Cooley–Tukey FFT vs Conventional FFT Algorithm
Understanding the difference between the Cooley–Tukey Fast Fourier Transform and the conventional FFT/DFT computation.
1. Conventional FFT Algorithm
The conventional FFT algorithm usually refers to the direct computation of the Discrete Fourier Transform (DFT). The DFT of a sequence of length \(N\) is:
\[
X[k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi nk / N}, \quad k = 0,1,\dots,N-1
\]
Direct DFT computation requires O(N²) operations (multiplications and additions) and can be slow for large sequences.
2. Cooley–Tukey FFT Algorithm
The Cooley–Tukey algorithm is a widely used FFT method based on divide-and-conquer. It splits a DFT of size \(N\) into smaller DFTs, which reduces computation to O(N log N).
For a sequence of length \(N\) (ideally a power of 2), the algorithm splits into even and odd indices:
\[
X[k] = \sum_{n=0}^{N/2-1} x[2n] e^{-j 2\pi k (2n)/N} + \sum_{n=0}^{N/2-1} x[2n+1] e^{-j 2\pi k (2n+1)/N}
\]
3. Key Differences
| Feature | Conventional DFT / FFT | Cooley–Tukey FFT |
|---|---|---|
| Complexity | O(N²) | O(N log N) |
| Method | Direct sum over all points | Divide-and-conquer, split into smaller DFTs |
| Optimal N | Any N | Most efficient when N = 2^m |
| Memory / Implementation | Simple, direct | Bit-reversal and twiddle factors required |
| Speed | Slower for large N | Much faster, widely used in practice |
4. Summary
Conventional FFT is the brute-force, direct computation of DFT and is slower for large sequences.
Cooley–Tukey FFT uses divide-and-conquer, reducing computational complexity and making FFT practical for large datasets.