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

Let us now introduce the normalized variables
x = e/Po and r = Pr/Po. Then equation (D35) becomes
For large n, we may replace (n - 1)! by
1
p(x) = e -(r + x) lo(2 V_) (D36)
n ! 2rr nn e 12n
n
N_
--n÷--
If we add n independent samples of the output the
resulting distribution will be the n-fold convolution of
p(x). The Fourier transform of p(x) is given by Campbell
arid Foster (ref. 1, pair 655.1) as
C(co) - 1 e-' +ico exp
giving
pnfy) =
1
-n(r - 1 + -- + y )
n -n- 1 12n 2
--y e ×
2rr
+ _ + _ + etc
1 ° n 2(n + 1)
Thus
(D39a)
[C(c°]n
-nF
(1 +leo) n exp
and by pair 650, op. cit., we find the inverse transform
to be
n-I
2
e-X In -
1 (2 vthr-x)
Actually equation (D39a) is in error by only 0.2% for
n = 1 and so may be used to compute pn(y) for any n.
The probability that y is less than a certain threshold
YT may be found by numerically integrating equation
(D39a) from y = 0 to y = YT" This integral gives the
probability that signal plus noise fails to exceed the
threshold YT and hence that the signal is not detected.
In the absence of a signal, r = 0, and equation (D39)
becomes
(D37)
If now we replace the sum x by the arithmetic mean
y = x/n, we obtain
n-I
Pn(Y) = n(_) 2 e-n(_ +y) in -1 (2n,,/ff)
(D38)
(ny) n - I e-nY
pn(y) = n (n- I)! (D40)
This may now be integrated from YT to infinity to
give the probability qn(YT ) that noise alone exceeds the
threshold YT" We find
n-I
Now
in _ _(2nvr_7) = (n_)n
SO
nnyn - le-n(r + y)
Pn('V) = (n - 1)!
Q
- l
nZry + 2(n + 1)
oo
_
k=O
+ n2ry 1m ×
(n2ry)
k!(n-l+k)!
( 1
1+ 3(n+2) +"
k
(D39)
qn(YT) = e -nyT _ (nyT)k (I341)
k=O k!
The threshold level required to give a specified p),a is
found from equation (D41). The signal-to-noise ratio, r,
needed to give a permissible probability of missing the
signal is then found by integrating equation (D39a) from
y = 0 toy = yT with trial values ofr.
GAUSSIAN
APPROXIMATIONS
From the central limit theorem we know thal as n -* o_
both (D38) and (D40) will approach gaussian distributions.
Reverting to unnormalized variables, the distribution
approached by the signal plus noise will be centered
at P = Po + Pr and will have a variance
193