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Micron Document

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<title>Haar-Wavelet</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Haar-Wavelet</h1>

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Das Haar-Wavelet ist das erste in der Literatur bekannt gewordene <a href="Wavelet" title="Wavelet">Wavelet</a> und wurde 1909 von <a href="Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a> eingeführt.<a href="#cite_note-1">[1]</a> Es ist außerdem das einfachste bekannte Wavelet und kann aus der Kombination zweier <a href="Rechteckfunktion" title="Rechteckfunktion">Rechteckfunktionen</a> gebildet werden.

Vorteilhaft am Haar-Wavelet ist die einfache Implementierbarkeit der zugehörigen <a href="Wavelet-Transformation" title="Wavelet-Transformation">Wavelet-Transformation</a> als <a href="Schnelle_Wavelet-Transformation" title="Schnelle Wavelet-Transformation">schnelle Wavelet-Transformation</a> (FWT). Der Nachteil des Haar-Wavelets ist, dass es <a href="Stetige_Funktion" title="Stetige Funktion">unstetig</a> und daher auch nicht <a href="Differentialrechnung" title="Differentialrechnung">differenzierbar</a> ist.

<div class="mw-heading mw-heading2"><h2 id="Die_Funktionen_der_Haar-Wavelet-Basis">Die Funktionen der Haar-Wavelet-Basis</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Skalierungsfunktion">Skalierungsfunktion</h3></div>

Die Skalierungsfunktion bzw. „Vater-Wavelet“-Funktion der Haar-Wavelet-Basis ist die <a href="Indikatorfunktion" title="Indikatorfunktion">Indikatorfunktion</a> des Intervalls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f99b30b4451167959e97802252ad13b87af5505.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle [0,1)}" loading="lazy">.

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<mtext>sonst</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\chi _{[0,1)}(x)={\begin{cases}1&0\leq x<1\\0&{\mbox{sonst}}\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3bd8e1f4f79a43d15c609e2fb6a82515e4eae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.577ex; height:6.176ex;" alt="{\displaystyle \phi (x)=\chi _{[0,1)}(x)={\begin{cases}1&0\leq x<1\\0&{\mbox{sonst}}\end{cases}}}" loading="lazy"></dd></dl>

Sie erfüllt die <a href="Funktionalgleichung" title="Funktionalgleichung">Funktionalgleichung</a>

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\phi (2x)+\phi (2x-1)={\sqrt {2}}\left(a_{0}\phi (2x)+a_{1}\phi (2x-1)\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e09ed2ae713094a316585273e435dacb5f359816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.018ex; height:3.176ex;" alt="{\displaystyle \phi (x)=\phi (2x)+\phi (2x-1)={\sqrt {2}}\left(a_{0}\phi (2x)+a_{1}\phi (2x-1)\right)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=a_{1}={\frac {1}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=a_{1}={\frac {1}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be38e86d7e0639b667c9d75b978de460e53c98ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.699ex; height:6.176ex;" alt="{\displaystyle a_{0}=a_{1}={\frac {1}{\sqrt {2}}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Waveletfunktion">Waveletfunktion</h3></div>

Die Waveletfunktion ist die „zusammengeschobene“ Differenz zweier aufeinanderfolgender Skalierungsfunktionen:

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)=\phi (2x)-\phi (2x-1)={\sqrt {2}}\left(b_{0}\phi (2x)+b_{1}\phi (2x-1)\right)={\begin{cases}1&0\leq x<1/2\\-1&1/2\leq x<1\\0&{\mbox{sonst}}\end{cases}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/feeb4828eaf4fb4eb7735d61e402c5533d17c5e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:80.067ex; height:8.509ex;" alt="{\displaystyle \psi (x)=\phi (2x)-\phi (2x-1)={\sqrt {2}}\left(b_{0}\phi (2x)+b_{1}\phi (2x-1)\right)={\begin{cases}1&0\leq x<1/2\\-1&1/2\leq x<1\\0&{\mbox{sonst}}\end{cases}}}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{0},b_{1})=({\tfrac {1}{\sqrt {2}}},-{\tfrac {1}{\sqrt {2}}})}">

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<annotation encoding="application/x-tex">{\displaystyle (b_{0},b_{1})=({\tfrac {1}{\sqrt {2}}},-{\tfrac {1}{\sqrt {2}}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a19d1f7de66c302defb87e86077a29f4308176bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.751ex; height:4.176ex;" alt="{\displaystyle (b_{0},b_{1})=({\tfrac {1}{\sqrt {2}}},-{\tfrac {1}{\sqrt {2}}})}" loading="lazy">.

