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Homework Assignment 2
Due: 16:00pm Tuesday, Feb. 14, 2023
Problem 1. Given g1(t) ? G1(f), g2(t) ? G2(f), please use the definitions of FT and inverse FT to
proof the the following FT properties.
a) The differentiation property: d
dt
g1(t)? j2πfG1(f).
b) The convolutional property: g1(t) ? g2(T )? G1(f)G2(f).
c) Parseval’s theorem: Eg =
∫∞
?∞ |g1(t)|2dt =
∫∞
?∞ |G1(f)|2df .
Problem 2. a) Find the energy spectral density of the signal g(t) = e?|t|.
b) Show that the signal g1(t) = e?|t?2| has the same energy spectral density as g(t).
Problem 3. Let gT0(t) be a periodic signal with period π. Over the period 0 ≤ t < π, it is defined by
gT0(t) = cos t. Find the Fourier transform of gT0(t) and draw the frequency spectrum.
Note: cosx cos y = 1
2
[cos(x? y) + cos(x+ y)],
sinx cos y = 1
2
[sin(x? y) + sin(x+ y)],∫
eax cos(bx)dx = e
ax
a2+b2
[a cos(bx) + b sin(bx)].
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
2j
[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.
1
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),
where
ge(t) =
1
2
[g(t) + g(?t)] , go(t) = 1
2
[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)?
Homework Assignment 2
Due: 16:00pm Tuesday, Feb. 14, 2023
Problem 1. Given g1(t) ? G1(f), g2(t) ? G2(f), please use the definitions of FT and inverse FT to
proof the the following FT properties.
a) The differentiation property: d
dt
g1(t)? j2πfG1(f).
b) The convolutional property: g1(t) ? g2(T )? G1(f)G2(f).
c) Parseval’s theorem: Eg =
∫∞
?∞ |g1(t)|2dt =
∫∞
?∞ |G1(f)|2df .
Problem 2. a) Find the energy spectral density of the signal g(t) = e?|t|.
b) Show that the signal g1(t) = e?|t?2| has the same energy spectral density as g(t).
Problem 3. Let gT0(t) be a periodic signal with period π. Over the period 0 ≤ t < π, it is defined by
gT0(t) = cos t. Find the Fourier transform of gT0(t) and draw the frequency spectrum.
Note: cosx cos y = 1
2
[cos(x? y) + cos(x+ y)],
sinx cos y = 1
2
[sin(x? y) + sin(x+ y)],∫
eax cos(bx)dx = e
ax
a2+b2
[a cos(bx) + b sin(bx)].
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
2j
[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.
1
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),
where
ge(t) =
1
2
[g(t) + g(?t)] , go(t) = 1
2
[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)?