The effect of a finite bandwidth of observation as seen by the
multiplier in the correlator, is to reduce the amplitude of the
visibility by a factor given by

, where is angular
size of the synthesized beam, is the center of the observing
band, is location of the point source relative to the field center
and is the bandwidth of the signal being correlated.

The distortion in the map due to the finite bandwidth of observation
can be visualized as follows. For continuum observations, the
visibility data integrated over the bandwidth is treated
as if the observations were made at a single frequency ,
the central frequency of the band. As a result the and
co-ordinates and the value of visibilities are correct only for
. The true co-ordinate at other frequencies in the band are
related to the recorded co-ordinates as

(11.4.14) |

Since the total weights used while mapping does not depend on the
frequency, the relation between the brightness distribution and
visibility at a frequency becomes

Hence the contribution of to the brightness distribution get
scaled by and the co-ordinates gets scaled by
. The effect of the scaling of the co-ordinates,
assuming a delta function for the *Dirty Beam*, is to smear a
point source at position into a line of length
in the radial direction. This will get
convolved with the *Dirty Beam* and the total effect can be found
by integrating the brightness distribution over the bandwidth as
given in Eq. 11.4.15

(11.4.17) |

Eq. 11.4.16 is equivalent to averaging large number of
maps made from monochromatic visibilities at . Since each of
such maps would scale by a different factor, the source away from the
center would move along the radial line from one map to another,
producing the radial smearing convolved with the *Dirty Beam*.
Since the source away from the center is elongated radially, its
side-lobes (because of the *Dirty Beam*) will also be elongated in
the radial direction. As a result the side-lobes of distant sources
will be elongated at the origin but not towards angle from the
vector joining the source and the origin.

The effect of bandwidth smearing can be reduced if the RF band is split into frequency channels with smaller channel widths. This effectively reduces the as seen by the mapping procedure and while gridding the visibilities then, the and can be computed separately for each channel and assigned to the correct -cell. The FX correlator used in GMRT provides up to 128 frequency channels over the bandwidth of observation.