This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.
Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.