Die Schreibweise mit Vorfaktor sorgt dafür, dass die Matrix

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<annotation encoding="application/x-tex">{\displaystyle H={\begin{pmatrix}a_{0}&a_{1}\\b_{0}&b_{1}\end{pmatrix}}={\frac {1}{\sqrt {2}}}\,{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5a909052fd7dfd5932151509a18f176b38a4e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.274ex; height:6.509ex;" alt="{\displaystyle H={\begin{pmatrix}a_{0}&a_{1}\\b_{0}&b_{1}\end{pmatrix}}={\frac {1}{\sqrt {2}}}\,{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}" loading="lazy"></dd></dl></dd></dl>

eine <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrix</a> ist. Dies ist Teil der Bedingungen, die orthogonale Wavelets erfordern.

<div class="mw-heading mw-heading3"><h3 id="Multiskalenanalyse">Multiskalenanalyse</h3></div>

Diese Funktion erzeugt die <a href="Multiskalenanalyse" title="Multiskalenanalyse">Multiskalenanalyse</a> der Stufenfunktionen. In dieser wird jeder Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1379612155e29953b9a3417c0ecaa37ccc8b61ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.244ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(\mathbb {R} )}" loading="lazy"> mit „endlicher Energie“ auf jeder Skala <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle J\in \mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6015ca2448a28410be5c8f7c43f6e63769be79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.862ex; height:2.176ex;" alt="{\displaystyle J\in \mathbb {Z} }" loading="lazy"> die folgende Projektion zugewiesen:

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<annotation encoding="application/x-tex">{\displaystyle f\mapsto P_{J}(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ad5bcc1f6513de6efce007cff5f1443cf9ad55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.745ex; height:2.843ex;" alt="{\displaystyle f\mapsto P_{J}(f)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{J}(f)(x)=\sum _{n\in \mathbb {Z} }\left(\int _{0}^{1}f\left(2^{-J}(n+t)\right)\,\mathrm {d} t\right)\cdot \phi (2^{J}x-n)}">

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<annotation encoding="application/x-tex">{\displaystyle P_{J}(f)(x)=\sum _{n\in \mathbb {Z} }\left(\int _{0}^{1}f\left(2^{-J}(n+t)\right)\,\mathrm {d} t\right)\cdot \phi (2^{J}x-n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39a2fb89d51ed2dc8880a6d006a8d6ff107c3f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.332ex; height:7.009ex;" alt="{\displaystyle P_{J}(f)(x)=\sum _{n\in \mathbb {Z} }\left(\int _{0}^{1}f\left(2^{-J}(n+t)\right)\,\mathrm {d} t\right)\cdot \phi (2^{J}x-n)}" loading="lazy">.</dd></dl>

Die Differenz zwischen zwei Skalen lässt sich dann durch das „Mutter-Wavelet“ bzw. die eigentliche Waveletfunktion ausdrücken:

