Digital Signal Processing Reference
In-Depth Information
Finally, it can be shown [211] that
H
q
noise
i
=
q
noise
i
h
S
Q
noise
i
Q
H
H H
noise
i
h
S
(5.74)
Therefore, the orthogonality relation (5.70) can also be expressed as
h
S
Q
noise
i
Q
H
noise
i
h
S
=
0,
i
=
0,
...
,
KP
−
L
−
K
(5.75)
q
noise
i
of the eigenvectors associated with the
noise subspace are required. Then, the estimate of the channel coefficient
vector
h
S
is obtained minimizing the following quadratic form:
In practice, only estimates
ˆ
H
q
noise
i
KP
−
L
−
K
2
H
h
S
Q
noise
h
S
J
quad
(
h
S
)
=
ˆ
=
(5.76)
i
=
0
where
Q
noise
is a
LP
LP
matrix given by
×
KP
−
L
−
K
H
noise
i
Q
noise
i
Q
Q
noise
=
(5.77)
i
=
0
and matrix
Q
noise
i
is defined by (5.71) through (5.73), replacing the eigenvec-
tors associated with the noise subspace with their estimates.
The minimization must be subject to appropriate constraints, in order
to avoid the trivial solution
h
S
=
0. In [211], the following criteria are
suggested.
•
Quadratic constraint: minimize
J
quad
(
1. The
optimal solution is given by the unit-norm eigenvector associated
with the smallest eigenvalue of
h
S
)
subject to
h
S
=
Q
noise
.
subject to
l
H
H
=
•
Linear constraint: minimize
J
quad
(
h
S
)
1, where
l
is a
1
vector.
LP
×
The first criterion can be considered to be the natural choice, although
it has a higher computational complexity due to the eigenvector estimation.
The second criterion, in spite of demanding a lower computational burden,
depends on the appropriate choice of an arbitrary vector
l
—the solution is
proportional to
Q
−
1
l
.
The quadratic form (5.76) can also be expressed in terms of the eigenvec-
tors associated with the signal subspace:
