The Fourier Transform and its Applications Problem Set Five Due Thursday, June 23
1. Evaluating integrals with the help of Fourier transforms Evaluate the following integrals using Parseval’s Theorem and one other method. (Yes, we expect you to evaluate the integral twice, and if you do it right you should get the same answer for both approaches (obviously)):
(a) ∞
−∞sinc4(t) dt
(b) ∞
−∞
2
1 + (2πt)2 sinc(2t) dt
(c) ∞
−∞t2sinc4(t) dt 2. Generalized Fourier transforms
(a) Find the Fourier transform of the signal shown below.
t f (t)
0 1
1
2 3
−1
−1
−3 −2
(b) Find the Fourier transform of the function f (t) = sin(2π|t|) plotted below.
(Simplify your expression as much as possible.)
1
−3 −2 −1 0 1 2 3
−1
−0.5 0 0.5 1
t
sin(2π|t|)
3. Consider the signal
f (t) = 1 + Λ(3t) ∗ III1/3(t).
Sketch the graph of f (t), find the Fourier transform Ff (s), and sketch its graph. Comment on what you see.
