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Fast Fourier Transform (FFT) (Section 4.11)

Fast Fourier Transform (FFT) (Section 4.11). CS474/674 – Prof. Bebis. DFT – Time Complexity. How much time does DFT take? . u=0,1,2,...,N-1. O(N 2 ) time. Fast Fourier Transform (FFT). FFT takes O(NlogN) time (assumes N=2 n ). Deriving FFT. Assume that N=2 n and let

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Fast Fourier Transform (FFT) (Section 4.11)

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  1. Fast Fourier Transform (FFT) (Section 4.11) CS474/674 – Prof. Bebis

  2. DFT – Time Complexity • How much time does DFT take? u=0,1,2,...,N-1 O(N2) time

  3. Fast Fourier Transform (FFT) • FFT takes O(NlogN) time (assumes N=2n)

  4. Deriving FFT • Assume that N=2n and let • Since N=2n, there exist M such that N=2M u=0,1,2,...,N-1

  5. Deriving FFT (cont’d) • Note that: • Therefore: or

  6. Deriving FFT (cont’d) • How can we compute F(u) for u=M,M+1,…,2M-1? • Note that x

  7. Deriving FFT (cont’d) • Thus:

  8. Deriving FFT (cont’d) • Therefore, an N-point transform can be computed using two N/2-point transforms! • Similarly, each N/2-point transform can be computed • using two N/4-point transforms etc.

  9. Example

  10. Example (cont’d)

  11. Implementation Details • The input must be provided in the required order at each level original order f(0) f(1) f(2) f(3) f(4) f(5) f(6) f(7) required order

  12. Implementation Details (cont’d) • Bit-wise reversal rule:

  13. Inverse FFT Forward DFT Inverse DFT • The inverse FFT can be computed using the same implementation • Use a flag for the sign of the exponential • Use F(u) instead of f(x) • Multiply by N

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