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Sec. 4–19 Study-Aid Examples 299
4–18 SUMMARY
The basic techniques used for bandpass signaling have been studied in this chapter. The
complex-envelope technique for representing bandpass signals and filters was found to be
very useful. A description of communication circuits with output analysis was presented for
filters, amplifiers, limiters, mixers, frequency multipliers, phase-locked loops, and detector
circuits. Nonlinear as well as linear circuit analysis techniques were used. The superheterodyne
receiving circuit was found to be fundamental in communication receiver design.
Generalized transmitters, receivers, and software radios were studied. Practical aspects of
their design, such as techniques for evaluating spurious signals, were examined.
4–19 STUDY-AID EXAMPLES
SA4–1 Voltage Spectrum for an AM Signal An AM voltage signal s(t) with a carrier
frequency of 1,150 kHz has a complex envelope g(t) = A c [1 + m(t)]. A c = 500 V, and the modulation
is a 1-kHz sinusoidal test tone described by m(t) = 0.8 sin (2p1,000t). Evaluate the voltage
spectrum for this AM signal.
Solution.
Using the definition of a sine wave from Sec. A–1,
m(t) = 0.8
j2 [ej2p1000t - e -j2p1000t ]
(4–118)
Using Eq. (2–26) with the help of Sec. A–5, we find that the Fourier transform of m(t) is †
M(f) = - j 0.4 d(f-1,000) + j 0.4 d(f + 1,000)
(4–119)
Substituting this into Eq. (4–20a) yields the voltage spectrum of the AM signal:
S(f) = 250 d(f-f c ) - j100 d(f-f c - 1,000) + j100 d (f-f c + 1,000)
+ 250 d(f + f c ) - j100 d(f + f c - 1,000) + j100 d(f + f c + 1,000)
(4–120)
See SA4_1.m for a plot of the AM signal waveform, and a plot of its spectrum which
was calculated by using the FFT. Compare this plot of the spectrum with that given by
Eq. (4–120).
SA4–2 PSD for an AM Signal
SA4–1.
Solution.
namely,
Compute the PSD for the AM signal that is described in
Using Eq. (2–71), we obtain the autocorrelation for the sinusoidal modulation m(t)
R m (t) = A2
2 cos v 0t = A2
4 [ejv 0t + e -jv 0t ]
(4–121)
† Because m(t) is periodic, an alternative method for evaluating M(f) is given by Eq. (2–109), where c -1 = j0.4,
c 1 =-j0.4, and the other c n ’s are zero.