Introduction
The Hopf Fibration formalism developed by Heinz Hopf in 1931 can be seen as the decomposition of a geometric space in subspaces called Hopf fibres. This kind of decomposition of a geometric space is closely related to the "sphere packing" problem. So, from a geometric point of view, the fibration S1-> S2 -> S3 can be considered as an arrangement of non-overlapping hyperspheres S1 within the containing space that is the hypersphere S3. This concept is considered here linking Hopf fibrations and Information theory in the design of Four-Dimensional Modulations. Here, our main focus is to find common representations for both Polarization Shift Keying (PolSK) and PDM-QAM constellations for coherent optical communications. For this purpose, Stokes vectors in the Poincaré sphere (S2) can be used to determine basis points for the Hopf Fibration.
The inverse Hopf map can link "points" in S2 to "circles" embedded in S3. This map is illustrated in the next figure:
Four-dimensional unit vectors were produced in structured constellations when we sampled discrete angular positions in the Hopf circles. One remarkable constellation is the 14PolSK-8QAM.
In the next figure, we illustrate (a) the stereographic projection of the 14PolSK-8QAM fibration, (b) the direct Hopf map of the 4D vectors.
Using the Cayley-Dickson form of quaternions we can split the 112 four-dimensional unit vectors of the 14PolSK-8QAM into two orthogonal 2D (QAM) spaces. In this way we find a PDM-QAM representation for the 14PolSK-8QAM scheme.
For more details please read our work published in:
F. A. Rodrigues, G. P. Temporão, and J. P. V Weid, “Discrete Hopf Fibrations in the Signal Space of Four-Dimensional Modulations,” in Advanced Photonics 2017 (IPR, NOMA, Sensors, Networks, SPPCom, PS), 2017, p. SpTu2F.3.
BibTex Citation:
@inproceedings{Rodrigues:17,
author = {Rodrigues, Fernando A and Tempor{\~{a}}o, Guilherme P and Weid, Jean Pierre V},
booktitle = {Advanced Photonics 2017 (IPR, NOMA, Sensors, Networks, SPPCom, PS)},
doi = {10.1364/SPPCOM.2017.SpTu2F.3},
keywords = {Fiber optics and optical communications; Coherent},
pages = {SpTu2F.3},
publisher = {Optical Society of America},
title = {{Discrete Hopf Fibrations in the Signal Space of Four-Dimensional Modulations}},
url = {http://www.osapublishing.org/abstract.cfm?URI=SPPCom-2017-SpTu2F.3},
year = {2017}
}