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APPENDIX
C
OPTIMUM DETECTION AND FILTERING
In communication systems we are frequently faced
with the problem of obtaining the best estimate of the
amplitude of a signal having a known shape, but which
has had noise added to it. Historically, this problem was
first analyzed t in connection with the detection of the
amplitudes and time of occurrence of radar pulses. We
begin by considering pulse detection. The results, however,
will be quite general, and applicable to a wide
variety
of signals.
Assume that a pulse f(t) recurs periodically and that
on each recurrence we wish to measure its amplitude as
accurately as possible in spite of added noise. One of the
most straightforward ways to do this is to multiply the
signal by a "gate" that lets the signal through, but
excludes noise before and after the pulse, and then
integrate the gated signal. Thus if nk(t ) is the particular
noise wave on the kth cycle, and g(t) is the gate, we
form the gated signal
s(t) = g(t) If(t) + nk(t)] (C1)
due to the signal, and
a component
qn(k) = L_g(t)nk(t) dt
(C4)
due to the noise. The integration actually occurs only
over the time for which g(t) 4= 0, but the limits may be
taken as infinite without affecting the result. We assume
that the various nk(t ) are uncorrelated and are samples
of a statistically stationary process. The mean square
value of qn(k) is then
_-A _k
where A p/k means the average over the index k.
Using Parseval's theorem, we may rewrite equation
(C4) as
(c5)
and integrate the result to obtain a charge
£
q(k) = g(t)[f(t) + nk(t)] dt (C2)
qs = v v
(C6)
v---.O0
Equation (C5) may be written as the product of two
independent integrals
This charge may be regarded as the sum of a
component
qn 2 = _]___ g(t)nk(t)dt g(r)nk(r)d
_'-..-o0
oo
qs
= J--oo g(t)f(t) dt (C3)
=-/-I.]_ 1 "g(t)g(r)nk(t)nk(T)dt d (C7)
By Claude Shannon, informal C(}lnlBulliCal i_ll Ill loll
The change of variable r = t + x, dr = dx/nay now be
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