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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mittenfrequenz</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>Mittenfrequenz</b> <i>f<sub>0</sub></i> ist das <a href="Geometrisches_Mittel" title="Geometrisches Mittel">geometrische Mittel</a> der unteren&nbsp;<i>f<sub>1</sub></i> und der oberen&nbsp;<i>f<sub>2</sub></i> <a href="Grenzfrequenz" title="Grenzfrequenz">Grenzfrequenz</a> (Übergangsfrequenz) eines <a href="Frequenzband" title="Frequenzband">Frequenzbands</a> mit einer bestimmten Filter<a href="Bandbreite" title="Bandbreite">bandbreite</a>, auch bekannt unter dem Begriff <a href="Bandpass" title="Bandpass">Bandpass</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Grundlagen">Grundlagen</h2></div>
<p>Die Mittenfrequenz der Filterbandbreite B&nbsp;=&nbsp;<i>f<sub>2</sub></i>&nbsp;−&nbsp;<i>f<sub>1</sub></i> wird berechnet aus:
</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_{0}={\sqrt {f_{1}\cdot f_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
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<mn>0</mn>
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f_{0}={\sqrt {f_{1}\cdot f_{2}}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df8fdc2cc21eb12ab9a52f0cc0cc974e6bed9fd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.682ex; height:3.509ex;" alt="{\displaystyle f_{0}={\sqrt {f_{1}\cdot f_{2}}}}" loading="lazy"></span></dd></dl>
<p>Oft wird fälschlicherweise mit dem <a href="Arithmetisches_Mittel" title="Arithmetisches Mittel">arithmetischen Mittel</a> gerechnet, obwohl die <a href="Frequenz" title="Frequenz">Frequenzen</a> in den Frequenzbändern <a href="Logarithmisch" class="mw-redirect" title="Logarithmisch">logarithmisch</a> zusammenhängen. Zum Beispiel ist die Mittenfrequenz der Telefonaudiofrequenzen von 300&nbsp;Hz bis 3400&nbsp;Hz <i>nicht</i> (3400&nbsp;Hz&nbsp;+&nbsp;300&nbsp;Hz)&nbsp;/&nbsp;2 = 1850&nbsp;Hz, sondern <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 {\sqrt {300\;{\rm {{Hz}\cdot 3400\;{\rm {Hz}}}}}}\approx 1010\;{\rm {Hz}}}">
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<mn>300</mn>
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<mo>⋅<!-- ⋅ --></mo>
<mn>3400</mn>
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<mo>≈<!-- ≈ --></mo>
<mn>1010</mn>
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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {300\;{\rm {{Hz}\cdot 3400\;{\rm {Hz}}}}}}\approx 1010\;{\rm {Hz}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c905e40257486bfdc45edb1e44f95793364c052c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.762ex; height:3.009ex;" alt="{\displaystyle {\sqrt {300\;{\rm {{Hz}\cdot 3400\;{\rm {Hz}}}}}}\approx 1010\;{\rm {Hz}}}" loading="lazy"></span>.
</p><p>Die Mittenfrequenz linear angeordneter Spektren, z.&nbsp;B. in der <a href="Antennentechnik" class="mw-redirect" title="Antennentechnik">Antennentechnik</a>, ist dennoch als arithmetisches Mittel zu berechnen:
</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_{0}={\frac {f_{1}+f_{2}}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
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<mn>0</mn>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>+</mo>
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<mi>f</mi>
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<mn>2</mn>
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<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle f_{0}={\frac {f_{1}+f_{2}}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96f71423a0524d6b36e42ffe4fbadea1e39ced8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.355ex; height:5.343ex;" alt="{\displaystyle f_{0}={\frac {f_{1}+f_{2}}{2}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<p>Durch die Definition der Mittenfrequenz sind die Verhältnisse der Grenzfrequenzen zur Mittenfrequenz gleich:
</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 {\frac {f_{1}}{f_{0}}}={\sqrt {\frac {f_{1}}{f_{2}}}}={\frac {f_{0}}{f_{2}}}}">
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<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<mn>1</mn>
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<mi>f</mi>
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<mo>=</mo>
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<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<msub>
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {f_{1}}{f_{0}}}={\sqrt {\frac {f_{1}}{f_{2}}}}={\frac {f_{0}}{f_{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f125ef243dedf41e2ad3acf030e978689798528d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.609ex; height:7.509ex;" alt="{\displaystyle {\frac {f_{1}}{f_{0}}}={\sqrt {\frac {f_{1}}{f_{2}}}}={\frac {f_{0}}{f_{2}}}}" loading="lazy"></span></dd></dl>
<p>Werden <i>f</i><sub>1</sub>, <i>f</i><sub>0</sub> und <i>f</i><sub>2</sub> auf einer logarithmischen Frequenzskala markiert, so befindet sich <i>f</i><sub>0</sub> streckenmäßig in der Mitte:
</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 \log {f_{0}}-\log {f_{1}}=\log {f_{2}}-\log {f_{0}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
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<msub>
<mi>f</mi>
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<mo>−<!-- − --></mo>
<mi>log</mi>
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<msub>
<mi>f</mi>
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<mn>1</mn>
