ECE102 Homework #6 solution

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1. (15 points) More properties of Fourier transform
(a) (10 points) Use Parseval’s theorem to evaluate the following integral:
Z ∞
−∞
sinc2
(t) cos(2πt)dt
Hint: Use the fact that cos(2u) = 2 cos2
(u) − 1
(b) (5 points) Suppose we apply Fourier transform four times to signal x(t) as shown below:
How is y(t) related to x(t)?
2. (31 points) Frequency Response
Parts (a), (b) and (c) are independent.
(a) (18 points) Consider the LTI system depicted in figure 1 whose response to an unknown
input, x(t), is
y(t) =
4e
−t − 4e
−4t

u(t)
Figure 1: System for Problem 2.
We know that for the same unknown input x(t), the intermediate signal, y1(t), is given
by:
y1(t) = 2e
−tu(t)