# Quadrature-amplitude modulation (QAM) code

## Description

Encodes into points into a subset of points lying on in \(\mathbb{R}^{2}\), here treated as \(\mathbb{C}\). Each pair of points is associated with a complex amplitude of an electromagnetic signal, and information is encoded into both the norm and phase of that signal [1; Ch. 16].

QAM schemes with \(q\) complex coordinates are often called \(q\)-QAM, and \(q\) is often a power of two in order to further concatenate with a binary code.

## Rate

Nearly achieves Shannon AWGN capacity for two-dimensional constellations in the limit of infinite signal to noise [2; Fig. 11.8].

## Realizations

Optical communication (e.g., Ref. [3]).Telephone-line modems: 1971 Codex 9600C and international standard V.29 used 16-QAM [4].

## Parent

## Child

- Pulse-amplitude modulation (PAM) code — PAM codes can be thought of as QAM codes restricted to the real line. A \(q\times q\)-QAM code is informationally equivalent to two \(q\)-PAM codes.

## Cousins

- Lattice-based code — QAM encodings often consist of lattice constellations, i.e., finite sets of points scooped out of an infinite 2D lattice.
- Gottesman-Kitaev-Preskill (GKP) code — Finite-energy GKP codes are quantum analogues of lattice-based QAM codes in that both use a subset of points on a lattice.
- Gray code — 2D Gray codes are often concatenated with \(n=1\) lattice-based QAM codes so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points.
- Niset-Andersen-Cerf code — The Niset-Andersen-Cerf code encodes two coherent states at a time with arbitrary complex values, making it analogous to a two-point QAM code. The code does not encode any quantum information since superpositions of the coherent states are not stored. However, analysis of the code is done via a quantum treatment.

## References

- [1]
- A. Lapidoth, A Foundation in Digital Communication (Cambridge University Press, 2017) DOI
- [2]
- R. E. Blahut, Modem Theory (Cambridge University Press, 2009) DOI
- [3]
- F. Buchali et al., “Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration”, Journal of Lightwave Technology 34, 1599 (2016) DOI
- [4]
- International Telecommunication Union-T, Recommendation V.29: 9600 Bits Per Second Modem Standardized For Use on Point-to-Point 4-Wire Leased Telephone-Tpe Circuits, 1993

## Page edit log

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

## Cite as:

“Quadrature-amplitude modulation (QAM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qam

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/analog/modulation/qam.yml.