Linear to Circular Conversion - Revision history

Dgary at 17:25, 27 June 2020

2020-06-27T17:25:24Z

← Older revision		Revision as of 17:25, 27 June 2020
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	\begin{align}		\begin{align}
	Stokes \, I &= \frac{RR^* + LL^}{2} = XX^ + YY^* \\		Stokes \, I &= \frac{RR^* + LL^}{2} = XX^ + YY^* \\
	Stokes \, V &= \frac{RR^* - LL^}{2} = i(XX^ - YY^*)		Stokes \, V &= \frac{RR^* - LL^}{2} = i(YX^ - XY^*)
	\end{align}		\end{align}
	</math>		</math>

Dgary at 13:30, 10 July 2017

2017-07-10T13:30:50Z

← Older revision		Revision as of 13:30, 10 July 2017
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	</math>		</math>
	</center>		</center>
	When I plot the quantities I, V, R and L as measured (Figure 1) for geosynchronous satellite Ciel-2, the results look reasonable, except that there are parts of the band where R and L are mis-assigned, and others where they do not separate well.		When I plot the quantities I, V, R and L as measured ('''Figure 1''') for geosynchronous satellite Ciel-2, the results look reasonable, except that there are parts of the band where R and L are mis-assigned, and others where they do not separate well.

	[[File:Figure1.jpg\|center\|thumb\|800px\|Figure 1: Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is pretty good separation of RCP and LCP at lower and higher frequencies in the range, but around 12.4 GHz the polarization has switched and is mis-assigned.]]		[[File:Figure1.jpg\|center\|thumb\|800px\|'''Figure 1:''' Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is pretty good separation of RCP and LCP at lower and higher frequencies in the range, but around 12.4 GHz the polarization has switched and is mis-assigned.]]

	The problem is that residual phase slope of Y with respect to X, caused by a difference in delay between the two channels. This can be seen in the upper panel of Figure 2, which shows the uncorrected phases of XY* and YX. To correct the phases, the RCP phase was fit by a linear least-squares routine, and then the phases were offset by π/2 for both XY and YX* according to:		The problem is that residual phase slope of Y with respect to X, caused by a difference in delay between the two channels. This can be seen in the upper panel of Figure 2, which shows the uncorrected phases of XY* and YX. To correct the phases, the RCP phase was fit by a linear least-squares routine, and then the phases were offset by π/2 for both XY and YX* according to:
	<center>		<center>
	:<math>\begin{align}		:<math>\begin{align}
	XY^{'} &= XY^e^{-i(\phi(v) + \frac{\pi}{2})} \\		XY^{'} &= XY^e^{-i(\phi(\nu) + \frac{\pi}{2})} \\
	YX^{'} &= YX^e^{ i(\phi(v) + \frac{\pi}{2})}		YX^{'} &= YX^e^{ i(\phi(\nu) + \frac{\pi}{2})}
	\end{align}		\end{align}
	</math>		</math>
	</center>		</center>
	where φ(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of Figure 2.		where φ(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of '''Figure 2'''.

	[[File:Linear_to_Circular_Conversion_Figure2.jpg\|center\|thumb\|800px\|Figure 2: Upper panel) Raw phases on XY* (blue) and YX* (green). Lower panel) Phases corrected by removing a delay slope of order 3 ns, and shifting the phase such that the XY* phases on RCP channels is -π/2. This adjustment automatically makes XY* phases on LCP channels become +π/2.]]		[[File:Linear_to_Circular_Conversion_Figure2.jpg\|center\|thumb\|800px\|'''Figure 2:''' Upper panel) Raw phases on XY* (blue) and YX* (green). Lower panel) Phases corrected by removing a delay slope of order 3 ns, and shifting the phase such that the XY* phases on RCP channels is -π/2. This adjustment automatically makes XY* phases on LCP channels become +π/2.]]

	When the corrected (primed) quantities are used in		When the corrected (primed) quantities are used in
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	</math>		</math>
	</center>		</center>
	the resulting channel separation is shown in Figure 3, which shows pretty much perfect channel separation, with the correct polarity.		the resulting channel separation is shown in '''Figure 3''', which shows pretty much perfect channel separation, with the correct polarity.

	[[File:Linear_to_Circular_Conversion_Figure3.jpg\|center\|thumb\|800px\|Figure 3: Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite, after correcting for delay. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is now PERFECT correspondence between RCP channels and the plot, and likewise for LCP channels.]]		[[File:Linear_to_Circular_Conversion_Figure3.jpg\|center\|thumb\|800px\|'''Figure 3:''' Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite, after correcting for delay. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is now PERFECT correspondence between RCP channels and the plot, and likewise for LCP channels.]]

