MIT 18 310C - The Finite Fourier Transform (8 pages)

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The Finite Fourier Transform



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The Finite Fourier Transform

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Pages:
8
School:
Massachusetts Institute of Technology
Course:
18 310c - Principles of Applied Mathematics

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23 The Finite Fourier Transform and the Fast Fourier Transform Algorithm 1 Introduction Fourier Series Early in the Nineteenth Century Fourier in studying sound and oscillatory motion conceived of the idea of representing periodic functions by their coefficients in an expansion as a sum of sines and cosines rather than their values He noticed that if for example you represented the shape of a vibrating string of length L fixed at its ends as y x ak sin 2 kx L the coefficients ak contained important and useful information about the quality of the sound that the string produces that was not easily accessible from the ordinary y f x representation of the shape of the string This kind of representation of a function is called a Fourier Series and there is a tremendous amount of mathematical lore about properties of such series and for what classes of functions they can be shown to exist One particularly useful fact about them is how we can obtain the coefficients ak from the function This follows from the orthogonality property of sines sin 2 kx L sin 2 jx L dx if the integral has limits 0 and L is 0 if k is different from j and is L 2 when k is j To see this notice that the product of these sines can be written as a constant multiple of the difference between cosines of 2 k j x L and 2 k j x L and each of these cosines has 0 integral over this range By multiplying the expression for y x above by 2 jx L and integrating the result from 0 to L we get then the expression aj 1 f x sin 2 jx L dx Fourier series represent only one of many alternate ways we can represent a function Whenever we can by introducing an appropriate weight function in the integral obtain a similar orthogonality relation among functions we can derive similar formulae for coefficients in a series 2 The Fourier Transform Given a function f defined for all real arguments we can give an alternative representation to it as an integral rather than as an infinite series as follows f x exp ikx g k dk where



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