Digital Signal Processing Reference
In-Depth Information
approaches of sparse-signal recovery is the Dantzig selector [
8
], which provides an
estimate of
ζ
as a solution to the following
1
-regularization problem:
subject to
H
(
y
−
z
)
∞
≤
min
z
∈C
LN
β
z
1
λ
·
σ,
(3.15)
where
λ
=
2log
(LN
β
)
is a control para
meter th
at ensures that the residual
(
y
=
√
tr
(
)/L
.
However, from the construction of
ζ
in (
3.14
), i.e., from
ζ
−
z
)
is within the noise level and
σ
ζ
0
,
ζ
1
,...,
ζ
L
−
1
]
T
=[
we observe that each
ζ
l
,l
1, is sparse with sparsity level
P
. Fur-
thermore, the system matrix
in (
3.14
) can be expressed as
=
0
,
1
,...,L
−
=[
0
1
...
L
−
1
]
,
(3.16)
where each block-matrix of dimension
LN
×
N
β
is orthogonal to any other block-
matrix, i.e.,
l
1
l
2
=
0
for
l
1
=
l
2
.
To exploit this additional structure in the sparse-recovery algorithm, we propose
a concentrated estimate
ζ
=[
ζ
0
,
ζ
1
,...,
ζ
L
−
1
]
T
which is obtained from the indi-
vidual solutions,
ζ
l
s, of the
L
small Dantzig selectors:
subject to
H
−
l
z
l
)
∞
≤
min
z
l
∈C
N
β
z
l
1
l
(
y
λ
l
·
σ
for
l
=
0
,
1
,...,L
−
1
,
(3.17)
where
λ
l
=
2log
(N
β
)
.As(
3.17
) exploits more prior structures of the sparse vec-
tor, it provides improved performances over (
3.15
) both in terms of computational-
time and estimation-accuracy [
48
].
3.4 Adaptive Waveform Design
In this section, we develop an adaptive waveform design technique based on a multi-
objective optimization (MOO) approach. To improve the detection performance,
we propose to maximize the Mahalanobis distance which quantifies the distance
between two distributions involved in the detection problem. However, in prac-
tice, the computation of the Mahalanobis distance requires estimations of the tar-
get scattering-response, target velocity, and noise covariance matrix. So, in addi-
tion to maximizing the Mahalanobis distance, we intend to increase the estimation-
accuracy by minimizing a weighted trace of the CRB matrix computed for the un-
known parameters. Furthermore, the formulation of sparse-measurement model al-
lows us to construct and solve another optimization problem that minimizes the
upper bound on the sparse-estimation error for improving the efficiency of sparse-
recovery. In the following, we first present in detail these three single-objective func-
tions and then describe the MOO problem.
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