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<span class="vector-toc-numb">2</span> <span>Resonance</span> </div> </a> <button aria-controls="toc-Resonance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Resonance subsection</span> </button> <ul id="toc-Resonance-sublist" class="vector-toc-list"> <li id="toc-Electrical_resonance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical_resonance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Electrical resonance</span> </div> </a> <ul id="toc-Electrical_resonance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acoustic_resonance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acoustic_resonance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Acoustic resonance</span> </div> </a> <ul id="toc-Acoustic_resonance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Early_multiplexing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_multiplexing"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Early multiplexing</span> </div> </a> <ul id="toc-Early_multiplexing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transmission_line_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transmission_line_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Transmission line theory</span> </div> </a> <ul id="toc-Transmission_line_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_filters" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Image_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Image filters</span> </div> </a> <ul id="toc-Image_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Network_synthesis_filters" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Network_synthesis_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Network synthesis filters</span> </div> </a> <ul id="toc-Network_synthesis_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_method_versus_synthesis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Image_method_versus_synthesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Image method versus synthesis</span> </div> </a> <ul id="toc-Image_method_versus_synthesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Realisability_and_equivalence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Realisability_and_equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Realisability and equivalence</span> </div> </a> <ul id="toc-Realisability_and_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Approximation</span> </div> </a> <button aria-controls="toc-Approximation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Approximation subsection</span> </button> <ul id="toc-Approximation-sublist" class="vector-toc-list"> <li id="toc-Butterworth_filter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Butterworth_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Butterworth filter</span> </div> </a> <ul id="toc-Butterworth_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Insertion-loss_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Insertion-loss_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Insertion-loss method</span> </div> </a> <ul id="toc-Insertion-loss_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elliptic_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elliptic_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Elliptic filters</span> </div> </a> <ul id="toc-Elliptic_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Other methods</span> </div> </a> <ul id="toc-Other_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_notable_developments_and_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_notable_developments_and_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Other notable developments and applications</span> </div> </a> <button aria-controls="toc-Other_notable_developments_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other notable developments and applications subsection</span> </button> <ul id="toc-Other_notable_developments_and_applications-sublist" class="vector-toc-list"> <li id="toc-Mechanical_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanical_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Mechanical filters</span> </div> </a> <ul id="toc-Mechanical_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributed-element_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributed-element_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Distributed-element filters</span> </div> </a> <ul id="toc-Distributed-element_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transversal_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transversal_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Transversal filters</span> </div> </a> <ul id="toc-Transversal_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matched_filter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matched_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Matched filter</span> </div> </a> <ul id="toc-Matched_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Filters_for_control_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Filters_for_control_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Filters for control systems</span> </div> </a> <ul id="toc-Filters_for_control_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modern_practice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modern_practice"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Modern practice</span> </div> </a> <ul id="toc-Modern_practice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown 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href="https://de.wikipedia.org/wiki/Analogfilter" title="Analogfilter – German" lang="de" hreflang="de" data-title="Analogfilter" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Filtro_anal%C3%B3gico" title="Filtro analógico – Spanish" lang="es" hreflang="es" data-title="Filtro analógico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Iragazki_analogiko" title="Iragazki analogiko – Basque" lang="eu" hreflang="eu" data-title="Iragazki analogiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Filtre_analogique_passif" title="Filtre analogique passif – French" lang="fr" hreflang="fr" data-title="Filtre analogique passif" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%B0%E0%A5%82%E0%A4%AA_%E0%A4%AB%E0%A4%BF%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%B0" title="अनुरूप फिल्टर – Hindi" lang="hi" hreflang="hi" data-title="अनुरूप फिल्टर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%BE%D0%B3%D0%BE%D0%B2%D1%8B%D0%B9_%D1%84%D0%B8%D0%BB%D1%8C%D1%82%D1%80" title="Аналоговый фильтр – Russian" lang="ru" hreflang="ru" data-title="Аналоговый фильтр" 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Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the history and development of passive linear analogue filters used in electronics. For linear filters in general, see <a href="/wiki/Linear_filter" title="Linear filter">Linear filter</a>. For electronic filters in general, see <a href="/wiki/Electronic_filter" title="Electronic filter">Electronic filter</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Filter used in signal processing on continuous-time signals</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size:130%"><a href="/wiki/Electronic_filter" title="Electronic filter">Linear analog<br />electronic filters</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)"><a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/Butterworth_filter" title="Butterworth filter">Butterworth filter</a></li> <li><a href="/wiki/Chebyshev_filter" title="Chebyshev filter">Chebyshev filter</a></li> <li><a href="/wiki/Elliptic_filter" title="Elliptic filter">Elliptic (Cauer) filter</a></li> <li><a href="/wiki/Bessel_filter" title="Bessel filter">Bessel filter</a></li> <li><a href="/wiki/Gaussian_filter" title="Gaussian filter">Gaussian filter</a></li> <li><a href="/wiki/Optimum_%22L%22_filter" title="Optimum "L" filter">Optimum "L" (Legendre) filter</a></li> <li><a href="/wiki/Linkwitz%E2%80%93Riley_filter" title="Linkwitz–Riley filter">Linkwitz–Riley filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)"><a href="/wiki/Composite_image_filter" title="Composite image filter">Image impedance filters</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/Constant_k_filter" title="Constant k filter">Constant k filter</a></li> <li><a href="/wiki/M-derived_filter" title="M-derived filter">m-derived filter</a></li> <li><a href="/wiki/General_mn-type_image_filter" title="General mn-type image filter">General image filters</a></li> <li><a href="/wiki/Zobel_network" title="Zobel network">Zobel network</a> (constant R) filter</li> <li><a href="/wiki/Lattice_phase_equaliser" title="Lattice phase equaliser">Lattice filter</a> (all-pass)</li> <li><a href="/wiki/Bridged_T_delay_equaliser" title="Bridged T delay equaliser">Bridged T delay equaliser</a> (all-pass)</li> <li><a href="/wiki/Composite_image_filter" title="Composite image filter">Composite image filter</a></li> <li><a href="/wiki/Mm%27-type_filter" title="Mm'-type filter">mm'-type filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)">Simple filters</div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/RC_circuit" title="RC circuit">RC filter</a></li> <li><a href="/wiki/RL_circuit" title="RL circuit">RL filter</a></li> <li><a href="/wiki/LC_circuit" title="LC circuit">LC filter</a></li> <li><a href="/wiki/RLC_circuit" title="RLC circuit">RLC filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar" style="padding-top:0;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Linear_analog_electronic_filter" title="Template:Linear analog electronic filter"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Linear_analog_electronic_filter" title="Template talk:Linear analog electronic filter"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_analog_electronic_filter" title="Special:EditPage/Template:Linear analog electronic filter"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Analogue <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filters</a></b> are a basic building block of <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> much used in <a href="/wiki/Electronics" title="Electronics">electronics</a>. Amongst their many applications are the separation of an audio signal before application to <a href="/wiki/Bass_(music)" class="mw-redirect" title="Bass (music)">bass</a>, <a href="/wiki/Mid-range_speaker" title="Mid-range speaker">mid-range</a>, and <a href="/wiki/Tweeter" title="Tweeter">tweeter</a> <a href="/wiki/Loudspeaker" title="Loudspeaker">loudspeakers</a>; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen <a href="/wiki/Radio_station" class="mw-redirect" title="Radio station">radio station</a> in a <a href="/wiki/Radio_receiver" title="Radio receiver">radio receiver</a> and rejection of others. </p><p>Passive linear electronic analogue filters are those filters which can be described with <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a> (linear); they are composed of <a href="/wiki/Capacitor" title="Capacitor">capacitors</a>, <a href="/wiki/Inductor" title="Inductor">inductors</a> and, sometimes, <a href="/wiki/Resistor" title="Resistor">resistors</a> (<a href="/wiki/Passive_component" class="mw-redirect" title="Passive component">passive</a>) and are designed to operate on continuously varying <a href="/wiki/Analogue_signal" class="mw-redirect" title="Analogue signal">analogue signals</a>. There are many <a href="/wiki/Linear_filter" title="Linear filter">linear filters</a> which are not analogue in implementation (<a href="/wiki/Digital_filter" title="Digital filter">digital filter</a>), and there are many <a href="/wiki/Electronic_filter" title="Electronic filter">electronic filters</a> which may not have a passive topology – both of which may have the same <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> of the filters described in this article. Analogue filters are most often used in wave filtering applications, that is, where it is required to pass particular frequency components and to reject others from analogue (<a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">continuous-time</a>) signals. </p><p>Analogue filters have played an important part in the development of electronics. Especially in the field of <a href="/wiki/Telecommunication" class="mw-redirect" title="Telecommunication">telecommunications</a>, filters have been of crucial importance in a number of technological breakthroughs and have been the source of enormous profits for telecommunications companies. It should come as no surprise, therefore, that the early development of filters was intimately connected with <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a>. Transmission line theory gave rise to filter theory, which initially took a very similar form, and the main application of filters was for use on telecommunication transmission lines. However, the arrival of <a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">network synthesis</a> techniques greatly enhanced the degree of control of the designer. </p><p>Today, it is often preferred to carry out filtering in the digital domain where complex algorithms are much easier to implement, but analogue filters do still find applications, especially for low-order simple filtering tasks and are often still the norm at higher frequencies where digital technology is still impractical, or at least, less cost effective. Wherever possible, and especially at low frequencies, analogue filters are now implemented in a <a href="/wiki/Electronic_filter_topology" title="Electronic filter topology">filter topology</a> which is <a href="/wiki/Active_component" class="mw-redirect" title="Active component">active</a> in order to avoid the wound components (i.e. inductors, transformers, etc.) required by <a href="/wiki/Passive_component" class="mw-redirect" title="Passive component">passive</a> topology. </p><p>It is possible to design linear analogue <a href="/wiki/Mechanical_filter" title="Mechanical filter">mechanical filters</a> using mechanical components which filter mechanical vibrations or <a href="/wiki/Acoustics" title="Acoustics">acoustic</a> waves. While there are few applications for such devices in mechanics per se, they can be used in electronics with the addition of <a href="/wiki/Transducer" title="Transducer">transducers</a> to convert to and from the electrical domain. Indeed, some of the earliest ideas for filters were acoustic resonators because the electronics technology was poorly understood at the time. In principle, the design of such filters can be achieved entirely in terms of the electronic counterparts of mechanical quantities, with <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> and <a href="/wiki/Heat_energy" class="mw-redirect" title="Heat energy">heat energy</a> corresponding to the energy in inductors, capacitors and resistors respectively. