Laplace Transform (1)

Hany FerdinandoDept. of Electrical Eng.

Petra Christian University

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Overview

Introduction Laplace Transform Convergence of Laplace Transform Properties of Laplace Transform Using table Inverse of Laplace Transform

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Introduction

It was discovered by Pierre-Simon Laplace, French Mathematician (1749-1827)

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Introduction

It transforms signal/system from time-domain to s-domain for continuous-time LTI system

It is analogous to Z Transform in discrete-time LTI system

It is similar to Fourier Transform, but ‘j’ is substituted by s

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Introduction

Laplace Transform is continuous sum of exponential function of the form est, where s = + j is complex frequency

Therefore, Fourier can be viewed as a special case in which s = j

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Laplace Transform

dtetfsF st)()(

j

j

stdtesFj

tf )(2

1)(

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Laplace Transform

For h(t) = e-at, find H(s) What is your assumption in finishing the

integration? If you do not have that assumption, then

what you can do? Is it important to have that assumption?

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Convergence…

The two-sided Laplace Transform exists if

dtetfsF st)()( is finite

Therefore,

dtetfdtetf tst )()( is finite

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Convergence…

Suppose there exists a real positive number R so that for some real and we know that

f(t) < R et for t > 0, and

f(t) < R et for t > 0

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Convergence…

0

)(

0

)(

0

)0

)

11)(

(()(

tt

tt

eeRsF

dteRdteRsF

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Convergence…

How did you make your assumption in order to solve the equation?

Can you solve it without that assumption?

The negative portion converges for < while the positive one converges for >

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Region of Convergence (RoC)

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Region of Convergence (RoC)

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Region of Convergence (RoC)

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Region of Convergence (RoC)

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Properties

Linearity Scaling

Time shift Frequency shift

(s)bF(s)aF(t)bf(t)af 2121

a

sF

a

1f(at)

τsF(s)eτ)f(t a)F(sf(t)e at

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Properties

Time convolution Frequency convolution

Time differentiation

jc

jc

2121 u)du(s(u)FF2ππ

1(t)(t)ff

sF(s)dt

df(t)

(s)(s)FF(t)f*(t)f 2121

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Properties

Time integration

Frequency differentation

βσα,0)max(,s

F(s)f(u)du

t

β,0)min(σα,s

F(s)f(u)du

t

n

nn

ds

F(s)df(t)t)(

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Properties

One-sided time differentiation

One-sided time integration

)0(...)0()0()()( )1()1(21 nnnn

n

n

ffsfssFsdt

tfd

t

s

f

s

sFduuf

)0()()(

)1(

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Using Standard Table

Use table from books both for transform and for its inverse

No RoC is needed Find the general form of the equation Properties of Laplace transform are

helpful You use that table also to find the

inverse

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Exercise

1)sin(2tf(t)

0.5ss

1F(s)

2

6ss

sF(s)

2

2)cos(5tef(t) 3t

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Next…

Signals and Linear Systems by Alan V. Oppenheim, chapter 9, p 603-616

Signals and System by Robert A. Gabel, chapter 6, p 373-394

The Laplace Transform is already discussed. It transforms continuous-time LTI system from

time-domain to s-domain. There are two types, one-sided (unilateral) and two-sided

Next, we will study the application of Laplace Transform in Electrical Engineering. Read the Electric Circuit handout to prepare yourself!