The fast Fourier transform, how and why it works

The Fast Fourier Transform (FFT) is a mathematical algorithm that decomposes complex signals into their individual frequency components, functioning like a prism separating light into colors. Originally developed during the Cold War to distinguish nuclear tests from earthquakes, the FFT achieves dramatic computational efficiency by recursively splitting data into smaller problems, reducing processing time from O(N squared) to O(N log N). This algorithm underpins modern digital communications, signal processing, and technologies like WiFi, MP3 compression, and MRI imaging.