Universidade Estadual de Campinas Faculdade de Engenharia El´ etrica e de Computac ¸˜ ao Departamento de Telem´ atica A New Mathematical Approach for the Design of Digital Communication Systems a Rodrigo G. Cavalcante, Henrique Lazari, Jo˜ ao D. Lima, and Reginaldo Palazzo Jr. a Acknowledgement : To FAPESP, CAPES and CNPq, Brazilian agencies, for supporting this research. 1
Content I - Introduction II - Embedding of Graphs in Surfaces III - GU Signals Sets in Homogeneous Spaces IV - Performance of Signal Sets in Riemannian Manifolds 2
I - Introduction Purpose: To show that the topological structure associated with each metric space (block diagram) should be considered in the design of a communication system. What should be the approach? The identification of the surface topology of each block diagram, starting with the graph associated with the DMC channel. Consequences: New mathematical concepts and approaches may be incorporated to the already known ones to achieve the goals of better performance and less complexity. 3
� ✁ � ✂ � ✁ ✂ ✁ � ✂ ✁ � ✁ ✁ ✂ ✂ � ✂ ✄ � ✄ ✁ ✂ DMC Channel - E d E 1 d 1 E 2 d 2 E 3 d 3 Source Channel Source Modulator Encoder Encoder Channel Source Channel Sink Demodulator Decoder Decoder E 2 d 2 E 3 d 3 E 1 d 1 Figure 1: Communication system model Current design is based on metric spaces (vector space structure). Proposal : design should also consider the topological structure associated with each metric space. 4
✞ ✡ ✆✝ ☎ ✂ ✠ � � � ✂ � ✠ ✟ ✞ ✆✝ ☎ Realization Abstract Approach Existence of GU Signal Sets in Homogeneous Spaces Channel DMC Riemannian Topology Topology Topological Geometry Geometric Properties Graph Invariants Invariants Genus Sectional Curvature g 1 g 2 P e g 1 P e g 2 K 1 K 2 P e K 1 P e K 2 ✂☛✡ ✂☛✡ Figure 2: Proposal overview 5
✄ ✄ ✄ Results: To extend the concept of geometrically uniform codes (Euclidean space) to other spaces with constant curvature (homogeneous spaces, in particular, to the hyperbolic space); To consider the performance analysis of a digital communication system in n -dimensional manifolds ; To show the best performance , among the spaces with constant curvature, is achieved when the curvature is negative . 6
✂ ✁ ✕ ✂ ✁ ✒✓ � ☞ ✂ ✏ ☎ � ✎ ✌ ✁ ☎ � ✞ � ✞ ✝ ✞ ☞ ✘ ✌ ✖ ✍ ✂ � ☞ ✆✝ ✆✝ ✒✓ ✔ ✓ ✔ ✓ ✕ ☎ ✂ Motivation for the Proposal : First Case C 2 8 2 K 2 S 8 T 6 2 T 4 3 T 2 8 8 ✎✑✏ ✆✗✖ S 2 C 2 2 K 2 2 Embedding Compact surface K m n Figure 3: Embeddings 7
Motivation for the Proposal : Second Case Topological Space Metric Space a b M−QAM b Torus g=1 a Sphere g=0 a a M−PSK Space Model Pe(QAM) < Pe(PSK) Plane Model Figure 4: Metric and Topological spaces 8
✬ ✒✓ ✫ ✦ ✮ ✍ ★ ✪ ✒✓ ★ ✦ ✧ ★ ✦ ✫ ✫ ✦ ★ ✍ ✒✓ ✒✓ ✜ ✙ ✥ ✚✛ ✣✤ ✢ ✥ ✜ ✢ ✯ ✣✤ ✥ ✣✤ ✢ ✍ ✢ ✣✤ ✥ ✍ Conform Homeomorphic Transformations Riemann Surfaces 2-D manifolds (conformal geometry) (Riemannian geometry) Algebraic Differential Combinatorial Geometry Topology χ Ω 2 2 g Euler charact Sectional curvature, K ✧✩★ Elliptic χ Ω g 0 0 K 0 ✧✭✬ Euclidean χ Ω g 1 0 K 0 χ Ω Hyperbolic g 2 0 K 0 ✧✭✯ Compact, Oriented Surfaces Homogeneous Spaces 9
✒✓ ✎ ✌ ✏ ✏ ✓ ✔ ✓ ✔ ✌ ✎ ✓ ✒ 0 0 1 1 2 2 C 8 3 3 sphere/2 endings 3 4 4 5 5 6 6 Catenoid 7 7 0 0 1 1 sphere/3 endings C 6 3 2 2 3 3 4 4 5 5 Trinoid Figure 5: Embeddings in Minimal Surfaces 10
★ ★ ✪ ✧ ✦ ★ ✍ ✓ Classification Theorem for Surfaces (with boundary): Every compact, connected surface is topologically equivalent to a sphere, or a connected sum of tori, or a connected sum of projective planes (with some finite number of discs removed) . Example 1 Ω χ Ω 2 T g 2 2 Edges identification a 1 b 4 b 1 a 2 a 4 b 2 b 3 a 3 Plane Model Space Model Figure 6: 2 Torus 11
