MSK modulation scheme

Minimum Shift Keying (MSK) is continuous-phase binary FSK in which the two tones are separated by the minimum spacing that still keeps them orthogonal, $\Delta f=f_1-f_2=\dfrac{1}{2T_b}$ (deviation ratio $h=0.5$).

Signal

$$s(t)=\sqrt{\frac{2E_b}{T_b}}\cos\left[2\pi f_c t+\theta(t)\right],\qquad \theta(t)=\theta(0)\pm\frac{\pi}{2T_b}t$$

• Bit 1: the phase increases by $\pi/2$ over the bit ($f_1=f_c+\tfrac{1}{4T_b}$).
• Bit 0: the phase decreases by $\pi/2$ over the bit ($f_2=f_c-\tfrac{1}{4T_b}$).
• The phase is continuous at bit boundaries, so there are no sudden jumps.

Expanding in quadrature form, MSK is equivalent to offset QPSK with half-sinusoid pulse shaping:

$$s(t)=\sqrt{\tfrac{2E_b}{T_b}}\cos\theta(t)\cos 2\pi f_ct-\sqrt{\tfrac{2E_b}{T_b}}\sin\theta(t)\sin 2\pi f_ct$$

Modulator

 data -> [S/P: odd/even bits, each 2Tb long]
            |  I bits             | Q bits (offset Tb)
            v                     v
     (X) cos(pi t/2Tb)     (X) sin(pi t/2Tb)
            |                     |
     (X) cos 2pi fc t      (X) sin 2pi fc t
            |                     |
            +-------->(+)<--------+
                       |
                     MSK out

Example

Data 1 1 0 1, starting phase $\theta(0)=0$:

The phase moves along straight lines (the phase trellis) and never jumps. For $T_b=1\ \mu\text{s}$ the tones are $f_c\pm 250$ kHz, a spacing of 500 kHz $=1/(2T_b)$.

Features

• Constant envelope, so efficient non-linear amplifiers can be used.
• The main lobe is 1.5 times wider than that of QPSK, but the sidelobes are much lower, falling as $1/f^4$ (less adjacent-channel interference).
• Same $P_e$ as BPSK/QPSK with coherent detection: $\tfrac12\text{erfc}\sqrt{E_b/N_0}$.
• Gaussian filtering of the data before MSK gives GMSK (used in GSM).