Minimum-shift keying

However, instead of square pulses as OQPSK uses, MSK encodes each bit as a half sinusoid. This results in a constant-modulus signal (constant envelope signal), which reduces problems caused by non-linear distortion. In addition to being viewed as related to OQPSK, MSK can also be viewed as a continuous-phase frequency-shift keyed (CPFSK) signal with a frequency separation of one-half the bit rate.

In MSK the difference between the higher and lower frequency is identical to half the bit rate. Consequently, the waveforms used to represent a 0 and a 1 bit differ by exactly half a carrier period. Thus, the maximum frequency deviation is δ = 0.5 f_m where f_m is the maximum modulating frequency. As a result, the modulation index m is 0.5. This is the smallest FSK modulation index that can be chosen such that the waveforms for 0 and 1 are orthogonal. A variant of MSK called Gaussian minimum-shift keying (GMSK) is used in the GSM mobile phone standard.

Mathematical representation

The resulting signal is represented by the formula:

${\displaystyle s(t)=a_I(t)\cos \left(\frac{\pi t}{2T}\right)\cos (2\pi f_{c}t)-a_Q(t)\sin \left(\frac{\pi t}{2T}\right)\sin \left(2\pi f_{c}t\right)}$

where ${\displaystyle a_I(t)}$ and ${\displaystyle a_Q(t)}$ encode the even and odd information respectively with a sequence of square pulses of duration 2T. ${\displaystyle a_I(t)}$ has its pulse edges on ${\displaystyle t=[-T,T,3T,\dots ]}$ and ${\displaystyle a_Q(t)}$ on ${\displaystyle t=[0,2T,4T,\dots ]}$. The carrier frequency is ${\displaystyle f_c}$.

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

${\displaystyle s(t)=\cos \left[2\pi f_{c}t+b_k(t)\frac{\pi t}{2T}+{\varphi }_k\right]}$

where b_k(t) is +1 when ${\displaystyle a_I(t)=a_Q(t)}$ and −1 if they are of opposite signs, and ${\displaystyle {\varphi }_k}$ is 0 if ${\displaystyle a_I(t)}$ is 1, and ${\displaystyle \pi }$ otherwise. Therefore, the signal is modulated in frequency and phase, and the phase changes continuously and linearly.