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_overview">Historical overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=1" title="Edit section: Historical overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three main stages in the history of <b>passive analogue filter development</b>: </p> <ol><li><b>Simple filters</b>. The frequency dependence of electrical response was known for capacitors and inductors from very early on. The resonance phenomenon was also familiar from an early date and it was possible to produce simple, single-branch filters with these components. Although attempts were made in the 1880s to apply them to <a href="/wiki/Telegraphy" title="Telegraphy">telegraphy</a>, these designs proved inadequate for successful <a href="/wiki/Frequency-division_multiplexing" title="Frequency-division multiplexing">frequency-division multiplexing</a>. Network analysis was not yet powerful enough to provide the theory for more complex filters and progress was further hampered by a general failure to understand the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> nature of signals.</li> <li><b><a href="/wiki/Composite_image_filter" title="Composite image filter">Image filters</a></b>. Image filter theory grew out of transmission line theory and the design proceeded in a similar manner to transmission line analysis. For the first time filters could be produced that had precisely controllable <a href="/wiki/Passband" title="Passband">passbands</a> and other parameters. These developments took place in the 1920s and filters produced to these designs were still in widespread use in the 1980s, only declining as the use of analogue telecommunications has declined. Their immediate application was the economically important development of frequency division multiplexing for use on <a href="/wiki/Long_line_(telecommunications)" title="Long line (telecommunications)">intercity and international lines</a>.</li> <li><b><a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></b>. The mathematical bases of network synthesis were laid in the 1930s and 1940s. After World War II, network synthesis became the primary tool of <a href="/wiki/Filter_design" title="Filter design">filter design</a>. Network synthesis put filter design on a firm mathematical foundation, freeing it from the mathematically sloppy techniques of image design and severing the connection with physical lines. The essence of network synthesis is that it produces a design that will (at least if implemented with ideal components) accurately reproduce the response originally specified in <a href="/wiki/Black_box" title="Black box">black box</a> terms.</li></ol> <p>Throughout this article the letters R, L, and C are used with their usual meanings to represent <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, <a href="/wiki/Inductance" title="Inductance">inductance</a>, and <a href="/wiki/Capacitance" title="Capacitance">capacitance</a>, respectively. In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a>, any 2-terminal<sup id="cite_ref-pole_1-0" class="reference"><a href="#cite_note-pole-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> combination of RLC elements and in some sections D is used for the rarely seen quantity <a href="/wiki/Elastance" title="Elastance">elastance</a>, which is the inverse of capacitance. </p> <div class="mw-heading mw-heading2"><h2 id="Resonance">Resonance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=2" title="Edit section: Resonance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Early filters utilised the phenomenon of <a href="/wiki/Resonance" title="Resonance">resonance</a> to filter signals. Although <a href="/wiki/Electrical_resonance" title="Electrical resonance">electrical resonance</a> had been investigated by researchers from a very early stage, it was at first not widely understood by electrical engineers. Consequently, the much more familiar concept of <a href="/wiki/Acoustic_resonance" title="Acoustic resonance">acoustic resonance</a> (which in turn, can be explained in terms of the even more familiar <a href="/wiki/Mechanical_resonance" title="Mechanical resonance">mechanical resonance</a>) found its way into filter design ahead of electrical resonance.<sup id="cite_ref-Lund24_2-0" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Resonance can be used to achieve a filtering effect because the resonant device will respond to frequencies at, or near, to the resonant frequency but will not respond to frequencies far from resonance. Hence frequencies far from resonance are filtered out from the output of the device.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electrical_resonance">Electrical resonance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=3" title="Edit section: Electrical resonance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Oudin_coil.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oudin_coil.png/260px-Oudin_coil.png" decoding="async" width="260" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oudin_coil.png/390px-Oudin_coil.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Oudin_coil.png/520px-Oudin_coil.png 2x" data-file-width="967" data-file-height="748" /></a><figcaption>A 1915 example of an early type of resonant circuit known as an <a href="/wiki/Oudin_coil" title="Oudin coil">Oudin coil</a> which uses Leyden jars for the capacitance.</figcaption></figure> <p>Resonance was noticed early on in experiments with the <a href="/wiki/Leyden_jar" title="Leyden jar">Leyden jar</a>, invented in 1746. The Leyden jar stores electricity due to its <a href="/wiki/Capacitance" title="Capacitance">capacitance</a>, and is, in fact, an early form of capacitor. When a Leyden jar is discharged by allowing a spark to jump between the electrodes, the discharge is oscillatory. This was not suspected until 1826, when <a href="/wiki/Felix_Savary" class="mw-redirect" title="Felix Savary">Felix Savary</a> in France, and later (1842) <a href="/wiki/Joseph_Henry" title="Joseph Henry">Joseph Henry</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> in the US noted that a steel needle placed close to the discharge does not always magnetise in the same direction. They both independently drew the conclusion that there was a transient oscillation dying with time.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> in 1847 published his important work on conservation of energy<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> in part of which he used those principles to explain why the oscillation dies away, that it is the resistance of the circuit which dissipates the energy of the oscillation on each successive cycle. Helmholtz also noted that there was evidence of oscillation from the <a href="/wiki/Electrolysis" title="Electrolysis">electrolysis</a> experiments of <a href="/wiki/William_Hyde_Wollaston" title="William Hyde Wollaston">William Hyde Wollaston</a>. Wollaston was attempting to decompose water by electric shock but found that both hydrogen and oxygen were present at both electrodes. In normal electrolysis they would separate, one to each electrode.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Helmholtz explained why the oscillation decayed but he had not explained why it occurred in the first place. This was left to <a href="/wiki/Sir_William_Thomson" class="mw-redirect" title="Sir William Thomson">Sir William Thomson</a> (Lord Kelvin) who, in 1853, postulated that there was inductance present in the circuit as well as the capacitance of the jar and the resistance of the load.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This established the physical basis for the phenomenon – the energy supplied by the jar was partly dissipated in the load but also partly stored in the magnetic field of the inductor.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>So far, the investigation had been on the natural frequency of transient oscillation of a resonant circuit resulting from a sudden stimulus. More important from the point of view of filter theory is the behaviour of a resonant circuit when driven by an external <a href="/wiki/Alternating_current" title="Alternating current">AC</a> signal: there is a sudden peak in the circuit's response when the driving signal frequency is at the resonant frequency of the circuit.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> heard of the phenomenon from <a href="/wiki/Sir_William_Grove" class="mw-redirect" title="Sir William Grove">Sir William Grove</a> in 1868 in connection with experiments on <a href="/wiki/Dynamo" title="Dynamo">dynamos</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and was also aware of the earlier work of <a href="/wiki/Henry_Wilde_(engineer)" title="Henry Wilde (engineer)">Henry Wilde</a> in 1866. Maxwell explained resonance<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> mathematically, with a set of differential equations, in much the same terms that an <a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a> is described today.<sup id="cite_ref-Lund24_2-1" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Heinrich Hertz</a> (1887) experimentally demonstrated the resonance phenomena<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> by building two resonant circuits, one of which was driven by a generator and the other was <a href="/wiki/Tuner_(radio)" title="Tuner (radio)">tunable</a> and only coupled to the first electromagnetically (i.e., no circuit connection). Hertz showed that the response of the second circuit was at a maximum when it was in tune with the first. The diagrams produced by Hertz in this paper were the first published plots of an electrical resonant response.<sup id="cite_ref-Lund24_2-2" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Acoustic_resonance">Acoustic resonance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=4" title="Edit section: Acoustic resonance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned earlier, it was acoustic resonance that inspired filtering applications, the first of these being a telegraph system known as the "<a href="/wiki/Harmonic_telegraph" class="mw-redirect" title="Harmonic telegraph">harmonic telegraph</a>". Versions are due to <a href="/wiki/Elisha_Gray" title="Elisha Gray">Elisha Gray</a>, <a href="/wiki/Alexander_Graham_Bell" title="Alexander Graham Bell">Alexander Graham Bell</a> (1870s),<sup id="cite_ref-Lund24_2-3" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="https://fr.wikipedia.org/wiki/Ernest_Mercadier" class="extiw" title="fr:Ernest Mercadier">Ernest Mercadier</a> and others. Its purpose was to simultaneously transmit a number of telegraph messages over the same line and represents an early form of <a href="/wiki/Frequency_division_multiplexing" class="mw-redirect" title="Frequency division multiplexing">frequency division multiplexing</a> (FDM). FDM requires the sending end to be transmitting at different frequencies for each individual communication channel. This demands individual tuned resonators, as well as filters to separate out the signals at the receiving end. The harmonic telegraph achieved this with electromagnetically driven tuned reeds at the transmitting end which would vibrate similar reeds at the receiving end. Only the reed with the same resonant frequency as the transmitter would vibrate to any appreciable extent at the receiving end.<sup id="cite_ref-Blanch425_17-0" class="reference"><a href="#cite_note-Blanch425-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Incidentally, the harmonic telegraph directly suggested to Bell the idea of the telephone. The reeds can be viewed as <a href="/wiki/Transducer" title="Transducer">transducers</a> converting sound to and from an electrical signal. It is no great leap from this view of the harmonic telegraph to the idea that speech can be converted to and from an electrical signal.<sup id="cite_ref-Lund24_2-4" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Blanch425_17-1" class="reference"><a href="#cite_note-Blanch425-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Early_multiplexing">Early multiplexing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=5" title="Edit section: Early multiplexing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:LeblancFDM.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/LeblancFDM.jpg/220px-LeblancFDM.jpg" decoding="async" width="220" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/LeblancFDM.jpg/330px-LeblancFDM.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/LeblancFDM.jpg/440px-LeblancFDM.jpg 2x" data-file-width="1000" data-file-height="1154" /></a><figcaption>Hutin and Leblanc's multiple telegraph filter of 1891 showing the use of resonant circuits in filtering.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>By the 1890s electrical resonance was much more widely understood and had become a normal part of the engineer's toolkit. In 1891 Hutin and Leblanc patented an FDM scheme for telephone circuits using resonant circuit filters.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Rival patents were filed in 1892 by <a href="/wiki/Michael_Pupin" class="mw-redirect" title="Michael Pupin">Michael Pupin</a> and <a href="/wiki/John_Stone_Stone" title="John Stone Stone">John Stone Stone</a> with similar ideas, priority eventually being awarded to Pupin. However, no scheme using just simple resonant circuit filters can successfully <a href="/wiki/Multiplexing" title="Multiplexing">multiplex</a> (i.e. combine) the wider bandwidth of telephone channels (as opposed to telegraph) without either an unacceptable restriction of speech bandwidth or a channel spacing so wide as to make the benefits of multiplexing uneconomic.<sup id="cite_ref-Lund24_2-5" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The basic technical reason for this difficulty is that the frequency response of a simple filter approaches a fall of 6 <a href="/wiki/Octave_(electronics)" title="Octave (electronics)">dB/octave</a> far from the point of resonance. This means that if telephone channels are squeezed in side by side into the frequency spectrum, there will be <a href="/wiki/Crosstalk" title="Crosstalk">crosstalk</a> from adjacent channels in any given channel. What is required is a much more sophisticated filter that has a flat frequency response in the required <a href="/wiki/Passband" title="Passband">passband</a> like a low-<a href="/wiki/Q_factor" title="Q factor">Q</a> resonant circuit, but that rapidly falls in response (much faster than 6 dB/octave) at the transition from passband to <a href="/wiki/Stopband" title="Stopband">stopband</a> like a high-Q resonant circuit.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>note 5<span class="cite-bracket">]</span></a></sup> Obviously, these are contradictory requirements to be met with a single resonant circuit. The solution to these needs was founded in the theory of transmission lines and consequently the necessary filters did not become available until this theory was fully developed. At this early stage the idea of signal bandwidth, and hence the need for filters to match to it, was not fully understood; indeed, it was as late as 1920 before the concept of bandwidth was fully established.