Laplace Transform: Definition
and Region of Convergence
Yao Wang
Polytechnic University
Some slides included are extracted from lecture notes from MIT open courseware
http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-003Fall2003/CourseHome/
Why do we need another
transform?
Fourier transform cannot handle large (and important)
classes of signals and unstable systems, i.e. when
Laplace Transform can be viewed as an extension of the
Fourier transform to allow analysis of broader class of
signals and systems (including unstable systems!)
EE3054, S08
Yao Wang, Polytechnic University
2
Eigen Function of LTI System
est is an eigenfunction of any LTI system
s= σ+ jω can be complex in general
Show on the board
H(s) is the Laplace transform of h(t)!
EE3054, S08
Yao Wang, Polytechnic University
3
The (Bilateral) Laplace
Transform
EE3054, S08
Yao Wang, Polytechnic University
4
Relation with Fourier
Transform
EE3054, S08
Yao Wang, Polytechnic University
5
Example 1
EE3054, S08
Yao Wang, Polytechnic University
6
Derive result on board, sketch ROC for
both a>0 and a<0
EE3054, S08
Yao Wang, Polytechnic University
7
Example 2
EE3054, S08
Yao Wang, Polytechnic University
8
Derive result on board
EE3054, S08
Yao Wang, Polytechnic University
9
Note: same X(s) may correspond to different x(t) depending on ROC!
EE3054, S08
Yao Wang, Polytechnic University
10
Example 3
EE3054, S08
Yao Wang, Polytechnic University
11
EE3054, S08
Yao Wang, Polytechnic University
12
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
13
EE3054, S08
Yao Wang, Polytechnic University
14
Finite duration signals that are absolutely integrable ->
ROC contains entire S-plane
EE3054, S08
Yao Wang, Polytechnic University
15
Importance of ROC
X(s) cannot uniquely define x(t)
Need ROC and X(s)!
EE3054, S08
Yao Wang, Polytechnic University
16
jω in the integral limit should be replaced by j∞
EE3054, S08
Yao Wang, Polytechnic University
17
EE3054, S08
Yao Wang, Polytechnic University
18
EE3054, S08
Yao Wang, Polytechnic University
19