Phase-shift keying (PSK) code 

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

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.
  • 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
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Zoo Code ID: psk

Cite as:
“Phase-shift keying (PSK) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/psk
BibTeX:
@incollection{eczoo_psk, title={Phase-shift keying (PSK) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/psk} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/modulation/psk.yml.