文档介绍：With Laplace transform, we expand the application
in which Fourier analysis can be used.
The Laplace transform provides us with a representation for signals as binations plex exponentials of the form with s=σ+ jω
The Laplace transform (拉普拉斯变换) is a generalization of the continuous-time Fourier transform.
INTRODUCTION
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1. The Laplace Transform
Let s = σ+ jω, and using X(s) to denote this integral, we obtain
For some signals which have not Fourier transforms, if we preprocess
them by multiplying with a real exponential signal , then they may have Fourier transforms.
The Laplace transform of x(t)
1) Development of The Laplace Transform
We will denote the transform relationship between x(t) and X(s) as
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The Laplace transform is an extension of the Fourier transform; the Fourier transform is a special case of the Laplace transform when σ= 0.
generally
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 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 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).
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Example Consider the signal
For convergence, we require that Re{s + α} > 0, or Re{s} > –α,
Thus,
region of convergence (ROC )
(收敛域)
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Example Consider the signal
For convergence, we require that Re{s + α} < 0, or Re{s} < –α,
Thus,
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–α
Re
Im
s-plane
–α
Re
Im
s-plane
ROC for Example
ROC for Example
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Example Consider the signal
Using Euler’s relation, we can write
Thus,
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Consequently,
Some useful LT pairs:
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2) the pole-zero plot (极零图)
Generally, the Laplace transform is rational, ., it is a ratio of polynomials in plex variable s:
The root