Post on 27-Mar-2015
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