Continuous-phase frequency-shift keying

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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 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 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.

[edit] Theory

If a finitely-valued digital signal to be transmitted (the message) is m(t), then the corresponding CPFSK signal is

s(t) = A_c \cos\left(2 \pi f_c t + D_f \int_{-\infty}^{t} m(\alpha) d \alpha\right)\,

where Ac represents the amplitude of the CPFSK signal, fc is the base carrier frequency, and Df 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.

[edit] References

Notation for the CPFSK waveform was taken from:

  • Leon W. Couch III, "Digital and Anlaog Communication Systems, 6th Edition", Prentice-Hall, Inc., 2001. ISBN 0-13-081223-4
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