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class="vector-toc-numb">1.3</span> <span>Line codes for baseband transmission</span> </div> </a> <ul id="toc-Line_codes_for_baseband_transmission-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Digital_television_and_OFDM_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Digital_television_and_OFDM_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Digital television and OFDM example</span> </div> </a> <ul id="toc-Digital_television_and_OFDM_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_chip_rate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_chip_rate"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Relationship to chip rate</span> </div> </a> <ul id="toc-Relationship_to_chip_rate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_bit_error_rate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_bit_error_rate"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Relationship to bit error rate</span> </div> </a> <ul id="toc-Relationship_to_bit_error_rate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Modulation</span> </div> </a> <button aria-controls="toc-Modulation-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 Modulation subsection</span> </button> <ul id="toc-Modulation-sublist" class="vector-toc-list"> <li id="toc-Binary_modulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binary_modulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Binary modulation</span> </div> </a> <ul id="toc-Binary_modulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-N-ary_modulation,_N_greater_than_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#N-ary_modulation,_N_greater_than_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span><i>N</i>-ary modulation, <i>N</i> greater than 2</span> </div> </a> <ul id="toc-N-ary_modulation,_N_greater_than_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Not_power_of_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Not_power_of_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Not power of 2</span> </div> </a> <ul id="toc-Not_power_of_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Data_rate_versus_error_rate" 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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">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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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role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Symbol_rate" title="Special:EditPage/Symbol rate">improve this article</a> by <a 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Symbol+rate%22">"Symbol rate"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Symbol+rate%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Symbol+rate%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Symbol+rate%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Symbol+rate%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Symbol+rate%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">October 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Rate of modulation of a digital signal</div> <p class="mw-empty-elt"> </p><p>In a digitally <a href="/wiki/Modulation" title="Modulation">modulated</a> signal or a <a href="/wiki/Line_code" title="Line code">line code</a>, <b>symbol rate</b>, <b>modulation rate</b> or <b>baud rate</b> is the number of symbol changes, waveform changes, or signaling events across the transmission medium per <a href="/wiki/Unit_of_time" title="Unit of time">unit of time</a>. The symbol rate is measured in <i><a href="/wiki/Baud" title="Baud">baud</a></i> (Bd) or <i>symbols per second</i>. In the case of a line code, the symbol rate is the pulse rate in pulses per second. Each symbol can represent or convey one or several <a href="/wiki/Bit" title="Bit">bits</a> of data. The symbol rate is related to the <i><a href="/wiki/Gross_bit_rate" class="mw-redirect" title="Gross bit rate">gross bit rate</a></i>, expressed in <i><a href="/wiki/Bits_per_second" class="mw-redirect" title="Bits per second">bits per second</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Symbols">Symbols</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=1" title="Edit section: Symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A symbol may be described as either a pulse in digital baseband transmission or a tone in passband transmission using modems. A symbol is a waveform, a state or a significant condition of the <a href="/wiki/Communication_channel" title="Communication channel">communication channel</a> that <i>persists</i>, for a fixed period of time. A sending device places symbols on the channel at a fixed and known symbol rate, and the receiving device has the job of detecting the sequence of symbols in order to reconstruct the transmitted data. There may be a direct correspondence between a symbol and a small unit of <a href="/wiki/Data" title="Data">data</a>. For example, each symbol may <a href="/wiki/Encoding" class="mw-redirect" title="Encoding">encode</a> one or several