Cryptography Reference
In-Depth Information
We then obtain the following solution:
−
1
f
∗
=
E
y
i
y
i
}
x
i−
Δ
y
i
}
{
E
{
(11.36)
Using the statistical properties of the estimated data
x
i
,wenotethat:
x
i−
Δ
y
i
}
x
i−
Δ
H
(
x
i
−
=
He
Δ
σ
x
E
{
=
E
{
x
i
)
}
(11.37)
where we have introduced the unit vector
e
Δ
with dimension
F
+
L
1
that
hasa1incoordinate
Δ
and 0 elsewhere. Denoting by
h
Δ
the
Δ
-th column
Δ
of matrix
H
, the previous expression can also be written:
E
x
i−
Δ
y
i
=
h
Δ
σ
x
−
(11.38)
In addition,
E
y
i
y
i
=
H
E
(
x
i
−
x
i
)
H
H
H
+
σ
w
I
x
i
)(
x
i
−
(11.39)
σ
x
)
HH
H
+
σ
x
h
Δ
h
Δ
+
σ
w
I
=(
σ
x
−
To summarize, the optimal form of the equalizer coecients can finally be writ-
ten:
f
∗
=
(
σ
x
−
σ
x
)
HH
H
+
σ
x
h
Δ
h
Δ
+
σ
w
I
−
1
h
Δ
σ
x
(11.40)
By bringing into play a simplified form of the matrix inversion lemma
8
,the
previous solution can then be written:
σ
x
1+
βσ
x
f
∗
f
∗
=
(11.41)
where we have introduced vector
f
and scalar quantity
β
defined as follows:
f
∗
=
(
σ
x
−
σ
x
)
HH
H
+
σ
w
I
−
1
β
=
f
T
h
Δ
h
Δ
and
(11.42)
By means of this new expression, we note that the computation of the
coecients of the equalizer is mainly based on the inversion of the matrix
(
σ
x
−
σ
x
)
HH
H
+
σ
w
I
, with dimensions
F
F
. Thismatrixhasarichstructure
since it is a Toeplitz matrix with Hermitian symmetry. Consequently, matrix in-
version can be performed eciently with the help of dedicated algorithms, with
a computation cost in
O
(
F
2
)
(see Chapter 4 in [11.21], for example). In order to
reduce even further the complexity of determining the coecients, the authors
of [11.33] have proposed a sub-optimal, but nevertheless ecient, method using
the
Fast Fourier Transform
, (or FFT), with a cost in
O
(
F
log
2
(
F
))
. However,
the number of coecients
F
must be a power of 2.
8
A
+
uu
H
−
1
=
A
−
1
−
A
−
1
uu
H
A
−
1
1+
u
H
A
−
1
u
×
