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<title>Continuous phase modulation</title>

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</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;">

<li><a href="Angle_modulation" title="Angle modulation">Angle modulation</a>

</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;">

</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;">

</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;">

</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="Multiplexing" title="Multiplexing">Multiplexing</a></li></ul></td>

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<p><b>Continuous phase modulation</b> (<b>CPM</b>) is a method for <a href="Frequency_modulation" title="Frequency modulation">modulation</a> of data commonly used in <a href="Wireless_modem" class="mw-redirect" title="Wireless modem">wireless modems</a>. In contrast to other coherent digital <a href="Phase_modulation" title="Phase modulation">phase modulation</a> techniques where the <a href="Carrier_wave" title="Carrier wave">carrier</a> phase

abruptly resets to zero at the start of every symbol (e.g. M-<a href="Phase-shift_keying" title="Phase-shift keying">PSK</a>), with CPM the carrier phase is modulated in a continuous manner. For instance, with <a href="QPSK" class="mw-redirect" title="QPSK">QPSK</a> the carrier instantaneously jumps from a sine to a cosine (i.e. a 90 degree <a href="Phase_shift" class="mw-redirect" title="Phase shift">phase shift</a>) whenever one of the two message <a href="Bit" title="Bit">bits</a> of the current symbol differs from the two message bits of the previous symbol. This discontinuity requires a relatively large percentage of the power to occur outside of the intended band (e.g., high fractional out-of-band power), leading to poor <a href="Spectral_efficiency" title="Spectral efficiency">spectral efficiency</a>. Furthermore, CPM is typically implemented as a constant-envelope <a href="Waveform" title="Waveform">waveform</a>, i.e., the transmitted carrier power is constant.

Therefore, CPM is attractive because the phase continuity yields high spectral efficiency, and the <a href="Constant_envelope" title="Constant envelope">constant envelope</a> yields excellent power efficiency. The primary drawback is the high implementation complexity required for an optimal receiver.

</p>

<div class="mw-heading mw-heading2"><h2 id="Phase_memory">Phase memory</h2></div>

<p>Each symbol is modulated by gradually changing the phase of the carrier from the starting value to the final value, over the symbol duration. The modulation and <a href="Demodulation" title="Demodulation">demodulation</a> of CPM is complicated by the fact that the initial phase of each symbol is determined by the cumulative total phase of all previous transmitted symbols, which is known as the <i>phase memory</i>.

Therefore, the optimal receiver cannot make decisions on any isolated symbol without taking the entire sequence of transmitted symbols into account. This requires a <a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum-likelihood</a> sequence estimator (MLSE), which is efficiently implemented using the <a href="Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Phase_trajectory">Phase trajectory</h2></div>

<p><a href="Minimum-shift_keying" title="Minimum-shift keying">Minimum-shift keying</a> (MSK) is another name for CPM with an excess bandwidth of 1/2 and a linear <i>phase trajectory</i>. Although this linear phase trajectory is continuous, it is not <i><a href="Smoothness" title="Smoothness">smooth</a></i> since the derivative of the phase is not continuous. The spectral efficiency of CPM can be further improved by using a smooth phase trajectory. This is typically accomplished by filtering the phase trajectory prior to modulation, commonly using a <a href="Raised_cosine" class="mw-redirect" title="Raised cosine">raised cosine</a> or a <a href="Gaussian_filter" title="Gaussian filter">Gaussian filter</a>. The raised cosine filter has zero crossings offset by exactly one symbol time, and so it can yield a <i>full-response</i> CPM waveform that prevents <a href="Intersymbol_interference" title="Intersymbol interference">intersymbol interference</a> (ISI).

</p>

<div class="mw-heading mw-heading2"><h2 id="Partial_response_CPM">Partial response CPM</h2></div>

<p>Partial-response signaling, such as duo-binary signaling, is a form of intentional ISI where

a certain number of adjacent symbols interfere with each symbol in a controlled manner.

A MLSE must be used to optimally demodulate any signal in the presence of ISI. Whenever

the amount of ISI is known, such as with any partial-response signaling scheme, MLSE can be used to determine the exact symbol sequence (in the absence of noise). Since the optimal demodulation of full-response CPM already requires MLSE detection, using partial-response signaling requires little additional complexity, but can afford a comparatively smoother phase trajectory, and thus, even greater spectral efficiency. One extremely popular form of partial-response CPM is <a href="GMSK" class="mw-redirect" title="GMSK">GMSK</a>, which is used by <a href="GSM" title="GSM">GSM</a> in most of the world's 2nd generation cell phones. It is also used in <a href="802.11" class="mw-redirect" title="802.11">802.11</a> FHSS, <a href="Bluetooth" title="Bluetooth">Bluetooth</a>, and many other proprietary wireless modems.

