Goertzel algorithm

The Goertzel algorithm is a technique in digital signal processing for efficiently evaluating individual terms of the discrete Fourier transform. Instead of computing a whole spectrum, it analyses one selectable frequency component of a discrete signal, applying a single real-valued coefficient at each iteration and using real-valued arithmetic when the input is real. This makes it a practical choice for jobs such as recognising the dual-tone multi-frequency tones produced by the keys of a traditional analog telephone keypad. Gerald Goertzel first described it in 1958. For a full spectrum it has a higher order of complexity than fast Fourier transform algorithms, but when only a small number of frequency components are needed it is more numerically efficient, and its simple structure suits small processors and embedded applications. It can also be run in reverse as a sinusoid synthesis function that needs only one multiplication and one subtraction per generated sample.