Minimum-shift keying

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Minimum-shift keying (MSK) is a type of continuous phase frequency-shift keying.

Similarly to OQPSK, MSK is encoded with bits alternating between quarternary components, with the Q component delayed by half the symbol period. However, instead of square pulses as OQPSK uses, MSK encodes each bit as a half sinusoid. This results in a constant-modulus signal, which reduces problems caused by non-linear distortion.

The resulting signal is represented by the formula

Failed to parse (Missing texvc executable; please see math/README to configure.): s(t) = a_{I}(t)\cos{\left(\frac{{\pi}t}{2T}\right)}\cos{(2{\pi}f_{c}t)}+a_{Q}(t)\sin{\left(\frac{{\pi}t}{2T}\right)}\sin{\left(2{\pi}f_{c}t\right)}

where Failed to parse (Missing texvc executable; please see math/README to configure.): a_{I}(t)

and Failed to parse (Missing texvc executable; please see math/README to configure.): a_{Q}(t)
encode the even and odd information respectively with a sequence of square pulses as of duration 2T.

Using the trigonometric identity, this can be rewritten in a form where the phase and frequency modulation is more obvious,

Failed to parse (Missing texvc executable; please see math/README to configure.): s(t) = \cos[2 \pi f_c t + b_k(t) \frac{\pi t}{2 T} + \phi_k]

where b_k(t) is +1 when Failed to parse (Missing texvc executable; please see math/README to configure.): a_{I}(t) = a_{Q}(t)

and -1 if they are of opposite signs, and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi_k
is 0 if Failed to parse (Missing texvc executable; please see math/README to configure.): a_{I}(t)
is 1, and Failed to parse (Missing texvc executable; please see math/README to configure.): \pi
otherwise. Therefore, the signal is modulated in frequency and phase, and the phase continuously and linearly changes.

A similar modulation scheme is Gaussian minimum-shift keying, which uses Gaussian instead of sinusoidal pulse shapes.

MSK, a proven and widely used technique, should not be confused with the completely unworkable Very minimum-shift keying (VMSK).

References

• Subbarayan Pasupathy, Minimal Shift Keying: A Spectrally Efficient Modulation, IEEE Communications Magazine, 1979de:Minimum Shift Keying

pl:MSK