Thursday Apr 15, 2021 ENGG 177 NAME:
OPEN BOOK Exam # 2


Problem 1. Use the Fourier transform differentiation and time-shift properties to find the
(25 pts) Fourier transform of the piecewise linear pulse:

() = ( − 1) + ( + 1)

Reduce your answer to the simplest form possible as a function of ().

Formulas: 1 − cos() = 2 sin2( 2⁄ ) sinc() = sin ()

cos() = +−
2
sin() = −−
2
Problem 2. The Complex Fourier coefficients of a periodic signal () are given
(25 pts) in the expansion interval = 0, 1, 2, 3 by:

0 = 1 = 0

= 1 − = 1, 2, 3

Given that the period is = sec, estimate the signal () for the expansion interval shown
above.

































Problem 3. Consider the integral
(25 pts)
() = � ()
−∞


where () = ( + 1) − ( − 1)

Find the Fourier transform () by using the differentiation and the integration in the time
domain properties. Reduce your answer to the simplest form possible as a function of ().

Formulas: sinc() = sin ()

sin() = −−
2

Problem 4. A discrete-time signal [] is given by
(25 pts)
[] = �1, = −1,0,10, = ℎ

Determine the Discrete Fourier Transform (Ω) of the convolution [] = [] ∗ [].

Use the DFT properties and reduce your answer for (Ω) to the simplest form possible as a
function of ().












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