Digital Signal Processing Reference
In-Depth Information
2D spatial-domain versions of (now classical) spectrum estimation techniques for
general idea is that we can obtain a higher resolution if the sidelobes generated
by strong sources are made small.
As an example, the “minimum variance distortionless response” (MVDR)
beamformer is defined such that, for pixel
p
, the response towards the direction
of interest
p
is unity, but signals from other directions are suppressed as much as
possible, i.e.,
w
H
Rw
w
H
a
w
(
p
)=
arg min
w
,
such that
(
p
)=
1
.
This problem can be solved in various ways. For example, after making a transfor-
mation
w
:
R
1
/
2
w
,
a
:
R
−
1
/
2
a
, the problem becomes
=
=
w
(
w
2
w
H
a
(
p
)=
arg min
w
,
such that
p
)=
1
.
To minimize the norm of
w
, it should be aligned to
a
, i.e.,
w
=
α
a
,andthe
solution is
w
=
a
/
(
a
H
a
)
. In terms of the original variables, the solution is then
R
−
1
a
(
p
)
w
(
p
)=
)
,
a
(
p
)
H
R
−
1
a
(
p
and the resulting image can thus be described as
1
H
Rw
I
MV DR
(
p
)=
w
(
p
)
(
p
)=
)
.
H
R
−
1
a
a
(
p
)
(
p
For a point-source model, this image will have a high resolution: two sources that
are closely spaced will be resolved. The corresponding beam responses to different
sources will in general be different: the beamshape is spatially varying.
The MVDR image is to be used instead of the dirty image
I
D
(
in the CLEAN
loop. Due to its high resolution, the location of sources is better estimated than
using the original dirty image (and the location estimate can be further improved by
searching for the true peak on a smaller grid in the vicinity of the location of the
maximum). A second modification to the CLEAN loop is also helpful: Suppose that
the location of the brightest source is
p
q
, then the corresponding power
p
)
α
q
should
H
2
. This can be done
be estimated by minimizing the residual
R
−
α
a
(
p
q
)
a
(
p
q
)
H
R
−
α
a
(
p
q
)
a
(
p
q
)
=
vec
(
R
)
−
α
[
a
(
p
q
)
⊗
a
(
p
q
)]
.
The optimal least squares solution for
α
