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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Time–frequency representation</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Time%E2%80%93frequency_analysis" title="Time–frequency analysis">Time–frequency analysis</a></div>

<p>A <b>time–frequency representation</b> (<b>TFR</b>) is a view of a <a href="Signal_processing" title="Signal processing">signal</a> (taken to be a function of time) represented over both time and <a href="Frequency" title="Frequency">frequency</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Time%E2%80%93frequency_analysis" title="Time–frequency analysis">Time–frequency analysis</a> means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD.

</p><p>TFRs are often complex-valued fields over time and frequency, where the <a href="Absolute_value#Complex_numbers" title="Absolute value">modulus</a> of the field represents either amplitude or "energy density" (the concentration of the <a href="Root_mean_square" title="Root mean square">root mean square</a> over time and frequency), and the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of the field represents phase.

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<div class="mw-heading mw-heading2"><h2 id="Background_and_motivation">Background and motivation</h2></div>

<p>A <a href="Signal_processing" title="Signal processing">signal</a>, as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of time, may be considered as a representation with perfect <i>time resolution</i>.

In contrast, the <a href="Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> of the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> (FT) of the signal may be considered as a representation with perfect <i>spectral resolution</i> but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in time, different events occur in the signal.

</p><p>TFRs provide a bridge between these two representations in that they provide <i>some</i> temporal information <i><b>and</b></i> <i>some</i> spectral information simultaneously. Thus, TFRs are useful for the representation and analysis of signals containing multiple time-varying frequencies.

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<div class="mw-heading mw-heading2"><h2 id="Formulation_of_TFRs_and_TFDs">Formulation of TFRs and TFDs</h2></div>

<p>One form of TFR (or TFD) can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such representations and formulations are known as <a href="Quadratic_function" title="Quadratic function">quadratic</a> or "bilinear" TFRs or TFDs (QTFRs or QTFDs) because the representation is quadratic in the signal (see <a href="Bilinear_time%E2%80%93frequency_distribution" title="Bilinear time–frequency distribution">Bilinear time–frequency distribution</a>). This formulation was first described by <a href="Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a> in 1932 in the context of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the <a href="Wigner%E2%80%93Ville_distribution" class="mw-redirect" title="Wigner–Ville distribution">Wigner–Ville distribution</a>, as it was shown in <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> that Wigner's formula needed to use the <a href="Analytic_signal" title="Analytic signal">analytic signal</a> defined in Ville's paper to be useful as a representation and for a practical analysis. Today, QTFRs include the <a href="Spectrogram" title="Spectrogram">spectrogram</a> (squared magnitude of <a href="Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a>), the <a href="Scaleogram" class="mw-redirect" title="Scaleogram">scaleogram</a> (squared magnitude of Wavelet transform) and the smoothed pseudo-Wigner distribution.

</p><p>Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms, also called "interferences". The cross-terms caused by the bilinear structure of TFDs and TFRs may be useful in some applications such as classification as the cross-terms provide extra detail for the recognition algorithm. However, in some other applications, these cross-terms may plague certain quadratic TFRs and they would need to be reduced. One way to do this is obtained by comparing the signal with a different function. Such resulting representations are known as linear TFRs because the representation is linear in the signal. An example of such a representation is the <i>windowed Fourier transform</i> (also known as the <a href="Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a>) which localises the signal by modulating it with a <a href="Window_function" title="Window function">window function</a>, before performing the Fourier transform to obtain the frequency content of the signal in the region of the window.

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<div class="mw-heading mw-heading2"><h2 id="Wavelet_transforms">Wavelet transforms</h2></div>

<p>Wavelet transforms, in particular the <a href="Continuous_wavelet_transform" title="Continuous wavelet transform">continuous wavelet transform</a>, expand the signal in terms of wavelet functions which are localised in both time and frequency. Thus the wavelet transform of a signal may be represented in terms of both time and frequency. Continuous wavelet transform analysis is very useful for identifying non-stationary signals in <a href="Time_series" title="Time series">time series</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> such as those related to climate<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or landslides.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>The notions of time, frequency, and amplitude used to generate a TFR from a wavelet transform were originally developed intuitively. In 1992, a quantitative derivation of these relationships was published, based upon a <a href="Stationary_phase_approximation" title="Stationary phase approximation">stationary phase approximation</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Linear_canonical_transformation">Linear canonical transformation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_canonical_transformation" title="Linear canonical transformation">Linear canonical transformation</a></div>

