Digital Signal Processing Reference
In-Depth Information
()
We need to pose several requirements on
M
or
M
to ensure identifiability.
(
)
→
=
, otherwise we
()=
First of all, in the first case we must have
M
M
cannot uniquely find
from
M
. Furthermore, for both cases we will require that
M
is a tall matrix (more rows than columns) and has full column rank, so that it has a
left inverse (this will allow to estimate
). This puts a limit on the number of sources
in the image (number of columns of
M
) in relation to the number of observations
(rows). If more snapshots (STIs) and/or multiple frequencies are available, as is the
case in practice, then
M
will become taller, and more sources can be estimated thus
increasing the resolution. If
M
is not tall, then there are some ways to generalize
this, e.g. via the context of compressive sampling where we can have
M
wide as
long as
For the moment, we will continue with the second formulation: one source per
pixel, fewer pixels than available correlation data.
5.2
Matrix Formulation of Imaging via Beamforming
on the covariance data
r
. We can stack all image values
I
over all pixels
p
into
a single vector
i
, and similarly, we can collect the weights
w
(
p
)
(
p
)
over all pixels
i
nt
o a single matrix
W
=[
w
(
p
1
)
,
w
(
p
2
)
, ···
]
=
R
H
vec
(
w
⊗
w
)
(
)
, so that we can write
i
BF
=(
H
r
W
◦
W
)
.
(24)
We saw before that the dirty image is obtained if we use the matched filter. In this
case, we have
W
1
=
J
A
,where
A
contains the array response vectors
a
(
p
)
for every
pixel
p
of interest. In this case, the image is
1
J
2
(
1
J
2
M
s
r
i
D
=
A
H
r
◦
A
)
=
.
(25)
=
The expected value of the image is obtained by using
r
M
:
1
J
2
M
s
M
1
J
2
(
1
J
2
(
M
s
M
s
)
s
+
M
s
M
n
)
n
.
i
D
=
=
The quality or “performance” of the image, or how close
i
D
is to
i
D
, is related to its
covariance,
1
J
4
M
s
C
w
M
s
i
D
)=
i
D
−
i
D
−
H
cov
(
E
{
(
i
D
)(
i
D
)
}
=
