Editing Phase-shift keying (section)

===Variants===

====Offset QPSK (OQPSK)====
[[File:Pi-by-O-QPSK Gray Coded.svg|thumb|Signal doesn't pass through the origin, because only one bit of the symbol is changed at a time.]]

''Offset quadrature phase-shift keying'' (''OQPSK'') is a variant of phase-shift keying modulation using four different values of the phase to transmit.  It is sometimes called ''staggered quadrature phase-shift keying'' (''SQPSK'').

[[File:Oqpsk phase plot.svg|thumb|Difference of the phase between QPSK and OQPSK]]

Taking four values of the phase (two [[bit]]s) at a time to construct a QPSK symbol can allow the phase of the signal to jump by as much as 180° at a time. When the signal is low-pass filtered (as is typical in a transmitter), these phase-shifts result in large amplitude fluctuations, an undesirable quality in communication systems. By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-period, the in-phase and quadrature components will never change at the same time. In the constellation diagram shown on the right, it can be seen that this will limit the phase-shift to no more than 90° at a time. This yields much lower amplitude fluctuations than non-offset QPSK and is sometimes preferred in practice.

The picture on the right shows the difference in the behavior of the phase between ordinary QPSK and OQPSK. It can be seen that in the first plot the phase can change by 180° at once, while in OQPSK the changes are never greater than 90°.

The modulated signal is shown below for a short segment of a random binary data-stream. Note the half symbol-period offset between the two component waves. The sudden phase-shifts occur about twice as often as for OQPSK (since the signals no longer change together), but they are less severe. In other words, the magnitude of jumps is smaller in OQPSK when compared to QPSK.

[[File:OQPSK timing diagram.png|frame|center|Timing diagram for offset-QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown the top and the total, combined signal at the bottom. Note the half-period offset between the two signal components.]]

====SOQPSK====
The license-free '''[[Pulse shaping|shaped]]-offset QPSK''' (SOQPSK) is interoperable with Feher-patented QPSK ('''FQPSK'''), in the sense that an integrate-and-dump offset QPSK detector produces the same output no matter which kind of transmitter is used.<ref>
{{cite document | last1 = Nelson | first1 = T. | last2 = Perrins | first2 = E. | last3 = Rice | first3 = M. | title=Common detectors for Tier 1 modulations |publisher=International Foundation for Telemetering |date=2005 |hdl=10150/604890}}
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{{cite conference | last1 = Nelson | first1 = T. | last2 = Perrins | first2 = E. | last3 = Rice | first3 = M. | doi = 10.1109/GLOCOM.2005.1578470 | title = Common detectors for shaped offset QPSK (SOQPSK) and Feher-patented QPSK (FQPSK) | book-title = GLOBECOM '05. IEEE Global Telecommunications Conference, 2005 | pages = 5 pp | year = 2005 | isbn = 0-7803-9414-3 | s2cid = 11020777 |url=https://www.researchgate.net/publication/4213516_Common_detectors_for_shaped_offset_QPSK_(SOQPSK)_and_Feher-patented_QPSK_(FQPSK)}}
</ref>

These modulations carefully shape the I and Q waveforms such that they change very smoothly, and the signal stays constant-amplitude even during signal transitions. (Rather than traveling instantly from one symbol to another, or even linearly, it travels smoothly around the constant-amplitude circle from one symbol to the next.) SOQPSK modulation can be represented as the hybrid of QPSK and [[Minimum-shift keying|MSK]]: SOQPSK has the same signal constellation as QPSK, however the phase of SOQPSK is always stationary.<ref>{{cite conference |first=Terrance J. |last=Hill |title=A non-proprietary, constant envelope, variant of shaped offset QPSK (SOQPSK) for improved spectral containment and detection efficiency |book-title=MILCOM 2000 Proceedings. 21st Century Military Communications. Architectures and Technologies for Information Superiority |publisher=IEEE |volume=1 |date=2000 |isbn=0-7803-6521-6 |pages=347–352 |doi=10.1109/MILCOM.2000.904973}}</ref><ref>{{cite journal |last=Li |first=Lifang |first2=M.K. |last2=Simon |title=Performance of coded offset quadrature phase-shift keying (OQPSK) and MIL-STD shaped OQPSK (SOQPSK) with iterative decoding |journal=Interplanetary Network Prog. Rep. |volume=42 |page=156 |date=2004 |url=https://ipnpr.jpl.nasa.gov/progress_report/42-156/156A.pdf}}</ref>

The standard description of SOQPSK-TG involves [[ternary signal|ternary symbols]].<ref>{{cite conference |last=Sahin |first=C. |last2=Perrins |first2=E. |title=The capacity of SOQPSK-TG |book-title=2011-MILCOM 2011 Military Communications Conference |publisher=IEEE |date=2011 |isbn=978-1-4673-0081-0 |pages=555–560 |doi=10.1109/MILCOM.2011.6127730}}</ref> SOQPSK is one of the most spread modulation schemes in application to [[Low Earth orbit|LEO]] satellite communications.<ref>{{cite journal |last=Saeed |first=N. |last2=Elzanaty |first2=A. |last3=Almorad |first3=H. |last4=Dahrouj |first4=H. |last5=Al-Naffouri |first5=T.Y. |last6=Alouini |first6=M.S. |title=Cubesat communications: Recent advances and future challenges |journal=IEEE Communications Surveys & Tutorials |volume=22 |issue=3 |pages=1839–62 |date=2020 |doi=10.1109/COMST.2020.2990499 |arxiv=1908.09501}}</ref>

====''&pi;''/4-QPSK====
[[File:Pi-by-4-QPSK Gray Coded.svg|thumb|right|Dual constellation diagram for π/4-QPSK. This shows the two separate constellations with identical Gray coding but rotated by 45° with respect to each other.]]

This variant of QPSK uses two identical constellations which are rotated by 45° (<math>\pi/4</math> radians, hence the name) with respect to one another. Usually, either the even or odd symbols are used to select points from one of the constellations and the other symbols select points from the other constellation. This also reduces the phase-shifts from a maximum of 180°, but only to a maximum of 135° and so the amplitude fluctuations of <math>\pi/4</math>-QPSK are between OQPSK and non-offset QPSK.

One property this modulation scheme possesses is that if the modulated signal is represented in the complex domain, transitions between symbols never pass through 0. In other words, the signal does not pass through the origin. This lowers the dynamical range of fluctuations in the signal which is desirable when engineering communications signals.

On the other hand, <math>\pi/4</math>-QPSK lends itself to easy demodulation and has been adopted for use in, for example, [[time-division multiple access|TDMA]] [[cellular telephone]] systems.

The modulated signal is shown below for a short segment of a random binary data-stream. The construction is the same as above for ordinary QPSK. Successive symbols are taken from the two constellations shown in the diagram. Thus, the first symbol (1 1) is taken from the "blue" constellation and the second symbol (0 0) is taken from the "green" constellation. Note that magnitudes of the two component waves change as they switch between constellations, but the total signal's magnitude remains constant ([[constant envelope]]). The phase-shifts are between those of the two previous timing-diagrams.

[[File:Pi-by-4-QPSK timing diagram.png|frame|center|Timing diagram for π/4-QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown the top and the total, combined signal at the bottom. Note that successive symbols are taken alternately from the two constellations, starting with the "blue" one.]]

====DPQPSK====
'''Dual-polarization quadrature phase shift keying''' (DPQPSK) or '''dual-polarization QPSK''' - involves the polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by a factor of 2. This is a cost-effective alternative to utilizing 16-PSK, instead of QPSK to double the spectral efficiency.