binary digits (bits). The data may also be represented by the transitions between symbols, or even by a sequence of many symbols. </p><p>The <i>symbol duration time</i>, also known as <a href="/wiki/Unit_interval_(data_transmission)" title="Unit interval (data transmission)">unit interval</a>, can be directly measured as the time between transitions by looking into an <a href="/wiki/Eye_diagram" class="mw-redirect" title="Eye diagram">eye diagram</a> of an <a href="/wiki/Oscilloscope" title="Oscilloscope">oscilloscope</a>. The symbol duration time <i>T</i><sub>s</sub> can be calculated 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 T_{s}={1 \over f_{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{s}={1 \over f_{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395e54f06eb034116601c7f8d86c0ffa78b54332" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.438ex; height:5.676ex;" alt="{\displaystyle T_{s}={1 \over f_{s}}}"></span></dd></dl> <p>where <i>f</i><sub>s</sub> is the symbol rate. </p><p>For example, a baud rate of 1 kBd = 1,000 Bd is synonymous to a symbol rate of 1,000 symbols per second. In case of a modem, this corresponds to 1,000 tones per second, and in case of a line code, this corresponds to 1,000 pulses per second. The symbol duration time is 1/1,000 second = 1 millisecond. </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_gross_bit_rate">Relationship to gross bit rate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=2" title="Edit section: Relationship to gross bit rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term baud rate has sometimes incorrectly been used to mean bit rate, since these rates are the same in old <a href="/wiki/Modem" title="Modem">modems</a> as well as in the simplest digital communication links using only one bit per symbol, such that binary "0" is represented by one symbol, and binary "1" by another symbol. In more advanced modems and data transmission techniques, a symbol may have more than two states, so it may represent more than one binary digit (a binary digit always represents one of exactly two states). For this reason, the baud rate value will often be lower than the gross bit rate. </p><p><i>Example of use and misuse of "baud rate"</i>: It is correct to write "the baud rate of my COM port is 9,600" if we mean that the bit rate is 9,600 bit/s, since there is one bit per symbol in this case. It is not correct to write "the baud rate of Ethernet is 100 <a href="/wiki/Megabaud" class="mw-redirect" title="Megabaud">megabaud</a>" or "the baud rate of my modem is 56,000" if we mean bit rate. See below for more details on these techniques. </p><p>The difference between baud (or signaling rate) and the data rate (or bit rate) is like a man using a single <a href="/wiki/Flag_semaphore" title="Flag semaphore">semaphore flag</a> who can move his arm to a new position once each second, so his signaling rate (baud) is one symbol per second. The flag can be held in one of eight distinct positions: Straight up, 45° left, 90° left, 135° left, straight down (which is the rest state, where he is sending no signal), 135° right, 90° right, and 45° right. Each signal (symbol) carries three bits of information. It takes three <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> digits to encode eight states. The data rate is three bits per second. In the Navy, more than one flag pattern and arm can be used at once, so the combinations of these produce many symbols, each conveying several bits, a higher data rate. </p><p>If <i>N</i> bits are conveyed per symbol, and the gross bit rate is <i>R</i>, inclusive of channel coding overhead, the symbol rate can be calculated 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 f_{s}={R \over N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{s}={R \over N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4b8c2abc80d307420ca3328bf37811df311212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.141ex; height:5.176ex;" alt="{\displaystyle f_{s}={R \over N}}"></span></dd></dl> <p>In that case <i>M</i> = 2<sup><i>N</i></sup> different symbols are used. In a modem, these may be sinewave tones with unique combinations of amplitude, phase and/or frequency. For example, in a <a href="/wiki/64QAM" class="mw-redirect" title="64QAM">64QAM</a> modem, <i>M</i> = 64. In a line code, these may be <i>M</i> different voltage levels. </p><p>By taking information per pulse <i>N</i> in bit/pulse to be the base-2-<a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of the number of distinct messages <i>M</i> that could be sent, <a href="/wiki/Ralph_Hartley" title="Ralph Hartley">Hartley</a><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> constructed a measure of the gross bit rate <i>R</i> 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 R=f_{s}\log _{2}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=f_{s}\log _{2}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fecaf9ab2831b0826b16264099cd06b6984ccd75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.67ex; height:2.843ex;" alt="{\displaystyle