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Signals and Systems Exam 2 - ECE 300 - Prof. Robert D. Throne, Exams of Electrical and Electronics Engineering

The exam questions for the signals and systems course (ece 300) at a university. The exam covers topics such as stability of systems, impulse responses, periodic signals, fourier series, and lti systems. Students are required to solve problems related to these topics and provide calculations and expressions in the exam. The document also includes some potentially useful relationships for reference.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Name __________________________________________________ CM____________
ECE 300
Signals and Systems
Exam 2
18 October 2007
NAME ________________________________________
This exam is closed-book in nature. You may use a calculator for simple
calculations, but not for things like integrals. Credit will not be given if your work
is not shown!
Problem 1 ________ / 25
Problem 2 ________ / 25
Problem 3 ________ / 25
Problem 4 ________ / 25
Exam 2 Total Score: _______ / 100
1
pf3
pf4
pf5
pf8

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ECE 300

Signals and Systems

Exam 2

18 October 2007

NAME ________________________________________

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

calculations, but not for things like integrals. Credit will not be given if your work

is not shown!

Problem 1 ________ / 25

Problem 2 ________ / 25

Problem 3 ________ / 25

Problem 4 ________ / 25

Exam 2 Total Score: _______ / 100

  1. Short Answer Questions (5 points each):

a) Is the system with impulse response ( ) ( )BIBO stable? Why or why not?

t h t = e u t

b) Is the system

( ) cos ( )

y t x t

⎟ BIBO stable? Why or why not?

c) What is the impulse response for the system

1

( ) ( 2)

t t y t e e x d

λ

− −

−∞

? Be sure to

include appropriate unit step functions.

d) Consider the two LTI systems shown below, with impulse responses shown. What is

the impulse response between x ( ) t and y t ( )?

1 ( )^ ( )

t h t e u t

= h 2^ ( ) t^^ =^2 δ^ (^ t −1)

x ( ) t v t ( ) y t ( )

e) Is the function ( ) cos(4 ) sin(6 )

2

x t t t

π = π + + π periodic? If yes, determine the

fundamental period.

  1. Assume periodic signal x ( ) t has Fourier series representation

k jkt

k

jk x t e jk

=∞

=−∞

x ( ) t is the input to an LTI system with transfer function given by

H j j

otherwise

⎪ <^ <

Determine the steady state output of the system. For full credit your answer

must be written in terms of cosines (and/or sines). Clearly indicate whether you are

writing your phase in degrees or in radians.

, y t ( )

  1. Assume x ( ) t is a periodic signal with period T 0 (^) = 3. x ( ) t is defined over one period as

t x t t

⎧^ −^ <^ ≤

⎩ <^ ≤

a) Determine the fundamental frequency ω 0.

b) Determine the average value of x ( ) t.

c) Determine the average power in the DC component of x ( ) t.

d) Determine an expression for the expansion coefficients,. X (^) k , where ( )

jk (^) ot x t X ek

ω

must write your expression in terms of the sinc function, and possibly a leading

nential term.

You

expo

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