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> For early radio, the concepts of Q-factor, <a href="/wiki/Selectivity_(radio)" title="Selectivity (radio)">selectivity</a> and tuning sufficed. This was all to change with the developing theory of <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a> on which <a href="/wiki/Image_filter" class="mw-redirect" title="Image filter">image filters</a> are based, as explained in the next section.<sup id="cite_ref-Lund24_2-6" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="Voice_frequency_telegraphy"></span> At the turn of the century as telephone lines became available, it became popular to add telegraph onto telephone lines with an earth return <a href="/wiki/Phantom_circuit" title="Phantom circuit">phantom circuit</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>note 6<span class="cite-bracket">]</span></a></sup> An <a href="/wiki/LC_circuit" title="LC circuit">LC filter</a> was required to prevent telegraph clicks being heard on the telephone line. From the 1920s onwards, telephone lines, or balanced lines dedicated to the purpose, were used for FDM telegraph at audio frequencies. The first of these systems in the UK was a <a href="/wiki/Siemens" title="Siemens">Siemens and Halske</a> installation between London and Manchester. <a href="/wiki/General_Electric_Company_plc" class="mw-redirect" title="General Electric Company plc">GEC</a> and <a href="/wiki/AT%26T_Corp." class="mw-redirect" title="AT&T Corp.">AT&T</a> also had FDM systems. Separate pairs were used for the send and receive signals. The Siemens and GEC systems had six channels of telegraph in each direction, the AT&T system had twelve. All of these systems used electronic oscillators to generate a different <a href="/wiki/Carrier_wave" title="Carrier wave">carrier</a> for each telegraph signal and required a bank of band-pass filters to separate out the multiplexed signal at the receiving end.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/L-carrier" title="L-carrier">L-carrier</a></div> <div class="mw-heading mw-heading2"><h2 id="Transmission_line_theory">Transmission line theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=6" title="Edit section: Transmission line theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Line_model_Ohm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Line_model_Ohm.svg/300px-Line_model_Ohm.svg.png" decoding="async" width="300" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Line_model_Ohm.svg/450px-Line_model_Ohm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Line_model_Ohm.svg/600px-Line_model_Ohm.svg.png 2x" data-file-width="1123" data-file-height="349" /></a><figcaption>Ohm's model of the transmission line was simply resistance.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Line_model_Kelvin.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Line_model_Kelvin.svg/300px-Line_model_Kelvin.svg.png" decoding="async" width="300" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Line_model_Kelvin.svg/450px-Line_model_Kelvin.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Line_model_Kelvin.svg/600px-Line_model_Kelvin.svg.png 2x" data-file-width="1123" data-file-height="349" /></a><figcaption>Lord Kelvin's model of the transmission line accounted for capacitance and the dispersion it caused. The diagram represents Kelvin's model translated into modern terms using <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> elements, but this was not the actual approach used by Kelvin.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Line_model_Heaviside.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Line_model_Heaviside.svg/300px-Line_model_Heaviside.svg.png" decoding="async" width="300" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Line_model_Heaviside.svg/450px-Line_model_Heaviside.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Line_model_Heaviside.svg/600px-Line_model_Heaviside.svg.png 2x" data-file-width="1123" data-file-height="361" /></a><figcaption>Heaviside's model of the transmission line. L, R, C and G in all three diagrams are the primary line constants. The infinitesimals δL, δR, δC and δG are to be understood as Lδ<i>x</i>, Rδ<i>x</i>, Cδ<i>x</i> and Gδ<i>x</i> respectively.</figcaption></figure> <p>The earliest model of the <a href="/wiki/Transmission_line" title="Transmission line">transmission line</a> was probably described by <a href="/wiki/Georg_Ohm" title="Georg Ohm">Georg Ohm</a> (1827) who established that resistance in a wire is proportional to its length.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>note 7<span class="cite-bracket">]</span></a></sup> The Ohm model thus included only resistance. <a href="/wiki/Latimer_Clark" class="mw-redirect" title="Latimer Clark">Latimer Clark</a> noted that signals were delayed and elongated along a cable, an undesirable form of distortion now called <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion</a> but then called retardation, and <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> (1853) established that this was due to the <a href="/wiki/Capacitance" title="Capacitance">capacitance</a> present in the transmission line.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>note 8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a> (1854) found the correct mathematical description needed in his work on early transatlantic cables; he arrived at an equation identical to the <a href="/wiki/Heat_equation" title="Heat equation">conduction of a heat pulse</a> along a metal bar.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> This model incorporates only resistance and capacitance, but that is all that was needed in undersea cables dominated by capacitance effects. Kelvin's model predicts a limit on the telegraph signalling speed of a cable but Kelvin still did not use the concept of bandwidth, the limit was entirely explained in terms of the dispersion of the telegraph <a href="/wiki/Symbol_rate" title="Symbol rate">symbols</a>.<sup id="cite_ref-Lund24_2-7" class="reference"><a href="#cite_note-Lund24-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The mathematical model of the transmission line reached its fullest development with <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>. Heaviside (1881) introduced series inductance and shunt <a href="/wiki/Electrical_conductance" class="mw-redirect" title="Electrical conductance">conductance</a> into the model making four <a href="/wiki/Distributed_elements" class="mw-redirect" title="Distributed elements">distributed elements</a> in all. This model is now known as the <a href="/wiki/Telegrapher%27s_equation" class="mw-redirect" title="Telegrapher's equation">telegrapher's equation</a> and the distributed-element parameters are called the <a href="/wiki/Primary_line_constants" title="Primary line constants">primary line constants</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>From the work of Heaviside (1887) it had become clear that the performance of telegraph lines, and most especially telephone lines, could be improved by the addition of inductance to the line.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <a href="/wiki/George_Ashley_Campbell" title="George Ashley Campbell">George Campbell</a> at <a href="/wiki/American_Telephone_%26_Telegraph" class="mw-redirect" title="American Telephone & Telegraph">AT&T</a> implemented this idea (1899) by inserting <a href="/wiki/Loading_coil" title="Loading coil">loading coils</a> at intervals along the line.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Campbell found that as well as the desired improvements to the line's characteristics in the passband there was also a definite frequency beyond which signals could not be passed without great <a href="/wiki/Attenuation" title="Attenuation">attenuation</a>. This was a result of the loading coils and the line capacitance forming a <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>, an effect that is only apparent on lines incorporating <a href="/wiki/Lumped-element_model" title="Lumped-element model">lumped components</a> such as the loading coils. This naturally led Campbell (1910) to produce a filter with <a href="/wiki/Ladder_topology" class="mw-redirect" title="Ladder topology">ladder topology</a>, a glance at the circuit diagram of this filter is enough to see its relationship to a loaded transmission line.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The cut-off phenomenon is an undesirable side-effect as far as loaded lines are concerned but for telephone FDM filters it is precisely what is required. For this application, Campbell produced <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass filters</a> to the same ladder topology by replacing the inductors and capacitors with <a href="/wiki/Resonator" title="Resonator">resonators</a> and anti-resonators respectively.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>note 9<span class="cite-bracket">]</span></a></sup> Both the loaded line and FDM were of great benefit economically to AT&T and this led to fast development of filtering from this point onwards.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Image_filters">Image filters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=7" title="Edit section: Image filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Composite_image_filters" class="mw-redirect" title="Composite image filters">composite image filters</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cambell_filter.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Cambell_filter.png/400px-Cambell_filter.png" decoding="async" width="400" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Cambell_filter.png/600px-Cambell_filter.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/ac/Cambell_filter.png 2x" data-file-width="724" data-file-height="178" /></a><figcaption>Campbell's sketch of the low-pass version of his filter from his 1915 patent<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> showing the now ubiquitous ladder topology with capacitors for the ladder rungs and inductors for the stiles. Filters of more modern design also often adopt the same ladder topology as used by Campbell. It should be understood that although superficially similar, they are really quite different. The ladder construction is essential to the Campbell filter and all the sections have identical element values. Modern designs can be realised in any number of topologies, choosing the ladder topology is merely a matter of convenience. Their response is quite different (better) than Campbell's and the element values, in general, will all be different.</figcaption></figure> <p>The filters designed by Campbell<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>note 10<span class="cite-bracket">]</span></a></sup> were named wave filters because of their property of passing some waves and strongly rejecting others. The method by which they were designed was called the image parameter method<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>note 11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Quad_40-0" class="reference"><a href="#cite_note-Quad-40"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Darl4pole_41-0" class="reference"><a href="#cite_note-Darl4pole-41"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> and filters designed to this method are called image filters.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>note 12<span class="cite-bracket">]</span></a></sup> The image method essentially consists of developing the <a href="/wiki/Transmission_constant" class="mw-redirect" title="Transmission constant">transmission constants</a> of an infinite chain of identical filter sections and then terminating the desired finite number of filter sections in the <a href="/wiki/Image_impedance" title="Image impedance">image impedance</a>. This exactly corresponds to the way the properties of a finite length of transmission line are derived from the theoretical properties of an infinite line, the image impedance corresponding to the <a href="/wiki/Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a> of the line.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>From 1920 <a href="/wiki/John_Renshaw_Carson" title="John Renshaw Carson">John Carson</a>, also working for AT&T, began to develop a new way of looking at signals using the <a href="/wiki/Operational_calculus" title="Operational calculus">operational calculus</a> of Heaviside which in essence is working in the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a>. This gave the AT&T engineers a new insight into the way their filters were working and led <a href="/wiki/Otto_Zobel" class="mw-redirect" title="Otto Zobel">Otto Zobel</a> to invent many improved forms. Carson and Zobel steadily demolished many of the old ideas. For instance the old telegraph engineers thought of the signal as being a single frequency and this idea persisted into the age of radio with some still believing that <a href="/wiki/Frequency_modulation" title="Frequency modulation">frequency modulation</a> (FM) transmission could be achieved with a smaller bandwidth than the <a href="/wiki/Baseband" title="Baseband">baseband</a> signal right up until the publication of Carson's 1922 paper.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Another advance concerned the nature of noise, Carson and Zobel (1923)<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> treated noise as a random process with a continuous bandwidth, an idea that was well ahead of its time, and thus limited the amount of noise that it was possible to remove by filtering to that part of the noise spectrum which fell outside the passband. This too, was not generally accepted at first, notably being opposed by <a href="/wiki/Edwin_Armstrong" class="mw-redirect" title="Edwin Armstrong">Edwin Armstrong</a> (who ironically, actually succeeded in reducing noise with <a href="/wiki/Frequency_modulation#Modulation_index" title="Frequency modulation">wide-band FM</a>) and was only finally settled with the work of <a href="/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> whose <a href="/wiki/Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">thermal noise power formula</a> is well known today.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Several improvements were made to image filters and their theory of operation by <a href="/wiki/Otto_Zobel" class="mw-redirect" title="Otto Zobel">Otto Zobel</a>. Zobel coined the term <a href="/wiki/Constant_k_filter" title="Constant k filter">constant k filter</a> (or k-type filter) to distinguish Campbell's filter from later types, notably Zobel's <a href="/wiki/M-derived_filter" title="M-derived filter">m-derived filter</a> (or m-type filter). The particular problems Zobel was trying to address with these new forms were impedance matching into the end terminations and improved steepness of roll-off. These were achieved at the cost of an increase in filter circuit complexity.