R=f_{s}\log _{2}(M)}"></span></dd></dl> <p>where <i>f</i><sub><i>s</i></sub> is the baud rate in symbols/second or pulses/second. (See <a href="/wiki/Hartley%27s_law" class="mw-redirect" title="Hartley's law">Hartley's law</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Modems_for_passband_transmission">Modems for passband transmission</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=3" title="Edit section: Modems for passband transmission"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Modulation is used in <a href="/wiki/Passband" title="Passband">passband</a> filtered channels such as telephone lines, radio channels and other <a href="/wiki/Frequency_division_multiplex" class="mw-redirect" title="Frequency division multiplex">frequency division multiplex</a> (FDM) channels. </p><p>In a digital modulation method provided by a <a href="/wiki/Modem" title="Modem">modem</a>, each symbol is typically a sine wave tone with a certain frequency, amplitude and phase. Symbol rate, baud rate, is the number of transmitted tones per second. </p><p>One symbol can carry one or several bits of information. In voiceband modems for the telephone network, it is common for one symbol to carry up to 7 bits. </p><p>Conveying more than one bit per symbol or bit per pulse has advantages. It reduces the time required to send a given quantity of data over a limited <a href="/wiki/Bandwidth_(computing)" title="Bandwidth (computing)">bandwidth</a>. A high <a href="/wiki/Spectral_efficiency" title="Spectral efficiency">spectral efficiency</a> in (bit/s)/Hz can be achieved; i.e., a high bit rate in bit/s although the bandwidth in hertz may be low. </p><p>The maximum baud rate for a passband for common modulation methods such as <a href="/wiki/QAM" class="mw-redirect" title="QAM">QAM</a>, <a href="/wiki/Phase-shift_keying" title="Phase-shift keying">PSK</a> and <a href="/wiki/OFDM" class="mw-redirect" title="OFDM">OFDM</a> is approximately equal to the passband bandwidth.<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> </p><p>Voiceband modem examples: </p> <ul><li>A <a href="/wiki/ITU-T_V.22bis" class="mw-redirect" title="ITU-T V.22bis">V.22bis</a> modem transmits 2400 bit/s using 1200 Bd (1200 symbol/s), where each <a href="/wiki/Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">quadrature amplitude modulation</a> symbol carries two bits of <a href="/wiki/Information_theory" title="Information theory">information</a>. The modem can generate <i>M</i>=2<sup>2</sup>=4 different symbols. It requires a bandwidth of 1200 Hz (equal to the baud rate). The <a href="/wiki/Carrier_frequency" class="mw-redirect" title="Carrier frequency">carrier frequency</a> is 1800 Hz, meaning that the lower cut off frequency is 1,800 − 1,200/2 = 1,200 Hz, and the upper cutoff frequency is 1,800 + 1,200/2 = 2,400 Hz.</li> <li>A <a href="/wiki/ITU-T_V.34" class="mw-redirect" title="ITU-T V.34">V.34</a> modem may transmit symbols at a baud rate of 3,420 Bd, and each symbol can carry up to ten bits, resulting in a gross bit rate of 3420 × 10 = 34,200 bit/s. However, the modem is said to operate at a net bit rate of 33,800 bit/s, excluding physical layer overhead.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Line_codes_for_baseband_transmission">Line codes for baseband transmission</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=4" title="Edit section: Line codes for baseband transmission"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In case of a <a href="/wiki/Baseband" title="Baseband">baseband</a> channel such as a telegraph line, a serial cable or a Local Area Network twisted pair cable, data is transferred using line codes; i.e., <a href="/wiki/Pulse_(signal_processing)" title="Pulse (signal processing)">pulses</a> rather than sinewave tones. In this case, the baud rate is synonymous to the pulse rate in pulses/second. </p><p>The maximum baud rate or pulse rate for a <a href="/wiki/Base_band" class="mw-redirect" title="Base band">base band</a> channel is called the <a href="/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>, and is double the bandwidth (double the cut-off frequency). </p><p>The simplest digital communication links (such as individual wires on a motherboard or the RS-232 serial port/COM port) typically have a symbol rate equal to the gross bit rate. </p><p>Common communication links such as 10 Mbit/s <a href="/wiki/Ethernet" title="Ethernet">Ethernet</a> (<a href="/wiki/10BASE-T" class="mw-redirect" title="10BASE-T">10BASE-T</a>), <a href="/wiki/USB" title="USB">USB</a>, and <a href="/wiki/FireWire" class="mw-redirect" title="FireWire">FireWire</a> typically have a data bit rate slightly lower than the baud rate, due to the overhead of extra non-data symbols used for <a href="/wiki/Self-synchronizing_code" title="Self-synchronizing code">self-synchronizing code</a> and <a href="/wiki/Error_detection" class="mw-redirect" title="Error detection">error detection</a>. </p><p>J. M. Emile Baudot (1845–1903) worked out a five-bit code for telegraphs which was standardized internationally and is