Fast Fourier Transform (FFT) (Section 4.11)

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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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## Fast Fourier Transform (FFT) (Section 4.11)

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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(N2) time

Fast Fourier Transform (FFT)
• FFT takes O(NlogN) time (assumes N=2n)
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)
• Note that:
• Therefore:

or

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

x

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
• 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)
• Bit-wise reversal rule:
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