563489578934

Sec. 6–8 Matched Filters 471
so
which is identical to Eq. (6–167).
r 0 (t 0 ) =
L
t 0
t 0 -T
r(l)s[t 0 - (t 0 - l)] dl
The correlation processor is often used as a matched filter for bandpass signals, as illustrated
in Example 6–15.
Example 6–15 MATCHED FILTER FOR DETECTION OF A BPSK SIGNAL
Referring to Fig. 6–18, let the filter input be a BPSK signal plus white noise. For example, this
might be the IF output of a BPSK receiver. The BPSK signal can be written as
s(t) = e +A cos v ct, nT 6 t … (n + 1)T for a binary 1
-A cos v c t, nT6 t … (n + 1)T for a binary 0
where f c is the IF center frequency, T is the duration of one bit of data, and n is an integer. The
reference input to the correlation processor should be either +A cos v c t or -A cos v c t, depending
on whether we are attempting to detect a binary 1 or a binary 0. Since these waveshapes are
identical except for the ;1 constants, we could just use cos v c t for the reference and recognize
that when a binary 1 BPSK signal is present at the input (no noise), a positive voltage
AT would be produced at the output. Similarly, a binary 0 BPSK signal would produce a negative
voltage - 1 2
AT at the output. Thus, for BPSK signaling plus white noise, we obtain the
matched filter shown in Fig. 6–19. Notice that this looks like the familiar product detector,
except that the LPF has been replaced by a gated integrator that is controlled by the bit-sync
clock. With this type of postdetection processing, the product detector becomes a matched
filter. However, to implement such an optimum detector, both bit sync and carrier sync are
needed. The technique, shown in Fig. 6–19, could be classified as a more general form of an
integrate-and-dump filter (first shown in Fig. 6–17).
1
2
Transversal Matched Filter
A transversal filter can also be designed to satisfy the matched-filter criterion. Referring to
Fig. 6–20, we wish to find the set of transversal filter coefficients {a i ; i = 1, 2,...,N} such
that s 2 0 (t 0 )/n 2 0 (t) is maximized. The output signal at time t = t 0 is
s 0 (t 0 ) = a 1 s(t 0 ) + a 2 s(t 0 - T) + a 3 s(t 0 - 2T)
+Á+a N s(t 0 - (N - 1)T)
or
N
s 0 (t 0 ) = a a k s(t 0 - (k - 1)T)
(6–168)
k=1