Fourier Transform -- from Wolfram MathWorld

Fourier Transform -- from Wolfram MathWorld - Fourier...

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Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld Classroom About Contribute to Send a Message to the Team Book 13,082 entries Last updated: Thu Sep 29 2011 Calculus and Analysis > Integral Transforms > Fourier Transforms > Interactive Entries > Interactive Demonstrations > Fourier Transform The Fourier transform is a generalization of the complex Fourier series in the limit as . Replace the discrete with the continuous while letting . Then change the sum to an integral , and the equations become (1) (2) Here, (3) (4) is called the forward ( ) Fourier transform, and (5) (6) is called the inverse ( ) Fourier transform. The notation is introduced in Trott (2004, p. xxxiv), and and are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202). Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency instead of the oscillation frequency . However, this destroys the symmetry, resulting in the transform pair (7) (8) (9) (10) To restore the symmetry of the transforms, the convention (11) (12) (13) (14) is sometimes used (Mathews and Walker 1970, p. 102). In general, the Fourier transform pair may be defined using two arbitrary constants and as (15) (16) The Fourier transform of a function is implemented as FourierTransform [ f , x , k ], and different choices of and can be used by passing the optional FourierParameters-> a , b option. By default, Mathematica takes FourierParameters as . Unfortunately, a number of other conventions are in widespread use. For example, is used in modern physics, is used in pure mathematics and systems engineering, is used in probability theory for the computation of the characteristic function , is used in classical physics, and is used in signal processing. In this work, following Bracewell (1999, pp. 6-7), it is always assumed that and unless otherwise stated. This choice often results in greatly simplified transforms of common functions such as 1, , etc. Since any function can be split up into
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Fourier Transform -- from Wolfram MathWorld - Fourier...

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