	This analysis can be performed on all antennas in a completely automated way, to produce the results of Fig. 3. The curves for each antenna are very similar, except that they show differences in amplitude (gain) as a function of frequency. When the curves are scaled to have the same Stokes I levels, the result is shown in Figure 4. Here the 13 antennas are shown in different colors. It is clear that matching Stokes I also causes an excellent match in the other polarizations. Remaining mismatch is likely due to not subtracting a no-signal background prior to scaling the Stokes I curves.		This analysis can be performed on all antennas in a completely automated way, to produce the results of '''Fig. 3'''. The curves for each antenna are very similar, except that they show differences in amplitude (gain) as a function of frequency. When the curves are scaled to have the same Stokes I levels, the result is shown in '''Figure 4'''. Here the 13 antennas are shown in different colors. It is clear that matching Stokes I also causes an excellent match in the other polarizations. Remaining mismatch is likely due to not subtracting a no-signal background prior to scaling the Stokes I curves.

	[[File:Linear_to_Circular_Conversion_Figure4.jpg\|center\|thumb\|800px\|Figure 4: Same as Fig. 3, but for all 13 atnennas, with the Stokes I curve for each antenna scaled so that they agree. The beacon frequency in marked in the lower panel.]]		[[File:Linear_to_Circular_Conversion_Figure4.jpg\|center\|thumb\|800px\|'''Figure 4:''' Same as Fig. 3, but for all 13 atnennas, with the Stokes I curve for each antenna scaled so that they agree. The beacon frequency in marked in the lower panel.]]

	Also marked in Figure 4 is the location of the beacon signal for Ciel-2, which is at 12.209 GHz. The beacon signal appears in precisely the correct bin (frequency subchannel), indicating that we have precise tuning (certainly not unexpected, but nice to see).		Also marked in '''Figure 4''' is the location of the beacon signal for Ciel-2, which is at 12.209 GHz. The beacon signal appears in precisely the correct bin (frequency subchannel), indicating that we have precise tuning (certainly not unexpected, but nice to see).

	Just to show that the approach is general, the results for a different communications satellite, Nimiq-5, is shown in Figure 5.		Just to show that the approach is general, the results for a different communications satellite, Nimiq-5, is shown in '''Figure 5'''.

	[[File:Linear_to_Circular_Conversion_Figure5.jpg\|center\|thumb\|800px\|Figure 4: Same as Fig. 5, but for the Nimiq-5 ~~commun-ications~~ satellite.]]		[[File:Linear_to_Circular_Conversion_Figure5.jpg\|center\|thumb\|800px\|'''Figure 5:''' Same as Fig. 4, but for the Nimiq-5 communications satellite.]]

Dgary at 13:46, 20 October 2016

2016-10-20T13:46:29Z

← Older revision		Revision as of 13:46, 20 October 2016
Line 73:		Line 73:

	[[File:Linear_to_Circular_Conversion_Figure5.jpg\|center\|thumb\|800px\|Figure 4: Same as Fig. 5, but for the Nimiq-5 commun-ications satellite.]]		[[File:Linear_to_Circular_Conversion_Figure5.jpg\|center\|thumb\|800px\|Figure 4: Same as Fig. 5, but for the Nimiq-5 commun-ications satellite.]]

	~~== Polarization Mixing Correction ==~~
	~~Due to relative feed rotation between az-al mounted antennas and equatorial mounted antennas~~

Dgary at 13:46, 20 October 2016

2016-10-20T13:46:07Z

← Older revision		Revision as of 13:46, 20 October 2016
Line 1:		Line 1:

	~~== Linear to Circular Conversion ==~~

	At EOVSA’s linear feeds, in the electric field the linear polarization, X and Y, relates to RCP and LCP (R and L) as:		At EOVSA’s linear feeds, in the electric field the linear polarization, X and Y, relates to RCP and LCP (R and L) as:
	<center>		<center>

