563489578934

Sec. 6–8 Matched Filters 467
Equation (6–160) shows that the impulse response of the matched filter (white-noise case)
is simply the known signal waveshape that is “played backward” and translated by an amount t 0
(as illustrated in Example 6–14). Thus, the filter is said to be “matched” to the signal.
An important property is the actual value of (SN) out that is obtained from the matched
filter. From Eq. (6–159), using Parseval’s theorem, as given by Eq. (2–41), we obtain
But
q
- q
a S q 2
N b ƒ S(f) ƒ
=
out L -q N 0 >2
df = 2 q
s 2 (t) dt
N 0 L -q
s 2 (t) dt = E s is the energy in the (finite-duration) input signal. Hence,
a S (6–161)
N b = 2E s
out N 0
This is a very interesting result. It states that (SN) out depends on the signal energy and PSD
level of the noise, and not on the particular signal waveshape that is used. Of course, the signal
energy can be increased to improve (SN) out by increasing the signal amplitude, the signal
duration, or both.
Equation (6–161) can also be written in terms of a time–bandwidth product and the
ratio of the input average signal power (over T seconds) to the average noise power. Assume
that the input noise power is measured in a band that is W hertz wide. We also know that
the signal has a duration of T seconds. Then, from Eq. (6–161),
a S (6–162)
N b = 2TW (E s>T)
= 2(TW) a S
out (N 0 W)
N b in
where (SN) in = (E s T)/(N 0 W). From Eq. (6–162), we see that an increase in the time–bandwidth
product (TW) does not change the output SNR, because the input SNR decreases
correspondingly. In radar applications, increased TW provides increased ability to resolve
(distinguish) targets, instead of presenting merged targets. Equation (6–161) clearly shows
that it is the input-signal energy with respect to N 0 that actually determines the (SN) out that is
attained [Turin, 1976].
Example 6–14 INTEGRATE-AND-DUMP (MATCHED) FILTER
Suppose that the known signal is the rectangular pulse, as shown in Fig. 6–16a:
s(t) = e 1, t 1 … t … t 2
0, t otherwise
(6–163)
The signal duration is T = t 2 - t 1 . Then, for the case of white noise, the impulse response required
for the matched filter is
h(t) = s(t 0 - t) = s(-(t - t 0 ))
(6–164)
C was chosen to be unity for convenience, and s(-t) is shown in Fig. 6–16b. From this figure, it is
obvious that, for the impulse response to be causal, we require that
t 0 Ú t 2
(6–165)