9.0 Laplace Transform 9.1 General Principles of Laplace Transform linear time-invariant Laplace Transform Eigenfunction Property y(t) = H(s)e st h(t)h(t)

9.0 Laplace Transform 9.1 General Principles of Laplace

Transform

linear time-invariant

Laplace TransformEigenfunction Property

y(t) = H(s)esth(t)x(t) = est

dehsH s

Chapters 3, 4, 5, 9, 10

jω

Chap 3 Chap 5Chap 4

Chap 10

Chap 9

Laplace TransformEigenfunction Property

– applies for all complex variables s

dtethjH

jstj

Fourier Transform

Laplace Transform

dtethjH

jstj

eigenfunction of all linear time-invariant systems with unit impulse response h(t)

Laplace Transform

A Generalization of Fourier Transform

dtetx

dtetxjX

tj

tj

e

t-

jsjs to fromjω

X( + jω)

tetx Fourier transform of

sXtx

jsdtetxsXL

st

,

Laplace Transform

Laplace Transform

𝜎 2+ 𝑗𝜔1

Laplace TransformA Generalization of Fourier Transform

– X(s) may not be well defined (or converged) for all s

– X(s) may converge at some region of s-plane, while x(t) doesn’t have Fourier Transform

– covering broader class of signals, performing more analysis for signals/systems

Laplace TransformRational Expressions and Poles/Zeros

sD

sNsX

roots zeros

roots poles

– Pole-Zero Plots

– specifying X(s) except for a scale factor

Laplace TransformRational Expressions and Poles/Zeros– Geometric evaluation of Fourier/Laplace transform

from pole-zero plots

jj

ii

s

sMsX

each term (s-βi) or (s-αj) represented by a vector with magnitude/phase

Poles & Zeros

x x

Region of Convergence (ROC)Property 1 : ROC of X(s) consists of strips parallel

to the jω -axis in the s-plane

– For the Fourier Transform of x(t)e-σt to converge

dtetx t

Property 2 : ROC of X(s) doesn’t include any poles

depending on σ only

Property 1, 3

Region of Convergence (ROC)Property 3 : If x(t) is of finite duration and absolutely

integrable, the ROC is the entire s-plane

dttxedtetx

dttxedtetx

TT

dttx

T

T

TT

T

t

T

T

TT

T

t

T

T

,0for

,0for

duration finite the: ,

2

1

22

1

2

1

12

1

2

1

21

Region of Convergence (ROC)Property 4 : If x(t) is right-sided (x(t)=0, t < T1),

and {s | Re[s] = σ0}ROC, then

{s | Re[s] > σ0}ROC, i.e., ROC

includes a right-half plane

dtetxedtetx

dtetx

t

T

T

T

t

t

T

0

1

101

1

1

0

1

for

01

Property 4

Region of Convergence (ROC)Property 5 : If x(t) is left-sided (x(t)=0, t > T2),

and {s | Re[s] = σ0}ROC, then

{s | Re[s] < σ0}ROC, i.e., ROC

includes a left-half plane

Property 5

Region of Convergence (ROC)Property 6 : If x(t) is two-sided, and {s | Re[s] =

σ0}ROC, then ROC consists of a strip

in s-plane including {s | Re[s] = σ0}

txtxtx

-tx-tx

txtxtx

LR

LR

LR

ROCROCROC

sidedleft : sided,right :

See Fig. 9.9, 9.10, p.667 of text

*note: ROC[x(t)] may not exist

Region of Convergence (ROC)A signal or an impulse response either doesn’t have a

Laplace Transform, or falls into the 4 categories of

Properties 3-6. Thus the ROC can be , s-plane, left-

half plane, right-half plane, or a signal strip

Region of Convergence (ROC)Property 7 : If X(s) is rational, then its ROC is

bounded by poles or extends to infinity– examples:

assas

tue

assas

tue

Lat

Lat

Re ROC ,1

Re ROC ,1

– partial-fraction expansion

See Fig. 9.1, p.658 of text

i i

i

assX

Region of Convergence (ROC)Property 8 : If x(s) is rational, then if x(t) is right-

sided, its ROC is the right-half plane to

the right of the rightmost pole. If x(t) is

left-sided, its ROC is the left-half plane

to the left of the leftmost pole.

