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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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dft time complexity
DFT – Time Complexity
  • How much time does DFT take?

u=0,1,2,...,N-1

O(N2) time

fast fourier transform fft
Fast Fourier Transform (FFT)
  • FFT takes O(NlogN) time (assumes N=2n)
deriving fft
Deriving FFT
  • Assume that N=2n and let
  • Since N=2n, there exist M such that N=2M

u=0,1,2,...,N-1

deriving fft cont d
Deriving FFT (cont’d)
  • Note that:
  • Therefore:

or

deriving fft cont d1
Deriving FFT (cont’d)
  • How can we compute F(u) for u=M,M+1,…,2M-1?
  • Note that

x

deriving fft cont d3
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.
implementation details
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

implementation details cont d
Implementation Details (cont’d)
  • Bit-wise reversal rule:
inverse fft
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