GEORGIA INSTITUTE OF TECHNOLOGYSCHOOL of ELECTRICAL & COMPUTER ENGINEERING

QUIZ #3

DATE: 20-Nov-12 COURSE: ECE 3084B

NAME: GT LOGIN:

LAST, FIRST

• Write your name on the front page ONLY. DETACH the LAST page, which contains a setof formulas for your convenience.

• No calculators.

• Closed book, but one page (81200 ⇥ 1100) of HAND-WRITTEN notes permitted. OK to write

on both sides.

• Unless stated otherwise, justify your reasoning clearly to receive any partial credit.

• You must write your answer in the space provided on the exam paper itself. Only theseanswers will be graded. Circle your answers, or write them in the boxes provided. If space isneeded for scratch work, use the backs of previous pages.

Problem Value Score

1 20

2 20

3 20

4(a) 20

4(b) 20

Problem Q2.1:

(a) Consider an LTI system with transfer function given by

H(s) =s3 � 5

s7 + 4s4 � 8.

Find the di↵erential equation that relates the input of this system, f(t), to its output, y(t).Please put all the terms containing the y(t) on the left hand side and all terms containingf(t) on the right hand side.

(b) Find the partial fraction expansion of the Laplace transform

Y (s) =12s

s2 + 5s+ 6.

Problem Q2.2:

(a) Consider an LTI system with the impulse response g(t) = �(t)� �(t� 6.02⇥ 1023). Find itstransfer function G(s). Can G(s) be expressed as a ratio of polynomials in s?

(b) Consider an LTI system with transfer function given by

H(s) =s

(s+ 42)2 + 64.

Find the step response of this system.

Problem Q2.3:

When listing values of c, a, !0 in the problems below, be sure to specify all values that satisfya given criteria. Think carefully when deciding whether your stated inequalities should be strict ornot (i.e. think about whether you should write � or versus > or <). Note that “none” may alsobe a possible answer. This problem contains many subtleties.

Consider an LTI system with transfer function given by

H(s) =5

(s+ c+ 60j)(s+ c� 60j)(s+ c+ 40j)(s+ c� 40j)

(a) For what values of c is the system BIBO stable?

(b) Suppose the input is given by f(t) = 7 exp(�at) step(t), where a is real (but could be positive,zero, or negative).

• If c < 0, for what values of a, if any, will the output grow without bound?

• If c = 0, for what values of a, if any, will the output grow without bound?

• If c > 0, for what values of a, if any, will the output grow without bound?

(c) Now suppose the input is given by f(t) = 7 sin(!0t+ ⇡/3)) step(t), where !0 > 0.

• If c < 0, for what values of !0, if any, will the output grow without bound?

• If c = 0, for what values of !0, if any, will the output grow without bound?

• If c > 0, For what values of !0, if any, will the output grow without bound?

Problem Q2.4:

The next three pages contain a problem I found on MIT’s OpenCourseWare website from adistance education course called 6.007 developed in 1987 by Prof. Alan Oppenheim. Do both parts(a) and (b) of this problem.

(a) To answer part (a), write the lower case letter indicating your chosen |H(j!)| plot foundon page P21-6 next to the number of the associated h(t) on page P21-5.

(b) To answer part (b), write the roman numeral indicating your chosen pole-zero plot foundon P21-7 next to the letter of the associated |H(j!)| plot found on page P21-6.

Incidentally, Prof. Oppenheim’s lectures for his 6.007 course are available on youtube. (Don’tyou wish you could watch youtube right now? I’m included a screenshot of one of his lectures belowfor inspiration. It appears that the speech recognition algorithms used by youtube’s automatictranscription service had some di�culties.)

Continuous-Time Second-Order Systems / Problems P21-5

P.21.10

Figures P21.10-1, P21.10-2, and P21.10-3 contain impulse responses h(t), frequency responses H(jw), and pole-zero plots, respectively.

(a) For each h(t), find the best matching IH(jw) |. (b) For each |H(jw) |, find the best matching pole-zero plot of H(s).

Consider entries with references to other plots, such as (3) and (5) or (b) and (h), as a pair.

Compare Compare with (2) with (1)

Compare Compare with (5) with (7)

t .h2 t

Compare area = K

with (3)

area = K

K Compare with (4)V

Figure P21.10-1

Signals and Systems P21-6

Compare with (d)

".."0/1

(c) (d) Compare with (a)

(e)

(g)

Compare wit (fW

Figure P21.10-2

J1'***o

Compare with (b)

co)

Continuous-Time Second-Order Systems / Problems P21-7

double pole

Im

(iii)

double pole --. j,

(iv) Im

double 'j zero

double zero

(vi)

(vii) (Viii)

Figure P21.10-3

Basic Definitions

Laplace Transform

L [ f (t)] = F(s) =Z •

0f (t)e�st

dt, L�1[F(s)] = f (t) =1

2p j

Zc+ j•

c� j•F(s)est

ds

Table of Laplace Transform Pairs

Signal Name Time-Domain: f (t) Laplace-Domain: F(s)

constant (step) 1, step(t)1s

impulse d 1

t

1s

2

polynomial t

k, k � 0k!

s

k+1

exponential e

at

1s�a

cosine cos(wt)s

s

2 +w2

sine sin(wt)w

s

2 +w2

e

�at cos(wt)s+a

(s+a)2 +w2

e

�at sin(wt)w

(s+a)2 +w2

Table of Laplace Transform Properties

Property Name Time-Domain: f (t) Laplace-Domain: F(s)

linearity a f1(t)+b f2(t) aF1(s)+bF2(s)

time-derivatived

dt

f (t) sF(s)� f (0)

integrationZ

t

0f (t)dt F(s)

s

delay f (t �T ) e

�sT

F(s)

modulation f (t)eat

F(s�a)

FVT limt!•

f (t), (limit exists) lims!0

sF(s)

IVT limt!0

f (t), (limit exists) lims!•

sF(s)

ECE3084: October 2012