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Fast Fourier Transform

Fast Fourier Transform. Agenda. Historical Introduction CFT and DFT Derivation of FFT Implementation. Historical Introduction. Continuous Fourier Transform (CFT). Given : Complex Fourier Coefficient: Fourier Series: (change of basis). Discrete Fourier Transform. Given :.

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Fast Fourier Transform

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  1. Fast Fourier Transform

  2. Agenda • Historical Introduction • CFT and DFT • Derivation of FFT • Implementation

  3. Historical Introduction

  4. Continuous Fourier Transform (CFT) • Given: • Complex Fourier Coefficient: • Fourier Series: (change of basis)

  5. Discrete Fourier Transform • Given:

  6. Discrete Fourier Transform • Fourier-Matrix • DFT of x: • Matrix-Vector-Product: • N^2 Multiplications • N(N-1) Additions • ArithmeticComplexity: O(N^2)

  7. Trigonometric Interpolation • Given: equidistantsamplesofi.e. • Goal: find such thatwith

  8. Trigonometric Interpolation Theorem: => Inverse DFT!

  9. Derivation of FFT

  10. FFT – LowerBound: (S. Winograd – ArithmeticComplexityofComputations 1980CBMS-NSF, Regional Conference Series in Applied Mathematics)

  11. FFT if N is prime Twoapproaches:

  12. Implementation (N = power of 2)

  13. RecursiveImplementation

  14. RecursiveImplementation • Stack-Problem: • Space Complexity: O(NlogN) • Better Approach: • Iterative Implementation => in place! (O(N))

  15. Iterative Implementation (N=8)

  16. Iterative Implementation (N=8)

  17. Iterative Implementation (N=8) • Bit-Inversion:

  18. Iterative Implementation Theorem: The bitinversionyieldstheresultofthepermutation graph.

  19. Iterative Implementation

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