Two Element Interferometers in Practice

(4.4.6) | |||

(4.4.7) | |||

So, in order to compensate for all time varying phase factors, it
is not sufficient to have
, one also needs to introduce
a time varying phase
. This additional
correction arises because the delay tracking is done at a frequency
different from . The introduction of the time varying phase
is called *fringe stopping*. Fringe stopping can be achieved in
a variety of ways. One common practice is to vary the instantaneous phase
of the local oscillator signal in arm of the interferometer by
the amount . Another possibility (which is the approach taken
at the GMRT), is to digitally multiply the signal from antenna by
a sinusoid with the appropriate instantaneous frequency.

Another consequence of doing delay tracking digitally is that
the geometric delay can be quantized only upto a step size which
is related to the sampling interval with which the signal was digitized.
In general therefore is not zero, and is called the
*fractional sampling time error*. Correction for this error will be
discussed in the Chapter 9.

The delay tracking and fringe stopping corrections apply for
a specific point in the sky, viz. the position . This point is
called the phase tracking center^{4.8}. Signals, such as terrestrial interference, which
enter from the far sidelobes of the antennas do not suffer the same
geometric delay as that suffered by the source. Consequently,
delay tracking and fringe stopping introduces a rapidly varying change
in the phase of these signals. On long baselines, where the fringe rate
is rapid, the terrestrial interference could hence get completely
decorrelated. While this may appear to be a terrific added bonus, in
principle, terrestrial interference is usually so much stronger than
the emission from cosmic sources, that even the residual correlation is
sufficient to completely swamp out the desired signal.

We end this chapter by re-establishing the connection between
what we have just done and the van Cittert-Zernike theorem. The
first issue that we have to appreciate is that the van Cittert-Zernike
theorem deals with the complex visibility,
. However, the quantity
that has been measured is
.
If one could also measure
, then
of course one could reconstruct the full complex visibility. This is
indeed what is done at interferometers. Conceptually, one has two
multipliers instead of the one in Figure 4.3. The second
multiplier is fed the same input as that in Figure 4.3,
except that an additional phase difference of is introduced
in each signal path. As can be easily verified, the output of this
multiplier is
. Such an
arrangement of two multipliers is called a *complex correlator*.
The two outputs are called the sine and cosine outputs respectively.
For quasi-sinsoidal processes, one has to introduce a phase
difference at each frequency present in the signal. The corresponding
transformation is called a *Hilbert transform ^{4.9}*.
How the complex correlator is achieved at the GMRT is described in
Chapter 9. The output of the complex correlator is hence
a single component of the Fourier transform of the source brightness
distribution

- Thompson, R. A., Moran, J. M. & Swenson, G. W. Jr., `Interferometry & Synthesis in Radio Astronomy', Wiley Interscience.
- R. A. Perley, F. R. Schwab, & A. H. Bridle, eds., `Synthesis Imaging in Radio Astronomy', ASP Conf. Series, vol. 6.

- ... frequency
^{4.7} - Note that it is important that the phase of the local oscillator signal be identical at the two antennas, i.e. the local oscillator signal has to be distributed in a phase coherent way to both antennas in the interferometer. Chapter 23 explains how this is acheived at the GMRT.
- ... center
^{4.8} - For maximum sensitivity, one would also point the antennas such that their primary beam maxima are also at .
- ... transform
^{4.9} - see
Chapter 1
- ...
distribution
^{4.10} - This is true only if the antenna dimensions are neglected. Strictly speaking, the measured visibility is an average over the visibilities in the range to where is the diameter of the antennas and b is the separation between their midpoints. As will be seen in Chapter 14 the fact that one has information on visibilities on scales smaller than is useful when attempting to image large regions of the sky.