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4.6 Canceling optical bench motion noise

There are now twelve Doppler data streams which have to be combined in an appropriate manner in order to cancel the noise from the laser as well as from the motion of the optical benches. As in the previous case of cancelling laser phase noise, here too, we keep the relevant terms only, namely those terms containing laser phase noise and optical bench motion noise. We then have the following expressions for the four data streams on spacecraft 1:
[ ] [ ] s1 = D3 p'2 + n0^n3 .D'2 - p1 - n0^n3 .D1 , (32) ' [ ] [ ' '] s1 = D2 p3 - n0^n2 .D3 - p1 + n0^n2 .D 1 , (33) ' ' t1 = p1 - p1 + 2n0^n2 .D 1 + m1, (34) t1'= p1 - p'1 - 2n0^n3 .D1 + m1. (35)
The other eight data streams on spacecraft 2 and 3 are obtained by cyclic permutations of the indices in the above equations. In order to simplify the derivation of the expressions cancelling the optical bench noises, we note that by subtracting Equation (35View Equation) from Equation (34View Equation), we can rewriting the resulting expression (and those obtained from it by permutation of the spacecraft indices) in the following form:
1- ' ' z1 =_ 2 (t1 - t1) = f1 - f1, (36)
where ' f1, f1 are defined as
f'1 =_ p'1 + n0^n2 .D'1, (37) f1 =_ p1- n0n^3 .D1,
The importance in defining these combinations is that the expressions for the data streams s i, s' i simplify into the following form:
s1 = D3f'2 - f1, ' ' (38) s1 = D2f3 - f1.
If we now combine the si, s' i, and zi in the following way,
j =_ s - D z = D f - f , j ' =_ s '+ z = D f - f , (39) 1 1 3 2 3 2 1 1 1 1 2 3 1 j2 =_ s2- D1z3 = D1f3 - f2, j2' =_ s2'+ z2 = D3f1 - f2, (40) j =_ s - D z = D f - f , j ' =_ s '+ z = D f - f , (41) 3 3 2 1 2 1 3 3 3 3 1 2 3
we have just reduced the problem of cancelling of six laser and six optical bench noises to the equivalent problem of removing the three random processes f1, f2, and f3 from the six linear combinations j i, j' i of the one-way measurements s i, s' i, and z i. By comparing the equations above to Equation (15View Equation) for the simpler configuration with only three lasers, analyzed in the previous Sections 4.1 to 4.4, we see that they are identical in form.
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