B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

220 ANGLE MODULATION AND DEMODULATION
(d) Find the phase deviation !l</J.
(e) Estimate the bandwidth of cpEM (t).
The signal bandwidth is the highest frequency in m(t) (or its derivative). In this case
B = 2000n /2n = 1000 Hz.
(a) The carrier amplitude is 10, and the power is
102
P= - = 50
2
(b) To find the frequency deviation !lf , we find the instantaneous frequency w;, given by
d
w; = -0(t) = W e + 15,000 cos 3000t + 20, 000n cos 2000nt
dt
The carrier deviation is 15,000 cos 3000t + 20, 000n cos 2000nt. The two sinusoids
will add in phase at some point, and the maximum value of this expression is 15, 000 +
20, 000n . This is the maximum carrier deviation !lw. Hence,
!lw
!lf = - = 12, 387.32 Hz
2n
!lf 12, 387.32
Cc) f3 = Ii =
1000
= 12.387
(d) The angle 0(t) = wt + (5 sin 3000t + 10 sin 2000nt). The phase deviation is the
maximum value of the angle inside the parentheses, and is given by !l</J = 15 rad.
(e) BEM = 2(!lf + B) = 26, 774.65 Hz
Observe the generality of this method of estimating the bandwidth of an anglemodulated
waveform. We need not know whether it is FM, PM, or some other kind
of angle modulation. It is applicable to any angle-modulated signal.
A Historical Note: Edwin H. Armstrong (1890-1954)
Today, nobody doubts that FM has a key place in broadcasting and communication. As recently
as the 1960s, however, the FM broadcasting seemed doomed because it was so uneconomical
in bandwidth usage.
The history of FM is full of strange ironies. The impetus behind the development of FM
was the desire to reduce signal transmission bandwidth. Superficial reasoning showed that it
was feasible to reduce the transmission bandwidth by using FM. But the experimental results
showed otherwise. The transmission bandwidth of FM was actually larger than that of AM.
Careful mathematical analysis by Carson showed that FM indeed required a larger bandwidth
than AM. Unfortunately, Carson did not recognize the compensating advantage of FM in
its ability to suppress noise. Without much basis, he concluded that FM introduces inherent
distortion and has no compensating advantages whatsoever. 1 In a later paper, he continues