Digital Signal Processing Reference
In-Depth Information
2Re
D
η
(
I
N
⊗
)
−
1
D
η
J
ηη
=
2Re
∂
(n)
∂v
x
x
H
A
H
−
1
A
∂
(n)
∂v
x
x
,
(3.27)
N
−
1
x
∂
(n)
∂v
y
x
∂
(n)
∂v
y
=
n
=
0
2Re
D
η
(
I
N
⊗
)
−
1
D
x
J
η
x
=
2Re
∂
(n)
∂v
x
x
H
A
H
−
1
A
(n)
,
N
−
1
x
∂
(n)
∂v
y
=
(3.28)
n
=
0
2Re
D
x
(
I
N
⊗
)
−
1
D
η
J
x
η
=
2Re
(n)
H
A
H
−
1
A
∂
(n)
∂v
x
x
,
N
−
1
x
∂
(n)
∂v
y
=
(3.29)
n
=
0
N
−
1
2Re
D
x
(
I
N
⊗
)
−
1
D
x
=
2Re
(n)
H
A
H
−
1
A
(n)
.
(3.30)
J
xx
=
n
=
0
Now, to obtain a scalar objective function that summarizes the CRB matrix, several
optimality criteria can be considered. For example,
A
-optimality criterion employs
the trace,
D
-optimality uses the determinant, and
E
-optimality computes the maxi-
mum eigenvalue of the CRB matrix [
6
, Chap. 7.5.2]. However, due to the different
physical characteristics of
η
and
x
, the variances of their estimators may differ in
several orders of magnitude and units. Therefore, we construct an objective function
to design the OFDM spectral-parameters that minimizes a weighted summation of
the traces of individual CRBs on
η
and
x
as
a
(
2
)
a
H
a
=
arg min
a
L
c
η
tr
(
CRB
ηη
)
+
c
x
tr
(
CRB
xx
)
subject to
=
1
,
(3.31)
∈C
where
c
η
and
c
x
are the weighting parameters.
3.4.3 Minimizing the Upper Bound on Sparse Error
Many functions of the system matrix
have been proposed to analyze the perfor-
mance of methods used to recover
ζ
from
y
, the most popular measure being the
restricted isometry constant (RIC). However, for a given arbitrary matrix, the com-
putation of RIC is extremely difficult. Therefore, in [
52
] we proposed a new, easily
computable measure,
1
-constrained minimal singular value (
1
-CMSV) of
,to
assess the reconstruction performance of an
1
-based algorithm. According to [
52
,
Def. 3], we define the
1
-CMSV of
as
s
ζ
2
ρ
s
(
)
=
min
2
,
for any
s
∈[
1
,LN
β
]
,
(3.32)
ζ
ζ
=
0
,s
1
(
ζ
)
≤
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