Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.

Laplace Transform

Douglas Wilhelm Harder

Department of Electrical and Computer Engineering

University of Waterloo

Copyright © 2008 by Douglas Wilhelm Harder. All rights reserved.

ECE 250 Data Structures and Algorithms

Laplace Transform

Outline

• In this talk, we will:– Definition of the Laplace transform– A few simple transforms– Rules– Demonstrations

• Classical differential equations

Laplace Transform

Background

tttt xyyy 12

tt eet 2

2

1

2

1y

1x t

Time Domain

Solve differential equation

• Laplace transforms

Laplace Transform

Background

tttt xyyy 12

tt eet 2

2

1

2

1y

s

t

sss

1X

23

1)H(

2

23

112 sss

1x t

Time Domain Frequency Domain

Solve algebraic equation

Laplace transform

Inverse Laplace transform

Laplace Transform

Definition

• The Laplace transform is

• Common notation:

s

dtett st

F

ff0

L

st

st

Gg

Ff

L

L st

st

Gg

Ff

Laplace Transform

Definition

• The Laplace transform is the functional equivalent of a matrix-vector product

0

fF dtets st

n

jjjii vm

1,Mv

n

jjjvu

1

vu

0

vuvu dttttt

Laplace Transform

Definition

• Notation:– Variables in italics t, s

– Functions in time space f, g

– Functions in frequency space F, G

– Specific limits

t

t

t

t

flim0f

flim0f

0

0

Laplace Transform

Existence

• The Laplace transform of f(t) exists if– The function f(t) is piecewise continuous– The function is bound by

for some k and M ktMet f

Laplace Transform

Example Transforms

• We will look at the Laplace transforms of:– The impulse function (t)

– The unit step function u(t)

– The ramp function t and monomials tn

– Polynomials, Taylor series, and et

– Sine and cosine

Laplace Transform

Example Transforms

• While deriving these, we will examine certain properties:– Linearity– Damping– Time scaling– Time differentiation– Frequency differentiation

Laplace Transform

Impulse Function

• The easiest transform is that of the impulse function:

1

δδ

0

0

s

st

e

dtettL

1δ t

• Next is the unit step function

s

es

es

dte

dtett

s

st

st

st

1

10

1

uu

0

0

0

0

L

11

00u

t

tt

st

1u

Laplace Transform

Unit Step Function

Laplace Transform

Integration by Parts

• Further cases require integration by parts

• Usually written as

b

a

b

a

b

a

dfgfgdgf

Laplace Transform

Integration by Parts

• Product rule

• Rearrange and integrate

t

dt

dttt

dt

dtt

dt

dgfgfgf

b

a

b

a

b

a

b

a

b

a

dtttdt

dtt

dtttdt

ddttt

dt

ddtt

dt

dt

ttdt

dtt

dt

dt

dt

dt

gfgf

gfgfgf

gfgfgf

Laplace Transform

Ramp Function

• The ramp function

2

0

0

00

0

111

10

111

u

se

ss

dtes

dtes

es

t

dttett

st

st

stst

st

L

2

1u

stt

t

t

ddf

f

st

st

es

te

1

g

ddg

Laplace Transform

Monomials

• By repeated integration-by-parts, it is possible to find the formula for a general monomial for n ≥ 0

1

!u

nn

s

nttL

1

!u

nn

s

ntt

Laplace Transform

Linearity Property

• The Laplace transform is linear

• If and then

sbsatbta

sbsatbta

GF)g()f(

GF)g()f(

L

st F)f( L st G)g( L

Laplace Transform

Initial and Final Values

• Given then

• Note sF(s) is the Laplace transform of f(1)(x)

st Ff

ss

ss

s

s

Flimf

Flim0f

0

Laplace Transform

Polynomials

• The Laplace transform of the polynomial follows:

n

kkk

n

k

kk s

katta

01

0

!uL

Laplace Transform

Polynomials

• This generalizes to Taylor series, e.g.,

1

1

1

!

!

1

u!

