Cryptography Reference
In-Depth Information
describing the convolution by the channel:
⎛
⎞
h
0
···
h
L−
1
0
···
0
⎝
.
⎠
0
h
0
h
L−
1
H
=
(11.32)
.
.
.
.
.
.
.
0
0
···
0
h
0
···
h
L−
1
With these notations, the interference cancellation step from the estimated signal
x
canthenbewrittenformally:
y
i
=
y
i
−
Hx
i
(11.33)
where the vector
x
i
=(
x
i
,...,x
i−
Δ+1
,
0
, x
i−
Δ
−
1
,...,x
i−F−L
+2
)
T
is of dimen-
sion
F
+
L
1. The component related to symbol
x
i−
Δ
is set to zero in order to
cancel only the ISI and not the signal of interest. At the output of the forward
filter, the expression of the equalized sample at instant
i
is given by:
−
z
i
=
f
T
y
i
=
f
T
[
y
i
−
Hx
i
]
(11.34)
It remains to determine the theoretical expression of the coecients of the
filter
f
minimizing the mean square error
E
2
. In the most general
case, these coecients vary in time. The corresponding solution, developed in
detail by Tüchler
et al.
[11.51, 11.50], leads to an equalizer whose coecients
must be recalculated for each received symbol. This equalizer represents what
can best be done currently for MMSE equalization in the presence of
a priori
information. On the other hand, the computation load associated with updating
the coecients symbol by symbol increases quadratically with the number
F
of
coecients, which again turns out to be too complex for real time implemen-
tations. The equalizer that we present here can be seen as a simplified, and
therefore sub-optimal, version of the solution cited above. The coecients of
filter
f
are calculated only once per frame (at each iteration) and then applied
to the whole block, which considerably decreases the implementation cost. On
the other hand, and despite this reduction in complexity, this equalizer retains
performance close to the optimal one
6
, which makes it an excellent candidate
for practical realizations. This solution was derived independently by several
authors, including [11.51] and [11.33].
With these hypotheses, the optimal form of the set of coecients
f
is obtained
using the projection theorem, which stipulates that the estimation error must
be orthogonal to the observations
7
:
E
(
z
i
−
{|
z
i
−
x
i−
Δ
|
}
x
i−
Δ
)
y
i
=
0
(11.35)
6
The degradation measured experimentally in comparison with Tüchler's original time-
varying solution is at most 1 dB, depending on the channel model considered.
7
We recall here that the notation
A
H
denotes the
Hermitian
(conjugate) transpose of ma-
trix
A
.