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<annotation encoding="application/x-tex">{\displaystyle P_{J+1}(f)(x)-P_{J}(f)(x)=\sum _{n\in \mathbb {Z} }\left(\int _{0}^{1}f\left(2^{-J-1}(2n+t)\right)\,\mathrm {d} t-\int _{0}^{1}f\left(2^{-J-1}(2n+1+t)\right)\,\mathrm {d} t\right)\cdot \psi (2^{J}x-n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f1b147a91d518076d54c217f59a4198cdd1c06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:101.967ex; height:7.009ex;" alt="{\displaystyle P_{J+1}(f)(x)-P_{J}(f)(x)=\sum _{n\in \mathbb {Z} }\left(\int _{0}^{1}f\left(2^{-J-1}(2n+t)\right)\,\mathrm {d} t-\int _{0}^{1}f\left(2^{-J-1}(2n+1+t)\right)\,\mathrm {d} t\right)\cdot \psi (2^{J}x-n)}" loading="lazy">.</dd></dl>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{j,k}(x)={{\sqrt {2}}\,}^{j}\phi ({2\,}^{j}x-k)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{j,k}(x)={{\sqrt {2}}\,}^{j}\phi ({2\,}^{j}x-k)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6112284c92d99d8fcc8bfbce31b071ede77d4677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.277ex; height:3.676ex;" alt="{\displaystyle \phi _{j,k}(x)={{\sqrt {2}}\,}^{j}\phi ({2\,}^{j}x-k)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{j,k}(x)={{\sqrt {2}}\,}^{j}\psi ({2\,}^{j}x-k)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{j,k}(x)={{\sqrt {2}}\,}^{j}\psi ({2\,}^{j}x-k)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e4aaa0c09a7fce16f8a865f81a1782c8c85406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.532ex; height:3.676ex;" alt="{\displaystyle \psi _{j,k}(x)={{\sqrt {2}}\,}^{j}\psi ({2\,}^{j}x-k)}" loading="lazy"> als Funktionen im <a href="Hilbertraum" title="Hilbertraum">Hilbertraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">

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<ul><li>alle diese Funktionen haben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-Norm 1,</li>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f83e3cf7f4e45eb1ca14af4a744fdc9cc693bc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.609ex; height:2.843ex;" alt="{\displaystyle \phi _{j,k}}" loading="lazy"> ist senkrecht zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{j,l}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/838d29e2035129c49de497f19715d7b297b3d89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.243ex; height:2.843ex;" alt="{\displaystyle \phi _{j,l}}" loading="lazy"> falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\not =l}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e3c64220d90a7934a6903d468b4dd0b64e1dccf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.627ex; height:2.843ex;" alt="{\displaystyle \psi _{i,k}}" loading="lazy"> ist senkrecht zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{j,l}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/258f587996312a7a63470c5ec3293b1d745b7824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.37ex; height:2.843ex;" alt="{\displaystyle \psi _{j,l}}" loading="lazy"> falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\not =j}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40e183b251354fde28bb4a4b055e31499ce83f85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\not =j}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\not =l}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e3c64220d90a7934a6903d468b4dd0b64e1dccf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.627ex; height:2.843ex;" alt="{\displaystyle \psi _{i,k}}" loading="lazy"> bilden eine <a href="Hilbertraumbasis" title="Hilbertraumbasis">Hilbertbasis</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">

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<div class="mw-heading mw-heading2"><h2 id="Schnelle_Haar-Wavelet-Transformation">Schnelle Haar-Wavelet-Transformation</h2></div>

Gegeben sei ein <a href="Digitales_Signal" class="mw-redirect" title="Digitales Signal">diskretes Signal</a> f, welches durch eine endliche oder quadratsummierbare Folge

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<annotation encoding="application/x-tex">{\displaystyle f=(\dots ,f_{-2},f_{-1},f_{0},f_{1},f_{2},f_{3},\dots )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cb9c2ff8c1d0d5cdfa364532027e8acb8fb87e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.975ex; height:2.843ex;" alt="{\displaystyle f=(\dots ,f_{-2},f_{-1},f_{0},f_{1},f_{2},f_{3},\dots )}" loading="lazy"></dd></dl>

dargestellt ist. Ihm ist als kontinuierliches Signal die Treppenfunktion

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<annotation encoding="application/x-tex">{\displaystyle F(x)=\dots +f_{-1}\phi _{0,-1}(x)+f_{0}\phi _{0,0}(x)+f_{1}\phi _{0,1}(x)+f_{2}\phi _{0,2}(x)+\dots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d07b975bb3a7946f33c037806f0dc0ff414a5760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.39ex; height:3.009ex;" alt="{\displaystyle F(x)=\dots +f_{-1}\phi _{0,-1}(x)+f_{0}\phi _{0,0}(x)+f_{1}\phi _{0,1}(x)+f_{2}\phi _{0,2}(x)+\dots }" loading="lazy"></dd></dl>

zugeordnet.