<sup id="cite_ref-Zobel_47-0" class="reference"><a href="#cite_note-Zobel-47"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Darl5_48-0" class="reference"><a href="#cite_note-Darl5-48"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>A more systematic method of producing image filters was introduced by <a href="/wiki/Hendrik_Bode" class="mw-redirect" title="Hendrik Bode">Hendrik Bode</a> (1930), and further developed by several other investigators including Piloty (1937–1939) and <a href="/wiki/Wilhelm_Cauer" title="Wilhelm Cauer">Wilhelm Cauer</a> (1934–1937). Rather than enumerate the behaviour (transfer function, attenuation function, delay function and so on) of a specific circuit, instead a requirement for the image impedance itself was developed. The image impedance can be expressed in terms of the open-circuit and short-circuit impedances<sup id="cite_ref-Zoc_49-0" class="reference"><a href="#cite_note-Zoc-49"><span class="cite-bracket">[</span>note 13<span class="cite-bracket">]</span></a></sup> of the filter as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle Z_{i}={\sqrt {Z_{o}Z_{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle Z_{i}={\sqrt {Z_{o}Z_{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b795528aab8c18902e10cf7bfe048b84a7d3330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.509ex;" alt="{\displaystyle \scriptstyle Z_{i}={\sqrt {Z_{o}Z_{s}}}}"></span>. Since the image impedance must be real in the passbands and imaginary in the stopbands according to image theory, there is a requirement that the <a href="/wiki/Zeros_and_poles" title="Zeros and poles">poles and zeroes</a> of <i>Z<sub>o</sub></i> and <i>Z<sub>s</sub></i> cancel in the passband and correspond in the stopband. The behaviour of the filter can be entirely defined in terms of the positions in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> of these pairs of poles and zeroes. Any circuit which has the requisite poles and zeroes will also have the requisite response. Cauer pursued two related questions arising from this technique: what specification of poles and zeroes are realisable as passive filters; and what realisations are equivalent to each other. The results of this work led Cauer to develop a new approach, now called network synthesis.<sup id="cite_ref-Darl5_48-1" class="reference"><a href="#cite_note-Darl5-48"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Belev851_50-0" class="reference"><a href="#cite_note-Belev851-50"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ECauer6_51-0" class="reference"><a href="#cite_note-ECauer6-51"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>This "poles and zeroes" view of filter design was particularly useful where a bank of filters, each operating at different frequencies, are all connected across the same transmission line. The earlier approach was unable to deal properly with this situation, but the poles and zeroes approach could embrace it by specifying a constant impedance for the combined filter. This problem was originally related to FDM telephony but frequently now arises in loudspeaker <a href="/wiki/Audio_crossover" title="Audio crossover">crossover filters</a>.<sup id="cite_ref-Belev851_50-1" class="reference"><a href="#cite_note-Belev851-50"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Network_synthesis_filters">Network synthesis filters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=8" title="Edit section: Network synthesis filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></div> <p>The essence of <a href="/wiki/Network_synthesis" title="Network synthesis">network synthesis</a> is to start with a required filter response and produce a network that delivers that response, or approximates to it within a specified boundary. This is the inverse of <a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">network analysis</a> which starts with a given network and by applying the various electric circuit theorems predicts the response of the network.<sup id="cite_ref-ECauer4_52-0" class="reference"><a href="#cite_note-ECauer4-52"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The term was first used with this meaning in the doctoral thesis of <a href="/wiki/Yuk-Wing_Lee" class="mw-redirect" title="Yuk-Wing Lee">Yuk-Wing Lee</a> (1930) and apparently arose out of a conversation with <a href="/wiki/Vannevar_Bush" title="Vannevar Bush">Vannevar Bush</a>.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> The advantage of network synthesis over previous methods is that it provides a solution which precisely meets the design specification. This is not the case with image filters, a degree of experience is required in their design since the image filter only meets the design specification in the unrealistic case of being terminated in its own image impedance, to produce which would require the exact circuit being sought. Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>The development of network analysis needed to take place before network synthesis was possible. The theorems of <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a> and others and the ideas of <a href="/wiki/Charles_Steinmetz" class="mw-redirect" title="Charles Steinmetz">Charles Steinmetz</a> (<a href="/wiki/Phasor_(sine_waves)" class="mw-redirect" title="Phasor (sine waves)">phasors</a>) and <a href="/wiki/Arthur_Kennelly" class="mw-redirect" title="Arthur Kennelly">Arthur Kennelly</a> (<a href="/wiki/Complex_impedance" class="mw-redirect" title="Complex impedance">complex impedance</a>)<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> laid the groundwork.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The concept of a <a href="/wiki/Port_(circuit_theory)" title="Port (circuit theory)">port</a> also played a part in the development of the theory, and proved to be a more useful idea than network terminals.<sup id="cite_ref-pole_1-1" class="reference"><a href="#cite_note-pole-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Darl5_48-2" class="reference"><a href="#cite_note-Darl5-48"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The first milestone on the way to network synthesis was an important paper by <a href="/wiki/Ronald_M._Foster" title="Ronald M. Foster">Ronald M. Foster</a> (1924),<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> <i>A Reactance Theorem</i>, in which Foster introduces the idea of a <a href="/wiki/Driving_point_impedance" class="mw-redirect" title="Driving point impedance">driving point impedance</a>, that is, the impedance that is connected to the generator. The expression for this impedance determines the response of the filter and vice versa, and a realisation of the filter can be obtained by expansion of this expression. It is not possible to realise any arbitrary impedance expression as a network. <a href="/wiki/Foster%27s_reactance_theorem" title="Foster's reactance theorem">Foster's reactance theorem</a> stipulates necessary and sufficient conditions for realisability: that the reactance must be algebraically increasing with frequency and the poles and zeroes must alternate.<sup id="cite_ref-Cauer1_58-0" class="reference"><a href="#cite_note-Cauer1-58"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Wilhelm_Cauer" title="Wilhelm Cauer">Wilhelm Cauer</a> expanded on the work of Foster (1926)<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> and was the first to talk of realisation of a one-port impedance with a prescribed frequency function. Foster's work considered only reactances (i.e., only LC-kind circuits). Cauer generalised this to any 2-element kind one-port network, finding there was an isomorphism between them. He also found ladder realisations<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>note 14<span class="cite-bracket">]</span></a></sup> of the network using <a href="/wiki/Thomas_Stieltjes" class="mw-redirect" title="Thomas Stieltjes">Thomas Stieltjes</a>' continued fraction expansion. This work was the basis on which network synthesis was built, although Cauer's work was not at first used much by engineers, partly because of the intervention of World War II, partly for reasons explained in the next section and partly because Cauer presented his results using topologies that required mutually coupled inductors and ideal transformers. Designers tend to avoid the complication of mutual inductances and transformers where possible, although transformer-coupled <a href="/wiki/Double-tuned_amplifier" title="Double-tuned amplifier">double-tuned amplifiers</a> are a common way of widening bandwidth without sacrificing selectivity.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Belev850_63-0" class="reference"><a href="#cite_note-Belev850-63"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Image_method_versus_synthesis">Image method versus synthesis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=9" title="Edit section: Image method versus synthesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Image filters continued to be used by designers long after the superior network synthesis techniques were available. Part of the reason for this may have been simply inertia, but it was largely due to the greater computation required for network synthesis filters, often needing a mathematical iterative process. Image filters, in their simplest form, consist of a chain of repeated, identical sections. The design can be improved simply by adding more sections and the computation required to produce the initial section is on the level of "back of an envelope" designing. In the case of network synthesis filters, on the other hand, the filter is designed as a whole, single entity and to add more sections (i.e., increase the order)<sup id="cite_ref-class_65-0" class="reference"><a href="#cite_note-class-65"><span class="cite-bracket">[</span>note 15<span class="cite-bracket">]</span></a></sup> the designer would have no option but to go back to the beginning and start over. The advantages of synthesised designs are real, but they are not overwhelming compared to what a skilled image designer could achieve, and in many cases it was more cost effective to dispense with time-consuming calculations.<sup id="cite_ref-Darl9_66-0" class="reference"><a href="#cite_note-Darl9-66"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> This is simply not an issue with the modern availability of computing power, but in the 1950s it was non-existent, in the 1960s and 1970s available only at cost, and not finally becoming widely available to all designers until the 1980s with the advent of the desktop personal computer. Image filters continued to be designed up to that point and many remained in service into the 21st century.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>The computational difficulty of the network synthesis method was addressed by tabulating the component values of a <a href="/wiki/Prototype_filter" title="Prototype filter">prototype filter</a> and then scaling the frequency and impedance and transforming the bandform to those actually required. This kind of approach, or similar, was already in use with image filters, for instance by Zobel,<sup id="cite_ref-Zobel_47-1" class="reference"><a href="#cite_note-Zobel-47"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> but the concept of a "reference filter" is due to <a href="/wiki/Sidney_Darlington" title="Sidney Darlington">Sidney Darlington</a>.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Darlington (1939),<sup id="cite_ref-Darl4pole_41-1" class="reference"><a href="#cite_note-Darl4pole-41"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> was also the first to tabulate values for network synthesis prototype filters,<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> nevertheless it had to wait until the 1950s before the Cauer-Darlington <a href="/wiki/Elliptic_filter" title="Elliptic filter">elliptic filter</a> first came into use.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>Once computational power was readily available, it became possible to easily design filters to minimise any arbitrary parameter, for example time delay or tolerance to component variation. The difficulties of the image method were firmly put in the past, and even the need for prototypes became largely superfluous.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Darl12_72-0" class="reference"><a href="#cite_note-Darl12-72"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Furthermore, the advent of <a href="/wiki/Active_filter" title="Active filter">active filters</a> eased the computation difficulty because sections could be isolated and iterative processes were not then generally necessary.<sup id="cite_ref-Darl9_66-1" class="reference"><a href="#cite_note-Darl9-66"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Realisability_and_equivalence">Realisability and equivalence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=10" title="Edit section: Realisability and equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Realisability (that is, which functions are realisable as real impedance networks) and equivalence (which networks equivalently have the same function) are two important questions in network synthesis. Following an analogy with <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>, Cauer formed the matrix equation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {[A]} =s^{2}\mathbf {[L]} +s\mathbf {[R]} +\mathbf {[D]} =s\mathbf {[Z]} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">A</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">L</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">R</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">D</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">Z</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {[A]} =s^{2}\mathbf {[L]} +s\mathbf {[R]} +\mathbf {[D]} =s\mathbf {[Z]} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d1e8662886952ad6ecea5cb3927e4b2bbdd368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.939ex; height:3.176ex;" alt="{\displaystyle \mathbf {[A]} =s^{2}\mathbf {[L]} +s\mathbf {[R]} +\mathbf {[D]} =s\mathbf {[Z]} }"></span></dd></dl> <p>where [<b>Z</b>],[<b>R</b>],[<b>L</b>] and [<b>D</b>] are the <i>n</i>x<i>n</i> matrices of, respectively, <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a>, <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, <a href="/wiki/Inductance" title="Inductance">inductance</a> and <a href="/wiki/Elastance" title="Elastance">elastance</a> of an <i>n</i>-<a href="/wiki/Mesh_analysis" title="Mesh analysis">mesh</a> network and <i>s</i> is the <a href="/wiki/Complex_frequency" class="mw-redirect" title="Complex frequency">complex frequency</a> operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle s=\sigma +i\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>i</mi> <mi>ω</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s=\sigma +i\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9db2868afe3e461db98029bdce00e8e36d8c0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.858ex; height:1.676ex;" alt="{\displaystyle \scriptstyle s=\sigma +i\omega }"></span>. Here [<b>R</b>],[<b>L</b>] and [<b>D</b>] have associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the <a href="/wiki/Driving_point_impedance" class="mw-redirect" title="Driving point impedance">driving point impedance</a> by the method of <a href="/wiki/Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a>; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{\mathrm {p} }(s)={\frac {\det \mathbf {[A]} }{s\,a_{11}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">A</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> </mrow> <mrow> <mi>s</mi> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{\mathrm {p} }(s)={\frac {\det \mathbf {[A]} }{s\,a_{11}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c744631cb7628de021c21e7cbad60509309b42b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.688ex; height:6.009ex;" alt="{\displaystyle Z_{\mathrm {p} }(s)={\frac {\det \mathbf {[A]} }{s\,a_{11}}}}"></span></dd></dl> <p>where <i>a<sub>11</sub></i> is the complement of the element <i>A<sub>11</sub></i> to which the one-port is to be connected. From <a href="/wiki/Stability_theory" title="Stability theory">stability theory</a> Cauer found that [<b>R</b>], [<b>L</b>] and [<b>D</b>] must all be <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite matrices</a> for <i>Z</i><sub>p</sub>(<i>s</i>) to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology.<sup id="cite_ref-ECauer4_52-1" class="reference"><a href="#cite_note-ECauer4-52"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> This work is also partly due to <a href="/wiki/Otto_Brune" title="Otto Brune">Otto Brune</a> (1931), who worked with Cauer in the US prior to Cauer returning to Germany.<sup id="cite_ref-Belev850_63-1" class="reference"><a href="#cite_note-Belev850-63"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> A well known condition for realisability of a one-port rational<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>note 16<span class="cite-bracket">]</span></a></sup> impedance due to Cauer (1929) is that it must be a function of <i>s</i> that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the <a href="/wiki/Poisson_integral" class="mw-redirect" title="Poisson integral">Poisson integral</a> representation of these functions. Brune coined the term <a href="/wiki/Positive-real" class="mw-redirect" title="Positive-real">positive-real</a> for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to <a href="/wiki/Sidney_Darlington" title="Sidney Darlington">Sidney Darlington</a> states that any positive-real function <i>Z</i>(<i>s</i>) can be realised as a lossless <a href="/wiki/Two-port_network" title="Two-port network">two-port</a> terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.<sup id="cite_ref-Belev850_63-2" class="reference"><a href="#cite_note-Belev850-63"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Darl7_75-0" class="reference"><a href="#cite_note-Darl7-75"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>As for equivalence, Cauer found that the group of real <a href="/wiki/Affine_transformation" title="Affine transformation">affine transformations</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {[T]} ^{T}\mathbf {[A]} \mathbf {[T]} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">T</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">A</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">T</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {[T]} ^{T}\mathbf {[A]} \mathbf {[T]} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59e3a0eb1cdb85270ed2580debff880baa0bd94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.58ex; height:3.343ex;" alt="{\displaystyle \mathbf {[T]} ^{T}\mathbf {[A]} \mathbf {[T]} }"></span></dd></dl> <dl><dd>where,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {[T]} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">[</mo> <mi mathvariant="bold">T</mi> <mo mathvariant="bold" stretchy="false">]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> <mo>⋯</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋅</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {[T]} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5843c3ee96f594963075b4131cc46dbf79ad3593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.472ex; height:12.509ex;" alt="{\displaystyle \mathbf {[T]} ={\begin{bmatrix}1&0\cdots 0\\T_{21}&T_{22}\cdots T_{2n}\\\cdot &\cdots \\T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}}}"></span></dd></dl> <p>is invariant in <i>Z</i><sub>p</sub>(<i>s</i>), that is, all the transformed networks are equivalents of the original.<sup id="cite_ref-ECauer4_52-2" class="reference"><a href="#cite_note-ECauer4-52"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Approximation">Approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=11" title="Edit section: Approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The approximation problem in network synthesis is to find functions which will produce realisable networks approximating to a prescribed function of frequency within limits arbitrarily set. The approximation problem is an important issue since the ideal function of frequency required will commonly be unachievable with rational networks. For instance, the ideal prescribed function is often taken to be the unachievable lossless transmission in the passband, infinite attenuation in the stopband and a vertical transition between the two. However, the ideal function can be approximated with a <a href="/wiki/Rational_function" title="Rational function">rational function</a>, becoming ever closer to the ideal the higher the order of the polynomial. The first to address this problem was <a href="/wiki/Stephen_Butterworth" title="Stephen Butterworth">Stephen Butterworth</a> (1930) using his <a href="/wiki/Butterworth_polynomials" class="mw-redirect" title="Butterworth polynomials">Butterworth polynomials</a>. Independently, Cauer (1931) used <a href="/wiki/Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>, initially applied to image filters, and not to the now well-known ladder realisation of this filter.<sup id="cite_ref-Belev850_63-3" class="reference"><a href="#cite_note-Belev850-63"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Butterworth_filter">Butterworth filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=12" title="Edit section: Butterworth filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Butterworth_filter" title="Butterworth filter">Butterworth filter</a></div> <p>Butterworth filters are an important class<sup id="cite_ref-class_65-1" class="reference"><a href="#cite_note-class-65"><span class="cite-bracket">[</span>note 15<span class="cite-bracket">]</span></a></sup> of filters due to <a href="/wiki/Stephen_Butterworth" title="Stephen Butterworth">Stephen Butterworth</a> (1930)<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> which are now recognised as being a special case of Cauer's <a href="/wiki/Elliptic_filter" title="Elliptic filter">elliptic filters</a>. Butterworth discovered this filter independently of Cauer's work and implemented it in his version with each section isolated from the next with a <a href="/wiki/Valve_amplifier" title="Valve amplifier">valve amplifier</a> which made calculation of component values easy since the filter sections could not interact with each other and each section represented one term in the <a href="/wiki/Butterworth_polynomials" class="mw-redirect" title="Butterworth polynomials">Butterworth polynomials</a>. This gives Butterworth the credit for being both the first to deviate from image parameter theory and the first to design active filters. It was later shown that Butterworth filters could be implemented in ladder topology without the need for amplifiers. Possibly the first to do so was William Bennett (1932)<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> in a patent which presents formulae for component values identical to the modern ones. Bennett, at this stage though, is still discussing the design as an artificial transmission line and so is adopting an image parameter approach despite having produced what would now be considered a network synthesis design. He also does not appear to be aware of the work of Butterworth or the connection between them.<sup id="cite_ref-Quad_40-1" class="reference"><a href="#cite_note-Quad-40"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Matt85_79-0" class="reference"><a href="#cite_note-Matt85-79"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Insertion-loss_method">Insertion-loss method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=13" title="Edit section: Insertion-loss method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The insertion-loss method of designing filters is, in essence, to prescribe a desired function of frequency for the filter as an attenuation of the signal when the filter is inserted between the terminations relative to the level that would have been received were the terminations connected to each other via an ideal transformer perfectly matching them. Versions of this theory are due to <a href="/wiki/Sidney_Darlington" title="Sidney Darlington">Sidney Darlington</a>, Wilhelm Cauer and others all working more or less independently and is often taken as synonymous with network synthesis. Butterworth's filter implementation is, in those terms, an insertion-loss filter, but it is a relatively trivial one mathematically since the active amplifiers used by Butterworth ensured that each stage individually worked into a resistive load. Butterworth's filter becomes a non-trivial example when it is implemented entirely with passive components. An even earlier filter which influenced the insertion-loss method was Norton's dual-band filter where the input of two filters are connected in parallel and designed so that the combined input presents a constant resistance. Norton's design method, together with Cauer's canonical LC networks and Darlington's theorem that only LC components were required in the body of the filter resulted in the insertion-loss method. However, ladder topology proved to be more practical than Cauer's canonical forms.<sup id="cite_ref-Darl8_80-0" class="reference"><a href="#cite_note-Darl8-80"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p><p>Darlington's insertion-loss method is a generalisation of the procedure used by Norton. In Norton's filter it can be shown that each filter is equivalent to a separate filter unterminated at the common end. Darlington's method applies to the more straightforward and general case of a 2-port LC network terminated at both ends. The procedure consists of the following steps: </p> <ol><li>determine the poles of the prescribed insertion-loss function,</li> <li>from that find the complex transmission function,</li> <li>from that find the complex <a href="/wiki/Reflection_coefficient#Telecommunications" title="Reflection coefficient">reflection coefficients</a> at the terminating resistors,</li> <li>find the driving point impedance from the short-circuit and open-circuit impedances,<sup id="cite_ref-Zoc_49-1" class="reference"><a href="#cite_note-Zoc-49"><span class="cite-bracket">[</span>note 13<span class="cite-bracket">]</span></a></sup></li> <li>expand the driving point impedance into an LC (usually ladder) network.</li></ol> <p>Darlington additionally used a transformation found by <a href="/wiki/Hendrik_Bode" class="mw-redirect" title="Hendrik Bode">Hendrik Bode</a> that predicted the response of a filter using non-ideal components but all with the same <i>Q</i>. Darlington used this transformation in reverse to produce filters with a prescribed insertion-loss with non-ideal components. Such filters have the ideal insertion-loss response plus a flat attenuation across all frequencies.<sup id="cite_ref-Darl9_66-2" class="reference"><a href="#cite_note-Darl9-66"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Elliptic_filters">Elliptic filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=14" title="Edit section: Elliptic filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Elliptic_filter" title="Elliptic filter">Elliptic filter</a></div> <p>Elliptic filters are filters produced by the insertion-loss method which use <a href="/wiki/Elliptic_rational_functions" title="Elliptic rational functions">elliptic rational functions</a> in their transfer function as an approximation to the ideal filter response and the result is called a Chebyshev approximation. This is the same Chebyshev approximation technique used by Cauer on image filters but follows the Darlington insertion-loss design method and uses slightly different elliptic functions. Cauer had some contact with Darlington and Bell Labs before WWII (for a time he worked in the US) but during the war they worked independently, in some cases making the same discoveries. Cauer had disclosed the Chebyshev approximation to Bell Labs but had not left them with the proof. <a href="/wiki/Sergei_Alexander_Schelkunoff" title="Sergei Alexander Schelkunoff">Sergei Schelkunoff</a> provided this and a generalisation to all equal ripple problems. Elliptic filters are a general class of filter which incorporate several other important classes as special cases: Cauer filter (equal <a href="/wiki/Ripple_(electrical)#Frequency-domain_ripple" title="Ripple (electrical)">ripple</a> in passband and <a href="/wiki/Stopband" title="Stopband">stopband</a>), Chebyshev filter (ripple only in passband), reverse Chebyshev filter (ripple only in stopband) and Butterworth filter (no ripple in either band).