As we discussed earlier, an aperture synthesis telescope
can be regarded as a collection of two element interferometers.
Hence, for understanding the sensitivity of such a telescope, it
is easier to first start with the case of a two element interferometer.
Consider such an interferometer composed of two antennas , (of
identical gains, but possibly different system temperatures), looking
at a point source of flux density S. We assume that the point source is at the
phase center^{5.1}and hence that in the absence of noise the visibility phase is zero.
Let the individual antenna gains^{5.2}be G and system temperatures be T and T. If
and are the noise voltages of antennas and
respectively,then
, and
. Similarly if and
are the voltages induced by the incoming radiation from
the point source,
. The
instantaneous correlator^{5.3} output is given by:

The mean^{5.4} of the correlator output is hence:

where we have assumed that the noise voltages of the two antennas are not correlated, and also of course that the signal voltages are not correlated with the noise voltages. is hence an unbiased estimator of the true visibility.

To determine the noise in the correlator output, we would need to compute the rms of for which we need to be able to work out:

where for ease of notation we have stopped explicitly specifying that
all voltages are functions of time. This quantity is not trivial to
work out in general. However, if we assume that all the random processes
involved are Gaussian
processes^{5.5} the complexity is considerably reduced because for Gaussian
random variables the fourth moment can then be expressed in terms of
products of the second moment. In particular^{5.6}, if
have a
joint gaussian distribution then:

Rather than directly computing , it is instructive first to consider the more general quantity

viz. the cross-correlation between the outputs of interferometers
and . We have:

The case we are currently interested in is , which from eqn(5.1.3) is:

To get the variance of we need to subtract the square of the mean of from the expression in eqn(5.1.4). Substituting for from eqn(5.1.1) we have:

Note that the angular brackets denote ensemble averaging. In real life of course one cannot do an ensemble average. Instead one does an average over time, i.e. we work in terms of a time averaged correlator output , defined as

As can easily be verified, . However, computing the second moment, viz., is slightly more tricky. It can be shown

where is the auto-correlation function of , and is the variance of . Now, if is a quasi-sinusoidal process with bandwidth , then the integral of will be negligible outside the coherence time . Further, if , then the factor in parenthesis in eqn(5.1.6) can be taken to be for . Hence we have:

where is the power spectrum

Putting all this together we get that the signal to noise ratio of a two element interferometer is given by:

There are two special cases which often arise in practice. The first is when the source is weak, i.e. . In this case the snr becomes

For a single dish with the collecting area equal to the sum of the collecting areas of antennas and (i.e. with gain G), and with system temperature the signal to noise would have been a factor of better

The other special case of interest is when the source is extremely
bright, i.e.
. In this case, the signal to
noise ratio is:

Having derived the signal to noise ratio for a two element interferometer,
let us now consider the case of an N element interferometer. This can
be considered as two element interferometers. Let us take
the case where the source is weak. Then from eqn(5.1.3) the
correlation between and is given by

The outputs are uncorrelated, even though these two interferometers have one antenna in common

This is the fundamental equation

For a complex correlator^{5.12}, the analysis that we have just done holds
separately for the cosine and sine channels of the correlator. If we
call the outputs of such a correlator and then
it can be shown that the noise in and is
uncorrelated. Further
since the time averaging can be regarded as the adding together of a
large number of independent samples (
),
from the central limit theorem, the statistics of the noise in
and
are well approximated as
Gaussian. It is then possible to derive the statistics of functions
of
and
, such as the visibility
amplitude (
) and the visibility
phase (
). For example, it
can be shown that the visibility amplitude has a Rice
distribution^{5.13}

For an extended source, the entire analysis that we have done continues to hold, with the exception that S should be treated as the correlated part of the source flux density. For example, at low frequencies, the Galactic background is often much larger than the receiver noise and one would imagine that the limiting case of large source flux density (i.e. eqn(5.1.11) is applicable. However, since this background is largely resolved out at even modest spacings, its only effect is an increase in the system temperature.

Finally we look at the noise in the image plane, i.e.
after Fourier transformation of the visibilities. Since most of
the astronomical analysis and interpretation will be based on
the image, it is the statistics in the image plane that is usually
of interest. The intensity at some point in the image
plane is given by:

where is the weight

In the absence of any sources, the visibilities are uncorrelated with one another, and hence, we have

Hence in the case that all the noise on each measurement is the same, and that the weights given to each visibility point is also the same, (i.e. uniform tapering), the correlation in the map plane has exactly the same shape as the dirty beam. Further the variance in image plane would then be , where is the noise on a single visibility measurement. This is equivalent to eqn(5.1.13), as indeed it should be.

Because the noise in the image plane has a correlation function shaped like the dirty beam, one can roughly take that the noise in each resolution element is uncorrelated. The expected statistics after simple image plane operations (like smoothing) can hence be worked out. However, after more complicated operations, like the various possible deconvolution operations, the statistics in the image plane are not easy to derive.

- ... center
^{5.1} - See Chapter 4.
- ... gains
^{5.2} - Here the gain is taken to be in units of Kelvin per Jansky of flux in the matched polarization
- ... correlator
^{5.3} - Here we are dealing with an ordinary
correlator, not the
*complex correlator*introduced in the chapter on two element interferometers. - ... mean
^{5.4} - Note that the average being taken over here
is
*ensemble*average, and*not*an average over time. - ...
processes
^{5.5} - Recall from the discussion of sensitivity of a single dish telescope that the central limit theorem ensures that the signal and noise statistics will be well approximated by a Gaussian. This of course does not include `systematics', like eg. interference, or correlator offsets because of bit getting stuck in the on or off mode etc.
- ... particular
^{5.6} - The derivation of this expression is particularly straightforward if one works with the moment generating function; see also the derivation sketched in Chapter 1.
- ...
shown
^{5.7} - Papoulis, `Probability, Random Variables & Stochastic Processes', Third Edition, Chapter 10
- ...
spectrum
^{5.8} - Where we have made the additional assumption that is a white noise process, i.e. that its spectrum is flat. The power spectrum for such processes is easily derived from noting that , and that for a quasi-sinosoidal proccess of bandwidth , the integrand is non zero only over an interval (including the negative frequencies).
- ... better
^{5.9} - As you can easily derive from eqns 5.1.1 and 5.1.3 by putting . Note that in this case eqn 5.1.1 becomes
- ... common
^{5.10} - This may seem counter intuitive, but note that the outputs are only uncorrelated, they are not independent.
- ... equation
^{5.11} - In some references, an efficiency factor is introduced to account for degradation of signal to noise ratio because of the noise introduced by finite precision digital correlation etc. This factor has been ignored here, or equivalently one can assume that it has been absorbed into the system temperature.
- ... correlator
^{5.12} - See the chapter on two element interferometers
- ...
distribution
^{5.13} - Papoulis, `Probability, Random Variables & Stochastic Processes', Third Edition, Chapter 6.
- ... weight
^{5.14} - As discussed in Chapter 11, this weight is in general a combination of weights chosen from signal to noise ratio considerations and from synthesized beam shaping considerations.