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| − | + | == Laplace Transform == | |
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| + | The laplace transform of a general signal <math> x(t) </math> is defined as | ||
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| + | <math> X(s) = \int_{-\infty}^{\infty}x(t)e^{-st}dt </math> | ||
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| + | The complex variable s can be written as <math> s = \sigma + jw</math> | ||
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| + | S means complex plane. So the laplace transform <math> X(s) </math> is on complex plane. | ||
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| + | However, the Fourier Transform <math> X(jw) </math> is on pure imaginary axis. | ||
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| + | Here is an example. | ||
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| + | <math> x(t) = -e^{-qt}u(-t) </math> | ||
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| + | <math> X(s) = -\int_{-\infty}^{\infty}e^{-qt}u(-t)e^{-st}dt</math> | ||
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| + | <math> = -\int_{-\infty}^{0}e^{-qt}e^{-st}dt</math> | ||
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| + | <math> = -\int_{-\infty}^{0}e^{-(q+s)t}dt</math> | ||
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| + | <math> = -\int_{-\infty}^{0}e^{-(q+a+jw)t}dt</math> | ||
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| + | <math> = -\int_{-\infty}^{0}e^{-(q+a)t}e^{-jwt}dt</math> | ||
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| + | if <math> q+a \geq 0, </math> integral diverges | ||
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| + | if <math> q+a < 0, X(s) = \frac{1}{q+a} </math> | ||
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| + | Then we can figure out where the laplace transform converges. | ||
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| + | In the example, the lapace transform only converges when <math> q+a < 0 </math> | ||
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| + | That region is called Region of converge, ROC. | ||
Latest revision as of 08:15, 23 November 2008
Laplace Transform
The laplace transform of a general signal $ x(t) $ is defined as
$ X(s) = \int_{-\infty}^{\infty}x(t)e^{-st}dt $
The complex variable s can be written as $ s = \sigma + jw $
S means complex plane. So the laplace transform $ X(s) $ is on complex plane.
However, the Fourier Transform $ X(jw) $ is on pure imaginary axis.
Here is an example.
$ x(t) = -e^{-qt}u(-t) $
$ X(s) = -\int_{-\infty}^{\infty}e^{-qt}u(-t)e^{-st}dt $
$ = -\int_{-\infty}^{0}e^{-qt}e^{-st}dt $
$ = -\int_{-\infty}^{0}e^{-(q+s)t}dt $
$ = -\int_{-\infty}^{0}e^{-(q+a+jw)t}dt $
$ = -\int_{-\infty}^{0}e^{-(q+a)t}e^{-jwt}dt $
if $ q+a \geq 0, $ integral diverges
if $ q+a < 0, X(s) = \frac{1}{q+a} $
Then we can figure out where the laplace transform converges.
In the example, the lapace transform only converges when $ q+a < 0 $
That region is called Region of converge, ROC.
