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FFT(Fast Fourier Transform)

FFT(Fast Fourier Transform). FFT. Coefficient representation: . How to evaluate A(x 0 )?. Horner’s rule: Point-value representation:. Thm1. Pf:. n-point interpolation: Lagrange’s formula:. Cof. rep. Point-value rep. Thm2. Lemma 3 (Cancellation Lemma)

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FFT(Fast Fourier Transform)

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  1. FFT(Fast Fourier Transform)

  2. FFT • Coefficient representation: How to evaluate A(x0)?

  3. Horner’s rule: • Point-value representation:

  4. Thm1 Pf:

  5. n-point interpolation: Lagrange’s formula:

  6. Cof. rep. Point-value rep.

  7. Thm2

  8. Lemma 3 (Cancellation Lemma) n, k, d: non-negative integers, • Cor. 4 n: even positive integer Pf:

  9. Lemma 5 (Halving lemma) Pf: • Lemma 6 (Summation lemma) Pf:

  10. DFT

  11. Interpolation at the complex roots of unity:

  12. Thm 7 Pf:

  13. FFT

  14. Thm 8 (Convolution thm) Componentwise product

  15. Efficient FFT implement

  16. FFT circuit S=1 S=2 S=3

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