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Polynomial and FFT

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Topics 1. Problem 2. Representation of polynomials 3. The DFT and FFT 4. Efficient FFT implementations 5. Conclusion

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Problem

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Representation of Polynomials Definition 2 For the polynomial (1), we have two ways of representing it:

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Coefficient Representation —— ( 秦九韶算法） Horner ’ s rule The coefficient representation is convenient for certain operations on polynomials. For example, the operation of evaluating the polynomial A(x) at a given point x 0

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Coefficient Representation —— adding and multiplication

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Point-value Representation By Horner ’ s rule, it takes Θ(n 2 ) time to get a point-value representation of polynomial (1). If we choose x k cleverly, the complexity reduces to n log n. Definition 3 The inverse of evaluation. The process of determining the coefficient form of a polynomial from a point value representation is called interpolation. Does the interpolation uniquely determine a polynomial? If not, the concept of interpolation is meaningless.

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Uniqueness of Interpolation

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Lagrange Formula We can compute the coefficients of A(x) by (4) in time Θ(n 2 ).

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拉格朗日［ Lagrange, Joseph Louis ， 1736-1813 ●法国数学家。 ● 涉猎力学，著有分析力学。 ● 百年以来数学界仍受其理论影响。

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Virtues of point value representation

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Fast multiplication of polynomials in coefficient form Can we use the linear-time multiplication method for polynomials in point-value form to expedite polynomial multiplication in coefficient form?

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Basic idea of multiplication

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If we choose “ complex roots of unity ” as the evaluation points carefully, we can produce a point-value representation by taking the Discrete Fourier Transform of a coefficient vector. The inverse operation interpolation, can be performed by taking the inverse DFT of point value pairs.

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Complex Roots of Unity

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Additive Group

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Properties of Complex Roots

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Fourier Transform Now consider generalization to the case of a discrete function :

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Discrete Fourier Transform

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Idea of Fast Fourier Transform

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Recursive FFT

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Complexity of FFT Property 4 By divide-and-conquer method, the time cost of FFT is T(n) = 2T(n/2)+Θ(n) =Θ(n log n).

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Interpolation

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Proof

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DFT n vs DFT -1 n

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Efficient FFT Implementation

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Butterfly Operation

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Iterative-FFT ITERATIVE-FFT (a) 1 BIT-REVERSE-COPY (a, A) 2 n ← length[a] // n is a power of 2. 3 for s ← 1 to lg n 4 do m ← 2 s 5 ω m ← e 2πi/m 6 for k ← 0 to n - 1 by m 7 do ω ← 1 8 for j ← 0 to m/2 - 1 9 do t ← ωA[k + j + m/2] 10 u ← A[k + j] 11 A[k + j] ← u + t 12 A[k + j + m/2] ← u - t 13 ω ← ω ω m

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conclusion Fourier analysis is not limited to 1- dimensional data. It is widely used in image processing to analyze data in 2 or more dimensions. Cooley and Tukey are widely credited with devising the FFT in the 1960 ’ s.

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