1u

01

01

0

s

s

s

k

k

ttk

te

n

kk

n

kk

n

k

kt LL

1

1u

stet

Laplace Transform

The Sine Function

• Sine requires two integration by parts:

ttss

dtetss

dtetss

stets

dtets

dtets

stets

dtettt

st

st

st

st

st

usin11

sin11

sin11

cos1

cos1

0

cos1

sin1

sinusin

22

022

02

0

0

00

0

L

L

1 of 2

Laplace Transform

The Sine Function

• Consequently:

1

1usin

1usin1

usin11

usin

2

2

22

stt

tts

ttss

tt

L

L

LL

1

1usin

2

stt

2 of 2

Laplace Transform

The Cosine Function

• As does cosine:

ttss

dtetss

dtetss

stetss

dtetss

dtets

stets

dtettt

st

st

st

st

st

ucos11

cos1

01

cos11

sin11

sin11

sin1

cos1

cosucos

2

022

02

0

0

00

0

L

L

1 of 2

Laplace Transform

The Cosine Function

• Consequently:

1

ucos

ucos1

ucos11

ucos

2

2

2

s

stt

stts

ttss

tt

L

L

LL

1

ucos2

s

stt

2 of 2

Laplace Transform

Periodic Functions

• If f(t) is periodic with period T then

• For example,

sT

Tst

e

dtet

t

1

f

f 0L

s

s

s

st

es

sse

e

dtet

t

11

1

cos

f2

0L

Laplace Transform

Periodic Functions

• Here cos(t) is repeated with period

tfL

tcos

tcosL

tf

• Consider f(t) below:

Laplace Transform

Periodic Functions

s

s

s

s

s

st

es

e

es

e

e

dte

t222

1

0

1

1

1

1

1f

L

tf tu

s

t1

u L tfL

Laplace Transform

Damping Property

• Time domain damping ⇔ frequency domain

shifting

as

dtet

dtetete

tas

statat

F

f

ff

0

)(

0

L

aste at Ff

Laplace Transform

Damping Property

• Damped monomials

A special case:

1

1

!u

!u

nnat

nn

as

ntte

s

ntt

as

te

st

at

1u

1u

• Consider cos(t)u(t)

1

ucos2

s

stt

Laplace Transform

Damping Property

• Time scale by = 2

22 2

1u2sin

stt

Laplace Transform

Damping Property

• Time scale by = ½

4

1221

1usin

stt

Laplace Transform

Damping Property

Laplace Transform

Time-Scaling Property

• Time domain scaling ⇔ attenuated frequency domain scaling

a

s

a

adea

da

e

dteatat

as

as

st

F1

)f(1

1)f(

)f()f(

0

0

0

L

dta

d

at

a

s

aat F

1f

• Time scaling of trigonometric functions:

22

2

1

11usin

s

sttL

22

2

1

1ucos

s

s

s

s

ttL

1

1usin

2

stt

1ucos

2

s

stt

22

usin

s

tt 22

ucos

s

stt

Laplace Transform


• Consider sin(t)u(t)

1

1usin

2

stt

Laplace Transform


• Time scale by = 2

22 2

1u2sin

stt

Laplace Transform


• Time scale by = ½

4122

11

usin

s

tt

Laplace Transform


Laplace Transform

Damping Property

• Damped time-scaled trigonometric functions are also shifted

22

22

usin

usin

astte

stt

at

22

22

ucos

ucos

as

astte

s

stt

at

Laplace Transform

Time Differentiation Property

• The Laplace transform of the derivative

0fF

f0f

ff

ff

0

00

0

11

ss

dtets

dtetset

dtett

st

stst

stL

Laplace Transform


• The general case is shown with induction:

0f0f

0f0f0f

Ff

12

23121

nn

nnn

nn

s

sss

sst

L

Laplace Transform


• If g(t) = f(t)u(t) then 0 = g(0+) = g(1)(0+) = ···

• Thus the formula simplifies:

• Problem:– The derivative is more complex

sst nn Fg L

ttttdt

dδ0fufg )1(

Laplace Transform


• Example: if g(t) = cos(t)u(t) theng(0–) = 0

g(1)(t) = sin(t)u(t) + (t)

Laplace Transform


• We will demonstrate that– The Laplace transform of a derivative is the

Laplace transform times s– The next six slides give examples that

f(1)(t) = g(t) implies sF(s) = G(s)

1 of 7

Laplace Transform

Differentiation of Polynomials

• We now have the following commutative diagram when n > 0

1

!

u

n

n

s

ns

ttsL

ttn n u1L tt ndtd uL

ns

n!

