Setting the OMC DCPD matrix - part II

This is a follow-up to alog 45734.

It answers the question of how to set the DCPD input matrix, with some more background.

Basics:

- We can measure two numbers (ratios):
     g = PD_A/PD_B coherent signal amplitude ratio (best done by looking at the PD_A/PD_B transfer function at low frequencies)
     h = PD_A/PD_B shot noise signal amplitude ratio (by matching the amplitude spectra - a transfer function does not work here)

- These are directly related to the OMC BS splitting ratio 'bA/bB' and the sensing chain gain ratio between the two photo diodes, gA/gB via
     g = gA/gB * bA/bB
     h = gA/gB * sqrt(bA/bB)
  in particular we have g/h = sqrt(bA/bB)

- I can write the photodiode signals as
     PDA = g * s  + h * nA
     PDB =     s  +     nB
  or, rescaling PDA to match the shot noise level of PDB, PDA_ = PDA/h
     PDA_ = g/h * s  + nA
     PDB_ =       s  + nB
  Here s is the signal as seen by PDB, and nA and nB are realizations of shot noise, scaled such that <nA^2>=<nB^2> (i.e. same power spectral density).
  They are independent, i.e. <nA*nB>=0.

- Since the two shot noise levels are equal and uncorrelated, we can now rotate the signal by any angle theta to maximize our readout SNR:

     / PDA~ \       / PDA_ \
    |        | = R |       | 
     \ PDB~ /       \ PDA_ /

  or in vector notation:
        x~     = R x_

  Note that the shot noise levels in PDA~ and PDB~ remain uncorrelated and equal during this rotation.
  Here R is the recular rotation matrix:

          / cos(theta)   sin(theta) \
    R  = |                           | 
          \-sin(theta)   cos(theta) /

- If we rotate all the signal into PDA~, PDA~ becomes DCPD_SUM, and PDB~ becomes DCPD_NULL. This is achieved for 

    theta = atan(h/g) = atan(sqrt(bB/bA))

- The matrix proposed in alog 45734 did exactly that, with
   i)  an additonal sign change for DCPD_NULL, i.e. DCPD_NULL = - PDA~
   ii) rescaling the overall matrix gain to some aribitrary, traditional value set by A+B=2 (see alog 45734 for details, also the attached MATLAB file).
   To be explicit, the alog 45734 matrix M is equal to

    M = gain * diag([1,-1]) * R * diag([1/h,-1]);

- The cross-power CP between the two photodiodes can be calculated for any rotation angle theta:
  With
          /   0     1/2 \
    C  = |               | 
          \ 1/2       0 /

  we get

    CP = < PDA~ * PDB~ > = < transpose(x~) C x~ = transpose(x_) transpose(R) C R x_  >
 
  or
                              / -sin(2*theta)   cos(2*theta) \
    CP = < 1/2 transpose(x_) |                                | x  >
                              \  cos(2*theta)   sin(2*theta) /

  or

    CP = cos(2*theta) < PDA_ * PDB_ >  + sin(2*theta)/2 * ( <  PDB_^2 > - <  PDA_^2 > )

  There are now two ways to choose the theta for this cross-power calculations:
    A) Stay in the original PDA_-PDB_ basis: theta=0deg
    B) Optimize the SNR ratio in the cross-power: theta = theta_SUMNUL - 45deg

    A) Guarantees that the cross-power measurement is independent of the measured gain-scaling - at least at high frequencies
       (Low frequencies require loop gain corrections anyway).
    B) This introduces a slight sensitivity to the measured shot noise ratio in the two diodes (sin(2*theta)/2 term).
       But since the loop gain correction needs to be done in the SUM_NULL basis anyway, that is a bit of a mute point,
       and should not matter for realistic OMC BS splitting ratios.

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Bottom line:
If we follow the procedure from alog 45734, the cross-power estimator should work as advertised.

Practical suggestion:
The process will be less mysterious if we put the shot-noise-matching gain '1/h' up front in the PDA filter modules.
Then we can use a pure rotation matrix for the PDA_PDB to SUM_NULL matrix M.