Pre-Phase A Report - Lisa - Nasa

4.3 Signal extraction 101
here that arm 1 is used and that the frequency of interest is well separated from any of
the harmonics.
Following approximately the notation of Giampieri et al. [108], except using units of meters,
the phase outputs of the diodes of spacecraft 1 and 2 are
s1(t) = p(t − 2L1/c) − p(t)+n1(t)+h(t) ,
s2(t) = p(t − 2L2/c) − p(t)+n2(t) − h(t) ,
(4.45)
where p(t) is the laser phase noise, L1 and L2 are the arm lengths, n1 and n2 are the
separate shot noises and any other noises that are not common to the two arms, and h is
the gravitational radiation signal. Transforming to the frequency domain we get
s1(f) =p(f)(e4πifL1/c −1) + n1(f)+h(f) ,
s2(f) =p(f)(e4πifL2/c −1) + n2(f) − h(f) .
An estimate of the clock signal is formed from arm 1 by
(4.46)
s1(f)
p(f) =
e4πif(L1+δL1)/c , (4.47)
−1
where our knowledge of the arm length L1 is in error by an amount δL1. Including the
definition of s1 we get
e
p(f) =p(f)
4πifL1/c −1
e4πif(L1+δL1)/c +
−1
Using the estimate p and the definition of s1 we define
We then form the difference
with
P = p(f)
n1
e 4πif(L1+δL1)/c −1 +
s1 = p [ e4πif(L1+δL1)/c −1] ,
s2 = p [ e4πif(L2+δL2)/c −1] .
h
e 4πif(L1+δL1)/c −1 . (4.48)
(4.49)
(s1 − s2) − (s1 − s2) =P + N + H, (4.50)
e4πifL1/c 4πifL2/c e
− e − 4πifL1/c −1
e4πif(L1+δL1)/c 4πif(L1+δL1)/c 4πif(L2+δL2)/c e − e
−1
N = n1(f) − n2(f) − n1(f) e4πif(L1+δL1)/c 4πif(L2+δL2)/c
− e
e4πif(L1+δL1)/c −1
H = h(f) 2 − e4πif(L1+δL1)/c − e4πif(L2+δL2)/c e4πif(L1+δL1)/c
.
−1
Expanding by using 4πfδL1/c ≪ 4πfL1/c ≪ 1and4πfδL2/c ≪ 4πfL2/c ≪ 1these
become
P ≈ p(f) L1δL2 − L2δL1
L1
4πif/c (4.51)
Corrected version 2.08 3-3-1999 9:33