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

x/eeoG + Ueo2
o =
(D42)
while that approached by the noise alone will be
centered at P ---Po and will have the variance
O"O -
x/- q-o
x/if-
(D43)
where /a = 1 if n discrete independent samples are
averaged and /a = 2/3 if sin x/x filters are used with
n = r/T. See eqs. (DI8) and (D29).
The threshold must exceed the mean, Po, of the noise
distribution by some multiple mo times Oo where mo is
chosen to give the desired Pfa' and is determined by
circuits following the photocell and simply count the
number of photons received. The statistical distribution
of this count will depend on the signal and background
signal strengths and on the number of Nvcluist intervals
over which the count is taken-that is, on the value of
n = (Br) where B is the bandwidth of the (ideal) optical
filter ahead of the photodetector and z is the integration
time.
Incoherent Radiation: Short Integration Time
If the integration time is short compared with lIB,
the probability p(n) of receiving n photons is the
exponential
distribution
,
p(n) - I +no ITTn-/o/
(o48)
Pfa = e-Z2 leclz (D44)
The mean, Po + Pr, of the distribution with signal
present must be some multiple m times o above the
threshold, where rn is chosen to give the allowable
probability of missing the signal, and is determined by
where _o = expectation count. The first moment of this
distribution is M, = _o, while the second moment is
M2 = _o (1 + 2fro). The mean square fluctuation in
count is therefore
002 = M2 -Mr 2 = no(l +no) (D49)
Prns = e -z2 /2dz (D45)
The limit of satisfactory operation is thus given by
If the count _ = fib + ns, where _b is a background
count and ns is the count due to the received radiation,
the signal-to-noise power ratio is "_s2/O2. Letting
-fro = r_(Po/hv) r, _s = rl(Ps/hu) r, we find from equation
(D49) that
Pr = mo°o +too = m o _+m
S _S 2 lOS 2
- = (DS0)
N _o(1 +_o) (hv/OPor+Po 2
or:
Pr (mS +vC_mo)+m4m2 + 2X/'-_mo +#n
which is exactly the result we would expect from (D9)
with
Po
n
(D46)
G(f) = [sin lrr/)/(rrr/)l 2
IfPfa = Pros' then mo = m and
Pr - 2