The Short-Time Fourier Transform

The Short-Time Fourier Transform (STFT) (or short-*term* Fourier
transform) is a powerful general-purpose tool for audio signal
processing [7,9,8]. It
defines a particularly useful class of *time-frequency
distributions* [43] 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:

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

Examples of the second case include estimating the decay-time-versus-frequency for vibrating strings [288] and body resonances [119], or measuring as precisely as possible the fundamental frequency of a periodic signal [106] based on tracking its many harmonics in the STFT [64].

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 [106], and this distinction is
lost. This is not a problem for strictly periodic signals.

- Mathematical Definition of the STFT
- Practical Computation of the STFT
- Summary of STFT Computation Using FFTs
- Two Dual Interpretations of the STFT
- The STFT as a Time-Frequency Distribution
- STFT in Matlab

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University