6 The Laplace Transform

0
( s 13 j ) lim t 1 1 t e s 1 3 j s 1 3 j
Thus,
e
(1 3 j ) t
e
(13 j ) t
1 u (t ) , Re{s} 1, s (1 3 j )
LT
1 u (t ) , Re{s} 1, s (1 3 j )
Example 6.2
X (s)
Consider the signal x(t ) e t u(t ).



e
t st
e u ( t )dt
0

0

e ( s )t dt
1 ( s )t e s
Thus,
lim t 1 1 ( s ) t e s s
Re{s} 2.
Pole-zero plot and ROC
6.2 The Region of Convergence for Laplace Transform
Property 1: The ROC of X(s) consists of stripes parallel to the jωaxis in the s-plane. Property 2: For rational Laplace transforms, the ROC does not contain any poles.
t LT
2s 2 5s 12 2 , Re{s} 1. ( s 2s 10)( s 2)
Some useful LT pairs:
cost u (t )
LT LT
s s
2 2
, Re {s} 0,
sint u (t ) , Re {s} 0 2 2 s
LT
LT e 2t u(t )
1 , Re{s} 2. s2
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Consequently,
e
2t
1 1 1 1 1 u (t ) e (cos 3t )u (t ) s 2 2 s (1 3 j ) 2 s (1 3 j )
–α
Re
–α
Re
ROC for Example 5.1
ROC for Example 6.2
j s-plane
0

Example 6.3
Consider the signal
x(t ) e 2t u(t ) e t (cos 3t )u(t ).
Using Euler’s relation, we can write
Consider the signal
x(t ) e t u(t ).
X (s)



e
t
u (t )e dt
0
st


0
e ( s )t dt
1 ( s )t e s
Thus,
t 1 ( s ) tlim 1 e s s
X (s) x(t )est dt
0

In practice, the term Laplace Transform means the unilateral Laplace Transform
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Example 6.1
The Laplace transform of x(t)
bilateral (or two-sided )
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unilateral (or one-sided ) Laplace Transform
Handle only causal signal
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Contents
6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Introduction The Laplace Transform The Region of Convergence for Laplace Transform The Inverse Laplace Transform Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot Properties of the Laplace Transform The Unilateral Laplace Transform System Function Analysis and characterization of LTI system using the Laplace transform Frequency Response of An LTIC System
For convergence, we require that Re{s + α} > 0, or Re{s} > –α ,
1 X (s) , Re {s} s
region of convergence (ROC ) (收敛域)
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Beijing Jiaot the infinity ( 无穷远点 ): in general, if the order of the denominator exceeds the order of the numerator by k, X(s) will have k zeros at infinity. Similarly, if the order of the numerator exceeds the order of the denominator by k, X(s) will have k poles at infinity.
Property 3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane.
Example 6.5
Let
e
x(t ) et u(t ) u(t T ) T 1 t st X ( s) e e dt 1 e ( s )T . 0 s n n ( 1 ) ( s ) T ( s )T
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Chapter 6 The Laplace Transform
The Laplace transform ( 拉 普 拉 斯 变 换 ) is a generalization of the continuous-time Fourier transform. The Laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form e st with s=σ + jω With Laplace transform, we expand the application in which Fourier analysis can be used.
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6.0 Introduction
拉普拉斯(Pierre-Simon Laplace,1749－1827)
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6.1 The Laplace Transform
For some signals which have not Fourier transforms, if we preprocess them by multiplying with a real exponential signal e t , then they may have Fourier transforms.
2t 1 (13 j )t 1 (13 j )t x(t ) e e e u(t ) 2 2
X (s)



e
(13 j ) t
u (t )e dt
st


0
e ( s 13 j )t dt
1 e ( s 13 j )t s 1 3 j
For convergence, we require that Re{s + α} < 0, or Re{s} < –α ,
1 e u(t ) , Re{s} s
t LT
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Im s-plane Im s-plane
X ( j ) x(t )e e
t
jt
dt
Let s = σ+ jω, and using X(s) to denote this integral, we obtain
X ( s)



x(t )e st dt
The Laplace transform is an extension of the Fourier transform; the Fourier transform is a special case of the Laplace transform when σ= 0.