Dgary: Move this page to a lower subsection

2016-10-20T13:45:46Z

Move this page to a lower subsection

New page

== Linear to Circular Conversion ==

At EOVSA’s linear feeds, in the electric field the linear polarization, X and Y, relates to RCP and LCP (R and L) as:
<center>
:<math>
\begin{align}
R = X + iY \\
L = X - iY
\end{align}
</math>
</center>
In terms of autocorrelation powers, we have the 4 polarization products XX*, YY*, XY* and YX*, where the * denotes complex conjugation. The quantities RR* and LL* are then
<center>
:<math>
\begin{align}
RR^* = (X + iY)(X + iY)^* &= XX^* - iXY^* + iYX^* + YY^* \\
LL^* = (X - iY)(X - iY)^* &= XX^* + iXY^* - iYX^* + YY^* \\
\end{align}
</math>
</center>
One problem is that there is generally a non-zero delay in Y with respect to X. This creates phase slopes in XY* and YX* from which we can determine the delay very accurately. As a check,
<center>
:<math>
\begin{align}
Stokes \, I &= \frac{RR^* + LL^*}{2} = XX^* + YY^* \\
Stokes \, V &= \frac{RR^* - LL^*}{2} = i(XX^* - YY^*)
\end{align}
</math>
</center>
For completeness:
<center>
:<math>\begin{align}
Stokes \, Q &= XX^* - YY^* \\
Stokes \, U &= XY^* - YX^* \\
P_{linear} &= \sqrt{U^2 + Q^2} \\
\Theta &= \frac{1}{2}\tan^{-1}{\frac{U}{Q}}
\end{align}
</math>
</center>
When I plot the quantities I, V, R and L as measured (Figure 1) for geosynchronous satellite Ciel-2, the results look reasonable, except that there are parts of the band where R and L are mis-assigned, and others where they do not separate well.

[[File:Figure1.jpg|center|thumb|800px|Figure 1: Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is pretty good separation of RCP and LCP at lower and higher frequencies in the range, but around 12.4 GHz the polarization has switched and is mis-assigned.]]

The problem is that residual phase slope of Y with respect to X, caused by a difference in delay between the two channels. This can be seen in the upper panel of Figure 2, which shows the uncorrected phases of XY* and YX*. To correct the phases, the RCP phase was fit by a linear least-squares routine, and then the phases were offset by π/2 for both XY* and YX* according to:
<center>
:<math>\begin{align}
XY^{*'} &= XY^*e^{-i(\phi(v) + \frac{\pi}{2})} \\
YX^{*'} &= YX^*e^{ i(\phi(v) + \frac{\pi}{2})}
\end{align}
</math>
</center>
where φ(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of Figure 2.

[[File:Linear_to_Circular_Conversion_Figure2.jpg|center|thumb|800px|Figure 2: Upper panel) Raw phases on XY* (blue) and YX* (green). Lower panel) Phases corrected by removing a delay slope of order 3 ns, and shifting the phase such that the XY* phases on RCP channels is -π/2. This adjustment automatically makes XY* phases on LCP channels become +π/2.]]

When the corrected (primed) quantities are used in
<center>
:<math>\begin{align}
RR^* &= XX^* - iXY^{*'} + iYX^{*'} + YY^* \\
LL^* &= XX^* + iXY^{*'} - iYX^{*'} + YY^*
\end{align}
</math>
</center>
the resulting channel separation is shown in Figure 3, which shows pretty much perfect channel separation, with the correct polarity.

[[File:Linear_to_Circular_Conversion_Figure3.jpg|center|thumb|800px|Figure 3: Amplitudes of Stokes I, Stokes V, RCP and LCP for Ciel-2 satellite, after correcting for delay. The red vertical lines show the center of RCP channels, while the green vertical lines show the center of LCP channels. There is now PERFECT correspondence between RCP channels and the plot, and likewise for LCP channels.]]

This analysis can be performed on all antennas in a completely automated way, to produce the results of Fig. 3. The curves for each antenna are very similar, except that they show differences in amplitude (gain) as a function of frequency. When the curves are scaled to have the same Stokes I levels, the result is shown in Figure 4. Here the 13 antennas are shown in different colors. It is clear that matching Stokes I also causes an excellent match in the other polarizations. Remaining mismatch is likely due to not subtracting a no-signal background prior to scaling the Stokes I curves.

[[File:Linear_to_Circular_Conversion_Figure4.jpg|center|thumb|800px|Figure 4: Same as Fig. 3, but for all 13 atnennas, with the Stokes I curve for each antenna scaled so that they agree. The beacon frequency in marked in the lower panel.]]

Also marked in Figure 4 is the location of the beacon signal for Ciel-2, which is at 12.209 GHz. The beacon signal appears in precisely the correct bin (frequency subchannel), indicating that we have precise tuning (certainly not unexpected, but nice to see).

Just to show that the approach is general, the results for a different communications satellite, Nimiq-5, is shown in Figure 5.

[[File:Linear_to_Circular_Conversion_Figure5.jpg|center|thumb|800px|Figure 4: Same as Fig. 5, but for the Nimiq-5 commun-ications satellite.]]

== Polarization Mixing Correction ==
Due to relative feed rotation between az-al mounted antennas and equatorial mounted antennas