Digital Signal Processing Reference
In-Depth Information
set of such Doppler shifts could be very large. However, restricting our operation to
a narrow region of interest (e.g., an urban canyon where the range is much greater
than the width) and a few classes of targets that have comparable velocities (e.g.,
cars/trucks within a city environment), we can limit the extent of viable Doppler
shifts to a smaller quantity.
In the following, we first convert the OFDM-measurement model of (
3.10
)toa
sparse model that accounts for a set of finely discretized Doppler shifts. Then, we
present an efficient sparse-recovery approach that employs a collection of multiple
small DS in order to utilize more prior structures of the sparse vector.
3.3.1 Sparse Model
Suppose we discretize the extent of feasible Doppler shifts into
N
β
grid points as
{
β
i
,i
=
0
,
1
,...,N
β
−
1
}
. Then, we can remodel (
3.8
)as
y
l
(n)
=
a
l
φ
l
(n)
T
ζ
l
+
e
l
(n),
(3.12)
where
•
φ
l
(n)
T
=[
φ
l
0
(n),φ
l
1
(n),...,φ
l(N
β
−
1
)
(n)
]
represents an equivalent sparsity-
based modeling of
φ
l
(n)
;
T
is an
N
β
×
•
ζ
l
=[
ζ
l
0
,ζ
l
1
,...,ζ
l(N
β
−
1
)
]
1 sparse vector, having
P(
N
β
)
nonzero entries corresponding to the true target scattering coefficients, i.e.,
x
lp
if
i
=
p,
ζ
li
=
(3.13)
0
otherwise.
Using the formulation of (
3.12
) and following the approach presented in Sect.
3.2.2
to obtain (
3.10
) from (
3.8
), we deduce a sparse-measurement model as
=
ζ
y
+
e
,
(3.14)
where
•
(
A
(
0
))
T
... (
A
(N
1
))
T
T
is an
LN
LN
β
sparse-measurement
matrix containing all the viable Doppler information in terms of the
L
=[
−
]
×
×
LN
β
dimensional matrices
(n)
blkdiag
(
φ
0
(n)
T
,
φ
1
(n)
T
,...,
φ
L
−
1
(n)
T
)
;
=
ζ
0
,
ζ
1
,...,
ζ
L
−
1
]
T
is an
LN
β
×
•
ζ
1 sparse-vector that has
LP
nonzero en-
tries representing the scattering coefficients of the target along all the
P
received
paths and
L
subcarriers.
=[
3.3.2 Sparse Recovery
The goal of a sparse-reconstruction algorithm is to estimate the vector
ζ
from the
noisy measurement
y
of (
3.14
) by exploiting the sparsity. One of the most popular
Search WWH ::
Custom Search