Editing Phase-shift keying (section)

====''&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.]]