<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Trellis coded modulation</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Trellis_coded_modulation"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Trellis_coded_modulation rootpage-Trellis_coded_modulation skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Trellis coded modulation</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">
<style data-mw-deduplicate="TemplateStyles:r1129693374">
/* start https://en.wikipedia.org/ */
.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 ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}
/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1246091330">
/* start https://en.wikipedia.org/ */
.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .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}}
/* end https://en.wikipedia.org/ */
</style><table class="sidebar nomobile nowraplinks skin-invert"><tbody><tr><th class="sidebar-title" style="background-color: #bdb"><a href="Passband" title="Passband">Passband</a> <a href="Signal_modulation" title="Signal modulation">modulation</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"></span></td></tr><tr><th class="sidebar-heading" style="background-color: #cfc;">
<a href="Signal_modulation#Analog_modulation_methods" title="Signal modulation">Analog modulation</a></th></tr><tr><td class="sidebar-content hlist" style="color: black;">
<ul><li><a href="Amplitude_modulation" title="Amplitude modulation">AM</a>
<ul><li><a href="Space_modulation" title="Space modulation">SM</a></li>
<li><a href="Single-sideband_modulation" title="Single-sideband modulation">SSB</a></li></ul></li>
<li><a href="Angle_modulation" title="Angle modulation">Angle modulation</a>
<ul><li><a href="Frequency_modulation" title="Frequency modulation">FM</a></li>
<li><a href="Phase_modulation" title="Phase modulation">PM</a></li></ul></li>
<li><a href="Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">QAM</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background-color: #cfc;">
<a href="Signal_modulation#Digital_modulation_methods" title="Signal modulation">Digital modulation</a></th></tr><tr><td class="sidebar-content hlist" style="color: black;">
<ul><li><a href="Amplitude-shift_keying" title="Amplitude-shift keying">ASK</a></li>
<li><a href="Amplitude_and_phase-shift_keying" title="Amplitude and phase-shift keying">APSK</a></li>
<li><a href="Continuous_phase_modulation" title="Continuous phase modulation">CPM</a></li>
<li><a href="Frequency-shift_keying" title="Frequency-shift keying">FSK</a></li>
<li><a href="Multiple_frequency-shift_keying" title="Multiple frequency-shift keying">MFSK</a></li>
<li><a href="Minimum-shift_keying" title="Minimum-shift keying">MSK</a></li>
<li><a href="On%E2%80%93off_keying" title="On–off keying">OOK</a></li>
<li><a href="Pulse-position_modulation" title="Pulse-position modulation">PPM</a></li>
<li><a href="Phase-shift_keying" title="Phase-shift keying">PSK</a></li>
<li><a href="Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">QAM</a></li>
<li><a href="Single-carrier_FDMA" title="Single-carrier FDMA">SC-FDE</a></li>
<li><a href="TC-PAM" title="TC-PAM">TC-PAM</a></li>
<li><a href="Wavelet_modulation" title="Wavelet modulation">WDM</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background-color: #cfc;">
<a href="Hierarchical_modulation" title="Hierarchical modulation">Hierarchical modulation</a></th></tr><tr><td class="sidebar-content hlist" style="color: black;">
<ul><li><a href="Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">QAM</a></li>
<li><a href="Wavelet_modulation" title="Wavelet modulation">WDM</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background-color: #cfc;">
<a href="Spread_spectrum" title="Spread spectrum">Spread spectrum</a></th></tr><tr><td class="sidebar-content hlist" style="color: black;">
<ul><li><a href="Chirp_spread_spectrum" title="Chirp spread spectrum">CSS</a></li>
<li><a href="Direct-sequence_spread_spectrum" title="Direct-sequence spread spectrum">DSSS</a></li>
<li><a href="Frequency-hopping_spread_spectrum" title="Frequency-hopping spread spectrum">FHSS</a></li>
<li><a href="Time-hopping" title="Time-hopping">THSS</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="background-color: #cfc;">
See also</th></tr><tr><td class="sidebar-content hlist" style="color: black;">
<ul><li>Capacity-approaching codes</li>
<li><a href="Demodulation" title="Demodulation">Demodulation</a></li>
<li><a href="Line_code" title="Line code">Line coding</a></li>
<li><a href="Modem" title="Modem">Modem</a></li>
<li><a href="Angle_modulation" title="Angle modulation">AnM</a></li>
<li><a href="Polar_modulation" title="Polar modulation">PoM</a></li>
<li><a href="Pulse-amplitude_modulation" title="Pulse-amplitude modulation">PAM</a></li>
