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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gabor-Transformation</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Gabor-Transformation</b> (nach <a href="Dennis_G%C3%A1bor" title="Dennis Gábor">Dennis Gábor</a>) ist eine spezielle (und in bestimmter Weise optimale) gefensterte <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a>. Sie ist eng verwandt mit der <a href="Wavelet" title="Wavelet">Wavelet</a>-Theorie und wird in vielen Bereichen der digitalen <a href="Digitale_Signalverarbeitung" title="Digitale Signalverarbeitung">Signal-</a> und <a href="Bildverarbeitung" title="Bildverarbeitung">Bildverarbeitung</a> eingesetzt. Sie ist ein Spezialfall der <a href="Kurzzeit-Fourier-Transformation" title="Kurzzeit-Fourier-Transformation">Kurzzeit-Fourier-Transformation</a>.
</p>

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

<p>Jede lokale Veränderung eines Signals <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> bewirkt eine Änderung der <a href="Fourier-Transformierte" class="mw-redirect" title="Fourier-Transformierte">Fourier-Transformierten</a> (FT) von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> über der gesamten Frequenzachse. So überdeckt zum Beispiel der Graph der FT der <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a> (Dirac-Funktion) den gesamten Frequenzbereich. Die FT enthält daher keine lokalen Informationen des Signals <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>. Dies bedeutet andererseits, dass die Information des <a href="Frequenzspektrum" title="Frequenzspektrum">Frequenzspektrums</a> den Zeitpunkt, in dem die Frequenz auftritt, nicht unmittelbar angibt. Eine Möglichkeit der Lokalisierung in der Zeit bietet die <a href="Kurzzeit-Fourier-Transformation" title="Kurzzeit-Fourier-Transformation">Kurzzeit-Fourier-Transformation</a> (<span style="font-style:normal;font-weight:normal"><a href="Englische_Sprache" title="Englische Sprache">englisch</a></span> <span lang="en-Latn" style="font-style:italic"><i>short-time Fourier transform</i></span>, kurz <i>STFT</i>), mit der der momentane Frequenzinhalt in einem Fenster <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> um den Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span> beschrieben werden kann. Dabei wird für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> üblicherweise eine schnell auf 0 abfallende Funktion gewählt, damit sie als Fenster wirkt.
</p><p><br>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\mathrm {Fen} }(\omega ,\tau )=\int \limits _{-\infty }^{+\infty }f(t)g(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t}">
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<mrow class="MJX-TeXAtom-ORD">
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<msup>
<mi>F</mi>
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<mi mathvariant="normal">F</mi>
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<mi>t</mi>
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<mi>τ<!-- τ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle F^{\mathrm {Fen} }(\omega ,\tau )=\int \limits _{-\infty }^{+\infty }f(t)g(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d0498ff86700e81eb94ac6eb7b87e7272582037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.783ex; height:9.009ex;" alt="{\displaystyle F^{\mathrm {Fen} }(\omega ,\tau )=\int \limits _{-\infty }^{+\infty }f(t)g(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t}" loading="lazy"></span></dd></dl>
<p>Die Fourier-Transformierte mit Fenster ist somit von zwei Parametern abhängig, der Frequenz <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> und dem Zentrum der Lokalisierung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span>. Man spricht deshalb auch von einer Darstellung im <a href="Zeitdom%C3%A4ne" class="mw-redirect" title="Zeitdomäne">Zeit</a>-/<a href="Frequenzdom%C3%A4ne" class="mw-redirect" title="Frequenzdomäne">Frequenzbereich</a>.
</p><p>Die STFT mit einer <a href="Gau%C3%9F-Funktion" class="mw-redirect" title="Gauß-Funktion">Gauß-Funktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\sigma }(t)}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
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<mi>σ<!-- σ --></mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{\sigma }(t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a0c4fff96df26029910a9e5f94a233b87717aa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.93ex; height:2.843ex;" alt="{\displaystyle g_{\sigma }(t)}" loading="lazy"></span> als <a href="Fensterfunktion" title="Fensterfunktion">Fensterfunktion</a> wurde von <a href="Dennis_G%C3%A1bor" title="Dennis Gábor">Dennis Gábor</a> 1946 verwendet:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\sigma }(t)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {t^{2}}{2\sigma ^{2}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
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<mi>σ<!-- σ --></mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
<mi>π<!-- π --></mi>
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<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
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<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{\sigma }(t)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {t^{2}}{2\sigma ^{2}}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65bd1085936c49a090f02183e5104961d8ceccce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.287ex; height:7.009ex;" alt="{\displaystyle g_{\sigma }(t)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {t^{2}}{2\sigma ^{2}}}}}" loading="lazy"></span></dd></dl>