Property 8

ROC

Region of Convergence (ROC)An expression of X(s) may corresponds to different

signals with different ROC’s.– an example:

21

1

sssX

– ROC is a part of the specification of X(s)


The ROC of X(s) can be constructed using these properties

Inverse Laplace Transform

jddsdsesXj

tx

dejXtx

dejXjXFetx

stj

j

tj

tjt

21

21

21

1

1

1

1

1

111

– integration along a line {s | Re[s]=σ1}ROC for a fixed σ1

Laplace Transform

( 合成 ?)

(basis?)X

X

( 分析 ?)X

[𝑒(𝜎1+ 𝑗 𝜔 ) 𝑡 ]∗=𝑒𝜎1 𝑡 ⋅𝑒− 𝑗 𝜔𝑡( 分析 )≠ �⃗�⋅𝑣

basis

( 合成 )

𝑥 (𝑡 )𝑒−𝜎 1𝑡= 12𝜋

− ∞

∞

𝑋 (𝜎 1+ 𝑗 𝜔)𝑒 𝑗𝜔𝑡 𝑑𝑡

( 分析 )

𝑋 (𝜎 1+ 𝑗 𝜔)=−∞

∞

[𝑥 (𝑡 )𝑒−𝜎1𝑡]⋅𝑒− 𝑗𝜔𝑡 𝑑𝑡

Practically in many cases : partial-fraction expansion works

Inverse Laplace Transform

1

m

i i

i

as

AsX

– ROC to the right of the pole at s = -ai

tueA tai

i

– ROC to the left of the pole at s = -ai

tueA tai

i

Known pairs/properties practically helpful

each termfor i

i

as

A

9.2 Properties of Laplace Transform

Linearity

222

111

ROC ,

ROC ,

ROC ,

RsXtx

RsXtx

RsXtx

L

L

L

212121 ROC , RRsbXsaXtbxtax L

Time Shift

RsXettx stL ROC ,00

Time Shift

Shift in s-plane

0

0

00

Reby shifted ROC

Re

ReROC ,0

s

Rsss

sRssXtxe Lts


– for s0 = jω0

RjsXtxe Ltj ROC ,00

shift along the jω axis

Shift in s-plane

𝑅𝑒 [𝑠0]

𝑠0= 𝑗𝜔0

Time Scaling (error on text corrected in class)

RsasaRa

sX

aatx L

ROC ,

1

– Expansion (?) of ROC if a > 1

Compression (?) of ROC if 1 > a > 0

reversal of ROC about jw-axis if a < 0

(right-sided left-sided, etc.)


RssRsXtx L ROC ,

Conjugation

real if

ROC ,

txsXsX

RsXtx L

– if x(t) is real, and X(s) has a pole/zero at

then X(s) has a pole/zero at

0

0

ss

ss

Convolution

212121 ROC , RRsXsXtxtx L

– ROC may become larger if pole-zero cancellation occurs

Differentiation

RssXdt

tdx L ROC ,

ROC may become larger if a pole at s = 0 cancelled

R

ds

sdXttx L ROC ,

(− 𝑗𝑡𝑥 (𝑡)𝐹↔

𝑑𝑑𝜔

𝑋 ( 𝑗𝜔 ))

Integration in time Domain

0Re ROC ,1

,

0Re ROC ,1

sss

tu

tutxdx

ssRsXs

dx

L

t

tL

(𝑢 (𝑡 )𝐹↔

1𝑗𝜔

+𝜋𝛿(𝜔))(

− ∞

𝑡

𝑥 (𝜏 ) 𝑑𝜏=𝑥 (𝑡 )∗𝑢(𝑡)𝐹↔

𝑋 ( 𝑗𝜔 )𝑑𝜔

+𝜋 𝑋 ( 𝑗0)𝛿(𝜔))

Initial/Final – Value Theorems

Theorem value-Final limlim

Theorem value-Initial lim

0ssXtx

ssXox

st

s

x(t) = 0, t < 0

x(t) has no impulses or higher order singularities at t = 0

Tables of Properties/Pairs

See Tables 9.1, 9.2, p.691, 692 of text

9.3 System Characterization with Laplace Transform

system function

transfer function

y(t)=x(t) ＊ h(t)h(t)

x(t)

X(s) Y(s)=X(s)H(s)H(s)

Causality

– A causal system has an H(s) whose ROC is a right-half plane

h(t) is right-sided

– For a system with a rational H(s), causality is equivalent to its ROC being the right-half plane to the right of the rightmost pole

– Anticausality

a system is anticausal if h(t) = 0, t > 0

an anticausal system has an H(s) whose ROC is a left-half plane, etc.

