Project Cyclops, A Design... - Department of Earth and Planetary ...

the component I r constitutes the signal. If, on the other
hand, we are measuring starlight (or incoherent radio
sources), we might have Pr = 0 and take I s = od_s to be
the signal. The accuracy with which either Ir or I o can
be measured depends on the magnitude of the fluctuations.
by:
The fluctuation noise power of the current i is given
-=_ -(7)=
with v over the band, this expression becomes simply
n(f) hk(Vo - .f) + ff(Vo +/91 = a _(vo)n(t). Thus the
power spectrum of the first term is proportional to the
low pass equivalent of the predetection filter.
The second term in the brackets in equation (D5)
arises from noise components mixing with one another.
All pairs of components separated by f beat to form an
output at frequency [. Assuming again that K(v) is
symmetrical about vo and that _k(u) is linear in v, the
power spectrum of this term will be proportional to the
autocorrelation of H0_), which we represent by H(f) * H(f)
or simply
H*H.
To these two components of equation (D5) we may
now add the shot noise power of the currentT, which on
a two-sided spectrum basis has the spectral density
qT = _(hv]_7). Thus we find for the total (two-sided)
power
spectrum
As will be shown later, Pi is Boltzmann distributed;
that is,
_(f) = _ (er +eo) + -- Preo
I__ m2 fnd[ H(f)
P(Pi) - Pe (D4)
+ Po2 fH*H"-"_"
1
(D6)
and therefore Pi 2 - Po 2 = Po 2. If, as we have assumed,
P0 is randomly polarized, then al z(t) = Po]4. At radio
frequencies, or at optical frequencies when a polarizing
filter is used, only the polarization corresponding to the
coherent signal would strike the detector. Thus a2(t) and
b2(t) would be zero and al 2(t) = Po/2. In general then,
al--_t) = Po/2m where m = 1,2 is the number of
orthogonal polarizations reaching the detector. Making
these substitutions, we find
2
Air 2 = ot2 [-- PrPo +Po 21 (DS)
m
To determine the effect on the noise power of
filtering the output, we need to know the power
spectrum of _'p. The first term in the brackets arises
from
d
components of the incoherent wave at frequencies
vo - f and v0 + f mixing with the coherent wave to
produce outputs at frequency f. If if(v) is the power
spectrum of the incoherent source, and K(v) = IF(v)l 2 is
the power transmission of the predetection filter, then the
power spectrum of the first term will be proportional to
[_(Vo - f)K(vo - I3 + _(Vo +f)K(vo +f)]. If the filter is
symmetrical about Vo then K(vo + f) = K(vo - f)- Hff),
and if if(v) is either constant or varies linearly
If G(f) is the (power) transmission of the postdetection
filter, then the total noise power in the output is
simply N = f G(D 7(.f) dr, while the signal power is either
S = a2G(0) Ps2 or S = a2G(0) Pr2 depending upon
whether we wish to measure Ps or detect Pr"
For the measurement of incoherent radiation in the
absence of coherent radiation, we set Pr = 0 in (6) and
obtain for the output signal-to-noise power ratio
s a(o) P/
N hv f G(H*H)df (D7)
Po _ f G df + Po
7? f H*H df
For the detection of the coherent signal in the
incoherent background:
S a(o)er 2
N hv 2 y GHdf
(Pr + Po) - f G df + - PrPo .
rl m f Hdf
2 f G(H,I-I)df
+ Po
f H*H df
+
(DS)
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