<div class="mw-heading mw-heading3"><h3 id="Vorwärtstransformation">Vorwärtstransformation</h3></div>

Aus dem diskreten Signal wird durch paarweises „Senkrechtstellen“ ein vektorwertiges Signal, die sogenannte Polyphasenzerlegung, erzeugt:

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<annotation encoding="application/x-tex">{\displaystyle f_{p}=\left(\dots ,\left({f_{-2} \atop f_{-1}}\right),\left({f_{0} \atop f_{1}}\right),\left({f_{2} \atop f_{3}}\right),\dots \right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/591b0d6cd1b5c74983c8528b408c9780ab1b0f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.643ex; height:6.176ex;" alt="{\displaystyle f_{p}=\left(\dots ,\left({f_{-2} \atop f_{-1}}\right),\left({f_{0} \atop f_{1}}\right),\left({f_{2} \atop f_{3}}\right),\dots \right)}" loading="lazy">.</dd></dl>

Dieser wird nun gliedweise mit der Haar-Transformationsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle H:={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80b534116322bb1751006f20d4d8568356078bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.372ex; height:6.509ex;" alt="{\displaystyle H:={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}" loading="lazy"> multipliziert

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<annotation encoding="application/x-tex">{\displaystyle \left({s \atop d}\right)=:Hf_{p}=\left(\dots ,\left({s_{-1} \atop d_{-1}}\right),\left({s_{0} \atop d_{0}}\right),\left({s_{1} \atop d_{1}}\right),\dots \right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/588edb0b317d6225970f51d1cb80e45ac1940539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.21ex; height:6.176ex;" alt="{\displaystyle \left({s \atop d}\right)=:Hf_{p}=\left(\dots ,\left({s_{-1} \atop d_{-1}}\right),\left({s_{0} \atop d_{0}}\right),\left({s_{1} \atop d_{1}}\right),\dots \right)}" loading="lazy">,</dd></dl>

dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}={\frac {f_{2k+1}+f_{2k}}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{k}={\frac {f_{2k+1}+f_{2k}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae456189397e2be67617b5ede1161966994ce6d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.155ex; height:6.343ex;" alt="{\displaystyle s_{k}={\frac {f_{2k+1}+f_{2k}}{\sqrt {2}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{k}={\frac {f_{2k+1}-f_{2k}}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{k}={\frac {f_{2k+1}-f_{2k}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3bf636b91130d99129e7d04ffe7e5e61c9bb731.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.273ex; height:6.343ex;" alt="{\displaystyle d_{k}={\frac {f_{2k+1}-f_{2k}}{\sqrt {2}}}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Rücktransformation">Rücktransformation</h3></div>

Wir erhalten ein Mittelwertsignal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> und ein Differenzsignal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">, aus denen durch einfache Umkehr der vorgenommenen Schritte das Ausgangssignal zurückgewonnen werden kann:

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<annotation encoding="application/x-tex">{\displaystyle f_{2k}={\frac {s_{k}-d_{k}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90448308871fc71f9c3bdb15727a3eef11227bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.302ex; height:6.343ex;" alt="{\displaystyle f_{2k}={\frac {s_{k}-d_{k}}{\sqrt {2}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2k+1}={\frac {s_{k}+d_{k}}{\sqrt {2}}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/476af87c4be46ccff5c86485558cea91e23659a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.402ex; height:6.343ex;" alt="{\displaystyle f_{2k+1}={\frac {s_{k}+d_{k}}{\sqrt {2}}}}" loading="lazy"></dd></dl>

Ist die Schwankung von Glied zu Glied im Ausgangssignal durch ein kleines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> jedoch durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon /{\sqrt {2}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fe17b3b8a377246ffcfc8d3985b45bd8892c5fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.205ex; height:3.176ex;" alt="{\displaystyle \epsilon /{\sqrt {2}}}" loading="lazy">. Ein glattes Signal wird also in ein immer noch glattes Signal halber Abtastfrequenz und in ein kleines Differenzsignal zerlegt. Dies ist der Ausgangspunkt für die <a href="Wavelet-Kompression" title="Wavelet-Kompression">Wavelet-Kompression</a>.

