Also known as: pi/4-DQPSK, π/4 DQPSK, differential QPSK
π/4-DQPSK (π/4-shifted differential quadrature phase-shift keying) is a four-symbol phase modulation in which the constellation is rotated by 45° (π/4) every symbol and information is carried in the change of phase rather than its absolute value.1 It is the air-interface modulation of TETRA, and was also used by the North American IS-136 (D-AMPS/NADC) and Japanese PDC cellular systems.
How it works
Start from ordinary QPSK, whose four symbols sit at 45°, 135°, 225° and 315°. π/4-DQPSK uses two such constellations: the base one and a copy rotated by 45° (at 0°, 90°, 180°, 270°). The transmitter alternates between them on every symbol, so the visible eight points are the union of both sets — but on any given symbol only four are legal.
Each dibit of input selects one of four phase increments — ±45° or ±135° — that are added to the phase of the previous symbol. Because every increment is an odd multiple of 45°, consecutive symbols always sit in the opposite set, guaranteeing a phase change at every symbol boundary and forcing the alternation. Two consequences follow directly from this rule:
- A transition on every symbol. There is no allowed “stay put” increment, so the phase always moves. That gives the receiver’s clock-recovery loop a reliable timing event every symbol, even during long constant-data runs.
- The trajectory never crosses the origin. The permitted ±45°/±135° steps route the signal around the centre of the IQ plane rather than through it, so the amplitude envelope never collapses to zero. A more constant envelope means the transmitter’s power amplifier can run closer to saturation efficiently without regrowing the spectrum, which matters for battery-powered handhelds.
Information lives in the difference between successive phases, so the receiver recovers dibits by differentially decoding — measuring each symbol’s phase relative to the one before it. A slowly varying absolute carrier-phase offset cancels in that subtraction, so no absolute phase reference is needed.
Variants
The closely related π/4-CQPSK (coherent) form uses the same 45°-shifted geometry but the phase changes carry the bits directly (Gray-mapped) rather than differentially; some P25-adjacent literature and GopherTrunk’s own CQPSK page treat the C4FM/CQPSK linear variant of P25 in this family. Plain DQPSK without the π/4 shift lacks the guaranteed per-symbol transition and can pass through the origin. π/4-DQPSK is nearly always paired with a root-raised-cosine pulse-shaping filter; TETRA uses a roll-off of 0.35.
In practice
TETRA carries 18 000 symbols per second at two bits each — 36 kbit/s gross — in a 25 kHz channel, and its four-slot TDMA structure rides on this π/4-DQPSK carrier. The modulation’s differential robustness and near-constant envelope are exactly the properties a professional mobile system wants: tolerance of a drifting carrier, efficient amplification, and a contained spectrum. The same reasoning drove its adoption in the first-generation digital cellular standards.
Relevance to SDR
A software receiver demodulates π/4-DQPSK by recovering symbol timing, sampling the complex symbol, computing the phase difference from the previous symbol, and mapping that difference to a dibit — no carrier phase lock required, which makes it forgiving to decode. The constellation of a healthy π/4-DQPSK signal shows the distinctive eight-point rosette. GopherTrunk’s scope tooling can display that constellation, and its DSP chain includes the RRC matched filter and differential-phase demodulation this modulation family needs; TETRA support status is tracked separately in the project’s protocol coverage.
Sources
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Phase-shift keying — π/4–QPSK — Wikipedia, for the differential π/4-shifted QPSK definition and constellation geometry. ↩