Cryptography Reference
In-Depth Information
It is particularly instructive to study the limiting form taken by the equalizer
in the classical case where the transmitted symbols are assumed to be equiprob-
able (which corresponds to the
1
st
iteration of the turbo equalizer). In this case,
σ
x
=0
and the equalizer coecients can be written:
f
∗
=
σ
x
HH
H
+
σ
w
I
−
1
h
Δ
σ
x
(11.43)
Here we can recognize the form of a classical linear MMSE equalizer with finite
length. Inversely, under the hypothesis of perfect
a priori
information on the
transmitted symbols, we have
σ
x
=
σ
x
. The equalizer then takes the following
form:
L
k
=0
|
−
1
σ
x
2
=
h
Δ
h
Δ
=
2
h
∗
Δ
f
=
with
h
h
k
|
(11.44)
2
+
σ
w
σ
x
h
and the equalized signal
z
i
can be written:
2
σ
x
x
i−
Δ
+
h
Δ
w
i
h
z
i
=
(11.45)
2
+
σ
w
σ
x
h
We recognize here the output of a classical MMSE interference canceller, fed
by a perfect estimation of the transmitted data. The equalized signal can be
decomposed as the sum of the useful signal
x
i−
Δ
, up to a scale factor that is char-
acteristic of the MMSE criterion, and a coloured noise term. In other words, the
equalizer suppresses all the ISI without raising the noise level and thus reaches
the theoretical matched-filter bound corresponding to ISI-free transmission.
To summarize, we see that the SISO MMSE linear equalizer adapts the
equalization strategy according to the reliability of the estimated data, measured
here by parameter
σ
x
.
To conclude this description of the equalizer, we point out that the interfer-
ence cancellation operation defined formally by Equation (11.33) has no physical
reality in the sense that it cannot be performed directly in this way using trans-
verse linear filters. In practice, we prefer to use one of the two architectures
presented in Figure 11.15, strictly equivalent from a theoretical point of view.
The coecient
g
Δ
appearing in implementation (1) is the central coecient
g
Δ
=
f
T
h
Δ
of the global filter formed by the cascade of the channel with filter
f
. In the case of implementation (2), we again find the classical structure of an
interference canceller type equalizer, operating here on the estimated signal
x
.
Filter
g
=
f
T
H
is given by the convolution of filter
f
with the impulse response
of the channel, the central coecient
g
Δ
then being forced to zero.
•
SISO demapping
The role of this module is to convert the equalized data
z
i
into extrinsic LLRs
on the interleaved coded bits, which will be then transmitted to the channel