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<mo>=</mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>f</mi>
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<mn>2</mn>
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<mo>−<!-- − --></mo>
<mi>log</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle \log {f_{0}}-\log {f_{1}}=\log {f_{2}}-\log {f_{0}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aabf0a702892a15e8936e80cba31dd5cdd06819.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.989ex; height:2.509ex;" alt="{\displaystyle \log {f_{0}}-\log {f_{1}}=\log {f_{2}}-\log {f_{0}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Verwendung_als_Näherung"><span id="Verwendung_als_N.C3.A4herung"></span>Verwendung als Näherung</h2></div>
<p>Die Bandbreite&nbsp;<i>f</i><sub>2</sub>&nbsp;−&nbsp;<i>f</i><sub>1</sub> ist häufig klein gegenüber der Mittenfrequenz. Dann kann man in guter Näherung das arithmetische Mittel zur Berechnung verwenden:
</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_{0}\approx {\frac {f_{1}+f_{2}}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>≈<!-- ≈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>f</mi>
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<mo>+</mo>
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<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle f_{0}\approx {\frac {f_{1}+f_{2}}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2400eae5f5a402126cbda31e6c32a5a8c265afd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.355ex; height:5.343ex;" alt="{\displaystyle f_{0}\approx {\frac {f_{1}+f_{2}}{2}}}" loading="lazy"></span></dd></dl>
<p>Bei vielen <a href="Mittelwellensender" title="Mittelwellensender">Mittelwellensendern</a> z.&nbsp;B. beträgt die Bandbreite nur 9&nbsp;kHz. Ein Sender, der mit 1500&nbsp;kHz angegeben ist, sendet hier also im Band von 1495,5&nbsp;kHz bis 1504,5&nbsp;kHz. Die <a href="N%C3%A4herungsformel" class="mw-redirect" title="Näherungsformel">Näherungsformel</a> ergibt
</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_{0}\approx 1500\,\mathrm {kHz} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
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<mn>0</mn>
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<mo>≈<!-- ≈ --></mo>
<mn>1500</mn>
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<annotation encoding="application/x-tex">{\displaystyle f_{0}\approx 1500\,\mathrm {kHz} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a8a271d781e03ef2bb59de50143687cade4d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.332ex; height:2.509ex;" alt="{\displaystyle f_{0}\approx 1500\,\mathrm {kHz} }" loading="lazy"></span>,</dd></dl>
<p>während man mit der genauen Formel ermittelt:
</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_{0}=1499{,}993\,\mathrm {kHz} }">
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<mi>f</mi>
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<mn>1499,993</mn>
<mspace width="thinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle f_{0}=1499{,}993\,\mathrm {kHz} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/028d27abad8344218845f8b490345fe7dc9e5a85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.466ex; height:2.509ex;" alt="{\displaystyle f_{0}=1499{,}993\,\mathrm {kHz} }" loading="lazy"></span>.</dd></dl>
<p>Der mit der Näherungsformel berechnete Wert ist stets zu groß. Wenn man die Bandbreite mit&nbsp;<i>B</i> bezeichnet, beträgt die Abweichung der Näherungsformel ungefähr
</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 \Delta f\approx {\frac {B^{2}}{8\cdot f_{0}}}}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Delta f\approx {\frac {B^{2}}{8\cdot f_{0}}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6c04557e802109a97a53d6a94471432c7082866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.184ex; height:6.176ex;" alt="{\displaystyle \Delta f\approx {\frac {B^{2}}{8\cdot f_{0}}}}" loading="lazy"></span>,</dd></dl>
<p>im angegebenen Beispiel also weniger als 7&nbsp;Hz.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Michael_Dickreiter" title="Michael Dickreiter">Michael Dickreiter</a>, Volker Dittel, Wolfgang Hoeg, Martin Wöhr (Hrsg.), „Handbuch der Tonstudiotechnik“, 8., überarbeitete und erweiterte Auflage, 2 Bände, Verlag: Walter de Gruyter, Berlin/Boston, 2014, ISBN 978-3-11-028978-7 oder e-ISBN 978-3-11-031650-6</li>
<li>Gregor Häberle, Heinz Häberle, Thomas Kleiber: <i>Fachkunde Radio-, Fernseh- und Funkelektronik.</i> 3. Auflage, Verlag Europa-Lehrmittel, Haan-Gruiten, 1996, ISBN 3-8085-3263-7</li>
<li>Karl Hermann Huber: <i>Filtern und Sieben von Tonfrequenzen.</i> 1. Auflage, Frech Verlag, Stuttgart, 1974, ISBN 3-7724-0162-7</li>
<li>Warren L. Stutzman: <i>Antenna Theory and Design</i>. 3. Auflage, Wiley Verlag, Weinheim, 2012, ISBN 978-0-470-57664-9</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Bandspreiztechnik" class="mw-redirect" title="Bandspreiztechnik">Bandspreiztechnik</a></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.sengpielaudio.com/Rechner-geommittel.htm">Mittenfrequenz als geometrisches Mittel der Grenzfrequenzen - im Vergleich zum arithmetischen Mittel</a></li>
<li><a rel="nofollow" class="external text" href="http://www.sengpielaudio.com/Rechner-bandbreite.htm">Umrechnung: 'Bandbreite in Oktaven' <i>N</i> in Gütefaktor <i>Q</i> und Gütefaktor <i>Q</i> in 'Bandbreite in Oktaven' <i>N</i></a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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