### [linrad] Re: Polarization question

```
Hi Leif,

I have been thinking more about the Linrad polarization
questions that I raised here two weeks ago.

I wrote:

Suppose they [the feedline lengths in X and Y channels]
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.

You replied:

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.

Let's work in terms of the Stokes Parameters, with complex
signals X and Y.

I = |X|^2 + |Y|^2 (total power)
Q = |X|^2 - |Y|^2 (horizontal linear component)
U = 2Re(X^* Y)(vertical linear component)
V = 2Im(X^* Y)(circular component)

L = sqrt(Q^2 + U^2) (linear polarized component)
Theta = 0.5*atan(U/Q)   (polarization angle)

If I shift the phase of X relative to Y (say, by inserting
an extra piece of cable in the X feedline), the values of U,
V, L, and Theta will surely change.

I don't know what you meant when writing From a practical
point of view the cables are well matched.  In my station,
at present, it would be a complete accident if the downlines
from the tower-mounted X and Y preamps were the same
electrical length.  They are just two pieces of coax that I

Moreover, my xpol yagis have the H elements located forward
of the V elements by some 10 inches or so (I forget the
exact amount) -- maybe 1/8 of a wavelength.  This will have
a very significant effect, as well, no?

So, it seems to me that the only way to get things
calibrated correctly is the one you outlined:

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.

On the other hand, a strong practical reason to use the +
configuration is that one wants to use the antenna for tropo
as well as EME -- and therefore wants the ability to
transmit a horizontal signal.  My array, therefore, is in
the + configuration.

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:-)

To me it seems much easier to do it in software, and perhaps
I will try this within MAP65.  Suppose the gains in the X
and Y channels are already matched.  Then, while receiving a
100% horizontally polarized signal, shouldn't it be
sufficient to multiply the complex signal for X (or Y) by a
complex constant e^(i*phi), with the phase shift phi
chosen so as to minimize Stokes Parameter V ?

-- Joe, K1JT

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### [linrad] Re: Polarization question

```Hi Joe,

I have been thinking more about the Linrad polarization
questions that I raised here two weeks ago.

I wrote:

Suppose they [the feedline lengths in X and Y channels]
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.

You replied:

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.

Let's work in terms of the Stokes Parameters, with complex
signals X and Y.

I = |X|^2 + |Y|^2 (total power)
Q = |X|^2 - |Y|^2 (horizontal linear component)
U = 2Re(X^* Y)(vertical linear component)
V = 2Im(X^* Y)(circular component)

L = sqrt(Q^2 + U^2) (linear polarized component)
Theta = 0.5*atan(U/Q)   (polarization angle)

If I shift the phase of X relative to Y (say, by inserting
an extra piece of cable in the X feedline), the values of U,
V, L, and Theta will surely change.Yes, of course, but any
difference in length of the cables will be insignificant as
compared to the differences in phase shifts that you have in
the amplifiers.

I don't know what you meant when writing From a practical
point of view the cables are well matched.  In my station,
at present, it would be a complete accident if the downlines
from the tower-mounted X and Y preamps were the same
electrical length.  They are just two pieces of coax that I
But I am sure that none of them is VERY long in terms of
wavelengths. It means that the phase shift vs frequency
the difference in cable lenghts introduce is very small and
negligible as compared to the large difference in phase vs
frequency that non-matched RF amplifiers  might introduce.
Over a narrow bandwidth, such as 2 MHz, even that can be neglected
if the RF amplifiers are reasonably similar. In the end, you
just have an unknown, but frequency independent phase shift between
the two RF channels.

Moreover, my xpol yagis have the H elements located forward
of the V elements by some 10 inches or so (I forget the
exact amount) -- maybe 1/8 of a wavelength.  This will have
a very significant effect, as well, no?
It just adds to the undeterminated phase difference and it
is independent of frequency over 2 MHz.

So, it seems to me that the only way to get things
calibrated correctly is the one you outlined:

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.

On the other hand, a strong practical reason to use the +
configuration is that one wants to use the antenna for tropo
as well as EME -- and therefore wants the ability to
transmit a horizontal signal.  My array, therefore, is in
the + configuration.
But that is a non-argument. I was using the X configuration for
10 years with only the option to put 50% of the power into each
polarisation. In phase for vertical and out of phase for horizontal.
(I also had +/- 90 degrees for circular, but although very good
for aurora I found it useless for EME) I actually lost some interesting
contacts because I could not put all the power in +45 or -45 degrees.
I did hear GW0KZG/MM from the red sea and got QRZ from him.
I knew 45 degrees would have given 3 dB more signal, but I could not
do it because my X configuration was set up in the simple way with
only H or V (or circular) as the emitted polarisation.

