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Performance of Communication Systems Corrupted by Noise Chap. 7
[s 01 (t 0 ) - s 02 (t 0 )] 2
= [s d(t 0 )] 2
2 2
s 0 s 0
s d (t 0 ) ! s 01 (t 0 ) - s 02 (t 0 ) is the difference signal sample value that is obtained by subtracting
the sample s 02 from s 01 . The corresponding instantaneous power of the difference output
signal at t = t 0 is s 2 d (t 0 ). As derived in Sec. 6–8, the linear filter that maximizes the instantaneous
output signal power at the sampling time t = t 0 when compared with the average output
noise power s 2 0 = n 2 0 (t) is the matched filter. For the case of white noise at the receiver input,
the matched filter needs to be matched to the difference signal s d (t) = s 1 (t) - s 2 (t). Thus, the
impulse response of the matched filter for binary signaling is
h(t) = C[s 1 (t 0 - t) - s 2 (t 0 - t)]
(7–18)
where s 1 (t) is the signal (only) that appears at the receiver input when a binary 1 is sent, s 2 (t)
is the signal that is received when a binary 0 is sent, and C is a real constant. Furthermore, by
using Eq. (6-161), the output peak signal to average noise ratio that is obtained from the
matched filter is
[s d (t 0 )] 2
s 0
2
= 2E d
N 0
N 0 /2 is the PSD of the noise at the receiver input, and E d is the difference signal energy at the
receiver input, where
T
E d = [s 1 (t) - s 2 (t)] 2 dt
L
0
(7–19)
Thus, for binary signaling corrupted by white Gaussian noise, matched-filter reception, and
by using the optimum threshold setting, the BER is
E
P e = Q¢ d
≤
(7–20)
C 2N 0
We will use this result to evaluate the P e for various types of binary signaling schemes
where matched-filter reception is used.
Results for Colored Gaussian Noise
and Matched-Filter Reception
The technique that we just used to obtain the BER for binary signaling in white noise can be
modified to evaluate the BER for the colored noise case. The modification used is illustrated
in Fig. 7–3. Here a prewhitening filter is inserted ahead of the receiver processing circuits.
The transfer function of the prewhitening filter is
H p (f) =
1
3 n (f)
(7–21)