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x(t) ? e ?2t u(t) ? e ? t (cos 3t)u(t).
Using Euler's relation, we can write
x(t)
?
???e?2t
?
1 ? (s?? ) lim t
e
? t??
1
s??
0 s??
s??
For convergence, we require that Re{s + α} > 0, or Re{s} > –α ,
Thus,
X(s) ? 1 ,
s??
Re{s} ? ??
region of convergence (ROC ) (收敛域)
generally
? X(? ? j? ) ?
?? ??
??x(t)e?? t ??e? j? tdt
That is, the laplace transform of x(t) can be interpreted as the
Fourier transform of x(t) after multiplication by a real exponential
signal. The real exponential e? ? t may be decaying or growing in time, depending on whether ? is positive or negative.
?In specifying the Laplace transform of a signal, both the algebraic
?
1
?
1 ? ( s? ? ) lim t
e
t? ??
s??
?? s ? ? s ? ?
For convergence, we require that Re{s + α} < 0, or Re{s} < –α ,
Thus,
? e?? tu(? t) ?? L? 1 , Re{s} ? ?? s??
Ch6. The Laplace Transform
Example 6.1 Consider the signal
x(t) ? e ?? t u(t).
? ? X (s) ? ? ? e?? tu(t)e ? st dt ? ? ? e? (s?? )t dt
??
0
??
1
??
e?(s?? )t ? ?
4
Ch6. The Laplace Transform
Example 6.2 Consider the signal
x(t) ? ? e?? tu(? t).
? ? X(s) ? ? ? ? e?? te? stu(? t)dt ? ? 0 e? (s?? )t dt
??
??
?
1
0
e? (s?? )t
in which Fourier analysis can be used.
1
Ch6. The Laplace Transform 1. The Laplace Transform
1) Development of The Laplace Transform
For some signals which have not Fourier transforms, if we preprocess
Ch6. The Laplace Transform
INTRODUCTION
?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 est with s=σ + jω ?With Laplace transform, we expand the application
5
Ch6. The Laplace Transform
Im s-plane
Im s-plane
–α
Re
Re –α
ROC for Example 6.1
ROC for Example 6.2
6
Ch6. The Laplace Transform
Example 6.3 Consider the signal
?The Laplace transform is an extension of the Fourier transform;
the Fourier transform is a special case of the Laplace transform
when σ= 0.
X(s) ? F{x(t)} ? X( j? ) s? j?
? X(s) ? ? ? x(t)e? st dt ??
TheFra Baidu bibliotekLaplace transform of x(t)
We will denote the transform relationship between x(t) and X(s) as
L
x(t) ? X(s) 2
Ch6. The Laplace Transform
them by multiplying with a real exponential signal e?? t , then they may
have Fourier transforms.
? ? ? F
x(t)e?? t
?
?? ??
??x(t)e?? t ??e? j? tdt
Let s = σ+ jω, and using X(s) to denote this integral, we obtain
expression and the range of values of s for which this expression is
valid are required.
?The range of values of s for which the integral in X(s) converges is referred to as the region of convergence (ROC). 3
Using Euler's relation, we can write
x(t)
?
???e?2t
?
1 ? (s?? ) lim t
e
? t??
1
s??
0 s??
s??
For convergence, we require that Re{s + α} > 0, or Re{s} > –α ,
Thus,
X(s) ? 1 ,
s??
Re{s} ? ??
region of convergence (ROC ) (收敛域)
generally
? X(? ? j? ) ?
?? ??
??x(t)e?? t ??e? j? tdt
That is, the laplace transform of x(t) can be interpreted as the
Fourier transform of x(t) after multiplication by a real exponential
signal. The real exponential e? ? t may be decaying or growing in time, depending on whether ? is positive or negative.
?In specifying the Laplace transform of a signal, both the algebraic
?
1
?
1 ? ( s? ? ) lim t
e
t? ??
s??
?? s ? ? s ? ?
For convergence, we require that Re{s + α} < 0, or Re{s} < –α ,
Thus,
? e?? tu(? t) ?? L? 1 , Re{s} ? ?? s??
Ch6. The Laplace Transform
Example 6.1 Consider the signal
x(t) ? e ?? t u(t).
? ? X (s) ? ? ? e?? tu(t)e ? st dt ? ? ? e? (s?? )t dt
??
0
??
1
??
e?(s?? )t ? ?
4
Ch6. The Laplace Transform
Example 6.2 Consider the signal
x(t) ? ? e?? tu(? t).
? ? X(s) ? ? ? ? e?? te? stu(? t)dt ? ? 0 e? (s?? )t dt
??
??
?
1
0
e? (s?? )t
in which Fourier analysis can be used.
1
Ch6. The Laplace Transform 1. The Laplace Transform
1) Development of The Laplace Transform
For some signals which have not Fourier transforms, if we preprocess
Ch6. The Laplace Transform
INTRODUCTION
?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 est with s=σ + jω ?With Laplace transform, we expand the application
5
Ch6. The Laplace Transform
Im s-plane
Im s-plane
–α
Re
Re –α
ROC for Example 6.1
ROC for Example 6.2
6
Ch6. The Laplace Transform
Example 6.3 Consider the signal
?The Laplace transform is an extension of the Fourier transform;
the Fourier transform is a special case of the Laplace transform
when σ= 0.
X(s) ? F{x(t)} ? X( j? ) s? j?
? X(s) ? ? ? x(t)e? st dt ??
TheFra Baidu bibliotekLaplace transform of x(t)
We will denote the transform relationship between x(t) and X(s) as
L
x(t) ? X(s) 2
Ch6. The Laplace Transform
them by multiplying with a real exponential signal e?? t , then they may
have Fourier transforms.
? ? ? F
x(t)e?? t
?
?? ??
??x(t)e?? t ??e? j? tdt
Let s = σ+ jω, and using X(s) to denote this integral, we obtain
expression and the range of values of s for which this expression is
valid are required.
?The range of values of s for which the integral in X(s) converges is referred to as the region of convergence (ROC). 3