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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hadamard-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>Hadamard-Transformation</b>, auch bezeichnet als <b>Walsh-Hadamard-Transformation</b>, <b>Hadamard-Rademacher-Walsh-Transformation</b>, <b>Walsh-Transformation</b> und als <b>Walsh-Fourier-Transformation</b>, ist eine <a href="Liste_von_Transformationen_in_der_Mathematik#Diskrete_Transformationen" title="Liste von Transformationen in der Mathematik">diskrete Transformation</a> aus dem Bereich der <a href="Fourier-Analysis" title="Fourier-Analysis">Fourier-Analysis</a>. Sie ist eine <a href="Orthogonale_Abbildung" title="Orthogonale Abbildung">orthogonal</a>-symmetrische, <a href="Involution_(Mathematik)" title="Involution (Mathematik)">selbstinverse</a> und <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare</a> Transformation und von der Struktur her verwandt mit der <a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">diskreten Fourier-Transformation</a> (DFT). Die Hadamard-Transformation bildet einen Satz 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 2^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/667d0154f26e56e3f7979803f08afac16b4dcb16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}" loading="lazy"></span> reellen oder komplexen Eingangswerten in einen <a href="Bildbereich" class="mw-redirect" title="Bildbereich">Bildbereich</a> aus überlagerten <a href="Walsh-Funktion" title="Walsh-Funktion">Walsh-Funktionen</a>, dem Walsh-Spektrum, ab.<sup id="cite_ref-Kunz1_1-0" class="reference"><a href="#cite_note-Kunz1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Die Transformation ist benannt nach den Mathematikern <a href="Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a>, <a href="Joseph_L._Walsh" title="Joseph L. Walsh">Joseph L. Walsh</a> und <a href="Hans_Rademacher" title="Hans Rademacher">Hans Rademacher</a>.
</p><p>Die Anwendungen der Hadamard-Transformation liegen im Bereich der <a href="Digitale_Signalverarbeitung" title="Digitale Signalverarbeitung">digitalen Signalverarbeitung</a> und <a href="Datenkompression" title="Datenkompression">Datenkompression</a> wie beispielsweise bei <a href="JPEG_XR" title="JPEG XR">JPEG XR</a> und <a href="H.264/MPEG-4_AVC" class="mw-redirect" title="H.264/MPEG-4 AVC">H.264/MPEG-4 AVC</a>.
</p>

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

<p>Die Hadamard-Transformation <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 \operatorname {H} _{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eb9435f48286de51f6af8d8ce1bf4b3a363ac12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.418ex; height:2.509ex;" alt="{\displaystyle \operatorname {H} _{m}}" loading="lazy"></span> wird aus einer <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 2^{m}\times 2^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
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</msup>
<mo>×<!-- × --></mo>
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
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<annotation encoding="application/x-tex">{\displaystyle 2^{m}\times 2^{m}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8726a2f88d3da9c6bb30db610c3f114401483e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.515ex; height:2.343ex;" alt="{\displaystyle 2^{m}\times 2^{m}}" loading="lazy"></span>-<a href="Hadamard-Matrix" title="Hadamard-Matrix">Hadamard-Matrix</a>, skaliert mit einem Normalisierungsfaktor, gebildet, welche eine Eingangsfolge <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 (x_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{n})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/012f44968fa86fe5e3827e9957d957b08f2d9e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle (x_{n})}" loading="lazy"></span> der Länge <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 2^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
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</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{m}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/667d0154f26e56e3f7979803f08afac16b4dcb16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.837ex; height:2.343ex;" alt="{\displaystyle 2^{m}}" loading="lazy"></span> mittels einer <a href="Matrix-Vektor-Multiplikation" class="mw-redirect" title="Matrix-Vektor-Multiplikation">Matrix-Vektor-Multiplikation</a> in eine Ausgangsfolge <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 (X_{k})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X_{k})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc39037affa37535f42c0670ac97165069a6662e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle (X_{k})}" loading="lazy"></span> transformiert.
</p><p>Die Hadamard-Transformation kann verschiedenartig definiert werden, unter anderem <a href="Rekursiv" class="mw-redirect" title="Rekursiv">rekursiv</a>, wobei von einer <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 1\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4bf91a527dc01af9ef6ace81199becf1308e00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}" loading="lazy"></span>-Hadamard-Transformation <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 \operatorname {H} _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/435920bee3006c7598eb24c37e3466995f3e8667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \operatorname {H} _{0}}" loading="lazy"></span> mit der Identität <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 \operatorname {H} _{0}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{0}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01668d18b2e0abf98a12e19cd55eb5fb6634963a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.058ex; height:2.509ex;" alt="{\displaystyle \operatorname {H} _{0}=1}" loading="lazy"></span> ausgegangen wird 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 \operatorname {H} _{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{m}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1eb9435f48286de51f6af8d8ce1bf4b3a363ac12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.418ex; height:2.509ex;" alt="{\displaystyle \operatorname {H} _{m}}" loading="lazy"></span> 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 m>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m&gt;0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/501173910e6da8425b4e9d44a4e8643620bc2464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>0}" loading="lazy"></span> festgelegt wird zu:
</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 \operatorname {H} _{m}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\operatorname {H} _{m-1}&amp;\operatorname {H} _{m-1}\\\operatorname {H} _{m-1}&amp;-\operatorname {H} _{m-1}\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<msub>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {H} _{m}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\operatorname {H} _{m-1}&amp;\operatorname {H} _{m-1}\\\operatorname {H} _{m-1}&amp;-\operatorname {H} _{m-1}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ceb0d7e0ffd47dfe7746898fa3ac41280d2cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.179ex; height:6.509ex;" alt="{\displaystyle \operatorname {H} _{m}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\operatorname {H} _{m-1}&amp;\operatorname {H} _{m-1}\\\operatorname {H} _{m-1}&amp;-\operatorname {H} _{m-1}\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>mit dem Normalisierungsfaktor <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 {\tfrac {1}{\sqrt {2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f539855d8bdbadfd437df59db26fb28500894993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.027ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}}" loading="lazy"></span>, der mitunter auch weggelassen wird.
