Differential PSK (DPSK)

Phase Ambiguity in Coherent PSK

In standard , the receiver needs the exact reference phase to interpret the transmitted bits. A phase-synchronization error rotates the entire constellation relative to the detector's decision points.

This effect is called phase ambiguity. In real communication systems, the phase can be shifted by the channel or by imperfections in the transmitter and receiver oscillators. The constellation can then rotate relative to the detector. In BPSK, a $180^{\circ}$ shift swaps the decisions for '0' and '1' until the ambiguity is removed.

The Differential Solution: Encoding Phase Changes

DPSK encodes information in the phase change between consecutive symbols. The receiver compares neighboring symbols, so a common phase shift can be omitted.

For consecutive complex received symbols $r_{k-1}$ and $r_k$ , a differential detector can estimate the phase change as $\Delta\phi_k=\arg(r_k r_{k-1}^{*})$ , where the asterisk denotes complex conjugation. In DBPSK, the decision depends on whether the change is closer to $0^{\circ}$ or $180^{\circ}$ .

In M-DPSK, one symbol carries $\log_2 M$ bits, and the detector assigns data from the phase difference between consecutive symbols.

Example: Differential BPSK (DBPSK)

In the simplest form, DBPSK, we use two phase shifts to represent the data:

Input Bit '0': Keep the phase the same as the previous symbol (a $0^{\circ}$ shift).
Input Bit '1': Change the phase by $180^{\circ}$ relative to the previous symbol.

The process requires an initial reference phase. The sequence $1011$ is transmitted with an initial phase of $0^{\circ}$ :

Input Bit	Rule	Previous Transmitted Phase	New Transmitted Phase
(Start)	Initial reference	none	$0^{\circ}$
1	Shift by $180^{\circ}$	$0^{\circ}$	$0^{\circ} + 180^{\circ} = 180^{\circ}$
0	Keep phase (shift by $0^{\circ}$ )	$180^{\circ}$	$180^{\circ} + 0^{\circ} = 180^{\circ}$
1	Shift by $180^{\circ}$	$180^{\circ}$	$180^{\circ} + 180^{\circ} = 360^{\circ} \equiv 0^{\circ}$
1	Shift by $180^{\circ}$	$0^{\circ}$	$0^{\circ} + 180^{\circ} = 180^{\circ}$

DPSK: Receiver Complexity and BER

Relative to coherent CPSK, DPSK offers lower receiver complexity with a measurable performance cost.

Advantages of DPSK

Tolerance to Phase Drift: A common, slowly changing phase shift can be reduced by comparing consecutive symbols, provided that the change during one symbol remains small.
Simplified Receiver: DPSK generally does not require recovery of the carrier's absolute phase. Demodulation compares the current symbol with the previous one and uses a delay line with a differential detector.

Disadvantages of DPSK

Worse BER Performance: With ideal differential detection, equally likely bits, and two-sided AWGN noise power spectral density $N_0/2$ , DBPSK has a $P_b=\frac{1}{2}\exp\left(-E_b/N_0\right)$ , where $E_b$ is the received energy per information bit. This is higher than coherent BPSK under the same convention, whose BER is $P_b=Q\left(\sqrt{2E_b/N_0}\right)$ . Comparing these expressions gives a penalty that depends on the target BER, not a fixed 3 dB. For typical BER targets, the difference is usually around 1 to 1.5 dB, depending on the operating point. This reflects noncoherent detection and the use of a noisy reference symbol. A single decision error can also affect two neighboring outputs. This effect is called .
Power Penalty: The BER expressions above imply that DBPSK needs a higher $E_b/N_0$ to reach the same BER as coherent BPSK. This is not universally twice the energy or a fixed 3 dB penalty. For typical BER targets, the difference is around 1 to 1.5 dB, and the exact value depends on the DPSK variant and detector.