### 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:
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 (35) from Equation (34), we can rewriting the resulting
expression (and those obtained from it by permutation of the spacecraft indices) in the following form:
where , are defined as
The importance in defining these combinations is that the expressions for the data streams ,
simplify into the following form:
If we now combine the , , and in the following way,
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 , , and from the six linear combinations
, of the one-way measurements , , and . By comparing the equations above to
Equation (15) 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.