</p>

<div class="mw-heading mw-heading2"><h2 id="Continuous-phase_frequency-shift_keying">Continuous-phase frequency-shift keying</h2></div>

<p><b>Continuous-phase frequency-shift keying</b> (CPFSK) is a commonly used variation of <a href="Frequency-shift_keying" title="Frequency-shift keying">frequency-shift keying</a> (FSK), which is itself a special case of analog <a href="Frequency_modulation" title="Frequency modulation">frequency modulation</a>. FSK is a method of modulating <a href="Digital_data" title="Digital data">digital data</a> onto a <a href="Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> <a href="Carrier_wave" title="Carrier wave">carrier wave</a>, encoding the information present in the data to variations in the carrier's <a href="Instantaneous_phase" class="mw-redirect" title="Instantaneous phase">instantaneous frequency</a> between one of two frequencies (referred to as the <a href="Space_frequency" class="mw-redirect" title="Space frequency">space frequency</a> and <a href="Mark_frequency" class="mw-redirect" title="Mark frequency">mark frequency</a>). In general, a standard FSK signal does not have <a href="Continuous_function" title="Continuous function">continuous</a> phase, as the modulated waveform cuts instantaneously between two sinusoids with different frequencies.

</p><p>As the name suggests, the phase of a CPFSK is in fact continuous; this attribute is desirable for signals that are to be transmitted over a <a href="Bandlimited" class="mw-redirect" title="Bandlimited">bandlimited</a> channel, as discontinuities in a signal introduce <a href="Wideband" title="Wideband">wideband</a> frequency components. In addition, some classes of amplifiers exhibit nonlinear behavior when driven with nearly discontinuous signals; this could have undesired effects on the shape of the transmitted signal.

</p>

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

<p>If a finitely valued digital signal to be transmitted (the message) is <i>m</i>(<i>t</i>), then the corresponding CPFSK signal is

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<annotation encoding="application/x-tex">{\displaystyle s(t)=A_{c}\cos \left(2\pi f_{c}t+D_{f}\int _{-\infty }^{t}m(\alpha )d\alpha \right)\,}</annotation>

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</math></span><img src="./411a493da55972817e993035e6a3afacedea1603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.412ex; height:6.343ex;" alt="{\displaystyle s(t)=A_{c}\cos \left(2\pi f_{c}t+D_{f}\int _{-\infty }^{t}m(\alpha )d\alpha \right)\,}" loading="lazy"></span></dd></dl>

<p>where <i>A<sub>c</sub></i> represents the amplitude of the CPFSK signal, <i>f<sub>c</sub></i> is the base <a href="Carrier_frequency" class="mw-redirect" title="Carrier frequency">carrier frequency</a>, and <i>D<sub>f</sub></i> is a parameter that controls the <a href="Frequency_deviation" title="Frequency deviation">frequency deviation</a> of the modulated signal. The <a href="Integral" title="Integral">integral</a> located inside of the <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a>'s argument is what gives the CPFSK signal its continuous phase; an integral over any finitely valued function (which <i>m</i>(<i>t</i>) is assumed to be) will not contain any discontinuities. If the message signal is assumed to be <a href="Causal" class="mw-redirect" title="Causal">causal</a>, then the limits on the integral change to a lower bound of zero and a higher bound of <i>t</i>.

</p><p>Note that this does not mean that <i>m</i>(<i>t</i>) must be continuous; in fact, most ideal digital data waveforms contain discontinuities. However, even a discontinuous message signal will generate a proper CPFSK signal.

</p>

<ul><li><a href="Minimum-shift_keying" title="Minimum-shift keying">Minimum-shift keying</a> (MSK)</li></ul>

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

<p>Notation for the CPFSK waveform was taken from:

</p>

<ul><li><i>Leon W. Couch II</i>, <i><b>"Digital and Analog Communication Systems, 6th Edition"</b></i>, <i>Prentice-Hall, Inc., 2001</i>. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-081223-4</bdi></li>

<li><a rel="nofollow" class="external autonumber" href="http://www.csee.wvu.edu/~mvalenti/documents/49IyerSeshadriCISS2k7.pdf">[1]</a> S. Cheng, R. Iyer Sehshadri, M.C. Valenti, and D. Torrieri, The capacity of noncoherent continuous-phase frequency shift keying, in <i>Proc. Conf. on Info. Sci. and Sys (CISS)</i>, (Baltimore, MD), Mar. 2007.</li>

<li><a rel="nofollow" class="external text" href="http://people.eecs.ku.edu/~esp/php/cpmdistance.php">CPM minimum distance calculator (MLSE/MLSD bound)</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-08-31" href="https://en.wikipedia.org/wiki/?title=Continuous_phase_modulation&oldid=1243276606">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.

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