Minimum-shift keying

MSK waveform can also be designed as OQPSK (i.e. in I/Q manner) with the sinusoidal pulse shaping.[4][5] Mapping changes in continuous phase. Each bit time, the carrier phase changes by ±90°.

The resulting signal is represented by the formula:[3]

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

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 ${\displaystyle t=[-T,T,3T,\ldots ]}$ and ${\displaystyle a_{Q}(t)}$ on ${\displaystyle t=[0,2T,4T,\ldots ]}$. 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,

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

where b_k(t) is +1 when ${\displaystyle a_{I}(t)=a_{Q}(t)}$ and −1 if they are of opposite signs, and ${\displaystyle \phi _{k}}$ 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.

Power spectral density of MSK, BPSK, and QPSK. The side-lobes of MSK are lower (−23 dB) than in both BPSK and QPSK cases (−10 dB). Therefore, the inter-channel interference is lower in MSK case. Moreover, the main lobe of the MSK signal is wider, which means more energy in the null-to-null bandwidth. However, this can be also the disadvantage where extremely narrow bandwidth is required (null-to-null bandwidth of QPSK is equal to 3dB-bandwidth, null-to-null bandwidth of the MSK signal is 1.5 times as large as the 3dB-bandwidth.[6]

Since the minimum symbol distance is the same as in the QPSK,[7][8] the following formula can be used for the theoretical bit-error ratio bound:

${\displaystyle P_{b}=Q\left({\sqrt {\frac {2E_{b}}{N_{0}}}}\right)={\frac {1}{2}}\operatorname {erfc} \left({\sqrt {\frac {E_{b}}{N_{0}}}}\right)}$

where ${\displaystyle E_{b}}$ is the energy per one bit, ${\displaystyle N_{0}}$ is the noise spectral density, ${\displaystyle Q(*)}$ denotes the Q-function and ${\displaystyle \operatorname {erfc} }$ denotes the complementary error function.

Power spectral densities of MSK and GMSK. Note that the decreasing of time-bandwidth ${\displaystyle BT}$ negatively influences bit-error-rate performance due to increasing intersymbol interference.[9]

Gaussian minimum-shift keying, or GMSK, 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, and typically has much narrower phase shift angles than most MSK modulation systems. This has the advantage of reducing sideband power, which in turn reduces out-of-band interference between signal carriers in adjacent frequency channels.[10]

However, the Gaussian filter increases the modulation memory in the system and causes intersymbol interference, making it more difficult to differentiate between different transmitted data values and requiring more complex channel equalization algorithms such as an adaptive equalizer at the receiver. GMSK has high spectral efficiency, but it needs a higher power level than QPSK, for instance, in order to reliably transmit the same amount of data. GMSK is most notably used in the Global System for Mobile Communications (GSM), in Bluetooth, in satellite communications,[11][12] and Automatic Identification System (AIS) for maritime navigation.

• Constellation diagram used to examine the modulation in signal space (not time)
• Gaussian frequency-shift keying

• Subbarayan Pasupathy, Minimum Shift Keying: A Spectrally Efficient Modulation, IEEE Communications Magazine, 1979
• R. de Buda, Fast FSK Signals and their Demodulation, Can. Elec. Eng. J. Vol. 1, Number 1, 1976
• F. Amoroso, Pulse and Spectrum Manipulation in the Minimum (Frequency) Shift Keying (MSK) Format, IEEE Trans.
• "Appendix D – Digital Modulation and GMSK" (PDF). University of Hull. 2001-03-13. Archived from the original (PDF) on 2004-09-24.

• Link Budget Analysis: Digital Modulation-Part 2-FSK (Atlanta RF)
• Elnoubi, S., Chahine, S. A., & Abdallah, H. (2004, March). BER performance of GMSK in Nakagami fading channels. In Radio Science Conference, 2004. NRSC 2004. Proceedings of the Twenty-First National (pp. C13-1). IEEE.
• Feher, K. (1993, July). FQPSK: A modulation-power efficient RF amplification proposal for increased spectral efficiency and capacity GMSK and Π/4-QPSK compatible PHY standard. In IEEE 802.11 Wireless Access Methods Phys. Layer Spec. Doc.