Signal Processing Tools for Radio Astronomy - Signal Processing Systems - page 423

a b

40

1 0

−5

30

0.5 −10

20

−15

10

0 −20

0

−25

−0.5 −30

−10

−35

−20

−1 −40

1 0.5 0 −0.5 −1

−30

−40 −30 −20 −10 0 10 20 30

East ← l → West

East ← x → West

Fig. 4 ( a ) Coordinates of the antennas in a LOFAR station, which defines the spatial sampling
function, and ( b ) the resulting dirty beam

B is a known function: it only depends on the locations of the telescopes, or

rather the set of telescope baselines z i (

( p )

.
An example of a set of antenna coordinates and the corresponding dirty beam is
shown in Fig. 4 . This is for a single low-band LOFAR station and a single STI and
frequency bin. The dirty beam has heavy sidelobes as high as

m ) − z j ( m )

10 dB. A resulting
dirty image is shown in Fig. 5 . In this image, we see the complete sky, in

−

)
coordinates, where the reference direction is pointing towards zenith. The strong
visible sources are Cassiopeia A and Cygnus A, also visible is the milky way, ending
in the north polar spur (NPS) and, weaker, Virgo A. In the South, the Sun is visible
as well. The image was obtained by averaging 25 STIs, each consisting of 10 s
data in 25 frequency channels of 156 kHz wide taken from the band 45-67 MHz,
avoiding the locally present radio interference. As this shows data from a single
LOFAR station, with a relatively small maximal baseline (65 m), the resolution is
limited and certainly not representative of the capabilities of the full LOFAR array.
The dirty beam is essentially a non-ideal point spread function due to finite
and non-uniform spatial sampling: we only have a limited set of baselines. The
dirty beam usually has a main lobe centered at p

( , m

0 , and many side lobes. If we
would have a large number of telescopes positioned in a uniform rectangular grid,
the dirty beam would be a 2-D sinc-function (similar to a boxcar taper in time-
domain sampling theory). The resulting beam size is inversely proportional to the
aperture (diameter) of the array. This determines the resolution in the dirty image.
The sidelobes of the beam give rise to confusion between sources: it is unclear
whether a small peak in the image is caused by the main lobe of a weak source,
or the sidelobe of a strong source. Therefore, attempts are made to design the array
such that the sidelobes are low. It is also possible to introduce weighting coefficients
(“tapers”) in ( 16 ) to obtain an acceptable beamshape.
Another aspect is the summation over m (STI intervals) in ( 17 ) , where the
rotation of the Earth is used to obtain essentially many more antenna baselines.

=