commonly called <a href="/wiki/Baudot_code" title="Baudot code">Baudot code</a>. </p><p>More than two voltage levels are used in advanced techniques such as <a href="/wiki/FDDI" class="mw-redirect" title="FDDI">FDDI</a> and 100/1,000 <a href="/wiki/Megabit" class="mw-redirect" title="Megabit">Mbit</a>/s Ethernet LANs, and others, to achieve high data rates. </p><p>1,000 Mbit/s Ethernet LAN cables use four wire pairs in <a href="/wiki/Full_duplex" class="mw-redirect" title="Full duplex">full duplex</a> (250 Mbit/s per pair in both directions simultaneously), and many bits per symbol to encode their data payloads. </p> <div class="mw-heading mw-heading3"><h3 id="Digital_television_and_OFDM_example">Digital television and OFDM example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=5" title="Edit section: Digital television and OFDM example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Digital_television" title="Digital television">digital television</a> transmission the symbol rate calculation is: </p> <dl><dd>symbol rate in symbols per second = (Data rate in bits per second × 204) / (188 × bits per symbol)</dd></dl> <p>The 204 is the number of bytes in a packet including the 16 trailing <a href="/wiki/Reed%E2%80%93Solomon_error_correction" title="Reed–Solomon error correction">Reed–Solomon error correction</a> bytes. The 188 is the number of data bytes (187 bytes) plus the leading packet <a href="/wiki/Syncword" title="Syncword">sync byte</a> (0x47). </p><p>The bits per symbol is the (modulation's power of 2) × (Forward Error Correction). So for example, in 64-QAM modulation 64 = 2<sup>6</sup> so the bits per symbol is 6. The Forward Error Correction (FEC) is usually expressed as a fraction; i.e., 1/2, 3/4, etc. In the case of 3/4 FEC, for every 3 bits of data, you are sending out 4 bits, one of which is for error correction. </p><p>Example: </p> <dl><dd>given bit rate = 18096263 <dl><dd>Modulation type = 64-QAM</dd> <dd>FEC = 3/4</dd></dl></dd></dl> <p>then </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 {\text{symbol rate}}={\cfrac {18096263}{6\cdot {\frac {3}{4}}}}~{\cfrac {204}{188}}={\cfrac {18096263}{6}}~{\cfrac {4}{3}}~{\cfrac {204}{188}}=4363638}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>symbol rate</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>18096263</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>204</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>188</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>18096263</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>204</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>188</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4363638</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{symbol rate}}={\cfrac {18096263}{6\cdot {\frac {3}{4}}}}~{\cfrac {204}{188}}={\cfrac {18096263}{6}}~{\cfrac {4}{3}}~{\cfrac {204}{188}}=4363638}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57231b9b488bc090040a8f8cccc378f6e9413a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:61.865ex; height:7.676ex;" alt="{\displaystyle {\text{symbol rate}}={\cfrac {18096263}{6\cdot {\frac {3}{4}}}}~{\cfrac {204}{188}}={\cfrac {18096263}{6}}~{\cfrac {4}{3}}~{\cfrac {204}{188}}=4363638}"></span></dd></dl> <p>In digital terrestrial television (<a href="/wiki/DVB-T" title="DVB-T">DVB-T</a>, <a href="/wiki/DVB-H" title="DVB-H">DVB-H</a> and similar techniques) <a href="/wiki/OFDM" class="mw-redirect" title="OFDM">OFDM</a> modulation is used; i.e., multi-carrier modulation. The above symbol rate should then be divided by the number of OFDM sub-carriers in view to achieve the OFDM symbol rate. See the <a href="/wiki/OFDM_system_comparison_table" class="mw-redirect" title="OFDM system comparison table">OFDM system comparison table</a> for further numerical details. </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_chip_rate">Relationship to chip rate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=6" title="Edit section: Relationship to chip rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some communication links (such as <a href="/wiki/GPS" class="mw-redirect" title="GPS">GPS</a> transmissions, <a href="/wiki/CDMA" class="mw-redirect" title="CDMA">CDMA</a> cell phones, and other <a href="/wiki/Spread_spectrum" title="Spread spectrum">spread spectrum</a> links) have a symbol rate much higher than the data rate (they transmit many symbols called <a href="/wiki/Chip_(CDMA)" title="Chip (CDMA)">chips</a> per data bit). Representing one bit by a chip sequence of many symbols overcomes <a href="/wiki/Co-channel_interference" title="Co-channel interference">co-channel interference</a> from other transmitters sharing the same frequency channel, including <a href="/wiki/Radio_jamming" title="Radio jamming">radio jamming</a>, and is common in <a href="/wiki/Military_radio" class="mw-redirect" title="Military radio">military radio</a> and <a href="/wiki/Mobile_phone" title="Mobile phone">cell phones</a>. Despite the fact that using more <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> to carry the same bit