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110
Signals and Spectra Chap. 2
the spectrum is continuous. The result for the random data case, as worked out in Chapter 3 where
m 1f2 is given by Eq. (3–41), is
(f) = 1 4 T bc sin p T b (f - f c)
p T b (f - f c )
2
d + 1 4 T bc sin p T b(f + f c )
2
d
p T b (f + f c )
(2–200)
when the data rate is R = 1/T b bits/sec. This is shown by the dashed curve in Fig. 2–23b.
The preceding derivation demonstrates that we can often use (deterministic) square-wave
test signals to help us analyze a digital communication system, instead of using a more complicated
random-data model.
The bandwidth for the BPSK signal will now be evaluated for each of the bandwidth definitions
given previously. To accomplish this, the shape of the PSD for the positive frequencies is
needed. From Eq. (2–200), it is
(f) = c sin p T b (f - f c)
p T b (f - f c )
2
d
(2–201)
Substituting Eq. (2–201) into the definitions, we obtain the resulting BPSK bandwidths as shown
in Table 2–4, except for the FCC bandwidth.
The relationship between the spectrum and the FCC bandwidth parameter is a little more
tricky to evaluate. To do that, we need to evaluate the decibel attenuation
A(f) = -10 log 10 c P 4kHz(f)
P total
d
(2–202)
where P 4kHz (f) is the power in a 4-kHz band centered at frequency f and P total is the total signal
power. The power in a 4-kHz band (assuming that the PSD is approximately constant across the
4-kHz bandwidth) is
P 4kHz (f) = 4000 (f)
(2–203)
and, using the definition of equivalent bandwidth, we find that the total power is
P total = B eq P(f c )
(2–204)
TABLE 2–4 BANDWIDTHS FOR BPSK SIGNALING WHERE THE BIT RATE IS R = 1/T b BITS/S.
Definition Used
Bandwidth
Bandwidths (kHz)
for R = 9,600
bits/s
1. Absolute bandwidth q q
2. 3-dB bandwidth 0.88R 8.45
3. Equivalent noise bandwidth 1.00R 9.60
4. Null-to-null bandwidth 2.00R 19.20
5. Bounded spectrum bandwidth (50 dB) 201.04R 1,930.0
6. Power bandwidth 20.56R 197.4