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:-)

To me it seems much easier to do it in software, and perhaps
I will try this within MAP65.
OK, but remember there is a performance penalty

Suppose the gains in the X
and Y channels are already matched.  Then, while receiving a
100% horizontally polarized signal, shouldn't it be
sufficient to multiply the complex signal for X (or Y) by a
complex constant e^(i*phi), with the phase shift phi
chosen so as to minimize Stokes ```

### [linrad] Re: Polarization question

```

Sirs! Leif  Joe,

In addition to all the interesting observations and suggestions
regarding the polarization (indication) challenges it is maybe
useful to add that once we ignite our perfectionist behavior and
begin to trim all of the feed lines and matching as near to ideal
as possible, it is good to keep in mind that the phase-shift
provided by a physical length of transmission line is only in
good relation to the velocity factor when the line is terminated
in its characteristic impedance (this being an additional source
of error if input impedances of e.g .intermediate amplifiers or
mixer stages are way off the nominal cable impedance). This
can be an issue if one chooses e.g. two completely different
builds of coax, like one with foam or teflon and the other with
solid PE insulation (velocity factors possibly of 0.82 and 0.66
respectively), both at 50 ohms impedance though.

Though Linrad would not have problems with the resultant phase
shift, the indicated value for the polarization for the same station
would be different, when jumping the communications frequency
from say 144.020 to e.g. 144.380. Now if 40 meter long coaxes
were used with an undefined termination somewhere near the
operator's desk, then in addition to the phase change with
frequency there would be the amplitude difference variation
between channels, that Leif was referring to. Thus precision
and symmetry in constructing (to eliminate error sources) is
always a bonus, and a win-win starting point over commercial
hardware, that generally shows considerable variation, when
not sold as matched pairs (like some transistor components).

Actually one can go quite far with phase differences. I have
recently constructed a couple of boxes with mechanical
switches (11-position decks) that can combine ANY two
antennas to approximately line up their signal phases.
Linrad can do without it for reception as the software by
itself can line up those phases. But for transmit it is nice
to be able to share full power into two antennas with the
proper phase for that particular propagation condition.
Changing the phase of two separate final amplifiers will
not be sufficient or optimal for many circumstances.

73, Zaba OH1ZAA/2

At 00:44 26.6.2007, SM5BSZ wrote:

Hi Joe,

I have been thinking more about the Linrad polarization
questions that I raised here two weeks ago.

I wrote:

Suppose they [the feedline lengths in X and Y channels]
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.

You replied:

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.

Let's work in terms of the Stokes Parameters, with complex
signals X and Y.

I = |X|^2 + |Y|^2 (total power)
Q = |X|^2 - |Y|^2 (horizontal linear component)
U = 2Re(X^* Y)(vertical linear component)
V = 2Im(X^* Y)(circular component)

L = sqrt(Q^2 + U^2) (linear polarized component)
Theta = 0.5*atan(U/Q)   (polarization angle)

If I shift the phase of X relative to Y (say, by inserting
an extra piece of cable in the X feedline), the values of U,
V, L, and Theta will surely change.Yes, of course, but any
difference in length of the cables will be insignificant as
compared to the differences in phase shifts that you have in
the amplifiers.

I don't know what you meant when writing From a practical
point of view the cables are well matched.  In my station,
at present, it would be a complete accident if the downlines
from the tower-mounted X and Y preamps were the same
electrical length.  They are just two pieces of coax that I
But I am sure that none of them is VERY long in terms of
wavelengths. It means that the phase shift vs frequency
the difference in cable lenghts introduce is very small and
negligible as compared to the large difference in phase vs
frequency that non-matched RF amplifiers  might introduce.
Over a narrow bandwidth, such as 2 MHz, even that can be neglected
if the RF amplifiers are reasonably similar. In the end, you
just have an unknown, but frequency independent phase shift between
the two RF channels.

Moreover, my xpol yagis have the H elements located forward
of the V elements by some 10 inches or so (I forget the
exact amount) -- maybe 1/8 of a wavelength.  This will have
a very significant effect, as well, no?
It just adds to the undeterminated phase difference and it
is independent of frequency over 2 MHz.

So, it seems to me that the only way to get things
```