<sup id="cite_ref-Darl8_80-1" class="reference"><a href="#cite_note-Darl8-80"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>Generally, for insertion-loss filters where the transmission zeroes and infinite losses are all on the real axis of the complex frequency plane (which they usually are for minimum component count), the insertion-loss function can be written as; </p> <dl><dd><span class="mwe-math-element"><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 {1}{1+JF^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>J</mi> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1+JF^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919bb2e85ec7d7d64ce32c4fde9899e31c82bf53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.179ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{1+JF^{2}}}}"></span></dd></dl> <p>where <i>F</i> is either an even (resulting in an <a href="/wiki/Antimetric_(electrical_networks)" class="mw-redirect" title="Antimetric (electrical networks)">antimetric</a> filter) or an odd (resulting in an symmetric filter) function of frequency. Zeroes of <i>F</i> correspond to zero loss and the poles of <i>F</i> correspond to transmission zeroes. <i>J</i> sets the passband ripple height and the stopband loss and these two design requirements can be interchanged. The zeroes and poles of <i>F</i> and <i>J</i> can be set arbitrarily. The nature of <i>F</i> determines the class of the filter; </p> <ul><li>if <i>F</i> is a Chebyshev approximation the result is a Chebyshev filter,</li> <li>if <i>F</i> is a maximally flat approximation the result is a passband maximally flat filter,</li> <li>if 1/<i>F</i> is a Chebyshev approximation the result is a reverse Chebyshev filter,</li> <li>if 1/<i>F</i> is a maximally flat approximation the result is a stopband maximally flat filter,</li></ul> <p>A Chebyshev response simultaneously in the passband and stopband is possible, such as Cauer's equal ripple elliptic filter.<sup id="cite_ref-Darl8_80-2" class="reference"><a href="#cite_note-Darl8-80"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p><p>Darlington relates that he found in the New York City library <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Jacobi</a>'s original paper on elliptic functions, published in Latin in 1829. In this paper Darlington was surprised to find foldout tables of the exact elliptic function transformations needed for Chebyshev approximations of both Cauer's image parameter, and Darlington's insertion-loss filters.<sup id="cite_ref-Darl9_66-3" class="reference"><a href="#cite_note-Darl9-66"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_methods">Other methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=15" title="Edit section: Other methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Darlington considers the topology of coupled tuned circuits to involve a separate approximation technique to the insertion-loss method, but also producing nominally flat passbands and high attenuation stopbands. The most common topology for these is shunt anti-resonators coupled by series capacitors, less commonly, by inductors, or in the case of a two-section filter, by mutual inductance. These are most useful where the design requirement is not too stringent, that is, moderate bandwidth, roll-off and passband ripple.<sup id="cite_ref-Darl12_72-1" class="reference"><a href="#cite_note-Darl12-72"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_notable_developments_and_applications">Other notable developments and applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=16" title="Edit section: Other notable developments and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mechanical_filters">Mechanical filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=17" title="Edit section: Mechanical filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mechanical_filter" title="Mechanical filter">Mechanical filter</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Norton_mechanical_filter.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Norton_mechanical_filter.png/250px-Norton_mechanical_filter.png" decoding="async" width="250" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Norton_mechanical_filter.png/375px-Norton_mechanical_filter.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Norton_mechanical_filter.png/500px-Norton_mechanical_filter.png 2x" data-file-width="715" data-file-height="498" /></a><figcaption>Norton's mechanical filter together with its electrical equivalent circuit. Two equivalents are shown, "Fig.3" directly corresponds to the physical relationship of the mechanical components; "Fig.4" is an equivalent transformed circuit arrived at by repeated application of <a href="/wiki/Equivalent_impedance_transforms#Transform_5.2" title="Equivalent impedance transforms">a well known transform</a>, the purpose being to remove the series resonant circuit from the body of the filter leaving a simple <i>LC</i> ladder network.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><a href="/wiki/Edward_Lawry_Norton" title="Edward Lawry Norton">Edward Norton</a>, around 1930, designed a mechanical filter for use on <a href="/wiki/Phonograph" title="Phonograph">phonograph</a> recorders and players. Norton designed the filter in the electrical domain and then used the <a href="/wiki/Mechanical%E2%80%93electrical_analogies" title="Mechanical–electrical analogies">correspondence of mechanical quantities to electrical quantities</a> to realise the filter using mechanical components. <a href="/wiki/Mass" title="Mass">Mass</a> corresponds to <a href="/wiki/Inductance" title="Inductance">inductance</a>, <a href="/wiki/Stiffness" title="Stiffness">stiffness</a> to <a href="/wiki/Elastance" title="Elastance">elastance</a> and <a href="/wiki/Damping" title="Damping">damping</a> to <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>. The filter was designed to have a <a href="/wiki/Butterworth_filter#Maximal_flatness" title="Butterworth filter">maximally flat</a> frequency response.<sup id="cite_ref-Darl7_75-1" class="reference"><a href="#cite_note-Darl7-75"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>In modern designs it is common to use quartz <a href="/wiki/Crystal_filter" title="Crystal filter">crystal filters</a>, especially for narrowband filtering applications. The signal exists as a mechanical acoustic wave while it is in the crystal and is converted by <a href="/wiki/Transducer" title="Transducer">transducers</a> between the electrical and mechanical domains at the terminals of the crystal.<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Distributed-element_filters">Distributed-element filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=18" title="Edit section: Distributed-element filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distributed-element_filter" title="Distributed-element filter">Distributed-element filter</a></div> <p>Distributed-element filters are composed of lengths of transmission line that are at least a significant fraction of a wavelength long. The earliest non-electrical filters were all of this type. <a href="/wiki/William_Herschel" title="William Herschel">William Herschel</a> (1738–1822), for instance, constructed an apparatus with two tubes of different lengths which attenuated some frequencies but not others. <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> (1736–1813) studied waves on a string periodically loaded with weights. The device was never studied or used as a filter by either Lagrange or later investigators such as Charles Godfrey. However, Campbell used Godfrey's results by <a href="/wiki/Mechanical%E2%80%93electrical_analogies" title="Mechanical–electrical analogies">analogy</a> to calculate the number of loading coils needed on his loaded lines, the device that led to his electrical filter development. Lagrange, Godfrey, and Campbell all made simplifying assumptions in their calculations that ignored the distributed nature of their apparatus. Consequently, their models did not show the multiple passbands that are a characteristic of all distributed-element filters.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup> The first electrical filters that were truly designed by distributed-element principles are due to <a href="/wiki/Warren_P._Mason" title="Warren P. Mason">Warren P. Mason</a> starting in 1927.<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transversal_filters">Transversal filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=19" title="Edit section: Transversal filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Transversal_filter&action=edit&redlink=1" class="new" title="Transversal filter (page does not exist)">Transversal filters</a> are not usually associated with passive implementations but the concept can be found in a Wiener and Lee patent from 1935 which describes a filter consisting of a cascade of <a href="/wiki/All-pass_filter" title="All-pass filter">all-pass sections</a>.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> The outputs of the various sections are summed in the proportions needed to result in the required frequency function. This works by the principle that certain frequencies will be in, or close to antiphase, at different sections and will tend to cancel when added. These are the frequencies rejected by the filter and can produce filters with very sharp cut-offs. This approach did not find any immediate applications, and is not common in passive filters. However, the principle finds many applications as an active delay line implementation for wide band <a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> filter applications such as television, radar and high-speed data transmission.<sup id="cite_ref-Darl11_88-0" class="reference"><a href="#cite_note-Darl11-88"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Matched_filter">Matched filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=20" title="Edit section: Matched filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Matched_filter" title="Matched filter">matched filter</a></div> <p>The purpose of matched filters is to maximise the <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> (S/N) at the expense of pulse shape. Pulse shape, unlike many other applications, is unimportant in radar while S/N is the primary limitation on performance. The filters were introduced during WWII (described 1943)<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> by Dwight North and are often eponymously referred to as "<a href="/wiki/Matched_filter" title="Matched filter">North filters</a>".<sup id="cite_ref-Darl11_88-1" class="reference"><a href="#cite_note-Darl11-88"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Filters_for_control_systems">Filters for control systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=21" title="Edit section: Filters for control systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Control systems have a need for smoothing filters in their feedback loops with criteria to maximise the speed of movement of a mechanical system to the prescribed mark and at the same time minimise overshoot and noise induced motions. A key problem here is the extraction of <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian signals</a> from a noisy background. An early paper on this was published during WWII by <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> with the specific application to anti-aircraft fire control analogue computers. Rudy Kalman (<a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a>) later reformulated this in terms of <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state-space</a> smoothing and prediction where it is known as the <a href="/wiki/Linear-quadratic-Gaussian_control" class="mw-redirect" title="Linear-quadratic-Gaussian control">linear-quadratic-Gaussian control</a> problem. Kalman started an interest in state-space solutions, but according to Darlington this approach can also be found in the work of Heaviside and earlier.<sup id="cite_ref-Darl11_88-2" class="reference"><a href="#cite_note-Darl11-88"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Modern_practice">Modern practice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=22" title="Edit section: Modern practice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>LC filters at low frequencies become awkward; the components, especially the inductors, become expensive, bulky, heavy, and non-ideal. Practical 1 H inductors require many turns on a high-permeability core; that material will have high losses and stability issues (e.g., a large temperature coefficient). For applications such as a mains filters, the awkwardness must be tolerated. For low-level, low-frequency, applications, RC filters are possible, but they cannot implement filters with complex poles or zeros. If the application can use power, then amplifiers can be used to make RC <a href="/wiki/Active_filter" title="Active filter">active filters</a> that can have complex poles and zeros. In the 1950s, <a href="/wiki/Sallen%E2%80%93Key_topology" title="Sallen–Key topology">Sallen–Key active RC filters</a> were made with <a href="/wiki/Vacuum_tube" title="Vacuum tube">vacuum tube</a> amplifiers; these filters replaced the bulky inductors with bulky and hot vacuum tubes. Transistors offered more power-efficient active filter designs. Later, inexpensive <a href="/wiki/Operational_amplifier" title="Operational amplifier">operational amplifiers</a> enabled other active RC filter design topologies. Although active filter designs were commonplace at low frequencies, they were impractical at high frequencies where the amplifiers were not ideal; LC (and transmission line) filters were still used at radio frequencies. </p><p>Gradually, the low frequency active RC filter was supplanted by the <a href="/wiki/Switched-capacitor_filter" class="mw-redirect" title="Switched-capacitor filter">switched-capacitor filter</a> that operated in the discrete time domain rather than the continuous time domain. All of these filter technologies require precision components for high performance filtering, and that often requires that the filters be tuned. Adjustable components are expensive, and the labor to do the tuning can be significant. Tuning the poles and zeros of a 7th-order elliptic filter is not a simple exercise. Integrated circuits have made digital computation inexpensive, so now low frequency filtering is done with digital signal processors. Such <a href="/wiki/Digital_filter" title="Digital filter">digital filters</a> have no problem implementing ultra-precise (and stable) values, so no tuning or adjustment is required. Digital filters also don't have to worry about stray coupling paths and shielding the individual filter sections from one another. One downside is the digital signal processing may consume much more power than an equivalent LC filter. Inexpensive digital technology has largely supplanted analogue implementations of filters. However, there is still an occasional place for them in the simpler applications such as coupling where sophisticated functions of frequency are not needed.<sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup> Passive filters are still the technology of choice at microwave frequencies.<sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Multiple_feedback_topology_(electronics)" class="mw-redirect" title="Multiple feedback topology (electronics)">Multiple feedback topology (electronics)</a></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Audio_filter" title="Audio filter">Audio filter</a></li> <li><a href="/wiki/Composite_image_filter" title="Composite image filter">Composite image filter</a></li> <li><a href="/wiki/Digital_filter" title="Digital filter">Digital filter</a></li> <li><a href="/wiki/Electronic_filter" title="Electronic filter">Electronic filter</a></li> <li><a href="/wiki/Linear_filter" title="Linear filter">Linear filter</a></li> <li><a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=24" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-pole-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-pole_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pole_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A terminal of a network is a connection point where current can enter or leave the network from the world outside. This is often called a <i>pole</i> in the literature, especially the more mathematical, but is not to be confused with a <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> of the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> which is a meaning also used in this article. A 2-terminal network amounts to a single impedance (although it may consist of many elements connected in a complicated set of <a href="/wiki/Mesh_analysis" title="Mesh analysis">meshes</a>) and can also be described as a one-port network. For networks of more than two terminals it is not necessarily possible to identify terminal pairs as ports.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">The resonant frequency is very close to, but usually not exactly equal to, the natural frequency of oscillation of the circuit</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/Oliver_Lodge" title="Oliver Lodge">Oliver Lodge</a> and some other English scientists tried to keep acoustic and electric terminology separate and promoted the term "syntony". However it was "resonance" that was to win the day. Blanchard, p.422</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">This image is from a later, corrected, US patent but patenting the same invention as the original French patent</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a href="/wiki/Q_factor" title="Q factor">Q factor</a> is a dimensionless quantity enumerating the <i><b>q</b></i>uality of a resonating circuit. It is roughly proportional to the number of oscillations, which a resonator would support after a single external excitation (for example, how many times a guitar string would wobble if pulled). One definition of Q factor, the most relevant one in this context, is the ratio of resonant frequency to bandwidth of a circuit. It arose as a measure of <a href="/wiki/Selectivity_(radio)" title="Selectivity (radio)">selectivity</a> in radio receivers</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Telegraph lines are typically <a href="/wiki/Unbalanced_line" title="Unbalanced line">unbalanced</a> with only a single conductor provided, the return path is achieved through an <a href="/wiki/Ground_(electricity)" title="Ground (electricity)">earth</a> connection which is common to all the telegraph lines on a route. Telephone lines are typically <a href="/wiki/Balanced_line" title="Balanced line">balanced</a> with two conductors per circuit. A telegraph signal connected <a href="/wiki/Common-mode_signal" title="Common-mode signal">common-mode</a> to both conductors of the telephone line will not be heard at the telephone receiver which can only detect voltage differences between the conductors. The telegraph signal is typically recovered at the far end by connection to the <a href="/wiki/Center_tap" title="Center tap">center tap</a> of a <a href="/wiki/Repeating_coil" title="Repeating coil">line transformer</a>. The return path is via an earth connection as usual. This is a form of <a href="/wiki/Phantom_circuit" title="Phantom circuit">phantom circuit</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">At least, Ohm described the first model that was in any way correct. Earlier ideas such as <a href="/wiki/Barlow%27s_law" title="Barlow's law">Barlow's law</a> from <a href="/wiki/Peter_Barlow_(mathematician)" title="Peter Barlow (mathematician)">Peter Barlow</a> were either incorrect, or inadequately described. See, for example. p.603 of; <br />*John C. Shedd, Mayo D. Hershey, "The history of Ohm's law", <i>The Popular Science Monthly</i>, pp.599–614, December 1913 ISSN 0161-7370.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><a href="/wiki/Werner_von_Siemens" title="Werner von Siemens">Werner von Siemens</a> had also noted the retardation effect a few years earlier in 1849 and came to a similar conclusion as Faraday. However, there was not so much interest in Germany in underwater and underground cables as there was in Britain, the German overhead cables did not noticeably suffer from retardation and Siemen's ideas were not accepted. (Hunt, p.65.)</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">The exact date Campbell produced each variety of filter is not clear. The work started in 1910, initially patented in 1917 (US1227113) and the full theory published in 1922, but it is known that Campbell's filters were in use by AT&T long before the 1922 date (Bray, p.62, Darlington, p.5)</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Campbell has publishing priority for this invention but it is worth noting that <a href="/wiki/Karl_Willy_Wagner" title="Karl Willy Wagner">Karl Willy Wagner</a> independently made a similar discovery which he was not allowed to publish immediately because <a href="/wiki/World_War_I" title="World War I">World War I</a> was still ongoing. (Thomas H. Lee, <i>Planar microwave engineering</i>, p.725, Cambridge University Press 2004 <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83526-7" title="Special:BookSources/0-521-83526-7">0-521-83526-7</a>.)</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">The term "image parameter method" was coined by Darlington (1939) in order to distinguish this earlier technique from his later "insertion-loss method"</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">The terms wave filter and image filter are not synonymous, it is possible for a wave filter to not be designed by the image method, but in the 1920s the distinction was moot as the image method was the only one available</span> </li> <li id="cite_note-Zoc-49"><span class="mw-cite-backlink">^ <a href="#cite_ref-Zoc_49-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zoc_49-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">The open-circuit impedance of a two-port network is the impedance looking into one port when the other port is open circuit. Similarly, the short-circuit impedance is the impedance looking into one port when the other is terminated in a short circuit. The open-circuit impedance of the first port in general (except for symmetrical networks) is not equal to the open-circuit impedance of the second and likewise for short-circuit impedances</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text">which is the best known of the filter topologies. It is for this reason that ladder topology is often referred to as Cauer topology (the forms used earlier by Foster are quite different) even though ladder topology had long since been in use in image filter design</span> </li> <li id="cite_note-class-65"><span class="mw-cite-backlink">^ <a href="#cite_ref-class_65-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-class_65-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">A class of filters is a collection of filters which are all described by the same <a href="/wiki/Class_(mathematics)" class="mw-redirect" title="Class (mathematics)">class of mathematical function</a>, for instance, the class of Chebyshev filters are all described by the class of <a href="/wiki/Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomials</a>. For realisable linear passive networks, the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> must be a ratio of <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a>. The order of a filter is the order of the highest order polynomial of the two and will equal the number of elements (or resonators) required to build it. Usually, the higher the order of a filter, the steeper the roll-off of the filter will be. In general, the values of the elements in each section of the filter will not be the same if the order is increased and will need to be recalculated. This is in contrast to the image method of design which simply adds on more identical sections</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text">A rational impedance is one expressed as a ratio of two finite polynomials in <i>s</i>, that is, a <a href="/wiki/Rational_function" title="Rational function">rational function</a> in <i>s</i>. The implication of finite polynomials is that the impedance, when realised, will consist of a finite number of meshes with a finite number of elements</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Lund24-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lund24_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lund24_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lund24_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Lund24_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Lund24_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Lund24_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Lund24_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Lund24_2-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text">Lundheim, p.24</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">L. J. Raphael, G. J. Borden, K. S. Harris, <i>Speech science primer: physiology, acoustics, and perception of speech</i>, p.113, Lippincott Williams & Wilkins 2006 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7817-7117-X" title="Special:BookSources/0-7817-7117-X">0-7817-7117-X</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Joseph Henry, "On induction from ordinary electricity; and on the oscillatory discharge", <i>Proceedings of the American Philosophical Society</i>, <b>vol 2</b>, pp.193–196, 17 June 1842</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Blanchard, pp.415–416</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Hermann von Helmholtz, <i>Uber die Erhaltung der Kraft (On the Conservation of Force)</i>, G Reimer, Berlin, 1847</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Blanchard, pp.416–417</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">William Thomson, "On transient electric currents", <i>Philosophical Magazine</i>, <b>vol 5</b>, pp.393–405, June 1853</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Blanchard, p.417</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">William Grove, "An experiment in magneto–electric induction", <i>Philosophical Magazine</i>, <b>vol 35</b>, pp.184–185, March 1868</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">James Clerk Maxwell, "<a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/pdf/10.1080/14786446808639993">On Mr Grove's experiment in magneto–electric induction</a>", <i>Philosophical Magazine</i>, <b>vol 35</b>, pp. 360–363, May 1868</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Blanchard, pp.416–421</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Heinrich Hertz, "Electric waves", p.42, The Macmillan Company, 1893</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Blanchard, pp.421–423</span> </li> <li id="cite_note-Blanch425-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Blanch425_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Blanch425_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Blanchard, p.425</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">M Hutin, M Leblanc, <i><a rel="nofollow" class="external text" href="https://patentimages.storage.googleapis.com/6b/a2/44/57665febdb7fde/US838545.pdf">Multiple Telegraphy and Telephony</a></i>, United States patent US0838545, filed 9 May 1894, issued 18 December 1906</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Maurice Hutin, Maurice Leblanc, "Êtude sur les Courants Alternatifs et leur Application au Transport de la Force", <i>La Lumière Electrique</i>, 2 May 1891</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Blanchard, pp.426–427</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Lundheim (2002), p. 23</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">K. G. Beauchamp, <i>History of telegraphy</i>, pp. 84–85, Institution of Electrical Engineers, 2001 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-85296-792-6" title="Special:BookSources/0-85296-792-6">0-85296-792-6</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Georg Ohm, <i>Die galvanische Kette, mathematisch bearbeitet</i>, Riemann Berlin, 1827</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Hunt, pp. 62–63</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Thomas William Körner, <i>Fourier analysis</i>, p.333, Cambridge University Press, 1989 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38991-7" title="Special:BookSources/0-521-38991-7">0-521-38991-7</a></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Brittain, p.39<br />Heaviside, O, <i>Electrical Papers</i>, <b>vol 1</b>, pp.139–140, Boston, 1925</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Heaviside, O, "Electromagnetic Induction and its propagation", <i>The Electrician</i>, 3 June 1887</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">James E. Brittain, "The Introduction of the Loading Coil: George A. Campbell and Michael I. Pupin", <i>Technology and Culture</i>, <b>Vol. 11</b>, No. 1 (Jan., 1970), pp. 36–57, The Johns Hopkins University Press <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3102809">10.2307/3102809</a></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Darlington, pp.4–5</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Bray, J, <i>Innovation and the Communications Revolution</i>, p 62, Institute of Electrical Engineers, 2002</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">George A, Campbell, <i>Electric wave-filter</i>, <span><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US1227113">U.S. patent 1,227,113</a></span>, filed 15 July 1915, issued 22 May 1917.</span> </li> <li id="cite_note-Quad-40"><span class="mw-cite-backlink">^ <a href="#cite_ref-Quad_40-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Quad_40-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.quadrivium.nl/history/history.html">"History of Filter Theory"</a>, Quadrivium, retrieved 26 June 2009</span> </li> <li id="cite_note-Darl4pole-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl4pole_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl4pole_41-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">S. Darlington, "<a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1939181257">Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics</a>", <i>Journal of Mathematics and Physics</i>, <b>vol 18</b>, pp.257–353, September 1939</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Matthaei, pp.49–51</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">Carson, J. R., "Notes on the Theory of Modulation" <i>Procedures of the IRE</i>, <b>vol 10</b>, No 1, pp.57–64, 1922 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJRPROC.1922.219793">10.1109/JRPROC.1922.219793</a></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">Carson, J R and Zobel, O J, "<a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/abstract/document/6769216/">Transient Oscillation in Electric Wave Filters</a>", <i>Bell System Technical Journal</i>, vol 2, July 1923, pp.1–29</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">Lundheim, pp.24–25</span> </li> <li id="cite_note-Zobel-47"><span class="mw-cite-backlink">^ <a href="#cite_ref-Zobel_47-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zobel_47-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Zobel, O. J.,<i>Theory and Design of Uniform and Composite Electric Wave Filters</i>, Bell System Technical Journal, Vol. 2 (1923), pp. 1–46.