CIRCULAR QAM CONSTELLATIONS

Circular Quadrature Amplitude Modulations (CQAM) were discovered in a recent work. The paper was published at the 2017 IEEE International Symposium on Information Theory (ISIT) and can be downloaded here.

Table of circular QAM (CQAM) modulations for probabilistic amplitude shaping over the prime field F_p. p²-CQAM constellations are given for p = 5, 7, 11, 13, 17, 43, 101. Points are drawn as small circles in red. Blue segments connect points located at minimum Euclidean distance. By construction, the inner radius of the p²-CQAM is ρ_in=1, for all p. The outer radius ρ_out varies slightly with p but lim p→∞ ρ_out = ρ_out(∞) ≈ 3.6.

Constellation Name	Postscript File	PNG File
5^2-CQAM	EPS File for 5^2-CQAM	PNG File for 5^2-CQAM
7^2-CQAM	EPS File for 7^2-CQAM	PNG File for 7^2-CQAM
11^2-CQAM	EPS File for 11^2-CQAM	PNG File for 11^2-CQAM
13^2-CQAM	EPS File for 13^2-CQAM	PNG File for 13^2-CQAM
17^2-CQAM	EPS File for 17^2-CQAM	PNG File for 17^2-CQAM
43^2-CQAM	EPS File for 43^2-CQAM	PNG File for 43^2-CQAM
101^2-CQAM	EPS File for 101^2-CQAM	PNG File for 101^2-CQAM

Open the encapsulated Postscript file for a better resolution.

For p=101, zoom in the 101²-CQAM constellation. Three waves can be observed. These waves correspond to a high density (packing density) neighborhood. Indeed, define the kissing number of a point as the number of neighbors located at minium Euclidean distance. In a lattice, this number is constant and does not depend on the considered point because of the group structure. CQAM are non-lattice constellations. The kissing number may vary from one point to another. A high kissing number (the max is 6 in dimension 2) indicates a high density. You notice the three waves correspond to a kissing number of 6. Between waves, the kissing number goes down to 4.

Asymptotic Figure of Merit of CQAM

The Figure of merit is defined as the ratio of the squared minimum Euclidean distance by the average energy per point. Assuming equiprobable points, for p>>1, the average energy per point is well approximated by the second order moment of the region between the unit circle and the outer circle of radius ρ_out(∞). We found that the figure of merit for bi-dimensional uniform CQAM with p² points asymptotically approaches 6.5/p². The asymptotic figure of merit for grid QAM is 6/p².

Hence, the circular symmetry of CQAM does not penalize its figure of merit.

Last modification made on Summer 2017.