Signals and SystemsFall 2003

Lecture #186 November 2003

• Inverse Laplace Transforms• Laplace Transform Properties• The System Function of an LTI System• Geometric Evaluation of Laplace Transfor

ms and Frequency Responses

Inverse Laplace Transform

})({

,)()(

t

st

etxF

ROCjsdtetxsX

Fix σ ROC and apply the inverse Fourier ∈

dejXetx tjt )(2

1)(

dejXtx tj )()(2

1)(

But s = σ + jω (σfixed) ds =⇒ jdω

j

j

stdsesXj

tx )(2

1)(

Inverse Laplace Transforms Via Partial FractionExpansion and Properties

Example:

3

5 ,

3

2

21)2)(1(

3)(

BA

s

B

s

A

ss

ssx

Three possible ROC’s — corresponding to three different signals

Recall sided-left )(}{ ,1

tueaseas

at

sided-right )(}{ ,1

tueaseas

at

ROC I: — Left-sided signal.

ttuee

tuBetuAetx

tt

tt

as es Diverg)(3

5

3

2

)()()(

2

2

ROC II: — Two-sided signal, has Fourier Transform.

ttuetue

tuBetuAetx

tt

tt

as erges Div)(3

5)(

3

2

)()()(

2

2

ROC III:— Right-sided signal.

tu(t)ee

tuBetuAetx

tt

tt

as erges Div3

5

3

2

)()()(

2

2

Properties of Laplace Transforms

• Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC

• For example:

Linearity

)()()()( 2121 sbXsaXtbxtax

ROC at least the intersection of ROCs of X1(s) and X2(s)

ROC can be bigger (due to pole-zero cancellation)

0)(0)()( Then

and )()( E.g.

21

21

sXtbxtax

batxtx

⇒ ROC entire s-

Time Shift

)( as ROCsame ),()( sXsXeTtx sT

? 2}{ ,2

:Example3

ses

e s

Ttttueses

e tsT

|)( 2}{ ,2

2

3 T

)3( 2}{ ,2

)3(23

tueses

e ts

Time-Domain Differentiation

j

j

j

j

stst dsessXjdt

tdxdsesX

jtx )(

2

1)( ,)(

2

1)(

ROC could be bigger than the ROC of X(s), if there is pole-zero cancellation. E.g.,

)( of ROC thecontaining ROC with),()(

sXssXdt

tdx

plane-s entire ROC1

1)( )(

0}{ ,1

)( )(

sst

dt

tdx

ses

tutx

s-Domain Differentiation

X(s)ds

sdXttx as ROCsame with,

)()(

) osimeilar t

is Derivation(

sdt

d

aseasasds

dtute at

}{ ,

)(

11)( E.g.

2

Convolution Property

x(t) y(t)=h(t)＊ x(t)h(t)

For

Then )()()(

)()(),()(),()(

sXsHsY

sHthsYtysXtx

• ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X

(s)

• ROC could be empty if there is no overlap between the two ROCsE.g.

x(t)=etu(t),and h(t)=-e-tu(-t)

• ROC could be larger than the overlap of the two.

)()(*)( tthtx

The System Function of an LTI System

x(t) y(t)h(t)

function system the)()( sHth

The system function characterizes the system⇓

System properties correspond to properties of H(s) and its ROC

A first example:

dtth )( stable is System

axis theincludes

of ROC

jω

H(s)

Geometric Evaluation of Rational Laplace Transforms

Example #1: A first-order zeroassX )(1

Example #2: A first-order pole

)(

11)(

12 sXas

sX

)( )(

)|)(|log log(or )(

1 )(

12

121

2

sXsX

sX(s)||XsX

sX

Still reason with vector, but remember to "invert" for poles

Example #3: A higher-order rational Laplace transform

)(

)( )(

1

1

jPj

iRi

s

sMsX

jPj

iRi

s

sMsX

1

1 )(

R

i

p

jji ssMsX

1 1

)()( )(

First-Order System

Graphical evaluation of H(jω)

1

}{,/1

/1

1

1)(

se

sssH

)(1

)( / tueth t

)(1)( / tuets t

/1

11

/1

/1 )(

jjjH

Bode Plot of the First-Order System

1/ 2/

1/ 4/

0 0

)(tan )(

1/ /1

1/ 2/1

0 1

)/1(

/1 )(

1/j

1/ )(

1

22

jH

jH

jH

Second-Order System

e(pole)e{s} ROC2

)(22

2

nn

n

sssH

10 complex poles— Underdamped

1

1

double pole at s = −ωn

— Critically damped

2 poles on negative real axis— Overdamped

Demo Pole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems

Bode Plot of a Second-Order System

Top is flat whenζ= 1/√2 = 0.707⇒a LPF for

ω < ωn

Unit-Impulse and Unit-Step Response of a Second- Order System

No oscillations whenζ ≥ 1

⇒ Critically (=) and over (>) damped.

First-Order All-Pass System

1. Two vectors have the same lengths

2.

)0( }{ ,)(

aaseas

assH

a

a

jH

0~

2/

0

2

)(

)(

2

22

21