Signal constellations in the hyperbolic plane: A proposal for new communication systems

Silva, Eduardo Brandani da; Firer, Marcelo; Costa, Sueli I. R.; Palazzo, Reginaldo

doi:10.1016/j.jfranklin.2005.09.001

Cited by 21 publications

(19 citation statements)

References 6 publications

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“…In [20], the authors construct Fuchsian groups suitable for signal constellation design. In [16], the authors consider the unit disk model of the hyperbolic plane as the signal space, and the noise is modeled as a hyperbolic Gaussian random variable. By using some results of hyperbolic geometry they construct a hyperbolic equivalent to QAM and PSK constellations and point out that, when the channel model is hyperbolic (this is the case e.g.…”

Section: Introductionmentioning

confidence: 99%

Fuchsian codes with arbitrarily high code rates

Blanco-Chacón

Hollanti

Alsina

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Recently, so-called Fuchsian codes have been proposed in [I. BlancoChacón et al., "Nonuniform Fuchsian codes for noisy channels", J. of the Franklin Institute 2014] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1; e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.

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Section: Introductionmentioning

confidence: 99%

Fuchsian codes with arbitrarily high code rates

Blanco-Chacón

Hollanti

Alsina

et al. 2016

Journal of Pure and Applied Algebra

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“…Hyperbolic codes and hyperbolic tessellations have been the subject of interest in recent years due to the promising applications in digital signal transmission and digital signal processing techniques (Blanco-Chácon et al 2014a, b;Silva et al 2006). For instance, in power and bandwidth efficient digital communication systems the design of signal constellations is attracting researchers' attention not only for the fact that hyperbolic geometry provides an infinite number of tessellations on H 2 which can be used in the design of signal constellations (Cavalcante et al 2005;Agustini 2002), but mainly for the fact that such tessellations provide a better performance than the corresponding digital communication system using Euclidean tessellations (Silva et al 2006) under the same constraints.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we are interested in partitioning a given region on the hyperbolic plane which by the action of a proper group on this region tessellates the whole hyperbolic plane. Hence, the resulting partitioning consists of hyperbolic polygons (Silva et al 2006).…”

Section: Introductionmentioning

confidence: 99%

A new method of coding geodesics on surfaces whose fundamental regions consist of the union of elementary triangles derived from the Farey series

Leskow

Palazzo

2015

Comp. Appl. Math.

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In this paper, a new method of coding geodesics is presented. This method generalizes the method proposed by Series and it is based on the set of hyperbolic tessellations whose corresponding fundamental regions are made up of elementary triangles derived from the Farey series, and it makes use of the concepts of continued fraction expansion and cutting sequences. These tessellations have interesting properties with applications in (lossless) source coding, topological quantum codes, design of digital signal constellations, study of second-order ordinary differential equations of the Fuchsian type besides coding of geodesics related to dynamical systems. This method promises to open up new perspectives of applications of the Farey series in digital signal processing. The union of these triangles provides a systematic procedure to obtain families of tessellations and one of the main characteristics of the Farey series is the easiness in obtaining and characterizing the triangles of the new tessellations. In applications related to hyperbolic tessellations, knowing the genus of the surface is quite important due to the fact that it is related to the characteristic of the transmission channel as well as to the algebraic-geometric code to be used in the channel encoder. We consider the analyses of the surfaces, with genus g = 1, showing that the results may be extended to any value of n (n > 0), as well as to any g ≥ 2. Finally, the proposed method for coding geodesics in the hyperbolic plane is presented and the new codes are tabulated.

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“…• Finally, we present an alternative method for constructing Fuchsian codes by certain parametrization of the integer tuples defining the Möbius transformations used for the code construction. Our interest in Fuchsian groups as a basis for code construction stems from a series of recent papers by Palazzo et al In [23,6,19,21], among others, various interesting connections between Fuchsian groups and signal constellation design are presented. In [23], the authors construct Fuchsian groups suitable for signal constellation construction.…”

Section: Introductionmentioning

confidence: 99%

Nonuniform Fuchsian codes for noisy channels

Blanco-Chacón

Remón

Hollanti

et al. 2014

Journal of the Franklin Institute

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We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.Peer ReviewedPostprint (author’s final draft

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Signal constellations in the hyperbolic plane: A proposal for new communication systems

Cited by 21 publications

References 6 publications

Fuchsian codes with arbitrarily high code rates

Fuchsian codes with arbitrarily high code rates

A new method of coding geodesics on surfaces whose fundamental regions consist of the union of elementary triangles derived from the Farey series

Nonuniform Fuchsian codes for noisy channels

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