<div class="mw-heading mw-heading3"><h3 id="Rekursive_Filterbank">Rekursive Filterbank</h3></div>

Wir können den Vorgang wiederholen, indem wir s zum Ausgangssignal erklären und mit obigem Vorgehen zerlegen, wir erhalten eine Folge von Zerlegungen

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/640a2ea5ea628ce0bc033ac60e178a516c0cd240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.179ex; height:2.676ex;" alt="{\displaystyle s^{k}}" loading="lazy"> hat ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy">-tel der ursprünglichen Abtastfrequenz und eine durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k/2}\epsilon }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7eeec82e5cef189dcac93051bde1e9eb30bc4b28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.307ex; height:2.676ex;" alt="{\displaystyle d^{k}}" loading="lazy"> hat ebenfalls ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy">-tel der ursprünglichen Abtastfrequenz und durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-k/2}\epsilon }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e60d7c3d0e4f423a7e052e5a1d445441f61ff9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.118ex; height:2.843ex;" alt="{\displaystyle 2^{-k/2}\epsilon }" loading="lazy"> beschränkte Glieder.

<div class="mw-heading mw-heading3"><h3 id="Interpretation">Interpretation</h3></div>

Als <a href="Digitales_Signal" class="mw-redirect" title="Digitales Signal">diskretes Signal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><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"> wird meist eine reelle Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{n})}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a32c571f35fd41f5ef78fecc130ac39cd2108c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle (f_{n})}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> mit endlicher Energie betrachtet,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }\,|f_{n}|^{2}<\infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb3c4d495fdd9725f35d83859ec8dda818630127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.088ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }\,|f_{n}|^{2}<\infty }" loading="lazy">.</dd></dl>

Unter diesen gibt es einige sehr einfache Folgen δn, Kronecker- oder Dirac-Delta genannt, eine für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in Z}">

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<annotation encoding="application/x-tex">{\displaystyle n\in Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a95c794b34fadbd50ee66b5e2d64ddf4caa6f15d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle n\in Z}" loading="lazy">. Für deren Folgenglieder gilt, dass das jeweils <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><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">-te den Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> hat, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{n}{}_{n}=1}">

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<annotation encoding="application/x-tex">{\displaystyle \delta ^{n}{}_{n}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd5361a1ddaf7ee1f1a52257974bcdf134fe2b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.752ex; height:2.676ex;" alt="{\displaystyle \delta ^{n}{}_{n}=1}" loading="lazy">, und alle anderen den Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{n}{}_{k}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \delta ^{n}{}_{k}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2382b823fc7d5e7a575ab166b2bca00fd8a8b72c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.622ex; height:2.676ex;" alt="{\displaystyle \delta ^{n}{}_{k}=0}" loading="lazy"> falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\not =n}">

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<annotation encoding="application/x-tex">{\displaystyle k\not =n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c51ec82c55a70c10dc45ddf04f46e334da11180c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.704ex; height:2.676ex;" alt="{\displaystyle k\not =n}" loading="lazy">.

Jetzt können wir jedes Signal trivial als <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihe</a> im <a href="Hilbertraum" title="Hilbertraum">Signalraum</a> schreiben

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<annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=-\infty }^{\infty }\,f_{n}\cdot \delta ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50a099ffe233e4829481541fce99087c71e0e3c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.647ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=-\infty }^{\infty }\,f_{n}\cdot \delta ^{n}}" loading="lazy"></dd></dl>

oder als Summe zweier Reihen

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<annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=-\infty }^{\infty }\,f_{2n}\cdot \delta ^{2n}+\sum _{n=-\infty }^{\infty }\,f_{2n+1}\cdot \delta ^{2n+1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/152507b8ce4e0bb84957150511ddfb51d07c854e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.246ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=-\infty }^{\infty }\,f_{2n}\cdot \delta ^{2n}+\sum _{n=-\infty }^{\infty }\,f_{2n+1}\cdot \delta ^{2n+1}}" loading="lazy">.</dd></dl>

