Fourier Transform and the Gibbs Phenomemon

A. Davey, H. Havener, L. Isherwood

CSUMS Fall 2011 UMass Dartmouth

December 1, 2011

A. Davey, H. Havener, L. Isherwood and the Gibbs Phenomemon Outline

I Introduction of Fourier

I Fourier Transform

I Discrete Fourier Transform

I Errors

I Goals

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Intoduction to

Series expansions are useful tools that allow complicated functions to be dealt with in terms we understand. One example of a series expansion is the Taylor series, which expresses functions in terms of polynomials.

I Example McLaurin expansion of the exponential function:

x x2 x3 x4 ex = 1 + + + + .... 1! 2! 3! 4! The Fourier series expresses a function in terms of

I General Fourier Series: ∞ ∞ 1 X X f (x) = a + a cos nx + b sin nx 2 0 n n n=1 n=1

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon The Fourier Transform

The Fourier transform is an Integral transform that takes a and moves it from the time domain to the frequency domain. The Fourier Transform is developed from the complex form of the Fourier Series, which is gotten from Euler’s Formula.

I Fourier Transform of f (x): Z ∞ F (s) = f (x)e−2πixs dx −∞ Where F (s) is the transform of f (x), and the frequency domain is given by s

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Fourier Transform on Discrete Sets

The Fourier Transform can be approximated for a discrete set of points by the Discrete Fourier Transform (DFT).

I DFT of f(x): N−1 X −2πi sn X¯s = f (x ¯n)e N n=0

The DFT requires the points vectorx ¯n to be equidistant.

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Matrix Formulation of the DFT

The DFT can be expressed as a matrix equation in the form X = W x¯ where W is an NxN matrix andx ¯ is a column vector of N points X is the DFT ofx ¯.

1 1 1 ... 1  1 ω ω2 . . . ω(N−1)  1   1 ω2 ω4 . . . ω2(N−1)  W = √   N . . . .  . . . .  1 ω(N−1) ω2(N−1) . . . ω(N−1)(N−1)

−2iπ Where ω = e N

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Accuracy in Calculating The Fourier Series

The DFT calculates the approximate value of Complex Fourier Coefficients at a point, so that when the Fourier series is reconstructed (by the Inverse Discrete Fourier Transform (IDFT)) using the calculated coefficients, it matches the function value at the original points. Since we cannot take an infinite sum, when we truncate the series for calculation there is some inherent error inherent error. This is shown by the truncated Fourier series expansion for a as it oscillates above and below the original function.

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Gibbs Phenomenon

The Gibbs phenomenon is a specific kind of error in numerical Fourier Series centered around discontinuities. Unlike most kinds of error, the maximum error due to the gibbs phenomenon does not decrease with the amount of points you

take. A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon Project goals

I Get working DFT Code in Matlab

I Learn about Post Processing techniques to minimize error from the gibbs phenomenon

I To see if post precessing effects aliasing error from

A. Davey, H. Havener, L. Isherwood Fourier Transform and the Gibbs Phenomemon