Fast Fourier Transform Lecture 6 Spoken Language Processing
Prof. Andrew Rosenberg
Slide 3
Overview Fast Fourier Transform Multiplying Polynomials
Relationship between polynomial multiplication and Fourier Analysis
1
Slide 4
Multiplying polynomials Assume two polynomials of degree d. The
Fast Fourier Transform is an efficient (divide-and-conquer) way to
do this. 2
Slide 5
Multiplying polynomials Assume two polynomials of degree d. 3
Calculating c k takes (d) steps. To calculate C(x) requires (d 2
)
Slide 6
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
Slide 8
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
Slide 9
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
Slide 10
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