<p>Diese spezielle STFT heißt <b>Gabor-Transformation</b>. Bezeichnet man das Ergebnis der Gabortransformation von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{f}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G_{f}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86fb9e8d359d1eacb9be564288c4f2f64ce44ab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.963ex; height:2.843ex;" alt="{\displaystyle G_{f}}" loading="lazy"></span> so ergibt wegen der Symmetrie von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\sigma }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>σ<!-- σ --></mi>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{\sigma }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a398e0fa4c084ac3e7a435b082d191eb73bc51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:2.009ex;" alt="{\displaystyle g_{\sigma }}" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G_{f}(\omega ,\tau )&amp;=\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(\tau -t)e^{\mathrm {i} \omega (\tau -t)}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast (g_{\sigma }(\tau )e^{\mathrm {i} \omega \tau }))\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast h(\tau ))\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G_{f}(\omega ,\tau )&amp;=\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(\tau -t)e^{\mathrm {i} \omega (\tau -t)}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast (g_{\sigma }(\tau )e^{\mathrm {i} \omega \tau }))\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast h(\tau ))\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00f8513de2b62dee0e130efd60a09f09c16bb751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.693ex; margin-bottom: -0.312ex; width:43.339ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}G_{f}(\omega ,\tau )&amp;=\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(t-\tau )e^{-\mathrm {i} \omega t}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(\tau -t)e^{\mathrm {i} \omega (\tau -t)}\mathrm {d} t\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast (g_{\sigma }(\tau )e^{\mathrm {i} \omega \tau }))\\&amp;=e^{-\mathrm {i} \omega \tau }(f(\tau )\ast h(\tau ))\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Im Zeitbereich stellt die Gaborfilterung daher bis auf den Faktor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-\mathrm {i} \omega \tau }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msup>
<mi>e</mi>
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<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle e^{-\mathrm {i} \omega \tau }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad052e7621daa0d58a0d6c8c30ea47500c93ec87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.924ex; height:2.676ex;" alt="{\displaystyle e^{-\mathrm {i} \omega \tau }}" loading="lazy"></span> eine <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> dar. Dieser Faktor bewirkt jedoch lediglich eine <a href="Phasenverschiebung" title="Phasenverschiebung">Phasenverschiebung</a> und kann daher bei Anwendungen, die nur die <a href="Amplitude" title="Amplitude">Amplitude</a> des Ergebnisses berücksichtigen, vernachlässigt werden (Siehe <a href="Gabor-Filter" title="Gabor-Filter">Gabor-Filter</a>).
</p><p>Da die Fouriertransformierte einer Gauß-Funktion wieder eine Gauß-Funktion ergibt, stellt das Ergebnis der Gabortransformation sowohl im Zeit- als auch im Frequenzraum lokale Information dar. Das Filter kann jede beliebige elliptische Region des Zeit- oder des Frequenzraums überdecken. Ferner erzielt die Gabortransformation – unabhängig von der Anordnung – maximale gleichzeitige Auflösung im Zeit- und Frequenzraum, das heißt die Gauß-Funktion erreicht als (einzige) Fensterfunktion das Minimum der <a href="Unsch%C3%A4rferelation" class="mw-redirect" title="Unschärferelation">Unschärferelation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{g}^{2}\cdot \sigma _{G}^{2}\geq {\tfrac {\pi }{2}}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{g}^{2}\cdot \sigma _{G}^{2}\geq {\tfrac {\pi }{2}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/993ef6cad3723017f5eaea0f192fb090aaa17683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.792ex; height:3.343ex;" alt="{\displaystyle \sigma _{g}^{2}\cdot \sigma _{G}^{2}\geq {\tfrac {\pi }{2}}}" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{g}^{2}}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
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<mi>σ<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{g}^{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d57ddec5db10d442f8f292c20bf28ec472b89e69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{g}^{2}}" loading="lazy"></span> die Varianz der Fensterfunktion im Zeitbereich (Zeitunschärfe) und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{G}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle \sigma _{G}^{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0abffe314a54cada563011e6c7d394504fb0bc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.851ex; height:3.176ex;" alt="{\displaystyle \sigma _{G}^{2}}" loading="lazy"></span> entsprechend die im Frequenzraum (Frequenzunschärfe) angibt. Daraus ergibt sich direkt der <a href="Reziproke" class="mw-redirect" title="Reziproke">reziproke</a> Zusammenhang zwischen den Unschärfen und damit ein wichtiger <a href="Trade-off" title="Trade-off">trade-off</a>. Das heißt, um die Auflösung im Zeitbereich zu verdoppeln, muss eine halbierte Auflösung im Frequenzraum in Kauf genommen werden, und umgekehrt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Laplace-Transformation" title="Laplace-Transformation">Laplace-Transformation</a></li>
<li><a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">Diskrete Fourier-Transformation</a></li>
<li><a href="Diskrete_Kosinustransformation" title="Diskrete Kosinustransformation">Diskrete Kosinustransformation</a></li>
<li><a href="Wavelet-Transformation" title="Wavelet-Transformation">Wavelet-Transformation</a></li>
<li><a href="Gabor-Filter" title="Gabor-Filter">Gabor-Filter</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Hans G. Feichtinger, Thomas Strohmer: „Gabor Analysis and Algorithms“, Birkhäuser, 1998; ISBN 0817639594</li>
<li>Hans G. Feichtinger, Thomas Strohmer: „Advances in Gabor Analysis“, Birkhäuser, 2003; ISBN 0817642390</li>
<li>Karlheinz Gröchenig: „Foundations of Time-Frequency Analysis“, Birkhäuser, 2001; ISBN 0817640223</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://www.nuhag.eu/">NuHAG homepage: viele weitere Links</a></li>
<li><a rel="nofollow" class="external text" href="http://www.univie.ac.at/nuhag-php/gaborserver/">Gabor Server</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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