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== Introduction ==
[[Image:Reassigned spectral surface of bass pluck.png|thumb|400px|
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The spectrogram was computed using a 65.7 ms Kaiser window with a shaping
parameter of 12.]]
Many signals of interest have a distribution of energy that
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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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signal <math>x(t)</math> shifted in time by <math>t</math>
and windowed by <math>h(t)</math>.
<math>x(t)</math> can be reconstructed from the moving window coefficients by
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&= \iint M_{\tau}(\omega) h( \tau - t ) e^{ j \left[ \phi_{\tau}(\omega) - \omega \tau+ \omega t \right] } d\omega d\tau
\end{matrix}</math></center>
For signals having magnitude spectra,
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\omega + \frac{\partial \phi(\tau, \omega)}{\partial \tau} .
\end{matrix}</math></center>
This phenomenon is known in such fields as optics as the
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\left[ \phi(t, \omega+ \frac{\Delta \omega}{2}) - \phi(t, \omega-\frac{\Delta \omega}{2}) \right]
\end{matrix}</math></center>
For sufficiently small values of <math>\Delta t</math> and
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phase.
<ref name = "improving" />
In the case of the spectrogram, the reassignment operations
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is some energy to reassign, and has no meaning when the
distribution is zero-valued.
=Separability=
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This is the property, in the frequency ___domain, that Nelson
called ''
<ref name = "crossspectral" />
and is required of all signals so analyzed. If this property is not met, then
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<references/>
== Further
S. A. Fulop and K. Fitz, ''A spectrogram for the twenty-first century'',
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Journal of the Acoustical Society of America, vol. 119, pp. 360 – 371, Jan 2006.
== External
* [http://tftb.nongnu.org/ TFTB — Time-Frequency ToolBox]
* [http://www.klingbeil.com/spear/ SPEAR - Sinusoidal Partial Editing Analysis and Resynthesis]
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* [http://musicalgorithms.ewu.edu/algorithms/roughness.html SRA - A web-based research tool for spectral and roughness analysis of sound signals] (supported by a Northwest Academic Computing Consortium grant to J. Middleton, Eastern Washington University)
[[Category:Signal processing]]
[[Category:Transforms]]
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