ELEC201 Signals and Systems Fall 2014

Midterm Exam 2

December 23nd , 2014

4 Problems, 120 Minutes

Closed-book and closed-notes exam; calculators, cellphones, PDAs are NOT allowed!

Clearly show all your work! Answers without justification will not receive any credit!

Name:
Student ID:
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Q1 Q2 Q3 Q4 TOTAL

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 Question1. [20P] Consider the signal, x(t), depicted below.

x(t)

t
-1 -0.5 0.5 1

For x(t), it is given that

Find P (ω), Y (ω) and Z(ω) for the signals depicted in the figures below.

p(t)

1
(a) [7P]

t
-1 -0.5 0.5 1

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 y(t)

1
(b) [7P]

t
-1 -0.5 0.5 1

z(t)

1
(c) [6P]
0.5 1
t
-1 -0.5

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 Question2. [25P]

(a) Let x[n] be the discrete time signal

(i) [5P] Find the Fourier series coefficients of x[n].

(ii) [5P] Find the Discrete Time Fourier Transform of x[n].

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 (b) [15P] Let x[n] be the sequence

x[n] = 2δ[n + 2] − δ[n + 1] + 3δ[n] − δ[n − 1] + 2δ[n − 2]

Evaluate following quantities without explicitly finding X(Ω).

Rπ
(ii) [5P] −π
X(Ω)dΩ

(iii) [5P] Phase of X(Ω), i.e., φX (Ω)

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 Question3. [30P] Consider the system given below:

s1(t) s3(t) s5(t)

x(t) sin(ω0 t/2) sin((ωc +ω0/2)t) y(t)

s2(t) s4(t) s6(t)

cos(ω0 t/2) cos((ωc +ω0/2)t)

Fourier transforms of x(t) and h(t) are shown below

X(ω) H(ω)

1 1

ω ω
-ω0 ω0 -ω0/2 ω0/2
(a) [15P] Write the expressions for S1 (ω) and S2 (ω) in terms of X(ω). Sketch S1 (ω), S2 (ω), S3 (ω), S4 (ω), S5 (ω), S6 (ω),
and Y (ω) on the appropriate boxes given below and the following page.

S1(ω) S3(ω)

ω ω

S2(ω) S4(ω)

ω ω

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 S5(ω)

S6(ω)

Y(ω)

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 (b) [10P] Design a system, for which the Fourier transform of the output will be as shown below, when the input is
x(t).

Z(ω)
2π2

ω
-ωc -ωc+ω0/2 ωc-ω0/2 ωc

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 (c) [5P] Using the systems in the previous parts, design an AM modulator.

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