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-FFT Recap (or, what am I expected to know?) - Learning Finite State Environments 15-451 Avrim Blum 11/25/03

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FFT Recap The basic result: Given vectors –A = (a 0, a 1, a 2,..., a n-1 ), and –B = (b 0, b 1,..., b n-1 ), the FFT allows us in O(n log n) time to compute the convolution –C = (c 0, c 1,..., c 2n-2 ) where c j = a 0 b j + a 1 b j-1 +... + a j b 0. I.e., this is polynomial multiplication, where A,B,C are vectors of coefficients.

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How does it work? Compute F -1 (F(A) ¢ F(B)), where “F” is FFT. F(A) is evaluation of A at 1, m – is principal m th root of unity. m = 2n-1. E.g., = e /m. Or use modular arithmetic. –Able to do this quickly with divide-and-conquer. F(A) ¢ F(B) gives C(x) at these points. We then saw that F -1 = (1/m)F’, where F’ is same as F but using -1.

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Applications (not on test) signal analysis, lots moresignal analysislots more Pattern matching with don’t cares: –Given text string X = x 1,x 2,...,x n. x i 2 {0..25} –Given pattern Y = y 1,y 2,...,y k. y i 2 {0..25} [ {*}. –Want to find instances of Y inside X. Idea [Adam Kalai based on Karp-Rabin]: –Pick random R: r 1,r 2,...,r k, r i 2 1..N. E.g, N=n 2 –Set r i = 0 if y k-i+1 = *. –Let T = r 1 y k +... + r k y 1. (can do mod p > N) –Now do convolution of R and X. See if any entries match T. Each entry at most 1/N chance of false positive.

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OK, on to machine learning...

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FFT1 The Fast Fourier Transform. FFT2 Outline and Reading Polynomial Multiplication Problem Primitive Roots of Unity (§10.4.1) The Discrete Fourier Transform.

FFT1 The Fast Fourier Transform. FFT2 Outline and Reading Polynomial Multiplication Problem Primitive Roots of Unity (§10.4.1) The Discrete Fourier Transform.

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