fast fourier transform lecture 6 spoken language processing prof. andrew rosenberg
TRANSCRIPT
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- Fast Fourier Transform Lecture 6 Spoken Language Processing Prof. Andrew Rosenberg
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- Overview Fast Fourier Transform Multiplying Polynomials Relationship between polynomial multiplication and Fourier Analysis 1
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- Multiplying polynomials Assume two polynomials of degree d. The Fast Fourier Transform is an efficient (divide-and-conquer) way to do this. 2
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- Multiplying polynomials Assume two polynomials of degree d. 3 Calculating c k takes (d) steps. To calculate C(x) requires (d 2 )
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- Fast Fourier Transform How to multiply polynomials efficiently. What this has to do with frequency decomposition. 4
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- Representing polynomials A d degree polynomial can be represented by d+1 points. any points determine a line 5 Coefficients Values Evaluation Interpolation
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- Multiplication algorithm Input: coefficients of two polynomials A(x) and B(x) and degree d Output: Their product C = AB Selection Pick points x 0, x 1, x n-1 where n >= 2d+1 Evaluation Compute A(x 0 ), A(x 1 ),..., A(x n-1 ) and B(x 0 ), B(x 1 ),..., B(x n-1 ) Multiplication Compute C(x k ) = A(x k )B(x k ) for all k = 0,..., n-1 Interpolation Recover C(x) = c 0 + c 1 x +... + c 2d x 2d 6
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- Divide and Conquer for multiplication How can we pick the n points used in evaluation? Can we pick them to make the algorithm more efficient? If we choose positive and negative pairs, the calculations overlap a lot. 7
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- Example representation as even powers generally 8 Where A e are the even coefficients and A o the odd. Degree of A e and A o are