# 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 and 1960s.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 wave forms used to represent a 0 and a 1 bit differ by exactly half a carrier period. Thus, the maximum frequency deviation is $\delta\$= 0.25 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 wave forms for 0 and 1 are orthogonal. A variant of MSK called GMSK is used in the GSM mobile phone standard.

## Mathematical representation

The resulting signal is represented by the formula $s(t) = a_{I}(t)\cos{\left(\frac{{\pi}t}{2T}\right)}\cos{(2{\pi}f_{c}t)}-a_{Q}(t)\sin{\left(\frac{{\pi}t}{2T}\right)}\sin{\left(2{\pi}f_{c}t\right)}$

where $a_{I}(t)$ and $a_{Q}(t)$ encode the even and odd information respectively with a sequence of square pulses of duration 2T. $a_{I}(t)$ has its pulse edges on $t=[-T,T,3T,...]$ and $a_{Q}(t)$ on $t=[0,2T,4T,...]$. The carrier frequency is $f_{c}$.

Using the trigonometric identity, this can be rewritten in a form where the phase and frequency modulation are more obvious, $
s(t) = \cos\left[2 \pi f_c t + b_k(t) \frac{\pi t}{2 T} + \phi_k\right]
$

where bk(t) is +1 when $a_{I}(t) = a_{Q}(t)$ and -1 if they are of opposite signs, and $\phi_k$ is 0 if $a_{I}(t)$ is 1, and $\pi$ otherwise. Therefore, the signal is modulated in frequency and phase, and the phase changes continuously and linearly.

##### Frequency Shift Keying

Frequency Shift Keying Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through discrete frequency...

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