2 of 7

Laplace Transform

Differentiation of Trigonometric Functions

• We now have the following commutative diagram

1

1

usin

2

ss

ttsL

tt ucosL ttdtd usinL

12 s

s

3 of 7

Laplace Transform



1

ucos

2

s

ss

ttsL

11

1

δusin

2

s

tttL ttdtd ucosL

12

2

s

s

4 of 7

Laplace Transform

Differentiation of Exponential Functions


ass

tes at

1

uL

1

δu

as

a

ttae atL te atdtd uL

as

s

5 of 7

Laplace Transform



22

usin

ss

ttsL

22

ucos

s

s

ttL ttdtd usin L

22 s

s

6 of 7

Laplace Transform



22

ucos

s

ss

ttsL

1

δusin

22

s

tttL ttdtd ucos L

22

2

s

s

7 of 7

Laplace Transform

Frequency Differentiation Property

• The derivative of the Laplace transform

)f(

f

f

fF

0

0

0

)1(

tt

dtett

dtetds

d

dtetds

ds

st

st

st

L stt )1(Ff

Laplace Transform


• Consider monomials

1

!

nn

s

nt

2

1 !1

nnn

s

nttt

2

21

!1

!1

!

n

nn

s

ns

nn

s

n

ds

d

Laplace Transform


• Consider a sine function

• We have that

but what is ?

1

1sin

2

st

2221

2

1

1

s

s

sds

d

tt sinL

1 of 3

Laplace Transform


• Applying integration by parts

00

000

cossin1

cossin1

sin1sin

dtettdtets

dtettts

etts

dtett

stst

ststst

00

000

sincos1

sincos1

cos1cos

dtettdtets

dtettts

etts

dtett

stst

ststst

2 of 3

Laplace Transform


• Substituting

0000

sincos1

sin1

sin dtettdtets

dtets

dtett stststst

1

1

1

2sin

1

21sin

sin

1

1

1

1

sin1

1

1

11sin

222

22

2

222

22

sds

d

s

stt

sss

stt

s

tt

ssss

tts

s

ssstt

L

L

L

LL

3 of 3

Laplace Transform

Time Integration Property

• The Laplace transform of an integral

s

s

des

tdde

tdde

dtedd

st

st

st

sttt

F

f1

f

f

ff

0

0

0

0 00

L

Laplace Transform

Time Integration Property

• We will demonstrate that– The Laplace transform of an integral is the

Laplace transform over s– The next six slides give examples that

implies

t

dt0

f)g( s

ss

FG

1 of 7

Laplace Transform

Integration of Polynomials


n

n

s

n

s

s

t

!1

L

11

1

nt

nL

tnd

0

L

1

!ns

n

2 of 7

Laplace Transform

Integration of Exponential Functions


ass

s

e at

11

L

assa

ea

at

111

11L

ass 1

t

a de0

L

3 of 7

Laplace Transform

Integration of Trigonometric Functions


1

11

sin

2

ss

s

tL

1

1

cos1

2

s

s

s

tL

112 ss

t

d0

sin L

4 of 7

Laplace Transform



)sin(tL

1

12 s

t

d0

cos L

1

1

cos

2

s

s

s

s

tL

5 of 7

Laplace Transform



22

1

sin

ss

s

tL

22

11

)cos(11

s

s

s

tL

22 ss

t

d0

sin L

6 of 7

Laplace Transform



22

1

cos

s

s

s

s

tL

22

1

)sin(1

s

tL

22

1

s

t

d0

cos L

7 of 7

Laplace Transform

The Convolution

• Define the convolution to be

• Then

dt

dtt

gf

gfgf

ssj

tt

sst

GF2

1gf

GFgf

Laplace Transform

Integration

• As a special case of the convolution

s

s

ss

sttsdt

F1F

uff0

LL

Laplace Transform

Summary

• We have seen these Laplace transforms:

1

2

!u

1u

1u

1δ

nn

s

ntt

stt

st

t

1

ucos

1

1usin

1

1u

2

2

s

stt

stt

stet

Laplace Transform

Summary

• We have seen these properties:– Linearity– Damping– Time scaling

– Time differentiation– Frequency differentiation– Time integration

sbsatbta GF)g()f( aste at Ff

a

s

aat F

1f

sstt nn Fuf

sttt nn )(Fuf

s

sd

t Ff

0

Laplace Transform

Summary

• In this topic:– We defined the Laplace transform– Looked at specific transforms– Derived some properties– Applied properties

Laplace Transform

References• Lathi, Linear Systems and Signals, 2nd Ed., Oxford

University Press, 2005.• Spiegel, Laplace Transforms, McGraw-Hill, Inc., 1965.• Wikipedia,

http://en.wikipedia.org/wiki/Laplace_Transform

Usage Notes

• These slides are made publicly available on the web for anyone to use

• If you choose to use them, or a part thereof, for a course at another institution, I ask only three things:– that you inform me that you are using the slides,– that you acknowledge my work, and– that you alert me of any mistakes which I made or changes which

you make, and allow me the option of incorporating such changes (with an acknowledgment) in my set of slides

Sincerely,

Douglas Wilhelm Harder, MMath

[email protected]

Laplace Transform Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008 by Douglas Wilhelm Harder.

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