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Polynomial and FFT

Polynomial and FFT. Topics. 1. Problem 2. Representation of polynomials 3. The DFT and FFT 4. Efficient FFT implementations 5. Conclusion. Problem. Representation of Polynomials. Definition 2 For the polynomial (1), we have two ways of representing it:.

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Polynomial and FFT

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  1. Polynomial and FFT

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

  3. Problem

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

  5. 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 x0

  6. Coefficient Representation—— adding and multiplication

  7. Point-value Representation • By Horner’s rule, it takes Θ(n2) time to get a point-value representation of polynomial (1). If we choose xkcleverly, the complexity reduces to n log n. • Definition 3The inverse ofevaluation. 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.

  8. Uniqueness of Interpolation

  9. Lagrange Formula We can compute the coefficients of A(x) by (4) in time Θ(n2).

  10. 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。

  11. Virtues of point value representation

  12. 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?

  13. Basic idea of multiplication

  14. Basic idea of multiplication • 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.

  15. Complex Roots of Unity

  16. Additive Group

  17. Properties of Complex Roots

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

  19. Discrete Fourier Transform

  20. Idea of Fast Fourier Transform

  21. Recursive FFT

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

  23. Interpolation

  24. Proof

  25. DFTnvs DFT-1n

  26. Efficient FFT Implementation

  27. Butterfly Operation

  28. 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 ← 2s 5 ωm ← e2πi/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 ωm

  29. 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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