Faculty of Engineering Sciences
Electronics and Communications Department
Fall 2012
ELE 353
Communication theory (1)
Sheet(7)
Frequency modulation & Phase modulation
Dr. Amr Abouel Magd Ahmed
2012 – 2013
Sheet ( 4 )
In the following problems, we will use the following expressions:
a) ФFM(t) is frequency modulated signal.
b) ФPM(t) is phase modulated signal.
c) ∆f is the peak frequency deviation.
d) KF is the frequency sensitivity of the modulator.
e) KP is the phase sensitivity of the modulator.
f) A is the amplitude of FM or PM signal.
(1)
Sketch ФFM(t) and ФPM(t) for the modulating signal m(t) shown in the following figure, given fc=10
MHz, kf = 105*2 π, and kp = 25*2 π
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(2)
A baseband signal m(t) is a periodic sawtooth signal shown in the figure below. Sketch ФFM(t) and
ФPM(t) for this signal, if Wc = 2 π X 106 rad/sec, kf = 2000 π, and kp = π /2. Explain why it is necessary
to use kp < π in this case.
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(3)
An angle modulated signal Over an interval │t│ ≤ 1 is given by ФEM(t) = 10 cos (13, 000t)
It is known that the carrier frequency Wc = 10,000 rad/sec.
(a) If this were a PM signal with kp = 1000, determine m(t) over the interval │t│ ≤ 1.
(b) If this were an FM signal with kf = 1000, determine m(t) over the interval │t│ ≤ 1.
(4)
For a practical modulating signal
m (t) = 2 cos 100t + 18 cos 2000 πt
a) Write expressions (do not sketch) for ФPM(t) and ФFM(t) when A = 10, fc= 106 HZ, kf =1000 π,
and kp = 1. For determining ФFM(t), use the indefinite integral of m(t), that is, take the value of the
integral at t = to be 0
(b) Estimate the bandwidths of ФFM(t) and ФPM(t).
(5)
An angle-modulated signal with carrier frequency Wc = 2 π X 106 rad/sec is described by the equation
ФEM(t) = 5 cos (wct + 20 sin 1000 π t + 10 sin 2000 π t)
(a) Find the power of the modulated signal.
(b) Find the peak frequency deviation ∆f.
(c) Find the phase deviation ∆Ф.
(d) Estimate the bandwidth of ФEM(t)
(6)
Estimate the bandwidth for ФPM(t) and ФFM(t) in Prob.1. Assume the bandwidth of m(t) to be the fifth
harmonic frequency of m(t).
(7)
Given m(t) = sin2000 π t , kf = 200,000 π , and kP = 10.
(a) Estimate the bandwidths of ФPM(t) and ФFM(t).
(b) Repeat part (a) if the message signal amplitude is doubled.
(c) Repeat part (a) if the message signal frequency is doubled.
(d) Comment on the sensitivity of FM and PM bandwidths to the spectrum of m(t).
(8)
Given m (t) = e-t2, fc = 104 Hz, kf = 6000 π, and kp = 8000 π .
(a) Find ∆f the peak frequency deviation for FM and PM.
(b) Estimate the bandwidths of the FM and PM waves.
Hint: Find M(w) and observe the rapid decay of this spectrum. Its 3-dB bandwidth is even smaller than
1 Hz (B « ∆f).