In vielen praktisch relevanten Signalklassen, z. B. bei <a href="%C3%9Cberabtastung" title="Überabtastung">überabgetasteten</a> <a href="Bandbreite" title="Bandbreite">bandbeschränkten</a> kontinuierlichen Signalen, sind Werte benachbarter Folgenglieder auch benachbart, d. h. im Allgemeinen liegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2n}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{2n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7249a5e28d117b6377690b66299388f8b2ecf257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.18ex; height:2.509ex;" alt="{\displaystyle f_{2n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{2n+1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1ca10e2cabb4943c72c00ac213fd76e062fa38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.28ex; height:2.509ex;" alt="{\displaystyle f_{2n+1}}" loading="lazy"> dicht beisammen, relativ zu ihrem Absolutbetrag. Dies wird in der obigen Reihen aber überhaupt nicht berücksichtigt. In Mittelwert und Differenz von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2n}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{2n}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f_{2n+1}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ac+bd={\frac {1}{2}}(a+b)(c+d)+{\frac {1}{2}}(a-b)(c-d)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b54fdc92e45012762a7179c2229d768b6a71cf06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.725ex; height:5.176ex;" alt="{\displaystyle ac+bd={\frac {1}{2}}(a+b)(c+d)+{\frac {1}{2}}(a-b)(c-d)}" loading="lazy"></dd></dl>

um benachbarte Glieder der ersten Reihe bzw. korrespondierende Glieder in der zweiten Zerlegung zusammenzufassen in (skalierten) Mittelwerten und Differenzen:

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<annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=-\infty }^{\infty }\,{\frac {f_{2n}+f_{2n+1}}{\sqrt {2}}}\cdot {\frac {\delta ^{2n}+\delta ^{2n+1}}{\sqrt {2}}}+\sum _{n=-\infty }^{\infty }\,{\frac {f_{2n}-f_{2n+1}}{\sqrt {2}}}\cdot {\frac {\delta ^{2n}-\delta ^{2n+1}}{\sqrt {2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/467dd687d0a5c9c05635fee8aa07d08af45b6cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.7ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=-\infty }^{\infty }\,{\frac {f_{2n}+f_{2n+1}}{\sqrt {2}}}\cdot {\frac {\delta ^{2n}+\delta ^{2n+1}}{\sqrt {2}}}+\sum _{n=-\infty }^{\infty }\,{\frac {f_{2n}-f_{2n+1}}{\sqrt {2}}}\cdot {\frac {\delta ^{2n}-\delta ^{2n+1}}{\sqrt {2}}}}" loading="lazy"></dd></dl>

Jetzt führen wir neue Bezeichnungen ein:

<ul><li>die neuen Basisfolgen</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle a^{n}:={\frac {\delta ^{2n}+\delta ^{2n+1}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5788ef12a1f241dccce2cb400b84f6cbc3ffffb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.159ex; height:6.676ex;" alt="{\displaystyle a^{n}:={\frac {\delta ^{2n}+\delta ^{2n+1}}{\sqrt {2}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}:={\frac {\delta ^{2n}-\delta ^{2n+1}}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n}:={\frac {\delta ^{2n}-\delta ^{2n+1}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e5fc9d2036835631c9cd24a2d8207bacfe1db05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.927ex; height:6.676ex;" alt="{\displaystyle b^{n}:={\frac {\delta ^{2n}-\delta ^{2n+1}}{\sqrt {2}}}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>mit den neuen transformierten Koeffizienten</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle s_{n}:={\frac {f_{2n}+f_{2n+1}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4757a0f071d6eb30ac261c874dd14068b1b66ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.191ex; height:6.343ex;" alt="{\displaystyle s_{n}:={\frac {f_{2n}+f_{2n+1}}{\sqrt {2}}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}:={\frac {f_{2n}-f_{2n+1}}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{n}:={\frac {f_{2n}-f_{2n+1}}{\sqrt {2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb2f290cafe75f0a719a75439d8c94612c1ac20f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.309ex; height:6.343ex;" alt="{\displaystyle d_{n}:={\frac {f_{2n}-f_{2n+1}}{\sqrt {2}}}}" loading="lazy">.</dd></dl></dd></dl>

