ECE 380 Introduction to Communication Systems
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 g1(t) ⇌ j2πfG1(f). dt
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: cosxcosy = 21[cos(x−y)+cos(x+y)],
sinxcosy = 21[sin(x−y)+sin(x+y)],
eax cos(bx)dx = eax [a cos(bx) + b sin(bx)]. a2 +b2
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