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Random Processes and Spectral Analysis Chap. 6
The maximum (SN) out is obtained when H(f) is chosen such that equality is attained. Using
Eq. (6–158), this occurs when A(f ) = KB * ( f), or
which reduces to Eq. (6–155) of the theorem.
From a practical viewpoint, it is realized that the constant K is arbitrary, since both the
input signal and the input noise would be multiplied by K, and K cancels out when (SN) out is
evaluated. However, both the output signal and the noise levels depend on the value of the
constant.
In this proof, no constraint was applied to assure that h(t) would be causal. Thus, the filter
specified by Eq. (6–155) may not be realizable (i.e., causal). However, the transfer function
given by that equation can often be approximated by a realizable (causal) filter. If the causal constraint
is included (in solving for the matched filter), the problem becomes more difficult, and a
linear integral equation must be solved to obtain the unknown function h(t) [Thomas, 1969].
Results for White Noise
H(f)2 n (f) = KS *(f)e-jvt 0
For the case of white noise, the description of the matched filter is simplified as follows:
For white noise, n (f) = N 0
2. Thus, Eq. (6–155) becomes
From this equation, we obtain the following theorem.
THEOREM. When the input noise is white, the impulse response of the matched filter
becomes
h(t) = Cs(t 0 - t)
(6–160)
where C is an arbitrary real positive constant, t 0 is the time of the peak signal output,
and s(t) is the known input-signal waveshape.
Proof. We have
h(t) = -1 [H(f)] = 2K S
N * (f) e -jvt 0
e jvt df
0 L-q
= 2K q
c S(f)e j2pf(t0-t) *
df d
N 0 L
-q
= 2K
N 0
[s(t 0 - t)] *
2 n (f)
H(f) = 2K
N 0
S * (f)e -jvt 0
But s(t) is a real signal; hence, let C = 2KN 0 , so that the impulse response is equivalent to
Eq. (6–160).
q