rate gives low <a href="/wiki/Channel_spectral_efficiency" class="mw-redirect" title="Channel spectral efficiency">channel spectral efficiency</a> in (bit/s)/Hz, it allows many simultaneous users, which results in high <a href="/wiki/System_spectral_efficiency" class="mw-redirect" title="System spectral efficiency">system spectral efficiency</a> in (bit/s)/Hz per unit of area. </p><p>In these systems, the symbol rate of the physically transmitted high-frequency signal rate is called <a href="/wiki/Chip_rate" class="mw-redirect" title="Chip rate">chip rate</a>, which also is the pulse rate of the equivalent <a href="/wiki/Base_band" class="mw-redirect" title="Base band">base band</a> signal. However, in spread spectrum systems, the term symbol may also be used at a higher layer and refer to one information bit, or a block of information bits that are modulated using for example conventional QAM modulation, before the CDMA spreading code is applied. Using the latter definition, the symbol rate is equal to or lower than the bit rate. </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_bit_error_rate">Relationship to bit error rate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=7" title="Edit section: Relationship to bit error rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The disadvantage of conveying many bits per symbol is that the receiver has to distinguish many signal levels or symbols from each other, which may be difficult and cause bit errors in case of a poor phone line that suffers from low signal-to-noise ratio. In that case, a modem or network adapter may automatically choose a slower and more robust modulation scheme or line code, using fewer bits per symbol, in view to reduce the bit error rate. </p><p>An optimal symbol set design takes into account channel bandwidth, desired information rate, noise characteristics of the channel and the receiver, and receiver and decoder complexity. </p> <div class="mw-heading mw-heading2"><h2 id="Modulation">Modulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=8" title="Edit section: Modulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many <a href="/wiki/Data_transmission" class="mw-redirect" title="Data transmission">data transmission</a> systems operate by the <a href="/wiki/Modulation" title="Modulation">modulation</a> of a <a href="/wiki/Carrier_wave" title="Carrier wave">carrier signal</a>. For example, in <a href="/wiki/Frequency-shift_keying" title="Frequency-shift keying">frequency-shift keying (FSK)</a>, the frequency of a tone is varied among a small, fixed set of possible values. In a <a href="/wiki/Synchronization" title="Synchronization">synchronous</a> data transmission system, the tone can only be changed from one frequency to another at regular and well-defined intervals. The presence of one particular frequency during one of these intervals constitutes a symbol. (The concept of symbols does not apply to asynchronous data transmission systems.) In a modulated system, the term <b>modulation rate</b> may be used synonymously with symbol rate. </p> <div class="mw-heading mw-heading3"><h3 id="Binary_modulation">Binary modulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=9" title="Edit section: Binary modulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the carrier signal has only two states, then only one <a href="/wiki/Bit" title="Bit">bit</a> of data (i.e., a 0 or 1) can be transmitted in each symbol. The <a href="/wiki/Bit_rate" title="Bit rate">bit rate</a> is in this case equal to the symbol rate. For example, a binary FSK system would allow the carrier to have one of two frequencies, one representing a 0 and the other a 1. A more practical scheme is <a href="/wiki/Phase-shift_keying" title="Phase-shift keying">differential binary phase-shift keying</a>, in which the carrier remains at the same frequency, but can be in one of two phases. During each symbol, the phase either remains the same, encoding a 0, or jumps by 180°, encoding a 1. Again, only one <a href="/wiki/Bit" title="Bit">bit</a> of data (i.e., a 0 or 1) is transmitted by each symbol. This is an example of data being encoded in the transitions between symbols (the change in phase), rather than the symbols themselves (the actual phase). (The reason for this in phase-shift keying is that it is impractical to know the reference phase of the transmitter.) </p> <div class="mw-heading mw-heading3"><h3 id="N-ary_modulation,_N_greater_than_2"><span id="N-ary_modulation.2C_N_greater_than_2"></span><i>N</i>-ary modulation, <i>N</i> greater than 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=10" title="Edit section: N-ary modulation, N greater than 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By increasing the number of states that the carrier signal can take, the number of bits encoded in each symbol can be greater than one. The bit rate can then be greater than the symbol rate. For example, a differential phase-shift keying system might allow four possible jumps in phase between symbols. Then two bits could be encoded at each symbol