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[[GMSK]], which is used by [[GSM]] in most of the world's 2nd generation cell phones. It is also used in [[802.11]] FHSS, [[Bluetooth]], and many other proprietary wireless modems.
== Continuous-phase frequency-shift keying ==
'''Continuous-phase frequency-shift keying''' (CPFSK) is a commonly-used variation of [[frequency-shift keying]] (FSK), which is itself a special case of analog [[frequency modulation]]. FSK is a method of modulating [[digital data]] onto a [[sinusoidal]] [[carrier wave]], encoding the information present in the data to variations in the carrier's [[instantaneous phase#Instantaneous frequency|instantaneous frequency]] between one of two frequencies (referred to as the [[space frequency]] and [[mark frequency]]). In general, a standard FSK signal does not have [[continuous function|continuous]] phase, as the modulated waveform switches instantaneously between two sinusoids with different frequencies.
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 [[bandlimited]] channel, as [[discontinuities]] in a signal introduce [[wideband]] 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.
=== Theory ===
If a finitely-valued digital signal to be transmitted (the message) is ''m''(''t''), then the corresponding CPFSK signal is
:<math>s(t) = A_c \cos\left(2 \pi f_c t + D_f \int_{-\infty}^{t} m(\alpha) d \alpha\right)\,</math>
where ''A<sub>c</sub>'' represents the amplitude of the CPFSK signal, ''f<sub>c</sub>'' is the base [[carrier frequency]], and ''D<sub>f</sub>'' is a parameter that controls the [[frequency deviation]] of the modulated signal. The [[integral]] located inside of the [[cosine]]'s argument is what gives the CPFSK signal its continuous phase; an integral over any finitely-valued function (which ''m''(''t'') is assumed to be) will not contain any discontinuities. If the message signal is assumed to be [[causal]], then the limits on the integral change to a lower bound of zero and a higher bound of ''t''.
Note that this does not mean that ''m''(''t'') must be continuous; in fact, most ideal digital data waveforms contain discontinuities. However, even a discontinuous message signal will generate a proper CPFSK signal.
==See also==
*[[Minimum-shift keying|MSK]]
== References ==
Notation for the CPFSK waveform was taken from:
*''Leon W. Couch III'', '''''"Digital and Analog Communication Systems, 6th Edition"''''', ''Prentice-Hall, Inc., 2001''. ISBN 0-13-081223-4
*[http://www.csee.wvu.edu/~mvalenti/documents/49IyerSeshadriCISS2k7.pdf] S. Cheng, R. Iyer Sehshadri, M.C. Valenti, and D. Torrieri, The capacity of noncoherent continuous-phase frequency shift keying, in ''Proc. Conf. on Info. Sci. and Sys (CISS)'', (Baltimore, MD), Mar. 2007.
* [http://people.eecs.ku.edu/~esp/php/cpmdistance.php CPM minimum distance calculator (MLSE/MLSD bound)]
[[Category:Wireless networking]]
[[Category:Radio modulation modes]]
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