Physics 728
Radio Astronomy:  Lecture #7
Prof. Dale E. Gary
NJIT

The Receiving System for Interferometry

Introduction

We have seen what the receivers and front end (feed, amplifier, mixer) must do to sensitively detect weak radio emission.  We have also looked at the way interferometry can be used to image the sky, and how pairs of antennas (baselines) measure Fourier components of the sky brightness distribution.  We now will go back and look at what special aspects of the receiving system are needed to do interferometry.
Phaselock
The first part of the receiver is the heterodyning to bring the RF (radio frequency) down to a manageable intermediate frequency (IF) that we can work with.  This requires a mixer and local oscillator (frequency reference), as shown in Figure 1, below:

Figure 1: Heterodyne receiver, which uses a local oscillator (LO)
operating at frequency wo, to tune to the desired radio frequency (RF)
and mix with RF at a wide band of frequencies, and strip off a lower
bandwidth section of intermediate frequency (IF) for further processing.

For interferometry, we must correlate the signals from two antennas, which requires a number of additional considerations.  The main one is to ensure that the receivers of the two antennas are operating at exactly the same frequency.  If one were on a frequency different by only 1 Hz, the resultant phase between the two would change by 360 degrees every second!  To control the frequency of the two, we must use a phase lock system, whose block diagram might look like Figure 2.


Figure 2: Adding a phase lock loop, which compares the LO output
with an external reference frequency and sends an error signal back
to the LO to keep it in perfect phase lock with the reference signal.
Each antenna receives the Ref signal from the same source, so all
receivers are locked to the same frequency.

Mixer 2 compares the LO to a frequency reference, which comes from the same frequency source for all antennas.  Any error in the phase results in an error signal that is fed back to the oscillator to adjust its frequency to maintain exact frequency tuning.

Correlation
The IF signal from each receiver looks like a noise signal.  Part of the waveform is really signal from the source, and part of it (perhaps the largest part) is noise.  If they both look the same, how do we tell the difference?  The source signal will be correlated between the two antennas, while the noise signal will not.  This is illustrated with the simulated waveforms from two antennas, below:

Figure 3: Two simulated voltage waveforms, with phase 30 degrees, with the
waveform for antenna 1 shifted by 800 time samples.  The noise level is 1/5 of
the signal level in this example.  The waveforms appear to have no relation to
one another, but when correlated they give the plot in the third panel (cosine
channel), which shows a good correlation (spike) at a time lag of 800 samples.
Shifting the antenna 1 waveform by 90 degrees and performing the correlation
again gives the result shown in the bottom panel (sine channel).  The combination
of the sine and cosine channels gives an amplitude of 0.268 and phase of
30.2 degrees.  The correct values are 0.25 and 30 degrees.


Figure 4: Two simulated voltage waveforms, with the same characteristics as for
Figure 3, but now the noise level 5 times higher and is now equal to the signal level.
Because the noise is uncorrelated, the correlated signal is hardly affected, and gives
and amplitude of 0.245 and phase of 30.83 degrees, compared to the correct
values of 0.25 and 30 degrees.

Given time varying voltages V1 and V2, the correlation is found by multiplying them, with one delayed by the geometrical delay tg= B . s/c, then averaging, i.e.

r = <V1(t)V2(t)>
where < > denotes the expectation value, found by averaging over some integration time.  Considering for the moment a monochromatic time-varying signal
V1(t) = v1 cos[2pn(t-tg)]
V2(t) = v2 cos[2pnt]
we have
r = <v1v2 cos[2pn(t-tg)] cos[2pnt]>
   = v1v2 <cos2(2pnt) cos(2pntg) + cos(2pnt) sin(2pnt) sin(2pntg)>
   = v1v2 cos(2pntg)
Since the geometrical delay tg changes due to the Earth's rotation, the relatively slowly varying cosine term causes the oscillations that represent the motion of the source through the interferometer fringe pattern.  In the old days of chart recorders, this fringe pattern was traced on paper and its amplitude and phase could be measured by hand.  The rate of fringe oscillations is called the natural fringe rate.  The fringe frequency is
nF = dw/dt = -We u cos d
where We is the Earth rotation rate, w = (Bl . s) is the spatial frequency corresponding to the projected baseline z-component given by the coordinate transformation from the previous lecture, u is the usual E, or y-component spatial frequency, and d is the declination of the phase center.  The geometry is as shown in Figure 5.

Figure 5: The geometry and block diagram leading to the measured cosine component
of the correlation.  Both the multiplier and the integrator are part of the device called the
correlator.  A refinement is shown in Figure 6.

A major refinement is to use a second correlator, and shift one of the signals by p/2, so that both sine and cosine components are measured simultaneously, as shown below.  The components in the dashed box in Figure 5 are indicated by each circled X in Figure 6.

Figure 6: Inserting a phase shift of p/2 in one of the antennas and doing a
second correlation allows both sine and cosine components to be measured
simultaneously.  These are recorded and become the complex visibility at
spatial frequency u,v corresponding to the projected baseline between the
antennas.

The quantities measured out of the correlator are the real and imaginary parts of the complex visibility measured with the baseline, whose normalization is obtained by the calibration procedure, which we have not yet discussed.  You may wonder how we accomplish the 90 degree phase shift over a finite IF bandwidth Dn.  This is done in the OVSA receivers by changing the phase of the reference signal used to phase lock the local oscillator.  Note that this is not equivalent to a time delay, which would shift the phase by different amounts for different frequencies, but rather it shifts the phase of each frequency separately.  Some systems use the digital correlator to do the phase shifting.  As it turns out, phase shifting, or synchronous detection, is also important for eliminating any DC offsets.  If we multiply two waveforms with a DC offset, the offsets will give a non-zero signal even when there is no correlation in the signals.  This is eliminated by periodically inverting the signal at the antenna, and then synchronously inverting the signal again at the correlator.  In this way, the signals stay correlated while any unwanted DC offset gets inverted periodically and averages to zero.

To discuss correlators further, we will use the NRAO Summer School lecture on cross correlators.