The Short-Time Fourier Transform

The Short-Time Fourier Transform (STFT) [8,10,9] is a powerful general-purpose tool for audio signal processing. It defines a particularly useful class of time-frequency distributions [38] which specify complex amplitude versus time and frequency for any signal. We are primarily concerned here with tuning the STFT parameters for the following applications:

1. Approximating the time-frequency analysis performed by the ear for purposes of spectral display.
2. Measuring model parameters in a short-time spectrum.

In the first case, applications of audio spectral display go beyond merely looking at the spectrum. They also provide a basis for audio signal processing tasks intended to imitate human perception, such as auditory scene recognition [25,192] or automatic transcription of music [110].

Examples of the second case include estimating the decay-time-versus-frequency for vibrating strings [265] and body resonances [104], or measuring as precisely as possible the fundamental frequency of a periodic signal [92] based on tracking its many harmonics in the STFT [56].

An interesting example for which cases 1 and 2 normally coincide is pitch detection (case 1) and fundamental frequency estimation (case 2). Here, ``fundamental frequency'' is defined as the lowest frequency present in a series of harmonic overtones, while ``pitch'' is defined as the perceived fundamental frequency; perceived pitch can be measured, for example, by comparing to a harmonic reference tone such as a sawtooth waveform. (Thus, by definition, the pitch of a sawtooth waveform is its fundamental frequency.) When harmonics are stretched so that they become slightly inharmonic, pitch perception corresponds to a (possibly non-existent) compromise fundamental frequency, the harmonics of which ``best fit'' the most audible overtones in some sense. The topic of ``pitch detection'' in the signal processing literature is often really about fundamental frequency estimation [92], and this distinction is lost. This is not a problem for strictly periodic signals.

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Subsections

• Mathematical Definition of the STFT
• Practical Computation of the STFT
• Summary of STFT Computation Using the FFT
• Two Dual Interpretations of the STFT
• STFT in Matlab

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Order a Hardcopy of Spectral Audio Signal Processing

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Next: Mathematical Definition of the STFT

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.