Editing Continuous phase modulation (section)

=== 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.