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Fast Fourier Transform (FFT) (Section 4.11) CS474/674 – Prof. Bebis
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DFT – Time Complexity How much time does DFT take? u=0,1,2,...,N-1 O(N 2 ) time
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Fast Fourier Transform (FFT) FFT takes O(NlogN) time (assumes N=2 n )
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Deriving FFT Assume that N=2 n and let Since N=2 n, there exist M such that N=2M u=0,1,2,...,N-1
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Deriving FFT (cont’d) Note that: Therefore: or
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Deriving FFT (cont’d) How can we compute F(u) for u=M,M+1,…,2M-1? Note that x
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Deriving FFT (cont’d) Thus:
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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.
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Example
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Example (cont’d)
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Implementation Details The input must be provided in the required order at each level f(0) f(1) f(2) f(3) f(4) f(5) f(6) f(7) required order original order
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Implementation Details (cont’d) Bit-wise reversal rule:
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Inverse FFT 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 Forward DFTInverse DFT
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