# lec10 - CS575 Parallel Processing Lecture ten Fast Fourier...

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Unformatted text preview: CS575 Parallel Processing Lecture ten: Fast Fourier Transform Wim Bohm, Colorado State University Except as otherwise noted, the content of this presentation is licensed under the Creative Commons Attribution 2.5 license. CS575dl lecture 10 2 Fourier Series Fourier proved the following All periodic functions (or a function that is non zero over a finite domain) can be approximated by linear combinations of sines and cosines Periodic function (period T): F(t+T) = F(t) 2200 t That approximation is the Fourier series . Compare this to Taylor approximation: A function can be approximated by a polynomial, which is a linear combination of basic polynomials x i In both cases there is a set of basis functions that allow us to approximate any function The more basis functions the closer the approximation Sinusoidal functions CS575dl lecture 10 3 π a a cos t a sin t 2 π 2a sin t a sin 1/2t 2 π 4 π 2 π a 2a a sin2t a π f(t) = m+acos(ωt+θ) m : mean value, the average height above abscissa a : amplitude, height of oscilation ω : angular frequency (radians/time), ω=2πf f : frequency (cycles/time), f = 1/T, T: period θ : phase angle or phase shift extent to which sinusoid is shifted horizontally Continuous Fourier Series For a function with period T, we can write where a k and b k describe the response of f to a sine or cosine with a certain frequency: CS575dl lecture 10 4 )) sin( ) cos( ( ) ( 1 t k t k t f b a a k k k ϖ ϖ + + = ∑ ∞ = ,... 2 , 1 ) sin( ) ( 2 ) cos( ) ( 2 = = = ∫ ∫ k dt t k t f T dt t k t f T T k T k b a ϖ ϖ Euler Euler relates sin and cos to the complex plane e iφ = cos φ + i sin φ Take φ = π cos π = -1, sin π = 0 which gets us Euler’s famous identity : e iπ = -1 CS575dl lecture 10 5 cos φ φ e iφ = cos φ + i sin φ 1 Re Im i sin φ r=1 CS575dl lecture 10 6 DFT and FFT DFT: discrete version of Fourier series in complex domain, written using Euler’s e iφ = cos φ + i sin φ notation FFT implements DFT in O(n.log(n)) time ) sin( ) cos( ) sin( ) cos( 1 : cos sin, 1 .. 1 .. 1 1 1 1 1 n k i n k n k i n k N notation in or N n for inverse the and N k for N F F f f f F e F f e f F k N n k n n N n n k n ik N n n n n ik N n n k ϖ ϖ ϖ ϖ ϖ ϖ + =- =- = =- = = ∑ ∑ ∑ ∑- =- =- =-- = Convolution: Polynomial multiplication A more intuitive approach to Fourier transforms, noticing that Fourier transform is a convolution of X and (co)sines Convolution: p(x) and q(x): polynomials of degree n, compute p(x)*q(x) Trivial solution O(n 2 ) Slightly better: divide and conquer p(x) = x n/2 *p r (x) + p l (x) q(x) = x n/2 *q r (x) + q l (x) we can do this with 3 multiplications of size n/ 2 and some additions, complexity CS575dl lecture 10 7 Transformation S 1 → S 1 S 1 is some representation of some domain e.g. polynomials in coefficient form, f is some function, e.g. convolution S 1 ↓ π π maps representation S 1 to representation S 2 S 2 , e.g. point-wise form (x,(p(x)) CS575dl lecture 10 8 f Transformation cont’...
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## This note was uploaded on 09/24/2009 for the course CS 525 taught by Professor Rjyosy during the Winter '09 term at Central Mich..

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lec10 - CS575 Parallel Processing Lecture ten Fast Fourier...

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