ECE 380 Introduction to Communication Systems
Homework Assignment 4 Due: 16:00pm Tuesday, March. 7, 2023
Problem 1. In conventional AM, the carrier is c(t) = Ac cos(2πfct) and the message signal is m(t) = sinc(t) + sinc2(t).
Let the modulation index be ka = 1 and fc ≫ 1.
a) Find the frequency-domain representation and draw the frequency spectrum of the modulated signal.
b) What is the bandwidth of the modulated signal.
Problem 2. (Haykin and Moher Problem 3.17) In lectures, we focused on
c(t) = Ac cos(2πfct) as the sinusoid carrier wave. Suppose we choose
c(t) = Ac sin(2πfct)
as the sinusoid carrier wave to modulate the following single-tone signal
m(t) = Am sin(2πfmt). a) Evaluate the spectrum of the new AM wave:
s(t) = Ac[1 + kam(t)] sin(2πfct).
b) Compare the result derived in Part a) with those shown in lectures and discuss.
Problem3. Supposethesignalg(t)=m(t)+cos(2πfct)isappliedtoanonlinearsystemwhoseoutput
Determine and sketch the spectrum of y(t) when M(f) is as shown in Figure 1, where W ≪ fc.
y(t) = g(t) + 12g2(t).
Problem 4. Consider the AM system shown in Fig. 2. The message signal is m(t) = 4 sinc(4t).
i). If fc = 2, sketch the frequency spectra of the signals at points (a), (b), (c), (d), and (e).
ii). Find the minimum value of fc for which the signal at point (e) is equal to the signal at point (b).
Problem 5. Show that the Hilbert transform of ej2πf0t is −jsgn(f0)ej2πf0t.
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ECE 380 Introduction to Communication Systems
Figure 1: Frequency spectrum of m(t) for Problem 3.
Figure 2: Fig. 2: Frequency spectrum for m(t) in Problem 4.
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