<li><a href="Pulse-code_modulation" title="Pulse-code modulation">PCM</a></li>
<li><a href="Pulse-density_modulation" title="Pulse-density modulation">PDM</a></li>
<li><a href="Pulse-width_modulation" title="Pulse-width modulation">PWM</a></li>
<li><a href="Delta-sigma_modulation" title="Delta-sigma modulation">ΔΣM</a></li>
<li><a href="Orthogonal_frequency-division_multiplexing" title="Orthogonal frequency-division multiplexing">OFDM</a></li>
<li><a href="Frequency-division_multiplexing" title="Frequency-division multiplexing">FDM</a></li>
<li><a href="Multiplexing" title="Multiplexing">Multiplexing</a></li></ul></td>
</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
/* start https://en.wikipedia.org/ */
.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}}
/* end https://en.wikipedia.org/ */
</style></td></tr></tbody></table>
<p><b>Trellis coded modulation</b> (<b>TCM</b>) is a <a href="Modulation" class="mw-redirect" title="Modulation">modulation</a> scheme that transmits information with high efficiency over band-limited channels such as <a href="Telephone_line" title="Telephone line">telephone lines</a>. <a href="Gottfried_Ungerboeck" title="Gottfried Ungerboeck">Gottfried Ungerboeck</a> invented trellis modulation while working for IBM in the 1970s, and first described it in a conference paper in 1976. It went largely unnoticed, however, until he published a new, detailed exposition in 1982 that achieved sudden and widespread recognition.
</p><p>In the late 1980s, <a href="Modem" title="Modem">modems</a> operating over <a href="Plain_old_telephone_service" title="Plain old telephone service">plain old telephone service</a> (<i>POTS</i>) typically achieved 9.6 <a href="Kbit/s" class="mw-redirect" title="Kbit/s">kbit/s</a> by employing four bits per symbol <a href="Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">QAM</a> modulation at 2,400 baud (symbols/second). This bit rate ceiling existed despite the best efforts of many researchers, and some engineers predicted that without a major upgrade of the public phone infrastructure, the maximum achievable rate for a POTS modem might be 14 kbit/s for two-way communication (3,429 baud × 4 bits/symbol, using QAM).
</p><p>14 kbit/s is only 40% of the theoretical maximum bit rate predicted by <a href="Shannon's_theorem" class="mw-redirect" title="Shannon's theorem">Shannon's theorem</a> for POTS lines (approximately 35 kbit/s).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ungerboeck's theories demonstrated that there was considerable untapped potential in the system, and by applying the concept to new modem standards, speed rapidly increased to 14.4, 28.8 and ultimately 33.6 kbit/s.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="A_new_modulation_method">A new modulation method</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1305420660">
/* start https://en.wikipedia.org/ */
.mw-parser-output .ambox{border:1px solid #a2a9b1;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}html body.mediawiki .mw-parser-output .metadata.ambox{border-left:10px solid #36c!important;background-color:var(--background-color-neutral-subtle,#fbfbfb)!important}html body.mediawiki .mw-parser-output .metadata.ambox-speedy{border-left:10px solid #b32424!important;background-color:var(--background-color-destructive-subtle,#fee7e6)!important}html body.mediawiki .mw-parser-output .metadata.ambox-delete{border-left:10px solid #b32424!important}html body.mediawiki .mw-parser-output .metadata.ambox-content{border-left:10px solid #f28500!important}html body.mediawiki .mw-parser-output .metadata.ambox-style{border-left:10px solid #fc3!important}html body.mediawiki .mw-parser-output .metadata.ambox-move{border-left:10px solid #9932cc!important}html body.mediawiki .mw-parser-output .metadata.ambox-protection{border-left:10px solid #a2a9b1!important}.mw-parser-output .ambox.mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}
/* end https://en.wikipedia.org/ */
</style>
<p>The name <i>trellis</i> derives from the fact that a state diagram of the technique closely resembles a <a href="Garden_trellis" class="mw-redirect" title="Garden trellis">trellis lattice</a>. The scheme is basically a <a href="Convolutional_code" title="Convolutional code">convolutional code</a> of rates (<i>r</i>, <i>r</i>+1). Ungerboeck's unique contribution is to apply the parity check for each <a href="Symbol_(data)" class="mw-redirect" title="Symbol (data)">symbol</a>, instead of the older technique of applying it to the bit stream then modulating the bits. He called the key idea <i>mapping by set partitions</i>. This idea groups symbols in a tree-like structure, then separates them into two limbs of equal size. At each "limb" of the tree, the symbols are further apart.
