ECE 380 Introduction to Communication Systems
Homework Assignment 6 Due: 16:00pm Tuesday, Match 14, 2021
Problem 1. A baseband signal m(t) is the periodic sawtooth signal shown in Fig. 1, where T0 = 1, A = 1.
(a) Sketch the FM wave for this signal m(t) if fc = 4 and kf = 10.
(b) Estimate the bandwidth of the FM wave. Assume the bandwidth of m(t) is defined by the fifth harmonic frequency of m(t).
Figure 1: Message signal in Problem 1.
Problem 2. Given m(t) = sin[2π(1000)t], kf = 100, 000 and kp = 10.
(a) Estimate the bandwidth of FM and PM waves using Carson’s rule.
(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).
Problem 3. (Haykin and Moher Problem 4.11) The sinusoidal wave m(t) = Am cos(2πfmt)
is applied to a phase modulator with phase sensitivity kp. The unmodulated carrier wave has frequency fc and amplitude Ac. Find the spectrum of the resulting phase-modulated (PM) wave, assuming that the maximum phase deviation β = kpAm is sufficiently small.
Note: Use the approximations sin x ≈ x and cos x ≈ 1 for |x| ≪ 1.
Problem 4. (Haykin and Moher Problem 4.24 modified) An FM wave is given as t
s(t) = Ac cos 2πfct + 2πkf
where the message bandwidth is W and the maximum frequency deviation is ∆fmax. Consider a
memoryless channel characterized by the following non-linear input-output relationship:
v0(t) = a1s(t) + a2s2(t) + a3s3(t), where v0(t) is the system output and s(t) is the input.
(a) By using the generalized Carson’s rule, show that if
fc > 3∆fmax + 2W, 1
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ECE 380 Introduction to Communication Systems
the effect of the non-linear distortion can be removed by band-pass filtering. In other words, by applying v0(t) to a band-pass filter, the FM wave s(t) can be recovered.
(b) How to design the pass-band of the filter in Part (a)?
Note: cos2 x = 21 [1 + cos(2x)]. cos3 x = 14 [3 cos(x) + cos(3x)].
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