chap12_8up - Discrete Fourier Transform Fast Fourier...

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Unformatted text preview: Discrete Fourier Transform Fast Fourier Transform Applications Scientific Computing: An Introductory Survey Chapter 12 – Fast Fourier Transform Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Outline 1 Discrete Fourier Transform 2 Fast Fourier Transform 3 Applications Michael T. Heath Scientific Computing 2 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Trigonometric Interpolation In modeling periodic or cyclic phenomena, sines and cosines are more appropriate functions than polynomials or piecewise polynomials Representation as linear combination of sines and cosines decomposes continuous function or discrete data into components of various frequencies Representation in frequency space may enable more efficient manipulations than in original time or space domain Michael T. Heath Scientific Computing 3 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Complex Exponential Notation We will use complex exponential notation based on Euler’s identity e iθ = cos θ + i sin θ where i = √- 1 Since e- iθ = cos(- θ ) + i sin(- θ ) = cos θ- i sin θ, we have cos(2 πkt ) = e 2 πikt + e- 2 πikt 2 and sin(2 πkt ) = i e- 2 πikt- e 2 πikt 2 Pure cosine or sine wave of frequency k is equivalent to sum or difference of complex exponentials of half amplitude and frequencies k and- k Michael T. Heath Scientific Computing 4 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Roots of Unity For given integer n , primitive n th root of unity is given by ω n = cos(2 π/n )- i sin(2 π/n ) = e- 2 πi/n n th roots of unity, called twiddle factors in this context, are given by ω k n or by ω- k n , k = 0 , . . . , n- 1 < interactive example > Michael T. Heath Scientific Computing 5 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Discrete Fourier Transform Discrete Fourier transform , or DFT , of sequence x = [ x , . . . , x n- 1 ] T is sequence y = [ y , . . . , y n- 1 ] T given by y m = n- 1 X k =0 x k ω mk n , m = 0 , 1 , . . . , n- 1 In matrix notation, y = F n x , where entries of Fourier matrix F n are given by { F n } mk = ω mk n For example, F 4 = 1 1 1 1 1 ω 1 ω 2 ω 3 1 ω 2 ω 4 ω 6 1 ω 3 ω 6 ω 9 = 1 1 1 1 1- i- 1 i 1- 1 1- 1 1 i- 1- i Michael T. Heath Scientific Computing 6 / 32 Discrete Fourier Transform Fast Fourier Transform Applications Inverse DFT Note that 1 n 1 1 1 1 1 ω- 1 ω- 2 ω- 3 1 ω- 2 ω- 4 ω- 6 1 ω- 3 ω- 6 ω- 9 1 1 1 1 1 ω 1 ω 2 ω 3 1 ω 2 ω 4 ω 6 1 ω 3 ω 6 ω 9 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 In general, F- 1 n = (1 /n ) F H n Inverse DFT...
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This note was uploaded on 10/16/2011 for the course MECHANICAL 581 taught by Professor Wasfy during the Fall '11 term at IUPUI.

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chap12_8up - Discrete Fourier Transform Fast Fourier...

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