Wir erhalten somit die Zerlegung der Haar-Wavelet-Transformation

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<annotation encoding="application/x-tex">{\displaystyle f=\sum _{n=-\infty }^{\infty }\,s_{n}\cdot a^{n}+\sum _{n=-\infty }^{\infty }\,d_{n}\cdot b^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a32cfbd1226e3c66fcdfc3097ba540c20087b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.898ex; height:6.843ex;" alt="{\displaystyle f=\sum _{n=-\infty }^{\infty }\,s_{n}\cdot a^{n}+\sum _{n=-\infty }^{\infty }\,d_{n}\cdot b^{n}}" loading="lazy">.</dd></dl>

und mittels des unendlichen euklidischen <a href="Skalarprodukt" title="Skalarprodukt">Skalarproduktes</a> können wir schreiben

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<annotation encoding="application/x-tex">{\displaystyle s_{n}=\langle f,\,a^{n}\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74428ccde4fa0602c389f28b33fa3df9e576ad99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.365ex; height:2.843ex;" alt="{\displaystyle s_{n}=\langle f,\,a^{n}\rangle }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{n}=\langle f,\,b^{n}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle d_{n}=\langle f,\,b^{n}\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f597056280e288814694c24a579635f991e119b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.251ex; height:2.843ex;" alt="{\displaystyle d_{n}=\langle f,\,b^{n}\rangle }" loading="lazy">.</dd></dl>

Die letzten drei Identitäten beschreiben eine „Conjugate Quadrature Filterbank (CQF)“, welche so auch für allgemeinere Basisfolgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2687184a7698e75db65a25bea7afd207bff3d03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f8f52cd26bb201e02c8d1b3619a3a682f44dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.343ex;" alt="{\displaystyle b^{n}}" loading="lazy"> definiert werden kann. Die Basisfolgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2687184a7698e75db65a25bea7afd207bff3d03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}" loading="lazy"> entstehen alle durch Verschiebung um das jeweilige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

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<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle a^{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83a894726dd8a070c22ea9b3b52e93b3840b7c41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{0}}" loading="lazy">, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f8f52cd26bb201e02c8d1b3619a3a682f44dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.343ex;" alt="{\displaystyle b^{n}}" loading="lazy"> durch Verschiebung aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{0}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5456c9d8bf077b24d003cf08da7a63403e24a8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.052ex; height:2.676ex;" alt="{\displaystyle b^{0}}" loading="lazy">. Weiteres dazu im Artikel <a href="Daubechies-Wavelets" title="Daubechies-Wavelets">Daubechies-Wavelets</a>.

Nun enthält die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=(s^{n})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=(s^{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/242c90e09f5b7096117592e89cf1f4760cc0f910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.307ex; height:2.843ex;" alt="{\displaystyle s=(s^{n})}" loading="lazy"> eine geglättete Version des Ausgangssignals bei halber <a href="Abtastrate" title="Abtastrate">Abtastrate</a>, man kann also auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> nach dieser Vorschrift zerlegen und dieses Vorgehen über eine bestimmte Tiefe rekursiv fortsetzen. Aus einem Ausgangssignal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{0}=f}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s^{0}=f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58f356ed2116edd6ae1ca13d76ffaa589feecb8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.522ex; height:3.009ex;" alt="{\displaystyle s^{0}=f}" loading="lazy"> werden also nacheinander die Tupel

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s^{1},d^{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62ce657684789e58ec85485f6b381164bdba7de7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.26ex; height:3.176ex;" alt="{\displaystyle (s^{1},d^{1})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{2},d^{2},d^{1})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s^{2},d^{2},d^{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f04909e09d417944895bf7e8bce026681495b6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.567ex; height:3.176ex;" alt="{\displaystyle (s^{2},d^{2},d^{1})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{3},d^{3},d^{2},d^{1})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s^{3},d^{3},d^{2},d^{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/649bb559573de4c34f2c27d54fcb2c45628e2ce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.873ex; height:3.176ex;" alt="{\displaystyle (s^{3},d^{3},d^{2},d^{1})}" loading="lazy">, …</dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><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"> endlich, also fast überall Null, mit Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, dann haben die Folgen in der Zerlegung im Wesentlichen, d. h. bis auf additive Konstanten, die Längen

