Goertzel algorithm

The Goertzel algorithm is a technique in digital signal processing (DSP) that provides a means for efficient evaluation of individual terms of the discrete Fourier transform (DFT), thus making it useful in certain practical applications, such as recognition of DTMF tones produced by the buttons pushed on a telephone keypad. The algorithm was first described by Gerald Goertzel in 1958.

Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a discrete signal. Unlike direct DFT calculations, the Goertzel algorithm applies a single real-valued coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum, the Goertzel algorithm has a higher order of complexity than fast Fourier transform (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications, though not limited to these.

The Goertzel algorithm can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample.

The main calculation in the Goertzel algorithm has the form of a digital filter, and for this reason the algorithm is often called a Goertzel filter. The filter operates on an input sequence ${\displaystyle x[n]}$ in a cascade of two stages with a parameter ${\displaystyle \omega _{0}}$, giving the frequency to be analysed, normalised to radians per sample.

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