Digital Signal Processing Reference
In-Depth Information
ionosphere
phase screen
(time varying)
geometric delays
station
beamformers
x
1
(
t
)
x
J
(
t
)
Fig. 9
A radio interferometer where stations consisting of phased array elements replace telescope
dishes. The ionosphere adds phase delays to the signal paths. If the ionospheric electron density
has the form of a
wedge
, it will simply shift the apparent positions of all sources
used for LOFAR, this effect is more pronounced. Generally it is first assumed that
the ionosphere is “constant” over about 10 km and about 10 s. A better model is to
model the ionospheric delay as a “wedge”, a linear function of the distance between
piercing points (the intersection of the direction vectors
p
q
with the ionospheric
to a shift in the apparent position of the sources. For larger distances, higher-order
functions are needed to model the spatial behavior of the ionosphere, and if left
uncorrected, the resulting image distortions are comparable to the distortions one
sees when looking at lights at the bottom of a swimming pool.
Previously, we described that the array response matrix
A
is really a function of
the source direction vectors
p
q
, and we wrote
A
is a suitable
parametrization of the
p
q
(typically two direction cosines per source). If a linear
model for the ionospheric disturbance is sufficient, then it is sufficient to replace
A
()
where the vector
(
)
differs from
()
by
A
,where
due to the shift in apparent direction of
each source.
The modified data model that captures the above effects is thus
(
)
B
H
A
(
)
H
H
R
=(
A
B
)˙
(
)+˙
.
(33)
s
n
