Minimum-shift keying

In digital modulation, minimum-shift keying (MSK) is a type of continuous-phase frequency-shift keying that was developed in the late 1950s by Collins Radio employees Melvin L. Doelz and Earl T. Heald. Similar to OQPSK, MSK is encoded with bits alternating between quadrature 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 (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 fm where fm 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:

s ( t ) = a I ( t ) cos ⁡ ( π t 2 T ) cos ⁡ ( 2 π f c t ) − a Q ( t ) sin ⁡ ( π t 2 T ) sin ⁡ ( 2 π f c t ) {\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 a I ( t ) {\displaystyle a_{I}(t)} and a Q ( t ) {\displaystyle a_{Q}(t)} encode the even and odd information respectively with a sequence of square pulses of duration 2T. a I ( t ) {\displaystyle a_{I}(t)} has its pulse edges on t = [ − T , T , 3 T , … ] {\displaystyle t=[-T,T,3T,\ldots ]} and a Q ( t ) {\displaystyle a_{Q}(t)} on t = [ 0 , 2 T , 4 T , … ] {\displaystyle t=[0,2T,4T,\ldots ]} . The carrier frequency is f c {\displaystyle f_{c}} . Using the trigonometric identity, this can be rewritten in a form where the phase and frequency modulation are more obvious,

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

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

Properties

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

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