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−  == Linear to Circular Conversion ==
 +  [[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:
 +  
   
−  :<math>
 +  [[Polarization Mixing Due to Feed Rotation]] ([[Polarization Mixing Correction (Old)]]) 
−  \begin{align}
 +  
−  R = X + iY \\
 +  
−  L = X  iY
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  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
 +  
−   +  
−  :<math>
 +  
−  \begin{align}
 +  
−  RR^* = (X + iY)(X + iY)^* &= XX^*  iXY^* + iYX^* + YY^* \\
 +  
−  LL^* = (X  iY)(X  iY)^* &= XX^* + iXY^*  iYX^* + YY^* \\
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  One problem is that there is generally a nonzero 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,
 +  
−   +  
−  :<math>
 +  
−  \begin{align}
 +  
−  Stokes \, I &= \frac{RR^* + LL^*}{2} = XX^* + YY^* \\
 +  
−  Stokes \, V &= \frac{RR^*  LL^*}{2} = i(XX^*  YY^*)
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  For completeness:
 +  
−   +  
−  :<math>\begin{align}
 +  
−  Stokes \, Q &= XX^*  YY^* \\
 +  
−  Stokes \, U &= XY^*  YX^* \\
 +  
−  P_{linear} &= \sqrt{U^2 + Q^2} \\
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−  \Theta &= \frac{1}{2}\tan^{1}{\frac{U}{Q}}
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  When I plot the quantities I, V, R and L as measured (Figure 1) for geosynchronous satellite Ciel2, the results look reasonable, except that there are parts of the band where R and L are misassigned, and others where they do not separate well.
 +  
−   +  
−  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 leastsquares routine, and then the phases were offset by π/2 for both XY* and YX* according to:
 +  
−   +  
−  :<math>\begin{align}
 +  
−  XY^{*}' &= XY^*e^{i(\phi(v) + \frac{\pi}{2})}
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−  YX^{*}' &= YX^*e^{ i(\phi(v) + \frac{\pi}{2})}
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  where φ(v) is the phase fit by the linear function. The corrected phases are shown in the lower panel of Figure 2.
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−   +  
−  When the corrected (primed) quantities are used in
 +  
−   +  
−  :<math>\begin{align}
 +  
−  RR^* &= XX^*  iXY^(*)' + iYX^(*)' + YY^*
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−  LL^* &= XX^* + iXY^(*)'  iYX^(*)' + YY^*
 +  
−  \end{align}
 +  
−  </math>
 +  
−   +  
−  == Polarization Mixing Correction ==
 +  
−  Due to relative feed rotation between azal mounted antennas and equatorial mounted antennas
 +  
Latest revision as of 12:54, 18 November 2016
Linear to Circular Conversion
Polarization Mixing Due to Feed Rotation (Polarization Mixing Correction (Old))