
i
are neither null nor perfect, filter
g
i
must not be fed directly
with the transmitted symbols otherwise the equalizer will converge towards the
solution of the genie interference canceller, which is not the aim searched for. To
enable the equalizer to converge towards the targeted solution, the idea here in-
volves providing filter
g
i
with soft estimates built from the transmitted symbols,
during the training periods:
(
x
i
)
its
=
σ
x
x
i
+
1
σ
x
η
i
−
(11.57)
where
σ
x
corresponds to the variance of the soft estimates
x
i
obtained from
(11.24) and
η
i
a zero-mean complex circularly-symmetric additive white Gaus-
sian noise with unit variance.
In the tracking period and in order to enable the equalizer to follow the
variations of the channel, it is possible to replace the transmitted symbols
x
i
in
relations (11.56) by the decisions
x
i
at the output of the equalizer, or by the
decisions on the estimated symbols
x
i
.
When the SISO MMSE equalizer is realized in adaptive form, we do not
explicitly have access to the channel impulse response, and the updating relation
of
g
i
does not enable
g
Δ
to be obtained since component
g
i,
Δ
is constrained to
be zero. To perform the SISO demapping operation, we must however estimate
both bias
g
Δ
on the data
z
n
provided by the equalizer and variance
σ
ν
of the
