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{{more footnotes|date=March 2023}}
The '''method of reassignment''' is a technique for sharpening a [[time-frequency representation]] (e.g. [[spectrogram]] or the [[short-time Fourier transform]]) by mapping the data to time-frequency coordinates that are nearer to the true [[Support (mathematics)|region of support]] of the analyzed signal. The method has been independently introduced by several parties under various names, including ''method of reassignment'', ''remapping'', ''time-frequency reassignment'', and ''modified moving-window method''.<ref name="hainsworth">{{Cite thesis |type=PhD |chapter=Chapter 3: Reassignment methods |title=Techniques for the Automated Analysis of Musical Audio |last=Hainsworth |first=Stephen |year=2003 |publisher=University of Cambridge |citeseerx=10.1.1.5.9579 }}</ref>
== Introduction ==
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[[Cohen's class distribution function|Cohen's class]] of bilinear time-frequency representations is a class of "smoothed" Wigner–Ville distributions, employing a smoothing kernel that can reduce sensitivity of the distribution to noise and suppresses cross-components, at the expense of smearing the distribution in time and frequency. This smearing causes the distribution to be non-zero in regions where the true Wigner–Ville distribution shows no energy.
The spectrogram is a member of Cohen's class. It is a smoothed Wigner–Ville distribution with the smoothing kernel equal to the Wigner–Ville distribution of the analysis window. The method of reassignment smooths the Wigner–Ville distribution, but then refocuses the distribution back to the true regions of support of the signal components. The method has been shown to reduce time and frequency smearing of any member of Cohen's class.<ref name="improving">
{{cite journal |author1=F. Auger |author2=P. Flandrin |name-list-style=amp |date=May 1995 |title=Improving the readability of time-frequency and time-scale representations by the reassignment method |journal=IEEE Transactions on Signal Processing |volume=43 |issue=5 |pages=1068–1089 |doi=10.1109/78.382394 |bibcode=1995ITSP...43.1068A |citeseerx=10.1.1.646.794 |s2cid=6336685 }}</ref><ref>P. Flandrin, F. Auger, and E. Chassande-Mottin, ▼
▲{{cite journal |author1=F. Auger |author2=P. Flandrin |name-list-style=amp |date=May 1995 |title=Improving the readability of time-frequency and time-scale representations by the reassignment method |journal=IEEE Transactions on Signal Processing |volume=43 |issue=5 |pages=1068–1089 |doi=10.1109/78.382394 |bibcode=1995ITSP...43.1068A |citeseerx=10.1.1.646.794 |s2cid=6336685 }}
''Time-frequency reassignment: From principles to algorithms'',
in Applications in Time-Frequency Signal Processing
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== The method of reassignment ==
Pioneering work on the method of reassignment was published by Kodera, Gendrin, and de Villedary under the name of ''Modified Moving Window Method''.<ref name=Kodera>{{cite journal |author1=K. Kodera |author2=R. Gendrin |author3=C. de Villedary |name-list-style=amp |date=Feb 1978 |title=Analysis of time-varying signals with small BT values |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=26 |issue=1 |pages=64–76 |doi=10.1109/TASSP.1978.1163047 }}</ref> Their technique enhances the resolution in time and frequency of the classical Moving Window Method (equivalent to the spectrogram) by assigning to each data point a new time-frequency coordinate that better-reflects the distribution of energy in the analyzed signal.<ref name=Kodera/>
In the classical moving window method, a time-___domain signal, <math>x(t)</math> is decomposed into a set of coefficients, <math>\epsilon( t, \omega )</math>, based on a set of elementary signals, <math>h_{\omega}(t)</math>, defined
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For sufficiently small values of <math>\Delta t</math> and <math>\Delta \omega,</math> and provided that the phase difference is appropriately "unwrapped", this finite-difference method yields good approximations to the partial derivatives of phase, because in regions of the spectrum in which the evolution of the phase is dominated by rotation due to sinusoidal oscillation of a single, nearby component, the phase is a linear function.
Independently of Kodera ''et al.'', Nelson arrived at a similar method for improving the time-frequency precision of short-time spectral data
spectrum.<ref name
Auger and Flandrin showed that the method of reassignment, proposed in the context of the spectrogram by Kodera et al., could be extended to any member of [[Cohen's class]] of time-frequency representations by generalizing the reassignment operations to
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where <math>W_{x}(t,\omega)</math> is the Wigner–Ville distribution of <math>x(t)</math>, and <math>\Phi(t,\omega)</math> is the kernel function that defines the distribution. They further described an efficient method for computing the times and frequencies for the reassigned spectrogram efficiently and accurately without explicitly computing the partial derivatives of
phase.<ref name
In the case of the spectrogram, the reassignment operations can be computed by
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