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Random Processes and Spectral Analysis Chap. 6
6–8 MATCHED FILTERS
General Results
In preceding sections of this chapter, we developed techniques for describing random
processes and analyzing the effect of linear systems on these processes. In this section, we
develop a technique for designing a linear filter to minimize the effect of noise while maximizing
the signal.
A general representation for a matched filter is illustrated in Fig. 6–15. The input signal
is denoted by s(t) and the output signal by s 0 (t). Similar notation is used for the noise. This
filter is used in applications where the signal may or may not be present, but when the signal
is present, its waveshape is known. (It will become clear in Examples 6–14 and 6–15, how
the filter can be applied to digital signaling and radar problems.) The signal is assumed to be
(absolutely) time limited to the interval (0, T) and is zero otherwise. The PSD, n ( f ), of the
additive input noise n(t) is also known. We wish to determine the filter characteristic such that
the instantaneous output signal power is maximized at a sampling time t 0 , compared with the
average output noise power. That is, we want to find h(t) or, equivalently, H(f), so that
a S (6–154)
N b = s 0 2 (t)
out n 2 0 (t)
is a maximum at t = t 0 . This is the matched-filter design criterion.
The matched filter does not preserve the input signal waveshape, as that is not its objective.
Rather, the objective is to distort the input signal waveshape and filter the noise so that at
the sampling time t 0 , the output signal level will be as large as possible with respect to the
RMS (output) noise level. In Chapter 7 we demonstrate that, under certain conditions, the
filter minimizes the probability of error when receiving digital signals.
THEOREM. The matched filter is the linear filter that maximizes (S/N) out =
s 2 of Fig. 6–15 and that has a transfer function given by † 0 (t 0 )>n 2 0 (t)
H(f) = K S*(f)
(6–155)
n (f) e-jvt 0
where S( f ) = [s(t)] is the Fourier transform of the known input signal s(t) of duration
T sec. n ( f ) is the PSD of the input noise, t 0 is the sampling time when (SN) out is evaluated,
and K is an arbitrary real nonzero constant.
r(t)=s(t)+n(t)
Matched filter
h(t)
H(f)
r 0 (t)=s 0 (t)+n 0 (t)
Figure 6–15
Matched filter.
† It appears that this formulation of the matched filter was first discovered independently by B. M. Dwork and
T. S. George in 1950; the result for the white-noise case was shown first by D. O. North in 1943 [Root, 1987].