<p><a href="Linear_canonical_transformation" title="Linear canonical transformation">Linear canonical transformations</a> are the <a href="Linear_transform" class="mw-redirect" title="Linear transform">linear transforms</a> of the time–frequency representation that preserve the <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a>. These include and generalize the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, <a href="Fractional_Fourier_transform" title="Fractional Fourier transform">fractional Fourier transform</a>, and others, thus providing a unified view of these transforms in terms of their action on the time–frequency domain.

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<ul><li><a href="Newland_transform" class="mw-redirect" title="Newland transform">Newland transform</a></li>

<li><a href="Reassignment_method" title="Reassignment method">Reassignment method</a></li>

<li><a href="Time%E2%80%93frequency_analysis_for_music_signals" title="Time–frequency analysis for music signals">Time–frequency analysis for music signals</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">E. Sejdić, I. Djurović, J. Jiang, "Time-frequency feature representation using energy concentration: An overview of recent advances," Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009.</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">B. Boashash, "Note on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech. and Signal Processing, vol. 36, issue 9, pp 1518–1521, Sept. 1988. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F29.90380">10.1109/29.90380</a></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFTorrenceCompo1998" class="citation journal cs1">Torrence, Christopher; Compo, Gilbert P. (January 1998). <a rel="nofollow" class="external text" href="http://journals.ametsoc.org/doi/10.1175/1520-0477(1998)0792.0.CO;2">"A Practical Guide to Wavelet Analysis"</a>. <i>Bulletin of the American Meteorological Society</i>. <b>79</b> (1): <span class="nowrap">61–</span>78. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0477%281998%29079%3C0061%3AAPGTWA%3E2.0.CO%3B2">10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-0007">0003-0007</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrinstedMooreJevrejeva2004" class="citation journal cs1">Grinsted, A.; Moore, J. C.; Jevrejeva, S. (2004-11-18). <a rel="nofollow" class="external text" href="https://npg.copernicus.org/articles/11/561/2004/">"Application of the cross wavelet transform and wavelet coherence to geophysical time series"</a>. <i>Nonlinear Processes in Geophysics</i>. <b>11</b> (5/6): <span class="nowrap">561–</span>566. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5194%2Fnpg-11-561-2004">10.5194/npg-11-561-2004</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1023-5809">1023-5809</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFTomásLiLopez-SanchezLiu2016" class="citation journal cs1">Tomás, R.; Li, Z.; Lopez-Sanchez, J. M.; Liu, P.; Singleton, A. (2016-06-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s10346-015-0589-y">"Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide"</a>. <i>Landslides</i>. <b>13</b> (3): <span class="nowrap">437–</span>450. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10346-015-0589-y">10.1007/s10346-015-0589-y</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10045%2F62160">10045/62160</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1612-5118">1612-5118</a>.</cite></span>

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<cite id="CITEREFDelprat,_N.,_Escudii,_B.,_Guillemain,_P.,_Kronland-Martinet,_R.,_Tchamitchian,_P.,_and_Torrksani,_B.1992" class="citation journal cs1">Delprat, N., Escudii, B., Guillemain, P., Kronland-Martinet, R., Tchamitchian, P., and Torrksani, B. (1992). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-01222729/document">"Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies"</a>. <i>IEEE Transactions on Information Theory</i>. <b>38</b> (2): <span class="nowrap">644–</span>664. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F18.119728">10.1109/18.119728</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://tfd.sourceforge.net/">DiscreteTFDs — software for computing time–frequency distributions</a></li>

<li><a rel="nofollow" class="external text" href="http://tftb.nongnu.org/">TFTB — Time–Frequency ToolBox</a></li>

<li><a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/3091384_Time-stretched_short-time_Fourier_transform/">Time stretched short time Fourier transform for time-frequency analysis of ultra wideband signals</a></li></ul></div><div><div class="zim-footer">

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