Causality

ROCis

right half Plane

right-sided CausalX

Causal

𝑋 (𝑠 )=∑𝑖

𝐴𝑖

𝑠+𝑎𝑖,

𝐴𝑖

𝑠+𝑎𝑖

→ 𝐴𝑖𝑒−𝑎𝑖 𝑡𝑢 (𝑡)

Stability

– A system is stable if and only if ROC of H(s) includes the jω -axis

h(t) absolutely integrable, or Fourier transform converges


– A causal system with a rational H(s) is stable if and only if all poles of H(s) lie in the left-half of s-plane

ROC is to the right of the rightmost pole

Stability

Systems Characterized by Differential Equations

N

kkk

M

k

kk

M

k

kk

N

k

kk

k

kM

kk

N

kk

k

k

sa

sb

sX

sYsH

sXsbsYsa

dt

txdb

dt

tyda

0

0

00

00

zeros

poles

System Function Algebra

– Parallel

h(t) = h1(t) + h2(t)

H(s) = H1(s) + H2(s)

– Cascade

h(t) = h1(t) ＊ h2(t)

H(s) = H1(s) ‧ H2(s)

System Function Algebra

– Feedback

x(t) +

－＋

h1(t)H1(s)

e(t)y(t)

h2(t)H2(s)

z(t)

sHsH

sH

sX

sYsH

sYsHsXsHsY

21

1

21

1

9.4 Unilateral Laplace Transform

Transform Laplace bilateral

Transform Laplace unilateral 0

dtetxsX

dtetxsX

st

stu

impulses or higher order singularities at t = 0 included in the integration

uL sXtx

– ROC for X(s)u is always a right-half plane

– a causal h(t) has H(s) = H(s)

– two signals differing for t < 0

but identical for t ≥ 0 have identical unilateral Laplace transforms

– similar properties and applications

9.4 Unilateral Laplace Transform

sHsH u

Examples• Example 9.7, p.668 of text

0b Transform, Laplace No

0b ,sReb ,211

sRe ,1

sRe ,1

)(

22

bbsb

bsbse

bbs

tue

bbs

tue

tuetueetx

Ltb

Lbt

Lbt

btbttb


Examples• Example 9.9/9.10/9.11, p.671-673 of text

)()()( ,1sRe2

)( )( ,2sRe

)( )( ,1sRe

21

11

211)(

2

2

2

tuetuetx

tueetx

tueetx

sssssX

tt

tt

tt

Examples• Example 9.9/9.10/9.11, p.671-673 of text


stable is )( plane half-left in the poles Allcausal is )(

polerightmost theofright the tois )( of 2 ,1at poles ,

1sRe ,21

3)()(

)(

1sRe ,21

1)(

3sRe ,3

1)(

)( )()()( 23

sH

sHsHROC

-s-sROCROCROCss

ssXsY

sH

sssY

ssX

tueetytuetx

HXY

ttt

plane half-left in the poles All

left-half plane

Problem 9.60, p.737 of text

poles no , all ,1

3

22

33

3

see

eesH

TtTtth

sTsT

sTsT

Echo in telephone communication


)22

(log1

,01

of zeros find To)2(22 2

Tm

Tj

Ts

ejee

sHmjsTsT

zeros

𝜋

2𝑇𝜋

2𝑇+

2𝜋𝑇


TtTt

sH

Ttee

jT

jT

s

tT

jTts

3by that cancels )(by generated Signal

eeigen valu ,0

2 when )1(1

)2

(log1for

3

22

log1

000

0


Tteee

jT

js

Tjmee

eee

nTte

TtT

jts

jmTs

Tsn

snTnT

n

nT

when

2-1for

2-1s ,1

poles find to

11sX

e tx

)21(

000

)2()1(

)1(0

T-

0

0

Documents

9.0 Laplace Transform 9.1 General Principles of Laplace Transform linear time-invariant Laplace Transform Eigenfunction Property y(t) = H(s)e st h(t)h(t)