EE3054

Signals and Systems

Laplace Transform: Properties and Transfer

Functions

Yao Wang

Polytechnic University

Some slides included are extracted from lecture notes from MIT open coursewarehttp://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-003Fall-

2003/CourseHome/

EE3054, S08 Yao Wang, Polytechnic University 2

Review: The (Bilateral)

Laplace Transform

EE3054, S08 Yao Wang, Polytechnic University 3

Relation with Fourier

Transform

EE3054, S08 Yao Wang, Polytechnic University 4

Note: same X(s) may correspond to different x(t) depending on ROC!

EE3054, S08 Yao Wang, Polytechnic University 5

Example 3

EE3054, S08 Yao Wang, Polytechnic University 6

EE3054, S08 Yao Wang, Polytechnic University 7

General trend of ROC

� ROCs are always vertical half planes or stripes, bounded by poles

� Right side signals -> ROC in right half plane

� Left side signals -> ROC in left half plane

� Double sided signals -> ROC in a central stripe, or does not exist

EE3054, S08 Yao Wang, Polytechnic University 8

Laplace transform of common

signals

EE3054, S08 Yao Wang, Polytechnic University 9

Go through on board

EE3054, S08 Yao Wang, Polytechnic University 10

EE3054, S08 Yao Wang, Polytechnic University 11

EE3054, S08 Yao Wang, Polytechnic University 12

EE3054, S08 Yao Wang, Polytechnic University 13

� proof

EE3054, S08 Yao Wang, Polytechnic University 14

Using Laplace transform to

evaluate convolution

� Example: ?)(*)(),()(),()( 210 === −−thtxtuethtuetx

tt

EE3054, S08 Yao Wang, Polytechnic University 15

Using Laplace transform to

evaluate convolution

� Example: ?)(*)(),()(,)( 210=== −−

thtxtuethetxtt

EE3054, S08 Yao Wang, Polytechnic University 16

Transfer Function of LTI

Systems

y(t)=x(t)*h(t) <-> Y(s)=X(s) H(s)

H(s)=Y(s)/X(s) is called the “transfer function” or “system function”

Properties of the system depend on H(s) and its ROC!

Frequency response ωω

jssHjH

== )()(

EE3054, S08 Yao Wang, Polytechnic University 17

Deducing system property

from H(s) and its ROC

� BIBIO Stability

� Causality

� h(t) right sided and starts at 0 or after

� -> ROC is right half plane

� Poles are on the LHP only

EE3054, S08 Yao Wang, Polytechnic University 18

Examples

� Pole locations and stability and causality

EE3054, S08 Yao Wang, Polytechnic University 19

Sketching Frequency Response

from Pole-Zero Locations

ωω

jssHjH

== )()(

EE3054, S08 Yao Wang, Polytechnic University 20

EE3054, S08 Yao Wang, Polytechnic University 21

Other properties: linearity

EE3054, S08 Yao Wang, Polytechnic University 22

Other properties: Time Shift

EE3054, S08 Yao Wang, Polytechnic University 23

Time domain differentiation

EE3054, S08 Yao Wang, Polytechnic University 24

Time domain differentiation:

higher order

)()(

sXsdt

txd k

k

k

↔

EE3054, S08 Yao Wang, Polytechnic University 25

S-domain differentiation

( )k

kk

ds

sXdtxt

)()( ↔−

( )3

21

)(2

.as

tuet

Exat

+↔−

EE3054, S08 Yao Wang, Polytechnic University 26

Time domain integration

{ }0 ROC),(1

)( >ℜ=↔∫∞−

(s) R sXs

dx

t

Iττ

EE3054, S08 Yao Wang, Polytechnic University 27

Differential equation

representation of LTI system

EE3054, S08 Yao Wang, Polytechnic University 28

Example 1

EE3054, S08 Yao Wang, Polytechnic University 29

Example 2

EE3054, S08 Yao Wang, Polytechnic University 30

Three representations of LTI

system

� DE (for implementation)

� Transfer function or frequency response

� Impulse response

� Should be able to go from one to another!

EE3054, S08 Yao Wang, Polytechnic University 31

Interconnection of LTI

systems

EE3054, S08 Yao Wang, Polytechnic University 32

Example

� Given H1(s), H2(s), and their ROCs, what is the H(s) and its ROC? What is the corresponding h(t)?

EE3054, S08 Yao Wang, Polytechnic University 33

Summary

� LT is an important tool for analyzing CT-LTI systems and CT-signals

� Analogies with ZT for DT signals and systems

EE3054, S08 Yao Wang, Polytechnic University 34

Readings

� Oppenheim and Willsky, Signals and Systems, Sec. 9.0—9.7 (Handout)