ECE 380 Introduction to Communication Systems
Homework Assignment 1 Due: 16:00pm Tuesday, Feb. 7, 2023
Problem 1. Find the inverse Fourier transforms of G(f ) for the spectra in Figure 1 (a) and (b). Note: G(f) = |G(f)|ej∠G(f).
Figure 1: Signals for Problem 1.
Problem 2. Find the Fourier transforms of the signals g1(t) and g2(t) in Figure 2 using either the definition of Fourier Transform or the properties of Fourier Transform together with the table of Fourier transform pairs posted on the course website.
Figure 2: Signals for Problem 2.
Problem 3. a) Prove the following result via properties of Fourier transform: For any signal g(t) with
Fourier transform G(f), we have
g(t)sin(2πfct)⇌ 1[G(f−fc)−G(f+fc)].
b) Using the result in (a), please find the Fourier transform of the time-domain signal
s(t) = [2 + cos(2πf0t)] sin(200πt), where f0 > 0. Draw the spectra.
Note: Consider different ranges of f0.
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ECE 380 Introduction to Communication Systems
Problem 4. (Haykin & Moher Problem 2.20 with revision) Any function g(t) can be split unambigu-
ously into an even part, ge(t), and an odd part, go(t), as shown by g(t) = ge(t) + go(t),
ge(t) = 21 [g(t) + g(−t)] , go(t) = 21 [g(t) − g(−t)] . a) Evaluate the even and odd parts of u(t).
b) What are the Fourier transforms of these two parts and the Fourier transform of u(t)?
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