Phase-Shift Keying (PSK)

Principle of phase-shift keying

Phase-Shift Keying (PSK) is a digital modulation method in which each symbol selects one of several discrete phases of a sinusoidal carrier. In a simplified model, a symbol waveform is $s_m(t)=A\cos(2\pi f_c t+\phi_m)$ for $0\le t<T_s$ . The amplitude $A$ and frequency $f_c$ remain fixed, while the phase $\phi_m$ identifies the data.

Constant amplitude produces a constant envelope, so slow gain changes do not directly change the encoded phase. The receiver still requires frequency and phase synchronization, while phase noise, phase offset, and multipath affect the symbol decision.

Visualizing PSK: constellation diagrams

A PSK scheme can be represented with a . Each phase used by the modulator corresponds to one point. In the ideal PSK model, constant amplitude places all points on a circle centered at the origin.

Binary phase-shift keying (BPSK)

BPSK is the binary form of PSK. Two phases represent the two states: '0' and '1'.

Phases: With a fixed symbol energy, two phases are selected at the maximum angular separation. They are $180^\circ$ , or $\pi$ radians, apart. One possible assignment is:
- Bit '0' → 180° phase
- Bit '1' → 0° phase
Data rate: Each BPSK symbol encodes 1 bit, so the symbol and bit rates have the same numerical value for the same time unit.
BER: For coherent detection of uncoded BPSK over an AWGN channel, the bit error rate is $P_b=Q(\sqrt{2E_b/N_0})$ . Here $E_b$ is the energy per bit, $N_0$ is the noise power spectral density, and $Q$ is the normal-tail function. The decision boundary is halfway between the two constellation points.

BPSK constellation diagram

The BPSK constellation consists of two points. With phases of 0° and 180°, they lie on the in-phase (I) axis at equal distances from the origin.

Quadrature phase-shift keying (QPSK)

QPSK, also called 4-PSK, uses four phases and encodes 2 bits per symbol. At the same symbol rate, pulse-shaping filter, and nominal bandwidth, it carries twice the raw bit rate of BPSK.

Phases: Four phases are evenly spaced around the circle by $90^\circ$ . One equivalent setting uses 45°, 135°, 225°, and 315°.
Data rate: Four states ( $2^2 = 4$ ) encode 2 bits per symbol. Each point maps to one dibit, such as '00', '01', '11', or '10'.
Gray coding: Dibits are assigned to phases using . Adjacent constellation points then differ by one bit. If noise makes the receiver choose neighboring '11' for '01', this produces a single bit error.

M-ary phase-shift keying (M-PSK)

PSK can be extended to $M$ different phases. With binary mapping, $M=2^N$ is commonly used so each symbol represents an integer number $N$ of bits. This scheme is called M-ary PSK.

An M-PSK scheme encodes $N = \log_2(M)$ bits per symbol when $M$ is a power of 2.
Examples include BPSK ( $M=2$ ), QPSK ( $M=4$ ), and 8-PSK ( $M=8$ , which encodes 3 bits per symbol).
The trade-off: Increasing $M$ raises the bits per symbol and the potential spectral efficiency. For equally spaced phases, the minimum separation is $2\pi/M$ , so a fixed symbol energy leaves the system more sensitive to noise and phase errors.
Practical limits: 8-PSK remains in use in selected systems, while higher orders are less common because of their small phase separation. When more bits per symbol are needed, QAM is often selected because it also uses amplitude levels.

Differential phase-shift keying (DPSK)

A coherent PSK receiver uses a synchronized local carrier for . It needs a stable phase reference to determine the phase of the incoming signal. DPSK replaces the absolute phase decision with a comparison between consecutive symbols.

In DPSK, information is represented by the phase change between consecutive symbols. Timing and frequency tracking remain part of the receiver.

Example (DBPSK):
- To send a '1', the phase is changed by 180°.
- To send a '0', the phase is kept the same (0° change).
Advantages: The receiver compares the current symbol phase with the previous one, which removes the need for an absolute phase reference and tolerates slow common phase drift.
Disadvantages: DPSK generally has a BER penalty relative to ideal coherent detection. A symbol decision error can also corrupt the phase reference used for the next comparison, creating error propagation.

DP-QPSK: QPSK on two polarizations

DP-QPSK is used in high-capacity systems, including fiber-optic links. It adds a second dimension of the electromagnetic wave: .

The principle is to transmit two independent QPSK signals simultaneously on the same carrier frequency, but using two different, orthogonal polarizations (e.g., vertical and horizontal).

Each QPSK branch carries 2 bits per symbol, so the two polarizations provide 4 bits per combined symbol without occupying a second frequency band. Comparing this rate with 16-QAM requires the same definition of a symbol and the same number of polarizations. Performance depends on polarization crosstalk and other link impairments.