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== The Laplace Transform == | == The Laplace Transform == | ||
| − | The Laplace Transform is a generalization of the Fourier Transform. Instead of considering only the imaginary axis, <math>j\omega\!</math>, (as the Fourier Transform does) the Laplace Transform considers all complex values represented by the general complex variable <math>s\!</math>. Take the following picture: | + | The Laplace Transform is a generalization of the Fourier Transform. Instead of considering only the imaginary axis, <math>j\omega\!</math>, (as the Fourier Transform does) the Laplace Transform considers all complex values represented by the general complex variable <math>s\!</math>. Take the following simple picture: |
| − | Fourier Transform: <math>x(t) --> X(\omega)\!</math> where <math>\omega\!</math> is a frequency. | + | Fourier Transform: <math>x(t) --> X(\omega)\!</math> where <math>\omega\!</math> is a frequency. |
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| + | Laplace Transform: <math>x(t) --> X(s)\!</math> where <math>s\!</math> is a complex variable. | ||
Revision as of 16:06, 24 November 2008
The Laplace Transform
The Laplace Transform is a generalization of the Fourier Transform. Instead of considering only the imaginary axis, $ j\omega\! $, (as the Fourier Transform does) the Laplace Transform considers all complex values represented by the general complex variable $ s\! $. Take the following simple picture:
Fourier Transform: $ x(t) --> X(\omega)\! $ where $ \omega\! $ is a frequency.
Laplace Transform: $ x(t) --> X(s)\! $ where $ s\! $ is a complex variable.
