A mathematical method that analyzes a sound sample and determines what frequencies and harmonics are present in the sound. The algorithm works by processing a group of samples representing a short period of time of the waveform being transformed; therefore, the output of the transform represents the average frequency content over the period of time being sampled. A tradeoff that the implementer must make is deciding on the length of the time period being sampled; shorter periods of time produce a more instantaneous view of the waveform, but the resulting analysis has less resolution than a transform performed on a longer sample. Computing devices that use the Fourier transform (in particular the fast Fourier transform algorithm) have almost totally taken over the spectrum analyzer market, since they are more precise and work faster than older devices that relied on sweeping a filter across the spectrum.
The Walsh function is another method of spectral analysis. It is seldom used in music, but is seen in other types of signal processing, such as speech recognition and radar target detection.