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

Comparing this result with (D27) we see that, as in
the incoherent detection case, going to continuous
averaging has reduced the Po 2 term in the denominator
by the factor 2[3.
Similarly,
1 _b 2/202
p(b) - e (D32)
STATISTICS OF THE SQUARE LAW DETECTOR
OUTPUT (CLASSICAL CASE)
So far we have considered only the signal-to-noise
power ratios of the square-law detector output. These
are useful in computing the accuracy with which a
received power can be measured; but to determine the
false alarm probability Pfa (of detecting a signal when
none is present) or the probability of missing a signal
Pms that is actually present, we need to know the actual
statistical distribution functions.
Again we assume a coherent power Pr and an
incoherent background power P0. We assume from here
on that the latter is present in only one polarization,
since this is true for the radio case and can be true for
the optical case with the use of a polarizing filter. Then
the instantaneous power at the detector input is
P = [.4 +a(t)] 2 +b2(t) (D30)
m
where as before Pr = A 2, Po = a2 (t) + b2(t), and a(t) and
b(t) are gaussian variables with zero mean and zero cross
correlation. The vector relations between a(t), b(t), their
resultant c(t), the coherent signal amplitude A, and the
total signal amplitude s(t) are shown in the Figure D-I.
QUADRATURE $
NOISE
AMPLITUDE
! Pb.
where o2 = a2(t) = b_(t) = Po/2. Since a and b are
independent, the joint probability density distribution
for a and b is
e-(a 2 + b2)/2o 2
p(a,b) = p(a)p(b) = (D33)
2znl 2
Since a2 + b2 = c2 = A z + s2 - 2sA cos 0, the
probability density function for s is found by integrating
equation (D33) over all 0, that is:
p(s)
-
l
s e
eo
dO
7rPo T.it (A 2 + s2 - 2sA cos 0)
A 2 + s2
2sA
/7' _ COS 0
2s Po 1 _" Po
-
"Do
e --
,r
J0 e dO
A 2 +,5 2
=-- e Io (D34)
eo 2s Po 2(/)
where I0 is the zero order modified Bessel function of
the first kind. We note in passing that p(s) is the
probability density function for the output of a linear
detector and that p(s)/27rs is the probability density
versus radius distribution for the Golay detector. (See
Chap. 1 i).
//_
O' A 0
D,.
IN -PHASE
NOISE
AMPLITUDE
If q(P) is the probability density function for the
instantaneous power P, we require that
Figure D-1. Noise and signal vectors. P has a two-dimensional
Gaussian distribution about 0, which
makes C have a Rayleigh distribution in amplitude,
and S have the distribution given by equation (D34).
q(?) de
= p(s)ds
with dP = 2s ds and therefore find
The probability density function for a is
1 e__a2/202
p(a) - (D31)
1 P0
q(P) = Po
--e 1° ---_o /
(D35)
192