</p><p>Analog wie bei der diskreten Fourier-Transformation (DFT) und der optimierten <a href="Schnelle_Fourier-Transformation" title="Schnelle Fourier-Transformation">schnellen Fourier-Transformation</a> (FFT) existiert auch eine schnelle Hadamard-Transformation, welche die Anzahl <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> der Operationen auf <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 n\cdot \operatorname {log} (n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\cdot \operatorname {log} (n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35dfe8b9bd4c6b630f98255f2b2c7295ab23d313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.25ex; height:2.843ex;" alt="{\displaystyle n\cdot \operatorname {log} (n)}" 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 n=2^{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>n</mi>
<mo>=</mo>
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<mi>m</mi>
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<annotation encoding="application/x-tex">{\displaystyle n=2^{m}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab14407b66fdd78b64c1eeec8dff77a7df81b101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.331ex; height:2.343ex;" alt="{\displaystyle n=2^{m}}" loading="lazy"></span> reduziert.
</p>
<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_zur_diskreten_Fouriertransformation">Zusammenhang zur diskreten Fouriertransformation</h2></div>
<p>Wie auch die Hadamard-Transformation lässt sich die <a href="Diskrete_Fourier-Transformation" title="Diskrete Fourier-Transformation">diskrete Fourier-Transformation</a> als Produkt einer Transformationsmatrix und eines Eingangsvektors formulieren. Sollen per DFT <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 N=2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mo>=</mo>
<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle N=2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/405d64b14536deffc3465f1e81b1b7fe9358ad2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=2}" loading="lazy"></span> Elemente im Zeitbereich in den Spektralbereich transformiert werden, so lautet die DFT-Matrix
</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_{2}={\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
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<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F_{2}={\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14e08d80cfed5c4408133d16b419340ec7b94d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.309ex; height:6.176ex;" alt="{\displaystyle F_{2}={\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Die Hadamard-Matrix ohne Skalierungsfaktor ist dann als <a href="Kronecker-Produkt" title="Kronecker-Produkt">Kronecker-Produkt</a> aus einzelnen <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 2\times 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>×<!-- × --></mo>
<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8a0e3400ffb97d67c00267ed50cddfe824cbe80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}" loading="lazy"></span> DFT-Matrizen konstruierbar:
</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 H_{2^{k}}={\underset {k{\text{ mal}}}{\underbrace {{\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes \ldots \otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}} }}}">
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<mrow class="MJX-TeXAtom-ORD">
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<mtd>
<mo>−<!-- − --></mo>
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<mo>⊗<!-- ⊗ --></mo>
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<mn>1</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<mo>]</mo>
</mrow>
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<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
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<mtd>
<mn>1</mn>
</mtd>
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<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mo>⊗<!-- ⊗ --></mo>
<mo>…<!-- … --></mo>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mo>[</mo>
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<mn>1</mn>
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<mo>]</mo>
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<mo>⏟<!-- ⏟ --></mo>
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</mrow>
<mrow>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&nbsp;mal</mtext>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{2^{k}}={\underset {k{\text{ mal}}}{\underbrace {{\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes \ldots \otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}} }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4497c03aced72b983f7d8f97715b5861e40de7d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; margin-right: -0.028ex; width:59.705ex; height:9.676ex;" alt="{\displaystyle H_{2^{k}}={\underset {k{\text{ mal}}}{\underbrace {{\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}\otimes \ldots \otimes {\begin{bmatrix}1&amp;1\\1&amp;-1\end{bmatrix}}} }}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Kathy J. Horadam: <cite style="font-style:italic">Hadamard Matrices and their Applications</cite>. Princeton University Press, Princeton NJ u. a. 2007, ISBN 978-0-691-11921-2.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Hadamard-Transformation&amp;rft.au=Kathy+J.+Horadam&amp;rft.btitle=Hadamard+Matrices+and+their+Applications&amp;rft.date=2007&amp;rft.genre=book&amp;rft.isbn=9780691119212&amp;rft.place=Princeton+NJ+u.+a.&amp;rft.pub=Princeton+University+Press" style="display:none">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-Kunz1-1"><span class="mw-cite-backlink"><a href="#cite_ref-Kunz1_1-0">↑</a></span> <span class="reference-text">Henry O. Kunz: <i>On the Equivalence Between One-Dimensional Discrete Walsh-Hadamard and Multidimensional Discrete Fourier Transforms.</i> In: <i>IEEE Transactions on Computers.</i> Bd. 28, Nr. 3, 1979, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220018-9340%22&amp;key=cql">0018-9340</a></span></span>, S. 267–268, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1109/TC.1979.1675334">10.1109/TC.1979.1675334</a></span>.</span>
</li>
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