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Sec. 6–8 Matched Filters 469
occur at t = t 0 and that the input signal waveshape has been distorted by the filter in order to peak
up the output signal at t = t 0 .
In applications to digital signaling with a rectangular bit shape, this matched filter is
equivalent to an integrate-and-dump filter, as we now illustrate. Assume that we signal with one
rectangular pulse and are interested in sampling the filter output when the signal level is maximum.
Then the filter output at t = t 0 is
q
r 0 (t 0 ) = r(t 0 ) * h(t 0 ) = r(l)h(t 0 - l) dl
L
When we substitute for the matched-filter impulse response shown in Fig. 6–16c, this equation
becomes
-q
r 0 (t 0 ) =
L
t 0
t 0 -T
r(l) dl
(6–166)
Thus, we need to integrate the digital input signal plus noise over one symbol period T (which
is the bit period for binary signaling) and “dump” the integrator output at the end of the symbol
period. This is illustrated in Fig. 6–17 for binary signaling. Note that for proper operation of
this optimum filter, an external clocking signal called bit sync is required. (See Chapter 3 for a
discussion of bit synchronizers.) In addition, the output signal is not binary, since the output
sample values are still corrupted by noise (although the noise has been minimized by the
matched filter). The output could be converted into a binary signal by feeding it into a comparator,
which is exactly what is done in digital receivers, as described in Chapter 7.
Correlation Processing
THEOREM. For the case of white noise, the matched filter may be realized by correlating
the input with s(t); that is,
r 0 (t 0 ) =
L
t 0
t 0 -T
r(t)s(t) dt
(6–167)
where s(t) is the known signal waveshape and r(t) is the processor input, as illustrated
in Fig. 6–18.
Proof.
The output of the matched filter at time t 0 is
t 0
r 0 (t 0 ) = r(t 0 ) * h(t 0 ) = r(l)h(t 0 - l) dl
L
-q
But from Eq. (6–160),
h(t) = e s(t 0 - t), 0 … t … T
0, elsewhere