interval, achieving a data rate of double the symbol rate. In a more complex scheme such as <a href="/wiki/16-QAM" class="mw-redirect" title="16-QAM">16-QAM</a>, four bits of data are transmitted in each symbol, resulting in a bit rate of four times the symbol rate. </p> <div class="mw-heading mw-heading3"><h3 id="Not_power_of_2">Not power of 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=11" title="Edit section: Not power of 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although it is common to choose the number of symbols to be a power of 2 and send an integer number of bits per baud, this is not required. Line codes such as <a href="/wiki/Bipolar_encoding" title="Bipolar encoding">bipolar encoding</a> and <a href="/wiki/MLT-3" class="mw-redirect" title="MLT-3">MLT-3</a> use three carrier states to encode one bit per baud while maintaining <a href="/wiki/DC_balance" class="mw-redirect" title="DC balance">DC balance</a>. </p><p>The <a href="/wiki/4B3T" title="4B3T">4B3T</a> line code uses three 3-ary modulated bits to transmit four data bits, a rate of 1.3<span style="text-decoration:overline;">3</span> bits per baud. </p> <div class="mw-heading mw-heading3"><h3 id="Data_rate_versus_error_rate">Data rate versus error rate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=12" title="Edit section: Data rate versus error rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Modulating a carrier increases the frequency range, or <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>, it occupies. Transmission channels are generally limited in the bandwidth they can carry. The bandwidth depends on the symbol (modulation) rate (not directly on the <a href="/wiki/Bit_rate" title="Bit rate">bit rate</a>). As the bit rate is the product of the symbol rate and the number of bits encoded in each symbol, it is clearly advantageous to increase the latter if the former is fixed. However, for each additional bit encoded in a symbol, the constellation of symbols (the number of states of the carrier) doubles in size. This makes the states less distinct from one another which in turn makes it more difficult for the receiver to detect the symbol correctly in the presence of disturbances on the channel. </p><p>The history of <a href="/wiki/Modem" title="Modem">modems</a> is the attempt at increasing the bit rate over a fixed bandwidth (and therefore a fixed maximum symbol rate), leading to increasing bits per symbol. For example, <a href="/wiki/ITU-T_V.29" class="mw-redirect" title="ITU-T V.29">ITU-T V.29</a> specifies 4 bits per symbol, at a symbol rate of 2,400 baud, giving an effective bit rate of 9,600 bits per second. </p><p>The history of <a href="/wiki/Spread_spectrum" title="Spread spectrum">spread spectrum</a> goes in the opposite direction, leading to fewer and fewer data bits per symbol in order to spread the bandwidth. In the case of GPS, we have a data rate of 50 bit/s and a symbol rate of 1.023 Mchips/s. If each chip is considered a symbol, each symbol contains far less than one bit (50 bit/s / 1,023 ksymbols/s ≈ 0.000,05 bits/symbol). </p><p>The complete collection of <i>M</i> possible symbols over a particular channel is called a <b>M-ary</b> <a href="/wiki/Modulation" title="Modulation">modulation</a> scheme. Most modulation schemes transmit some integer number of bits per symbol <i>b</i>, requiring the complete collection to contain <i>M</i> = 2<sup><i>b</i></sup> different symbols. Most popular modulation schemes can be described by showing each point on a <a href="/wiki/Constellation_diagram" title="Constellation diagram">constellation diagram</a>, although a few modulation schemes (such as <a href="/wiki/Multiple_frequency-shift_keying" title="Multiple frequency-shift keying">MFSK</a>, <a href="/wiki/Dual-tone_multi-frequency" class="mw-redirect" title="Dual-tone multi-frequency">DTMF</a>, <a href="/wiki/Pulse-position_modulation" title="Pulse-position modulation">pulse-position modulation</a>, <a href="/wiki/Spread_spectrum" title="Spread spectrum">spread spectrum</a> modulation) require a different description. </p> <div class="mw-heading mw-heading2"><h2 id="Significant_condition">Significant condition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=13" title="Edit section: Significant condition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Telecommunication" class="mw-redirect" title="Telecommunication">telecommunication</a>, concerning the <a href="/wiki/Modulation" title="Modulation">modulation</a> of a <a href="/wiki/Carrier_(telecommunication)" class="mw-redirect" title="Carrier (telecommunication)">carrier</a>, a <b>significant condition</b> is one of the <a href="/wiki/Signal" title="Signal">signal</a>'s parameters chosen to represent <a href="/wiki/Information" title="Information">information</a>.