### [linrad] Re: Polarization question

```

Sir Joe!

As far as I have understood Linrad tries to combine the incoming
spectrum of two

antennas in order to reach a signal vector maximum for each spectral component.
Thus it does not seem to matter (at least for that summation process) what the
absolute length of feed lines or relative signal amplitude in each
channel is.

I have also the full WSE-set since 2005 or so, but I have never even
attempted to
power it up (that will be one of this Summer's projects). So far I
have run Linrad as

a home-brew single-channel I/Q direct conversion receiver, and man-made noise
suppression has already been quite impressive without calibration. I
have plans
to expand to a multi-path take-off angle analyzer with the two
channels activated.

It is probably already 7 years ago (at the 2000 VHF/UHF/SHF-meeting
in Finland)

that the three of us: SM5BSZ/OZ1RH/OH1ZAA discussed all night (probably up to
4 - 5 AM) about the early Linrad and other properties of weak signal
detection. It was
one of the most interesting discussions that I have ever experienced
during my ham
career. Most probably we did not define everything very accurately at
that time, as
at least one of the topics was how deep we could go into the noise
and I had the impression at that time that it went beyond certain
physical barriers

(my perceived limits). But maybe we were not too clear about the distinction
between noise and interference... well it is too long ago anyway to recollect.

I think that I had a question about the two-channel case for a
situation where the

two pre-amps would have a markedly different noise figure, and my understanding
was that Linrad could do some miracles in overcoming the noise issue probably
by correlating. Leif may have some good additional remarks on these matters.

73, Zaba  OH1ZAA / NNoY

P.S. These issues deal with the summation of (un)correlated noise

At 17:02 10.6.2007, K1JT wrote:

Hi Leif (or anyone else who knows),

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.

Is this the correct way to look at it?  Can you enlighten me any further?

-- 73, Joe, K1JT

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### [linrad] Re: Polarization question

```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.
Sure:-)

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
//re_A=C1*re_X+C2*re_Y-C3*im_Y
//im_A=C1*im_X+C2*im_Y+C3*re_Y
//re_B=C1*re_Y-C2*re_X-C3*im_X
//im_B=C1*im_Y-C2*im_X+C3*re_X
//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)
// **

73

Leif

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### [linrad] Re: Polarization question

```

Thanks Leif for all explications!

As such it is clear that in stable conditions (no polarization
change) a signal

emanating from one source must be able to show a fixed polarization state
irrespective of its spectral content, but any offset from the real
value is due

to feed line length differences or disparity in pre-amp phase curves or gain
differences (as a function of frequency). And there another source within the
same spectral range would probably result in an error signal, as one
cannot show two different polarization states with a single graphical line,
i.e. at each instant just one phase would be shown, but it is a transformed
(restless) plot of two independent signals?

Actually my plan is to measure the phase difference between two horizontal
antennas (with known radiation patterns) at different heights. With just one
take-off angle (incoming signal) Linrad would be able to show the phase
difference between these two antennas, and the if the amplitude is also
available, then from the math with the radiation pattern, the
angle(s) of arrival

would be solved for up to dual-path openings. Of course the ever changing
ionosphere makes it a dynamic challenge, also due to phase reversals of
the skipping wave. However, it would make home-operating a much deeper
experience from the scientific point of view (compared to regular
59-operating).

Recent experiments have involved the phase / out-of-phase combining of any
two given antennas on 50 MHz (still without Linrad). That adds a lot of extras
too in operating. With little more definition and Linrad this will
provide lots of fun.

73, Zaba  OH1ZAA/2

P.S. Most likely Linrad can do many more miracles in man-made noise removal
than with Gaussian noise. Assumedly it is to the detriment
of total sensitivity
if one of two pre-amps in e.g. a X-polarized setup would
have a markedly
higher noise-figure than the other pre-amplifier. Also some
operators claim
a much higher noise level than on a single horizontal yagi,
if a X-yagi setup
is configured for plain horizontal polarization (i.e.
DX-station signals will be
added as linear vectors, but e.g. galactic noise as n^2 from
two perpendicular
antennas; and Linrad can't help that?) [Thoughts back to
2000 5AM discussions]

From UKSMG 50 MHz Announcements:   (a week ago)

If your PC-clock is running on time, then old Winrad versions have
stopped working...
I2PHD's new Winrad 1.25 is available (User Guide 1.2 is not yet up to
date with the changes) ---
SM5BSZ's latest Linrad is at version 2-34 --- Last week we
reactivated OH3MHA at KP20XW
in an industrial environment with sharp pulsing, so probably
be very effective (i.e. mandatory for noise-free reception)  73,
Zaba OH1ZAA/OHoMZA ---

- Sunday, June 03 2007 at 08:25 (GMT)

At 19:54 10.6.2007, SM5BSZ wrote:

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.
Sure:-)

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 ```

### [linrad] Re: Polarization question

```
Leif, Zaba, and all,

Many thanks for the clear explanations!

-- Joe, K1JT

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```