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Artoria2e5 (talk | contribs) Milked most of the footnotes to their worth; now it's refimprove again. |
Artoria2e5 (talk | contribs) There's an article on arXiv written by one of the old contributors to this article. Among other things, it's got better refs and explanations for equations. |
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\end{align}</math>
where <math>M_{t}(\omega)</math> is the magnitude, and <math>\phi_{\tau}(\omega)</math> the phase, of <math>X_{t}(\omega)</math>, the Fourier transform of the signal <math>x(t)</math> shifted in time by <math>t</math> and windowed by <math>h(t)</math>.<ref name=Fitz09>{{citation |last1=Fitz |first1=Kelly R. |last2=Fulop |first2=Sean A. |title=A Unified Theory of Time-Frequency Reassignment |date=2009 |doi=10.48550/arXiv.0903.3080}}<!-- Yes, the author also contributed to this article, so a lot of the equations are the same. But this one seems to be more thorough in the explanation, even though the licensing is not good enough for us to copy. --></ref>{{rp|4}}
<math>x(t)</math> can be reconstructed from the moving window coefficients by<ref name=Fitz09/>{{rp|8}}
:<math>\begin{align}
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==Separability==
The short-time Fourier transform can often be used to estimate the amplitudes and phases of the individual components in a ''multi-component'' signal, such as a quasi-harmonic musical instrument tone. Moreover, the time and frequency reassignment operations can be used to sharpen the representation by attributing the spectral energy reported by the short-time Fourier transform to the point that is the local center of gravity of the complex energy distribution.<ref>K. Fitz, L. Haken, On the use of time-frequency reassignment in additve sound modeling, Journal of the Audio Engineering Society 50 (11) (2002) 879 – 893.</ref>
For a signal consisting of a single component, the instantaneous frequency can be estimated from the partial derivatives of phase of any short-time Fourier transform channel that passes the component. If the signal is to be decomposed into many components,
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