Hi Joe, Zaba and all,

> Ideally, I suppose, with an xpol antenna one would like to have 
> phase-matched preamps for the two polarizations and equal 
> electrical lengths for the two Rx feedlines.

> Suppose they are not well matched.  In other words, suppose that 
> the complex gains and signal delays in the two polarization 
> channels are not equal.  Will Linrad's polarization-matching 
> capability be compromised?  As far as I can see, it still works 
> well even with poorly matched feedlines.  I suppose this must 
> mean that Linrad solves for a differential complex gain, and 
> that over a fairly narrow bandwidth a different delay can be 
> treated as a phase shift.
>From a practical point of view the cables are well matched. I think
it is a safe assumption to guess that the length differences will 
be very small and of no concern.

Much more important would be the phase shift through very
narrow filters in case you have such at the output of
the preamplifier. A filter with 1 MHz bandwidth has a
delay of something like 1us and in case the other channel
has a filter with 1.5 MHz bandwidth, it would have much 
less delay and that could be a problem.

When the phase is not matched, the polarisation indicator
will not show the correct polarisation. That will not
cause any loss of sensitivity, only make it more difficult
to decide what tx polarisation to use.

When it comes to the amplitude balance and similarity in NF
I no longer remember how important it is for sensitivity.
Not very important in relation to how important it is for
getting correct polarisation readings in any case.

The procedure to follow is like this:

Connect everything. Compare the noise floor levels and insert
an attenuator (or tweak the second RF amp tuning) for the noise
floors of the two channels to become equal. 

Listen to a linearly polarised signal that arrives with similar 
strength in both polarisations. (This is the strongest reason
why the X configuration is so much better than the + configuration.
It is easy to find a pure H-pol signal. Finding a 45 degree 
terrestrial signal is virtually impossible due to ground reflections 
so a + configured system has to be calibrated on EME signals.) 
Change cable lengths until the signal appears close to linear
on the pol meter. Fine tune by tweaking the second RF amplifiers.
(will affect both amplitude and phase, but there are two second
RF amplifiers so it should be possible to find both amplitude and
phase matching.

It would of course be easy to add parameters for amplitude
and phase balance, but I have not done it since I found
it easy to do in hardware:-)

Once the phase and amplitudes are properly set it will be
a good idea to set the polarisation to + and -45 degrees
with respect to the real orientations of the elements.
The noise floor should ideally be exactly the same
regardless of the phase. (plus, minus, circular or anything
between.) In real life one might see significant differences
in the noise floor and that would be caused by mutual
coupling between the elements (to some extent the noise 
would then be correlated.) It is a good idea to verify 
that the isolation between polarisations is at least 20 dB 
in transmit mode:-)

The procedure Linrad uses is to compute the powers in both channels
as well as the complex correlation. Then like this:

// *****************************************
//Now we have x2,y2 (real values) and xy (complex).
//For explanation purposes, assume im_xy == 0, which corresponds to linear
//polarization. The signal vill then be polarised in a plane.
//a = angle between polarisation plane and the horisontal antenna.
//Assume that the noise level n is the same in the two antennas, and that
//the noise is uncorrelated.
//We then find:
//            x2 = cos(a)**2 + n**2
//            y2 = sin(a)**2 + n**2
//            xy = sin(a)*cos(a)
//From this we find: x2 * y2 - xy*xy = n**2 + n**4
//Neglect n**4:
//cos(a)=sqr( x2 - ( x2 * y2 - xy*xy) )
//sin(a)=sqr( y2 - ( x2 * y2 - xy*xy) )
//The transformation formula to use for rotating the polarization
//plane to produce new signals A and B, where A has all the signal and B
//only noise, will then be:
// A = X * cos(a) + Y * sin(a)
// B = Y * cos(a) - X * sin(a)
//Extending to im_xy != 0 the transformation becomes
//C1 = cos(a)
//C2 = sin(a) * re_xy / sqr( re_xy**2 + im_xy**2)
//C3 = sin(a) * im_xy / sqr( re_xy**2 + im_xy**2)
// **************************************



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