The Fourier Transforms: Difference between revisions

Revision as of 20:07, 11 October 2007

The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.

Properties of a Fourier Transform:

$f (t) * g (t)$	$\overset{ℱ}{⟺} \sqrt{2 π} \cdot F (ω) \cdot G (ω)$	(unitary convention)
	$\overset{ℱ}{⟺} F (ω) \cdot G (ω)$	(non-unitary convention)
	$\overset{ℱ}{⟺} F (f) \cdot G (f)$	(ordinary frequency)

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-The Fourier transform was named after Joseph Fourier, a French mathematician.
+The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.
+== Headline text ==
+Properties of a Fourier Transform:
+;Convolution
+::::{|
+|<math>f(t)* g(t) \,</math>
+|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
+\sqrt{2\pi}\cdot F(\omega)\cdot G(\omega) \,</math>
+| &nbsp; &nbsp; (unitary convention)
+|-
+|
+|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
+F(\omega)\cdot G(\omega) \,</math>
+| &nbsp; &nbsp; (non-unitary convention)
+|-
+|
+|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
+F(f)\cdot G(f) \,</math>
+| &nbsp; &nbsp; (ordinary frequency)
+|}