## Description

A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\).

For example, such a constellation could be \begin{align} \{1,e^{i\frac{2\pi}{q}},\cdots,e^{i\frac{2\pi}{q}(q-1)}\}~. \tag*{(1)}\end{align} Each point is typically associated with a complex amplitude of an electromagnetic signal, and information is encoded into the phase of that signal.

Concatenating PSK with \(q\)-ary codes yields a natural non-binary way of digitizing the analog AGWN channel [1,2].

## Rate

Nearly achieves Shannon AWGN capacity for one-dimensional constellations in the limit of infinite signal to noise [3; Fig. 11.7].

## Realizations

Telephone-line modems: 1967 Milgo 4400/48 and international standard V.27 used 8-PSK [4].

## Parents

- Polygon code — The PSK constellation forms a \(q\)-gon.
- Polyphase code — A polyphase code can be thought of as a concatenation of a \(q\)-ary outer code with a PSK inner code. When the outer code is trivial, the construction reduces to a PSK code.
- Modulation scheme — PSK is a modulation whose constellation consists of points arranged equidistantly on a circle.

## Children

- Binary PSK (BPSK) code — BPSK is also known as 2-PSK.
- Quadrature PSK (QPSK) code — QPSK is also known as 4-PSK.

## Cousins

- Gray code — 1D Gray codes are often concatenated with PSKs so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points.
- Cat code — PSK (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over classical (quantum) channels.
- Hyperbolic sphere packing — Hyperbolic PSK constellations may yield improved performance over Euclidean ones [5].
- PSK c-q code — PSK (PSK c-q) codes are used to transmit classical information using single-mode coherent states distributed on a circle over classical (quantum) channels.

## References

- [1]
- Massey, J. L. "Convolutional codes over rings." Fourth Joint Swedish-Soviet International Workshop on Information Theory. 1989.
- [2]
- Massey, J. L. "Ring convolutional codes for phase modulation." presented at the IEEE Int. Symp. on Information Theory, San Diego, CA, Jan. 14-19. 1990.
- [3]
- R. E. Blahut, Modem Theory (Cambridge University Press, 2009) DOI
- [4]
- International Telecommunication Union-T, Recommendation V.27ter: 4800/2400 Bits Per Second Modem Standardized For Use in the General Switched Telephone Network, 1984
- [5]
- Silva, E. B., and R. Palazzo Jr. "M-PSK signal constellations in hyperbolic space achieving better performance than the M-PSK signal constellations in Euclidean space." 1999 IEEE Information Theory Workshop, Metsovo, Greece. 1999.

## Page edit log

- Victor V. Albert (2022-11-07) — most recent

## Cite as:

“Phase-shift keying (PSK) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/psk