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {N}{2}},{\tfrac {N}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/394c017c7d6c54e0656cad5334ff63f5af9bd9b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.4ex; height:4.843ex;" alt="{\displaystyle \left({\tfrac {N}{2}},{\tfrac {N}{2}}\right)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {N}{4}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {N}{4}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7043362f97571e7a966926d74445eecd2a116fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.729ex; height:4.843ex;" alt="{\displaystyle \left({\tfrac {N}{4}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\tfrac {N}{8}},{\tfrac {N}{8}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\tfrac {N}{8}},{\tfrac {N}{8}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e56d2450860162f46029e747a57d142f6ebe8cb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.059ex; height:4.843ex;" alt="{\displaystyle \left({\tfrac {N}{8}},{\tfrac {N}{8}},{\tfrac {N}{4}},{\tfrac {N}{2}}\right)}" loading="lazy">, …</dd></dl>

so dass die Gesamtzahl wesentlicher Koeffizienten erhalten bleibt. Die Folgen in der Zerlegung eignen

sich meist besser zur Weiterverarbeitung wie Kompression oder Suche nach bestimmtem Merkmalen als

das rohe Ausgangssignal.

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

Die Polyphasenzerlegung des Ausgangssignals kann auch zu einer anderen Blockgröße s als 2 erfolgen, von der entsprechenden Haar-Matrix ist zu fordern, dass sie eine <a href="Orthogonale_Matrix" title="Orthogonale Matrix">orthogonale Matrix</a> ist und ihre erste Zeile nur aus Einträgen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {s}}}">

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {s}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b44fb3b762b7dc0d88244412ed4031d033e0d1da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.351ex; height:3.009ex;" alt="{\displaystyle 1/{\sqrt {s}}}" loading="lazy"> besteht. Diese Anforderung erfüllen die Matrizen der <a href="Diskrete_Kosinustransformation" title="Diskrete Kosinustransformation">diskreten Kosinustransformation</a> und die der <a href="Walsh-Hadamard-Transformation" class="mw-redirect" title="Walsh-Hadamard-Transformation">Walsh-Hadamard-Transformation</a>.

Die Haar-Wavelet-Transformation entspricht einer <a href="Diskrete_Kosinustransformation" title="Diskrete Kosinustransformation">diskreten Kosinustransformation</a> zur Blockgröße <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2127cfe51e8f7f982b180284ca0912f37ff38356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=2}" loading="lazy">, welche im Bild=Pixelrechteck nacheinander in horizontaler und vertikaler Richtung angewandt wird.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Viola-Jones-Methode" title="Viola-Jones-Methode">Viola-Jones-Methode</a></li></ul>

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

<ul><li><a href="Alfr%C3%A9d_Haar" title="Alfréd Haar">Alfréd Haar</a>: Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen 69, 331–371, 1910, <a href="https://doi.org/10.1007/BF01456927" class="extiw external" title="doi:10.1007/BF01456927">doi:10.1007/BF01456927</a>, insbesondere Kapitel 3 (ab S. 361).</li></ul>

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

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Haar_wavelet?uselang=de">Commons: Haar-Wavelet</a> – Sammlung von Bildern, Videos und Audiodateien</div>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Alfred Haar: <cite style="font-style:italic">Zur Theorie der orthogonalen Funktionensysteme</cite>. In: <cite style="font-style:italic"><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></cite>. 69. Jahrgang, Nr. 3, 1910, S. 331–371, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01456326">10.1007/BF01456326</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Haar-Wavelet&rft.atitle=Zur+Theorie+der+orthogonalen+Funktionensysteme&rft.au=Alfred%26%2332%3BHaar&rft.date=1910&rft.doi=10.1007%2FBF01456326&rft.genre=journal&rft.issue=3&rft.jtitle=Mathematische+Annalen&rft.pages=331-371&rft.volume=69.+Jahrgang" style="display:none">

</li>

</ol></div><div><div class="zim-footer">

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