How Fast Fourier Transform Works

How FFT Works (High-Level Idea)

FFT exploits the symmetry and periodicity properties of the DFT to reduce redundant calculations. The most common FFT algorithm is the Cooley-Tukey algorithm, which divides the computation into smaller sub-problems:

• Divide the input signal into smaller parts (even and odd indices).
• Conquer: Recursively compute the DFT for these smaller parts.
• Combine: Use symmetry to efficiently merge the results.

The DFT directly requires O(N^2) computations for N points, which can become computationally expensive for large datasets. FFT reduces this computational complexity to O(N logN), making it much faster.