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asshownin AppendixD, if thenoisein successive
samples isuncorrelated, theprobability thatthisaverage
exceeds agiventhresholdvalueofy =YT is
n-1
poO, T) = e -YT _ yTkk! (35)
k=O
If both signal and noise are present, the probability
that the same threshold is not exceeded is:
n-!
pl(yT) = n + Y) ln_l(2nx/"_)dy
foYT (Y) 2 e -n(r
(36)
I i i T I I q
a
bJ
o
x
LI3
Figure 5-5. Probability that signal plus noise fails to
exceed power P.
Z
Figure 5-5 shows pl(x) plotted against 10 log x for
various values of r expressed in decibels. By noting the
value of pl(x) for 10 log x = 4.053 dB and plotting this
versus the corresponding value of r, we obtain the curve
shown in Figure 5-6. We see that if the probability of
failing to exceed the threshold is to be 0.07865, then r,
the input signal to noise ratio must be 6.06 (i.e., 7.83
dB).
Thus for a square law detector with short time
averaging (r _ 1/B), the limiting range is
IJ
..3
dr Pelf (0.165 - b.
R = -- - (34)
f I 1/2
4 _B
.01 I I I t I I
3 4 5 6 7 B 9 I0
_o log _
where we have written 0.165 for (6.06) -n .
The n-bbM Average. lfy = -- _ x i is the average of
n
i=0
n samples of the output of a square law detector, then,
1
n
Figure 5-6. Probability of missing signal.
The same graphical method can be used with these
expressions that was illustrated with equations (31)and
(33) for the case n = 1. (This is the procedure used in
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