</span> </li> <li id="cite_note-Darl5-48"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl5_48-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl5_48-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Darl5_48-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Darlington, p.5</span> </li> <li id="cite_note-Belev851-50"><span class="mw-cite-backlink">^ <a href="#cite_ref-Belev851_50-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Belev851_50-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Belevitch, p.851</span> </li> <li id="cite_note-ECauer6-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-ECauer6_51-0">^</a></b></span> <span class="reference-text">Cauer et al., p.6</span> </li> <li id="cite_note-ECauer4-52"><span class="mw-cite-backlink">^ <a href="#cite_ref-ECauer4_52-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ECauer4_52-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ECauer4_52-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Cauer et al., p.4</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text">Karl L. Wildes, Nilo A. Lindgren, <i>A century of electrical engineering and computer science at MIT, 1882–1982</i>, p.157, MIT Press, 1985 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-23119-0" title="Special:BookSources/0-262-23119-0">0-262-23119-0</a></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text">Matthaei, pp.83–84</span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080216002510/http://www.ieee.org/web/aboutus/history_center/biography/kennelly.html">Arthur E. Kennelly, 1861 – 1939</a> IEEE biography, retrieved 13 June 2009</span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text">Darlington, p.4</span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text">Foster, R M, "A Reactance Theorem", <i>Bell System Technical Journal</i>, <b>vol 3</b>, pp.259–267, 1924</span> </li> <li id="cite_note-Cauer1-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cauer1_58-0">^</a></b></span> <span class="reference-text">Cauer et al., p.1</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text">Darlington, pp.4–6</span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text">Cauer, W, "Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit" ("The realisation of impedances of specified frequency dependence"), <i>Archiv für Elektrotechnic</i>, <b>vol 17</b>, pp.355–388, 1926 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01662000">10.1007/BF01662000</a></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text">Atul P. Godse, U. A. Bakshi, <i>Electronic Circuit Analysis</i>, p.5-20, Technical Publications, 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-8431-047-1" title="Special:BookSources/81-8431-047-1">81-8431-047-1</a></span> </li> <li id="cite_note-Belev850-63"><span class="mw-cite-backlink">^ <a href="#cite_ref-Belev850_63-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Belev850_63-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Belev850_63-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Belev850_63-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Belevitch, p.850</span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text">Cauer et al., pp.1,6</span> </li> <li id="cite_note-Darl9-66"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl9_66-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl9_66-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Darl9_66-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Darl9_66-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Darlington, p.9</span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text">Irwin W. Sandberg, <a href="/wiki/Ernest_S._Kuh" title="Ernest S. Kuh">Ernest S. Kuh</a>, "Sidney Darlington", <i>Biographical Memoirs</i>, <b>vol 84</b>, page 85, National Academy of Sciences (U.S.), National Academies Press 2004 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-309-08957-3" title="Special:BookSources/0-309-08957-3">0-309-08957-3</a></span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text">J. Zdunek, "The network synthesis on the insertion-loss basis", <i>Proceedings of the Institution of Electrical Engineers</i>, p.283, part 3, <b>vol 105</b>, 1958</span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text">Matthaei et al., p.83</span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text">Michael Glynn Ellis, <i>Electronic filter analysis and synthesis</i>, p.2, Artech House 1994 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89006-616-7" title="Special:BookSources/0-89006-616-7">0-89006-616-7</a></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text">John T. Taylor, Qiuting Huang, <i>CRC handbook of electrical filters</i>, p.20, CRC Press 1997 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8493-8951-8" title="Special:BookSources/0-8493-8951-8">0-8493-8951-8</a></span> </li> <li id="cite_note-Darl12-72"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl12_72-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl12_72-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Darlington, p.12</span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text">Cauer et al., pp.6–7</span> </li> <li id="cite_note-Darl7-75"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl7_75-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl7_75-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Darlington, p.7</span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text">Darlington, pp.7–8</span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text">Butterworth, S, "On the Theory of Filter Amplifiers", <i>Wireless Engineer</i>, <b>vol. 7</b>, 1930, pp. 536–541</span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text">William R. Bennett, <i>Transmission network</i>, <span><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US1849656">U.S. patent 1,849,656</a></span>, filed 29 June 1929, issued 15 March 1932</span> </li> <li id="cite_note-Matt85-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-Matt85_79-0">^</a></b></span> <span class="reference-text">Matthaei et al., pp.85–108</span> </li> <li id="cite_note-Darl8-80"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl8_80-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl8_80-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Darl8_80-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Darlington, p.8</span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text">Vasudev K Aatre, <i>Network theory and filter design</i>, p.355, New Age International 1986, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-85226-014-8" title="Special:BookSources/0-85226-014-8">0-85226-014-8</a></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text">Matthaei et al., p.95</span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text">E. L. Norton, "Sound reproducer", <span><a rel="nofollow" class="external text" href="https://patents.google.com/patent/USUS1792655">U.S. patent US1792655</a></span>, filed 31 May 1929, issued 17 February 1931</span> </li> <li id="cite_note-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-84">^</a></b></span> <span class="reference-text">Vizmuller, P, <i>RF Design Guide: Systems, Circuits, and Equations</i>, pp.81–84, Artech House, 1995 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89006-754-6" title="Special:BookSources/0-89006-754-6">0-89006-754-6</a></span> </li> <li id="cite_note-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-85">^</a></b></span> <span class="reference-text">Mason, pp. 409–410</span> </li> <li id="cite_note-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-86">^</a></b></span> <span class="reference-text">Fagen & Millman, p. 108</span> </li> <li id="cite_note-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-87">^</a></b></span> <span class="reference-text">N Wiener and Yuk-wing Lee, <i>Electric network system</i>, United States patent US2024900, 1935</span> </li> <li id="cite_note-Darl11-88"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darl11_88-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darl11_88-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Darl11_88-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Darlington, p.11</span> </li> <li id="cite_note-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-89">^</a></b></span> <span class="reference-text">B. S. Sonde, <i>Introduction to System Design Using Integrated Circuits</i>, pp.252–254, New Age International 1992 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-224-0386-7" title="Special:BookSources/81-224-0386-7">81-224-0386-7</a></span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text">D. O. North, <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/1444313/;jsessionid=118A488879B363BB792A940E39D45622?arnumber=1444313">"An analysis of the factors which determine signal/noise discrimination in pulsed carrier systems"</a>, <i>RCA Labs. Rep. PTR-6C</i>, 1943</span> </li> <li id="cite_note-91"><span class="mw-cite-backlink"><b><a href="#cite_ref-91">^</a></b></span> <span class="reference-text">Nadav Levanon, Eli Mozeson, <i>Radar Signals</i>, p.24, Wiley-IEEE 2004 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-47378-2" title="Special:BookSources/0-471-47378-2">0-471-47378-2</a></span> </li> <li id="cite_note-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-92">^</a></b></span> <span class="reference-text">Jack L. Bowers, "R-C bandpass filter design", <i>Electronics</i>, <b>vol 20</b>, pages 131–133, April 1947</span> </li> <li id="cite_note-93"><span class="mw-cite-backlink"><b><a href="#cite_ref-93">^</a></b></span> <span class="reference-text">Darlington, pp.12–13</span> </li> <li id="cite_note-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-94">^</a></b></span> <span class="reference-text"><a href="/wiki/Lars_Wanhammar" title="Lars Wanhammar">Lars Wanhammar</a>, <i>Analog Filters using MATLAB</i>, pp. 10–11, Springer, 2009 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387927670" title="Special:BookSources/0387927670">0387927670</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=26" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Vitold_Belevitch" title="Vitold Belevitch">Belevitch, V</a>, "Summary of the history of circuit theory", <i>Proceedings of the IRE</i>, vol. 50, iss. 5, pp. 848–855, May 1962 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJRPROC.1962.288301">10.1109/JRPROC.1962.288301</a>.</li> <li>Blanchard, J, "The History of Electrical Resonance", <i>Bell System Technical Journal</i>, vol. 23, pp. 415–433, 1944.</li> <li>Cauer, E; Mathis, W; Pauli, R, <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf">"Life and work of Wilhelm Cauer (1900–1945)"</a>, <i>Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)</i>, Perpignan, June, 2000.</li> <li>Darlington, S, "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors", <i>IEEE Transactions on Circuits and Systems</i>, vol. 31, pp. 3–13, 1984 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCS.1984.1085415">10.1109/TCS.1984.1085415</a>.</li> <li>Fagen, M D; Millman, S, <i>A History of Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980)</i>, AT&T Bell Laboratories, 1984 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0932764061" title="Special:BookSources/0932764061">0932764061</a>.</li> <li>Godfrey, Charles, <a rel="nofollow" class="external text" href="https://archive.org/stream/londonedinburgh5451898lon#page/356/mode/2up">"On discontinuities connected with the propagation of wave-motion along a periodically loaded string"</a>, <i>Philosophical Magazine</i>, ser. 5, vol. 45, no. 275, pp. 356–363, April 1898.</li> <li>Hunt, Bruce J, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=23rBH11Q9w8C"><i>The Maxwellians</i></a>, Cornell University Press, 2005 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8014-8234-8" title="Special:BookSources/0-8014-8234-8">0-8014-8234-8</a>.</li> <li>Lundheim, L, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110724183448/http://www.iet.ntnu.no/groups/signal/people/lundheim/Page_020-029.pdf">"On Shannon and Shannon's formula"</a>, <i>Telektronikk</i>, vol. 98, no. 1, pp. 20–29, 2002.</li> <li>Mason, Warren P, <a rel="nofollow" class="external text" href="https://archive.org/stream/bellsystemtechni20amerrich#page/404/mode/2up">"Electrical and mechanical analogies"</a>, <i>Bell System Technical Journal</i>, vol. 20, no. 4, pp. 405–414, October 1941.</li> <li>Matthaei, Young, Jones, <i>Microwave Filters, Impedance-Matching Networks, and Coupling Structures</i>, McGraw-Hill 1964.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analogue_filter&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Fry, T C, <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1929-35-04/S0002-9904-1929-04747-5/">"The use of continued fractions in the design of electrical networks"</a>, <i>Bulletin of the American Mathematical Society</i>, volume 35, pages 463–498, 1929 (full text available).</li></ul> <p class="mw-empty-elt"> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Analogue_filter&oldid=1237702303">https://en.wikipedia.org/w/index.php?title=Analogue_filter&oldid=1237702303</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Linear_filters" title="Category:Linear filters">Linear filters</a></li><li><a href="/wiki/Category:Filter_theory" title="Category:Filter theory">Filter theory</a></li><li><a href="/wiki/Category:Analog_circuits" title="Category:Analog circuits">Analog circuits</a></li><li><a href="/wiki/Category:History_of_electronic_engineering" title="Category:History of electronic engineering">History of electronic engineering</a></li><li><a href="/wiki/Category:Electronic_design" title="Category:Electronic design">Electronic design</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_June_2020" title="Category:Use dmy dates from June 2020">Use dmy dates from June 2020</a></li><li><a href="/wiki/Category:Good_articles" title="Category:Good articles">Good articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 July 2024, at 03:05<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; 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