Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Signals and Systems Exam 2: Solving Problems Involving Periodic Signals and LTI Systems - , Exams of Electrical and Electronics Engineering

The second exam for the ece 300 signals and systems course. The exam covers topics such as determining closed-form expressions for periodic signals, computing fourier series coefficients, analyzing the relationship between input and output of lti systems, and solving graphical convolution problems. Students are required to use calculus and trigonometry to find numerical values and algebraic relationships.

Typology: Exams

Pre 2010

Uploaded on 08/13/2009

koofers-user-04m
koofers-user-04m 🇺🇸

10 documents

1 / 10

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Name __________________________________________________ CM____________
ECE 300
Signals and Systems
Exam 2
30 April, 2009
NAME ________________________________________
This exam is closed-book in nature. You may use a calculator for simple
calculations during the exam, but not for integration. Do not write on the back of
any page, use the extra pages at the end of the exam.
Problem 1 ________ / 20
Problem 2 ________ / 25
Problem 3 ________ / 15
Problem 4 ________ / 20
Problem 5 ________ / 20
Exam 2 Total Score: _______ / 100
1
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Signals and Systems Exam 2: Solving Problems Involving Periodic Signals and LTI Systems - and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

ECE 300

Signals and Systems

Exam 2

30 April, 2009

NAME ________________________________________

This exam is closed-book in nature. You may use a calculator for simple

calculations during the exam, but not for integration. Do not write on the back of

any page, use the extra pages at the end of the exam.

Problem 1 ________ / 20

Problem 2 ________ / 25

Problem 3 ________ / 15

Problem 4 ________ / 20

Problem 5 ________ / 20

Exam 2 Total Score: _______ / 100

1) (20 points) The spectrum of periodic signal x ( ) t is shown below. The period of this

signal is T 0 (^) = 3 seconds and all angles are multiples of 45 degrees.

-4 -3 -2 -1 0 1 2 3 4

0

1

2

Magnitude

Harmonic

-4 -3 -2 -1 0 1 2 3 4

0

100

200

Phase (degrees)

Harmonic

a) Determine a closed-form expression for x ( ) t in terms of cosines.

b) Sketch the single-sided power spectrum for this signal as power versus harmonic. Be

sure to label all significant points (values) on your graph.

c) Compute the average power of this signal.

d) Compute the average value of this signal.

3) (15 points) Assume the periodic signal x ( ) t with the Fourier series representation

x jk (^) ot k k

x t ce

ω

is the input to an LTI system described by the differential equation

y t  ( ) + ay t ( ) = dx t ( − b )

Since the system is LTI the output will be periodic with Fourier series representation

jk (^) ot k

y

k

y t ce

ω

a) Determine an algebraic relationship between

x c k and

y c k

b) Determine the (continuous frequency) transfer function H ( j ω )relating the input and

output.

4) (20 points) The periodic signal x ( ) t has the Fourier series representation

j t

k

k

k x t e kj

=

=−

x ( ) t is the input to an LTI system (a band reject or notch filter) with the transfer function

3 2 | 0 ( 0 3.

j e and H j

ω

− ⎧ (^) ≤ ≤ ≤ = <

The steady state output of the system can be written as y t ( ) = ax t ( − b ) + d cos( et + f )

, e

Determine numerical values for the parameters (^) a b d , , and f

Some Potentially Useful Relationships

T 2 2

T T

E lim x t dt x t dt

∞ →∞ − −∞

T 2

T T

P lim x t d 2T

∞ (^) →∞ −

t

jx e = cos x + jsin x j = − 1

(^1) jx jx cos x e e 2

− = ⎡ + ⎤ ⎣ ⎦

(^1) jx jx sin x e e 2 j

− = ⎡ − ⎤ ⎣ ⎦

cos x cos 2x 2 2

sin x cos 2x 2 2

0 0 0

t t T T rect u t t u t t T 2

⎜ ⎟ =^ ⎜ −^ +^ ⎟ −^ ⎜ −^ −

⎝ ⎠ ⎝ ⎠ ⎝^2