The Short-Time Fourier Transform

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Time-Frequency Displays

Mathematical Definition of the STFT

A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals

An overtone is a sinusoidal component of a waveform of greater frequency than its fundamental frequency. — Click for http://en.wikipedia.org/wiki/Overtone

Pitch is the perceived fundamental frequency of a sound. — Click for http://en.wikipedia.org/wiki/Pitch_(music)

Click for https://scienceworld.wolfram.com/physics/Harmonic.html

For a tone which posesses a perfectly harmonic overtone series, the nth harmonic is generally defined to oscillate at nf₀Hz, where f₀ is the fundamental frequency. — Click for http://en.wikipedia.org/wiki/Harmonic

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Every periodic signal has a fundamental frequency given by the inverse of the period. If the period is in units of seconds, then the fundamental frequency is in units of Hertz (cycles per second). According to Fourier theory, every periodic signal can be expressed as a sum of sinusoids at frequencies given by integer multiples of the fundamental frequency. For integers greater than 1, these frequencies are called harmonic frequencies, and the sinusoids at harmonic frequencies are typically called harmonics. — Click for https://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics

A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals

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T60 is the time (in seconds) to decay by 60 dB, e.g., from 0 dB to -60 dB. It is a typical 'cut-off time' for impulse responses that decay exponentially. — Click for https://ccrma.stanford.edu/~jos/mdft/Audio_Decay_Time_T60.html

Audition is the act of listening. — Click for http://www.google.com/search?q=auditory

The spectrum of a signal gives the distribution of signal energy as a function of frequency. — Click for http://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Time-Frequency Displays

Mathematical Definition of the STFT

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.

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 [26,209] or automatic transcription of music [125].

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.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

The Short-Time Fourier Transform

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University