ee3054 signals and systems laplacetransform: properties and …yao/ee3054/laplace_ii.pdf · 2008....
TRANSCRIPT
![Page 1: EE3054 Signals and Systems LaplaceTransform: Properties and …yao/EE3054/Laplace_II.pdf · 2008. 4. 29. · Oppenheim and Willsky, Signals and Systems, Sec. 9.0—9.7 (Handout) Title:](https://reader035.vdocument.in/reader035/viewer/2022081621/6128d06684afbe35103ba33e/html5/thumbnails/1.jpg)
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/
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EE3054, S08 Yao Wang, Polytechnic University 2
Review: The (Bilateral)
Laplace Transform
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EE3054, S08 Yao Wang, Polytechnic University 3
Relation with Fourier
Transform
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EE3054, S08 Yao Wang, Polytechnic University 4
Note: same X(s) may correspond to different x(t) depending on ROC!
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EE3054, S08 Yao Wang, Polytechnic University 5
Example 3
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EE3054, S08 Yao Wang, Polytechnic University 6
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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
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EE3054, S08 Yao Wang, Polytechnic University 8
Laplace transform of common
signals
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EE3054, S08 Yao Wang, Polytechnic University 9
Go through on board
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EE3054, S08 Yao Wang, Polytechnic University 13
� proof
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EE3054, S08 Yao Wang, Polytechnic University 14
Using Laplace transform to
evaluate convolution
� Example: ?)(*)(),()(),()( 210 === −−thtxtuethtuetx
tt
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EE3054, S08 Yao Wang, Polytechnic University 15
Using Laplace transform to
evaluate convolution
� Example: ?)(*)(),()(,)( 210=== −−
thtxtuethetxtt
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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
== )()(
![Page 17: EE3054 Signals and Systems LaplaceTransform: Properties and …yao/EE3054/Laplace_II.pdf · 2008. 4. 29. · Oppenheim and Willsky, Signals and Systems, Sec. 9.0—9.7 (Handout) Title:](https://reader035.vdocument.in/reader035/viewer/2022081621/6128d06684afbe35103ba33e/html5/thumbnails/17.jpg)
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
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EE3054, S08 Yao Wang, Polytechnic University 18
Examples
� Pole locations and stability and causality
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EE3054, S08 Yao Wang, Polytechnic University 19
Sketching Frequency Response
from Pole-Zero Locations
ωω
jssHjH
== )()(
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EE3054, S08 Yao Wang, Polytechnic University 21
Other properties: linearity
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EE3054, S08 Yao Wang, Polytechnic University 22
Other properties: Time Shift
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EE3054, S08 Yao Wang, Polytechnic University 23
Time domain differentiation
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EE3054, S08 Yao Wang, Polytechnic University 24
Time domain differentiation:
higher order
)()(
sXsdt
txd k
k
k
↔
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EE3054, S08 Yao Wang, Polytechnic University 25
S-domain differentiation
( )k
kk
ds
sXdtxt
)()( ↔−
( )3
21
)(2
.as
tuet
Exat
+↔−
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EE3054, S08 Yao Wang, Polytechnic University 26
Time domain integration
{ }0 ROC),(1
)( >ℜ=↔∫∞−
(s) R sXs
dx
t
Iττ
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EE3054, S08 Yao Wang, Polytechnic University 27
Differential equation
representation of LTI system
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EE3054, S08 Yao Wang, Polytechnic University 28
Example 1
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EE3054, S08 Yao Wang, Polytechnic University 29
Example 2
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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!
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EE3054, S08 Yao Wang, Polytechnic University 31
Interconnection of LTI
systems
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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)?
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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
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EE3054, S08 Yao Wang, Polytechnic University 34
Readings
� Oppenheim and Willsky, Signals and Systems, Sec. 9.0—9.7 (Handout)