Reassignment method: Difference between revisions

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{{refimprovemore citations needed|date=March 2023}}

[[Image:Reassigned spectrogral surface of bass pluck.png|thumb|400px|

Reassigned spectral surface for the onset of an acoustic bass tone having a sharp pluck and a fundamental frequency of approximately 73.4 Hz. Sharp spectral ridges representing the harmonics are evident, as is the abrupt onset of the tone. The spectrogram was computed using a 65.7 ms Kaiser window with a shaping parameter of 12.]]

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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>{{cite arXiv |last1=Fitz |first1=Kelly R. |last2=Fulop |first2=Sean A. |title=A Unified Theory of Time-Frequency Reassignment |date=2009 |class=cs.SD |eprint=0903.3080 }} &ndash; this preprint manuscript is written by a previous contributor to this Wikipedia article; see [[Special:Diff/239438445|their contribution]].</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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&= \frac{\partial }{\partial t} \arg\{ X(t,\omega_{0}) \}

\end{align}</math>

Just as each bandpass filter in the short-time Fourier transform filterbank may pass at most a single complex exponential component, two temporal events must be sufficiently separated in time that they do not lie in the same windowed segment of the input signal. This is the property of separability in the time ___domain, and is equivalent to requiring that the time between two events be

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== Extensions ==

=== Consensus complex reassignmetreassignment ===

Gardner and Magnasco (2006) argues that the [[auditory nerve]]s may use a form of the reassignment method to process sounds. These nerves are known for preserving timing (phase) information better than they do for magnitudes. The authors come up with a variation of reassignment with complex values (i.e. both phase and magnitude) and show that it produces sparse outputs like auditory nerves do. By running this reassignment with windows of different bandwidths (see discussion in the section above), a "consensus" that captures multiple kinds of signals is found, again like the auditory system. They argue that the algorithm is simple enough for neurons to implement.<ref name=Gar06>{{cite journal |last1=Gardner |first1=Timothy J. |last2=Magnasco |first2=Marcelo O. |title=Sparse time-frequency representations |journal=Proceedings of the National Academy of Sciences |date=18 April 2006 |volume=103 |issue=16 |pages=6094–6099 |doi=10.1073/pnas.0601707103|doi-access=free |pmid=16601097 |pmc=1431718 |bibcode=2006PNAS..103.6094G }}</ref>

=== Synchrosqueezing transform ===

{{empty section|date=January 2024}}

<ref name=Meignen19>{{cite journal |last1=Meignen |first1=Sylvain |last2=Oberlin |first2=Thomas |last3=Pham |first3=Duong-Hung |title=Synchrosqueezing transforms: From low- to high-frequency modulations and perspectives |journal=Comptes Rendus Physique |date=July 2019 |volume=20 |issue=5 |pages=449–460 |doi=10.1016/j.crhy.2019.07.001|bibcode=2019CRPhy..20..449M }}</ref>

== References ==

{{realistReflist}}

== Further reading ==

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[[Category:Time–frequency analysis]]

[[Category:Transforms]]

[[Category:Data compression]]