<sup id="cite_ref-FS-1037C_3-0" class="reference"><a href="#cite_note-FS-1037C-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A significant condition could be an electric current (voltage, or <a href="/wiki/Electric_power" title="Electric power">power</a> level), an optical power level, a <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> value, or a particular <a href="/wiki/Frequency" title="Frequency">frequency</a> or <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>. The duration of a significant condition is the <a href="/wiki/Time" title="Time">time</a> interval between successive significant instants.<sup id="cite_ref-FS-1037C_3-1" class="reference"><a href="#cite_note-FS-1037C-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A change from one significant condition to another is called a <i><a href="/wiki/Signal_transition" title="Signal transition">signal transition</a>.</i> Information can be transmitted either during the given time interval, or encoded as the presence or absence of a change in the received signal.<sup id="cite_ref-MIL-STD-188-200_4-0" class="reference"><a href="#cite_note-MIL-STD-188-200-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Significant conditions are recognized by an appropriate device called a receiver, demodulator, or decoder. The decoder translates the actual signal received into its intended <a href="/wiki/Truth_value" title="Truth value">logical value</a> such as a binary digit (0 or 1), an alphabetic character, a mark, or a space. Each <a href="/w/index.php?title=Significant_instant&action=edit&redlink=1" class="new" title="Significant instant (page does not exist)">significant instant</a> is determined when the appropriate device assumes a condition or state usable for performing a specific function, such as recording, processing, or <a href="/wiki/Gating_(telecommunication)" title="Gating (telecommunication)">gating</a>.<sup id="cite_ref-FS-1037C_3-2" class="reference"><a href="#cite_note-FS-1037C-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <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=Symbol_rate&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Data_signaling_rate" title="Data signaling rate">Data signaling rate</a></li> <li><a href="/wiki/List_of_interface_bit_rates" title="List of interface bit rates">List of interface bit rates</a></li> <li><a href="/wiki/Pulse-code_modulation" title="Pulse-code modulation">Pulse-code modulation</a></li></ul> <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=Symbol_rate&action=edit&section=15" title="Edit section: References"><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"> <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">.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><cite id="CITEREFD._A._Bell1962" class="citation book cs1">D. A. Bell (1962). <i>Information Theory; and its Engineering Applications</i> (3rd ed.). New York: Pitman.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory%3B+and+its+Engineering+Applications&rft.place=New+York&rft.edition=3rd&rft.pub=Pitman&rft.date=1962&rft.au=D.+A.+Bell&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymbol+rate" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://web.cs.ucdavis.edu/~liu/289I/Material/book-goldsmith.pdf">Goldsmith A. Wireless communications. – Stanford University, 2004.</a> - p. 140, 326</span> </li> <li id="cite_note-FS-1037C-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FS-1037C_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FS-1037C_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FS-1037C_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">"Federal Standard 1037C"</a>. National Communications System. July 7, 1996.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Federal+Standard+1037C&rft.date=1996-07-07&rft_id=http%3A%2F%2Fwww.its.bldrdoc.gov%2Ffs-1037%2Ffs-1037c.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymbol+rate" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-MIL-STD-188-200-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-MIL-STD-188-200_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://assist.daps.dla.mil/quicksearch/basic_profile.cfm?ident_number=35582">"System Design and Engineering Standard for Tactical Communications"</a>. <i>Mil-STD-188-200</i>. <a href="/wiki/United_States_Department_of_Defense" title="United States Department of Defense">United States Department of Defense</a>. May 28, 1983.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mil-STD-188-200&rft.atitle=System+Design+and+Engineering+Standard+for+Tactical+Communications&rft.date=1983-05-28&rft_id=http%3A%2F%2Fassist.daps.dla.mil%2Fquicksearch%2Fbasic_profile.cfm%3Fident_number%3D35582&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymbol+rate" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symbol_rate&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.heyrick.co.uk/ricksworld/digibox/symfec.html">What is the Symbol rate?</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://archive.today/20121205233200/http://www.compkarori.com/dbase/bu07sh.htm">"On the origins of serial communications and data encoding"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.compkarori.com/dbase/bu07sh.htm">the original</a> on December 5, 2012<span class="reference-accessdate">. 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