Indigital modulation,minimum-shift keying (MSK) is a type ofcontinuous-phasefrequency-shift keying that was developed in the late 1950s byCollins Radio employees Melvin L. Doelz and Earl T. Heald.^[1] Similar toOQPSK, MSK is encoded with bits alternating betweenquadrature components, with theQ component delayed by half thesymbol period.

However, instead of square pulses as OQPSK uses, MSK encodes each bit as a halfsinusoid.^[2][3] 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 isδ = 0.5f_m wheref_m is the maximum modulating frequency. As a result, the modulation indexm is 0.5. This is the smallest FSKmodulation index that can be chosen such that the waveforms for 0 and 1 areorthogonal. A variant of MSK calledGaussian minimum-shift keying (GMSK) is used in theGSMmobile phone standard.

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The resulting signal is represented by the formula:^{[3][failed verification]}

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

where${\displaystyle a_I(t)}$ and${\displaystyle a_Q(t)}$ encode the even and odd information respectively with a sequence of square pulses of duration2T.${\displaystyle a_I(t)}$ has its pulse edges on${\displaystyle t=[-T,T,3T,\dots ]}$ and${\displaystyle a_Q(t)}$ on${\displaystyle t=[0,2T,4T,\dots ]}$. Thecarrier frequency is${\displaystyle f_c}$.

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

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

whereb_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.

Since theminimum symbol distance is the same as in theQPSK,^[7][6] the following formula can be used for the theoreticalbit-error ratio bound:

${\displaystyle P_b=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)=\frac{1}{2}\mathrm{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right)}$

where${\displaystyle E_b}$ is the energy per one bit,${\displaystyle N_0}$ is the noise spectral density,${\displaystyle Q(\ast )}$ denotes theQ-function and${\displaystyle \mathrm{erfc}}$ denotes thecomplementary error function.

Gaussian minimum-shift keying, or GMSK, is similar to standard minimum-shift keying (MSK); however, the digital data stream is first shaped with aGaussian 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 reducingsideband power, which in turn reduces out-of-band interference between signal carriers in adjacent frequency channels.^[9]

However, the Gaussian filter increases the modulation memory in the system and causesintersymbol interference, making it more difficult to differentiate between different transmitted data values and requiring more complex channel equalization algorithms such as anadaptive equalizer at the receiver. GMSK has highspectral efficiency, but it needs a higherpower level thanQPSK, for instance, in order to reliably transmit the same amount ofdata. GMSK is most notably used in theGlobal System for Mobile Communications (GSM), inBluetooth, in satellite communications,^[10][11] and theAutomatic 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 fromthe original(PDF) on 2005-10-24.

• 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.

• Quantized radio modulation modes

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