Robert McGwier wrote:
But the discussion here is not about the Microwave phased arrays.  Given 
the rate at which we are signaling and the demodulator requirements that 
will entail,  the spin modulation is a minor factor on the ACP.  For the 
very low bit rate SMS text messaging,  and the 70cm uplink,  where we 
were planning absolutely no phasing whatsoever,  it becomes a major 
factor and might even be a deal killer without greatly increased 
complexity in the 70cm and 2m system.  Essentially,  you would have to 
compute and predict a phase offset from the (say) 70 cm patch and apply 
the correct again before we HELAPS or while we HELAPS on 2m, given 
information from the phased array for other bands and multiply each 
sample by the time varying phasor correction in the SDX to compensate.
I think you are over estimating the complexity of the computation. For a non-phased antenna, the amplitude of the phase correction needed is (he pk-to-pk value is twice this): 

   2pi * sin(angle between boresite and earth) * (distance phase center is offset from spin axis/lambda)


The path that the element makes w.r.t. fixed observer is an ellipse.  I seem to recall that any ellipse can be expressed as the sum of two circles of radius R and r, where R*(1+e) = the semi-major axis and r is the semi-minor axis r=R*(1-e)  (where e is eccentricity). Let the center of the smaller circle "ride" the larger circle, let the larger R circle spin one turn per orbit prograde (the direction the satellite moves) and the t circle spin retrograde (in opposite direction). Pick the spot on the small circle (x,y)  so that R+r is the apogee and R-r is perigee. Then the equation of motion of the spot will be

               x = R*cos(Wt ) + r*cos(-Wt)    = (R+r) cos (Wt)     and
               y = R*sin (Wt) + r*sin  (-Wt)    = (R-r) sin (Wt)   ,
                  where W (really omega) = 2*pi*(spin rate)

This is sometimes called the equation of a central ellipse and the method dates back to Eudoxus of Cnidus (c. 400-347 B.C.)  and then to Hipparchus of Rhodes (c. 190-120 B.C.) (http://astro.isi.edu/games/kepler.html -- see Fig 1) although it is sometimes attributed to Ptolemy (c. 85-165).

So what the ancient Greeks tell you is that you can tweak one LO in the system (either TX or RX) to have a phase offset that is the sum of two contra-rotating phasors running at the spin rate, with the phase, amplitude and eccentricity determined by geometry determined from the S2 interferometer. No math higher than trig is needed.

73, Tom