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Continuous phase modulation
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== 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 cuts 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.
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