✄ ★ ✍ ✍ ✄ ✄ The majority of DMC channels of practical interest are embedded in compact surfaces with genus g 3 . 1 2 This motivates us to construct signal sets matched to groups , equivalently, GU signal sets ; The design of GU signal sets is strongly dependent on the existence of regular tessellations in homogeneous spaces. Homogeneous spaces are important for the rich algebraic and geometric properties , so far not fully explored in the context of communication and coding theory. The algebraic structures provide the means for systematic devices implementations whereas the geometric properties are relevant with respect to the efficiency of demodulation and decoding processes. 12
✦ ✍ ✍ ✦ ✰ ✎ ✲ ✌ ✍ ✦ ✮ ✧ ✍ ✮ ✦ ✎ ☞ ✱ ✧ ★ ✌ ✦ ✧ ✪ ✪ ✧ ✦ ✦ ✪ ✵ ✧ ✵ ✧✱ ✦ ✪ ☞ ✦ ✧ ✰ ✦ ✪ ✍ ✦ ✧ ✦ ✌ ✪ ★ ✧✱ ✧ ✲ ✍ ☞ ✧ II - Embedding of Graphs in Surfaces Minimum genus of an oriented surface is, [Ringel] : i for m g m K m m 2 n 2 4 n 2 n ✧✩★ where denotes the least integer greater than or equal to the real a number a . Maximum genus of an oriented surface is, [Ringeisen] : ii for m g M K m m 1 n 1 2 n 1 n where denotes the greatest integer less than or equal to the real a number a . Minimum genus of a non-oriented surface is, [Ringel] : iii g ˜ K m m 2 n 2 2 n ✎✴✳ Theorem 1 (Ringeisen) If a graph G has a 2-cell embedding in surfaces of genus g m and g M , then for every integer g , g m g M , G has a 2-cell g embedding in a surface of genus g . 13
✰ ✏ ✶ ✹ ☞ ✶ ✔ ✄ ✓ ☞ ✦ ✧ ✸ ✶ ✷ ☞ ★ ✧ ✦ ✦ ✧ ✏ ✲ ✶ ✳ ✄ ✶ ✦ ✧ ★ Assumption : All embeddings are 2-cell embeddings of K m n preserving the Euler characteristic of Ω . Needed Elements : α A model Ω α 1 R i spanned by the minimum embedding of mn i the graph K m n in an oriented compact surface Ω . The cardinality of the set of models mn , and Ω α Ω α mn : K m α n mn The number of regions associated with mn is constant and depends on χ . Ω 14
✻ ★ ✳ ✶ ✍ ✏ ✦ ✧ ✶ ★ ✾ ✿ ❂ ✸ ❁ ❀ ❀ ✹ ✻ ★ ✹ ✶ ✪ ✏ ✶ ✺ ✻ ✶ ✺ ✪ ★ ✪ ¿From this, and Theorem 1, we have Proposition 1 If Ω gT (a g -torus), then the number of regions of mn is 2 2 g m n mn α mn mn ✍✽✼ α Proposition 2 Let Ω α and 1 R i j , then i j is always an even mn mn j integer greater than or equal to 4. Lemma 1 The cardinality of the set α is equal to the number of positive integer solutions of the following equations ∑ i and α 2 mn 4 R 4 6 R 6 8 R 8 0 R 4 2 i ✻✑❀ where R k denotes the number of regions with k edges. 15
✍ ❀ ❀ ✍ ✲ ✍ ✰ ☞ ✍ ❀ ❀ ✌ ✍ ✧ ✲ ✎ ✦ ✦ ✧ ✎ ☞ ❀ ❀ ✍ ✌ ✰ ✌ ✦ ✌ ☞ ✧ ★ ✎ ☞ ✍ ✎ Definition 1 A channel class C m n is the set consisting of all channels with m vertices in X and n vertices in Y . Channel class C m ; i P Q C m p 1 p m q 1 q n n n . A type of soft-decision channel ; Channel class C m ii p q n . A type of hard-decision channel . Channel class C m iii p 16
❊ ✧ ✦ ✻ ✧ ✧ ✦ ✦ ✪ ✓ ✔ ✎ ✌ ❃ ❄ ✧ ❅ ✦ ✸ ✓ ✔ ✎ ❑ ✌ ❋ ✌ ✎ ✧ ❈ ✦ ❅ ✧ ■❏ ✻ ✸ ✦ ❃ ❋ ❃ ❈ ❋ ❈ ❉ ❈ ❈ ❈ ❇ ★ ❑ ☞ ✷ ❆ ❃ ✦ ✪ ✎ ✦ ❃ ❃ ✦ ✧ ✻ ✦ ✧ ✓ ✔ ✦ ✧ ✍ ✪ ✌ ✧ ☞ ❄ ❅ ✧❄ ✸ ✦ ❑ ❋ ❄ ✦ ❀ ✻ ✍ ✲ ✧ ✧ ✦ ✦ ❅ ✪ ❀ ❀ ✧ ★ ✧ ❅ ✪ ■❏ ✦ ✸ ✧ ❑ ✦ ✷ ✰ ✧ Let Ω α denote the set of surfaces corresponding to the embeddings of C m , that is, Ω α P Q n Ω α Ω α , Ω 1 α , , Ω α 1 1 1 where Ω i denotes surface Ω with j regions and i discs removed. j Lemma 2 If C m α and C m β are minimum embeddings, p g m T p gP ˜ then the set of surfaces for the 2-cell embedding of the C m channel is p α β 2 2 1 α β if α is even g m i T 2 i g ˜ j P j ❄❍● i 0 j 0 S m p α β 1 2 1 α β if α is odd. g m i T 2 i g ˜ j P j ❄▲● i 0 j 0 where the corresponding surfaces are denoted by T (torus) and P (projective plane). 17
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