[evlatests] Right/Left self correlations

[Barry Clark]

Lately people have been looking at the product between the left and
right polarizations of the same antenna.  I thought I might make a
clarifying remark about what we expect to see.  Things are a bit
confusing for these products, because the correlators are being fed
with a zero fringe rate; in true cross correlations a non-zero fringe
rate automatically sorts out a lot of things, sidebands in particular.
For the zero fringe rate case, we see only the output of the cosine
correlators - the sine is set to zero.

Consider the simplest case, where the leakage term is just a single
real number.  The real correlations, in lag space, are then just
the real leakage factor times the autocorrelations, for both negative
and positive lags.  Since the lag function is real and symmetric, the
spectrum is even about zero frequency.  So the fourier transform the
CBE does produces a spectrum which is real and even about zero
frequency.  The CBE discards the negative frequency half, which, unlike
cross correlations, has power in it.

The next simplest case is when the leakage term is a pure imaginary.
Once again, only the cosine correlators have data, but the zero lag
is clearly zero correlation, and in fact, this will result in
correlations which are an odd function of lag, so the fourier
transform will produce a spectrum with 90 degree phase, and
antisymmetric about zero frequency.

Generalizing, if the leakage is a single complex factor times the 
spectrum, then the real part is symmetric about zero frequency, and
the complex part antisymmetric.  So the CBE discarded spectrum is
identical to that it outputs, except for reversed phase.

Finally, in a case approaching reality, suppose the leakage has a
different delay.  It is a summation of little frequecy slices of the
case above - the amplitudes are the same in the discarded and output
spectra, but the phases are reversed, and since we are talking delay,
phases vary linearly with frequency.

Since, unlike true cross correlation, there is equal power in the
positive and negative frequency spectra, we have to decide which one
is the "real" spectrum to output.  To be consistent you pick one or
the other, depending on whether there are an odd or even number of upper
sideband mixers.  Since the station board sideband flippers are just 
upper sideband mixers, one should pick positive or negative frequencies
depending on the setting of the sideband flippers.  Since the CBE
doesn't do that, the reversal of delay direction for subbands with the
flippers set (usually the odd numbered subbands) is a natural and
expected consequence.