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introducedandtheorderof theintegrating andaveragingoperations
exchanged togive:
"qn2"f_£g(t)g(t+x)_Ink(t)nk(t+x) ]
dt dx
(C8)
The average over k of nk(t) nk(t + x) is the
autocorrelation, A(x), of the noise and is independent of
t. Thus,
qn 2 = A(x) (t)g(t + x)d dx
(C9)
In the presence of white noise the best gate is the
signal itself. From,equation (C2) we see that what we
should do is cross correlate the received signal with the
expected waveform of the noise-free signal. In function
space this amounts to finding the projection of s(t) on
f(t).
From equations (C6) and (CI1) and using the
optimum gate equation (CI2), we obtain the output
signal-to-noise power ratio
= -- = 2 dv (C14)
(S) q$ f°°lF(p)[2
N- opt qn 2 J__oo _u)
If if(v) is constant over the signal spectrum, this ratio
is expressed as
=L_A(x)_°°lG(v)]2ei21rVxdvldx
(lO)
opt
(C15)
where equation (CI0) follows from equation (9) by
applying Parseval's theorem to the second integral. Now
A(x) is the transform of the power spectrum of the
noise, while the quantity in brackets in equation (C I 0) is
the inverse transform of IG(v)l 2. Applying Parseval's
theorem once more equation (CI0) becomes
-- ¢(v)
%7: = _ IG(v)l2 av (c_1)
where if(u) is the one-sided power spectral density of the
noise. Since our integrals extend from --_ to +_o in
frequency, we must use the two sided density if(v)/2.
The problem may now be stated as follows: Find the
G(v) that minimizes qn 2 as given by equation (Cll)
under the constraint that qs as given by equation (C6)
is held constant. This is a straightforward problem in the
calculus of variations and yields the result
G(v) = m --
_(v)
(C12)
where m is an arbitrary real constant. If the
noise is "white" that is, if Pn(V) = constant, then
(;(v) = constant X F(v) and
where E is the pulse energy, and, for thermal noise,
= kT.
If, prior to gating, the received signal and noise are
first passed through a filter having the transmission K(v),
then in all subsequent expressions F(u) is replaced by
F(u) K(u) and _b(v) by _b(u) [K(u)I 2. Equation (El 2) in
particular then becomes
F(v)
G(v) K(v) = m -- (CI6)
_(v)
and we observe that it is only the product of the gate
spectrum and the conjugate of the filter transmission
that is specified. Deficiencies in either can be compensated
for by the the other. A case of particular interest
occurs if we set G(v) = constant. This corresponds to a
gate that is a 6-function at t = 0. We then have
and for white
noise
F(v)
K(v) = constant --
¢(v)
(el 7)
K(v) = constant X F(v) (c 18)
g(t) = constant × J(t) (C13) As given by equation (C 17) or (CI8) K(v) is called a
matched filter and gives the highest ratio of peak signal
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