</p><p>Though hard to visualize in multiple dimensions, a simple one-dimension example illustrates the basic procedure. Suppose the symbols are located at [1, 2, 3, 4, ...]. Place all odd symbols in one group, and all even symbols in the second group. (This is not quite accurate, because Ungerboeck was looking at the two dimensional problem, but the principle is the same.) Take every other symbol in each group and repeat the procedure for each tree limb. He next described a method of assigning the encoded bit stream onto the symbols in a very systematic procedure. Once this procedure was fully described, his next step was to program the algorithms into a computer and let the computer search for the best codes. The results were astonishing. Even the most simple code (4 state) produced error rates nearly one one-thousandth of an equivalent uncoded system. For two years Ungerboeck kept these results private and only conveyed them to close colleagues. Finally, in 1982, Ungerboeck published a paper describing the principles of trellis modulation.
</p><p>A flurry of research activity ensued, and by 1984 the <a href="International_Telecommunication_Union" title="International Telecommunication Union">International Telecommunication Union</a> had published a standard, V.32,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> for the first trellis-modulated modem at 9.6 kilobit/s (2,400 baud and 4 bits per symbol). Over the next several years further advances in encoding, plus a corresponding symbol rate increase from 2,400 to 3,429 baud, allowed modems to achieve rates up to 34.3 kilobits/s (limited by maximum power regulations to 33.8 kilobits/s). Today, the most common trellis-modulated V.34 modems use a 4-dimensional set partition—achieved by treating two two-dimensional symbols as a single lattice. This set uses 8, 16, or 32 state convolutional codes to squeeze the equivalent of 6 to 10 bits into each symbol the modem sends (for example, 2,400 baud × 8 bits/symbol = 19,200 bit/s).
</p>
<div class="mw-heading mw-heading2"><h2 id="Relevant_papers">Relevant papers</h2></div>
<ul><li>G. Ungerboeck, "Channel coding with multilevel/phase signals," <i>IEEE Trans. Inf. Theory</i>, vol. IT-28, pp. 55–67, 1982.</li>
<li>G. Ungerboeck, "Trellis-coded modulation with redundant signal sets part I: introduction," <i>IEEE Communications Magazine</i>, vol. 25-2, pp. 5–11, 1987.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Modem#Narrowband.2FVarious_Encoding_Methods" title="Modem">Modems</a>, for the history of various encoding modulations from 0.3 to 56 kbit/s</li>
<li><a href="Convolutional_code#Trellis_diagram" title="Convolutional code">Trellis diagram</a>, in the article about <a href="Convolutional_code" title="Convolutional code">convolutional codes</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}
/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFForney1984" class="citation journal cs1">Forney, G. David; et al. (September 1984). "Efficient modulation for band-limited channels". <i>IEEE Journal on Selected Areas in Communications</i>. <b>2</b> (5): <span class="nowrap">632–</span>647. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fjsac.1984.1146101">10.1109/jsac.1984.1146101</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13818684">13818684</a>.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.itu.int/ITU-T/recommendations/rec.aspx?rec=2752&lang=en">"ITU-T Recommendation database"</a>.</cite></span>
</li>
</ol></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://complextoreal.com/wp-content/uploads/2013/01/tcm.pdf">TCM tutorial</a></li>
<li><a rel="nofollow" class="external text" href="http://www.ieeeghn.org/wiki/index.php/Oral-History:Gottfried_Ungerboeck">Oral-History:Gottfried Ungerboeck</a>, Engineering and Technology History Wiki (IEEE Global History Network)</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2024-04-25" href="https://en.wikipedia.org/wiki/?title=Trellis_coded_modulation&oldid=1220791801">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>