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Sec. 7–4 Noncoherent Detection of Bandpass Binary Signals 515
Frequency-Shift Keying
A noncoherent receiver for detection of frequency-shift-keyed (FSK) signals is shown in
Fig. 7–11. The input consists of an FSK signal as described by Eq. (7–39) plus white
Gaussian noise with a PSD of N 0 /2. A sketch of the spectrum of the FSK signal and noise was
given in Fig. 7–8b, where the bandpass filter bandwidth is B p . It is assumed that the frequency
shift, 2∆F = f 1 - f 2 , is sufficiently large so that the spectra of s 1 (t) and s 2 (t) have negligible
overlap.
The BER for this receiver is obtained by evaluating Eq. (7–8). For signal alone at the
receiver input, the output of the summing junction is r 0 (t) = +A when a mark (binary 1) is
transmitted, and r 0 (t) = -A when a space (binary 0) is transmitted. Because of this symmetry
and because the noise out of the upper and lower receiver channels is similar, the optimum
threshold is V T = 0. Similarly, it is realized that the PDF of r 0 (t) conditioned on s 1 and that the
PDF of r 0 (t) conditioned on s 2 are similar. That is,
f(r 0 |s 1 ) = f(-r 0 |s 2 )
(7–59)
Substituting Eq. (7–59) into Eq. (7–8), we find the BER to be
or
0
q
P e = 1 2 f(r 0 |s 1 ) dr 0 + 1 2 f(r 0 |s 2 ) dr 0
L L
-q
0
q
P e = f(r 0 |s 2 ) dr 0
L
0
(7–60)
As shown in Fig. 7–11, r 0 (t) is positive when the upper channel output v U (t) exceeds the
lower channel output v L (t). Thus,
P e = P(v U 7 v L |s 2 )
(7–61)
Upper channel (mark)
FSK
input
Bandpass filter
centered at f 1
(B p = effective bandwidth)
Lower channel (space)
Envelope
detector
v U (t)
+
r 0 (t)
Sample
and
hold
r 0 (t 0 )
Threshold
device
~ m m ~
V T = 0
r 0
Digital
output
Bandpass filter
centered at f 2
(B p = effective bandwidth)
Envelope
detector
v L (t)
Analog output
Figure 7–11
Noncoherent detection of FSK.