Minimum-shift keying

This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed. Find sources: "Minimum-shift keying" – news · newspapers · books · scholar · JSTOR (April 2020) (Learn how and when to remove this message)

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 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.

Since the minimum symbol distance is the same as in the QPSK,^[7][6] 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.

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.^[9]

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,^[10][11] and Automatic Identification System (AIS) for maritime navigation.

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