Minimum-shift keying

Mapping changes in continuous phase

In digital modulation, Minimum-shift keying (MSK) is a type of continuous phase frequency-shift keying. Similar 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.

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

Mathematical representation

The resulting signal is represented by the formula

$s(t)=a_{I}(t)\cos {\left({\frac {{\pi }t}{2T}}\right)}\cos {(2{\pi }f_{c}t)}-a_{Q}(t)\sin {\left({\frac {{\pi }t}{2T}}\right)}\sin {\left(2{\pi }f_{c}t\right)}$

where $a_{I}(t)$ and $a_{Q}(t)$ 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,

$s(t)=\cos[2\pi f_{c}t+b_{k}(t){\frac {\pi t}{2T}}+\phi _{k}]$

where b_k(t) is +1 when $a_{I}(t)=a_{Q}(t)$ and -1 if they are of opposite signs, and $\phi _{k}$ is 0 if $a_{I}(t)$ is 1, and $\pi$ otherwise. Therefore, the signal is modulated in frequency and phase, and the phase continuously and linearly changes.

Gaussian minimum-shift keying

In digital communication, Gaussian minimum shift keying or GMSK is a continuous-phase frequency-shift keying modulation scheme. It is similar to standard minimum-shift keying (MSK); however the digital data stream is first shaped with a Gaussian filter before being applied to a frequency modulator. This has the advantage of reducing sideband power, which in turn reduces out-of-band interference between signal carriers in adjacent frequency channels. However, the Gaussian filter increases the modulation memory in the system and causes intersymbol interference, making it more difficult to discriminate between different transmitted data values and requiring more complex channel equalization algorithms such as an adaptive equalizer at the receiver.

Applications

GMSK is most notably used in the Global System for Mobile Communications (GSM).

Spectral efficiency

GMSK has high spectral efficiency, but it needs a higher power level than QPSK, for instance, in order to transmit the same amount of data reliably.

References

• Subbarayan Pasupathy, Minimum Shift Keying: A Spectrally Efficient Modulation, IEEE Communications Magazine, 1979
• Document from the University of Hull giving a thorough description of GMSK.
• Another good discussion (University of Toronto)
• PSK/GMSK comparison & illustrations (Aerospace)