Bandwidth Smearing

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
$sin(\pi l \Delta \nu / \nu_o \theta)/(\pi l \Delta\nu / \nu_o \theta)$, where $\theta$ is angular size of the synthesized beam, $\nu_o$ is the center of the observing band, $l$ is location of the point source relative to the field center and $\Delta \nu$ 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 $\Delta \nu$ is treated as if the observations were made at a single frequency $\nu_o$, the central frequency of the band. As a result the $u$ and $v$ co-ordinates and the value of visibilities are correct only for $\nu_o$. The true co-ordinate at other frequencies in the band are related to the recorded co-ordinates as

$$(u,v)=\left({\nu_o u_\nu \over \nu}, {\nu_o v_\nu \over \nu}\right).$$ (11.4.14)

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

$$V(u,v)=V\left({\nu_o u_\nu \over \nu}, {\nu_o v_\nu \over \n... ...ight)^2 I\left({l \nu \over \nu_0},{m \nu \over \nu_0}\right).$$ (11.4.15)

Hence the contribution of $V(u,v)$ to the brightness distribution get scaled by $(\nu/\nu_o)^2$ and the co-ordinates gets scaled by $(\nu/\nu_o)$. 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 $(l,m)$ into a line of length $(\Delta \nu/\nu_o)\sqrt{l^2 + m^2}$ 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

$$I^d(l,m)=\left[ {\int\limits_0^\infty \vert H_{RF}(\nu)\ve... ...s_0^\infty \vert H_{RF}(\nu)\vert^2 d\nu}} \right]*DB_o(l,m),$$ (11.4.16)

where $H_{RF}(\nu)$ is the band-shape function of the RF band and $DB_o$ is the Dirty Beam at frequency $\nu_o$. If one represents the synthesized beam as a gaussian function of standard deviation $\sigma_b=\theta_b/\sqrt{8 ln 2}$ and the bandpass represented by a rectangular function of width $\Delta \nu$, the fractional reduction in the strength of a source located at a radial distance $r=\sqrt{l^2+m^2}$ is given by

$$R_b=1.064{\theta_b\nu_o \over {r\Delta\nu}}erf\left(0.833{r\Delta\nu \over {\theta_b\nu_o}}\right).$$ (11.4.17)

Eq. 11.4.16 is equivalent to averaging large number of maps made from monochromatic visibilities at $\nu$. 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 $90^o$ 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 $\Delta \nu$ as seen by the mapping procedure and while gridding the visibilities then, the $u$ and $v$ can be computed separately for each channel and assigned to the correct $uv$-cell. The FX correlator used in GMRT provides up to 128 frequency channels over the bandwidth of observation.