CIRCULAR QAM CONSTELLATIONS
Circular Quadrature Amplitude Modulations
(CQAM) were discovered in a recent work by J.J. Boutros, F. Jardel, and C.
MŽasson. The paper was submitted to the 2017 IEEE International Symposium on
Information Theory (ISIT) and can be downloaded from arXiv at arXiv:1701.07976v1
Table of circular QAM (CQAM) modulations for probabilistic amplitude shaping over the prime field F_{p}. p^{2}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 p^{2}CQAM is ρ_{in}=1, for all p. The outer radius ρ_{out} varies slightly with p but lim p→° ρ_{out}_{ }= ρ_{out}(°) Å 3.6.
Constellation
Name 
Postscript File 
PNG File 
5^{2}CQAM


7^{2}CQAM


11^{2}CQAM


13^{2}CQAM


17^{2}CQAM


43^{2}CQAM


101^{2}CQAM

Open the encapsulated Postscript file for a better resolution.
For p=101, zoom in the 101^{2}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 nonlattice 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 bidimensional uniform CQAM with p^{2} points asymptotically approaches 6.5/p^{2}. The asymptotic figure of merit for grid QAM is 6/p^{2}.
Hence, the circular symmetry of CQAM does not penalize its figure of merit.
Last modification made on February 2, 2017.