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)].