Cryptography Reference
In-Depth Information
the soft estimate
x
i
is then strictly equal to the transmitted symbol
x
i
(perfect
estimate). To summarize, the value of the soft estimate
x
i
evolves as a function
of the reliability of the
a priori
information provided by the decoder. This
explains the name of "soft" (or probabilistic) estimate for
x
i
.Byconstruction,
the estimated data
x
i
are random variables. In particular, we can show (see
[11.33] for example) that they satisfy the following statistical properties:
=0
(11.28)
E
x
i
x
j
=
E
x
i
x
j
=
σ
x
δ
i−j
(11.29)
The parameter
σ
x
here denotes the variance of estimated data
x
i
. In practice,
this quantity can be estimated using the sample variance estimator on a frame
of
N
symbols as follows:
E
{
x
i
}
N−
1
i
=0
|
1
N
2
σ
x
=
x
i
|
(11.30)
We easily verify that under the hypothesis of equiprobable
a priori
symbols,
σ
x
=0
. Conversely, we obtain
σ
x
=
σ
x
in the case of perfect
a priori
informa-
tion on the transmitted symbols. To summarize, the variance of the estimated
data offers a measure of the reliability of the estimated data. This parameter
plays a major role in the behaviour of the equalizer.
Calculating the linear equalizer coecients
As explained above, the equalization step can be seen as the cascad-
ing of an interference cancellation operation followed by a filtering operation.
The filter coecients are optimized so as to minimize the mean square error
E
•
2
between the equalized symbol
z
i
at instant
i
and symbol
x
i−
Δ
transmitted at instant
i
{|
z
i
−
x
i−
Δ
|
}
Δ
. The introduction of a delay
Δ
enables the anti-
causality of the solution to be taken into account. Here we will use a matrix
formalism to derive the optimal form of the equalizer coecients. Indeed, dig-
ital filters always have a finite number of coe
cients in practice. The matrix
formalism takes this aspect into account and thus enables us to establish the
optimal coecients under the constraint of a finite-length implementation.
Here we consider a filter with
F
coecients:
f
=(
f
0
,...,
f
F−
1
)
. The channel
impulse response and the noise variance are assumed to be known, which requires
prior estimation of these parameters in practice. Starting from the expression
(11.3) and grouping the
F
last samples observed at the output of the channel
up until instant
i
in the form of a vector column
y
i
,wecanwrite:
−
y
i
=
Hx
i
+
w
i
(11.31)
with
y
i
=(
y
i
, ..., y
i−F
+1
)
T
,
x
i
=(
x
i
, ..., x
i−F−L
+2
)
T
and
w
i
=(
w
i
,...,
w
i−F
+1
)
T
.Matrix
H
,ofdimensions
F
1)
, is a Toeplitz matrix
5
×
(
F
+
L
−
5
The coecients of the matrix are constant along each of the diagonals.
