Motivation Preliminaries Problems Relation Lattice Coding I: From Theory To Application Amin Sakzad Dept of Electrical and Computer Systems Engineering Monash University amin.sakzad@monash.edu Oct. 2013 Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Motivation 1 Preliminaries 2 Definitions Three examples Problems 3 Sphere Packing Problem Covering Problem Quantization Problem Channel Coding Problem Relation 4 Probability of Error versus VNR Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Motivation I: Geometry of Numbers Initiated by Minkowski and studies convex bodies and integer points in R n . 1 Diophantine Approximation, 2 Functional Analysis Examples Approximating real numbers by rationals, sphere packing problem, covering problem, factorizing polynomials, etc. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Motivation II: Telecommunication 1 Channel Coding Problem, 2 Quantization Problem Examples Signal constellations, space-time coding, lattice-reduction-aided decoders, relaying protocols, etc. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition A set Λ ⊆ R n of vectors called discrete if there exist a positive real number β such that any two vectors of Λ have distance at least β . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition A set Λ ⊆ R n of vectors called discrete if there exist a positive real number β such that any two vectors of Λ have distance at least β . Definition An infinite discrete set Λ ⊆ R n is called a lattice if Λ is a group under addition in R n . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Every lattice is generated by the integer combination of some linearly independent vectors g 1 , . . . , g m ∈ R n , i.e., Λ = { u 1 g 1 + · · · + u m g m : u 1 , . . . , u m ∈ Z } . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Every lattice is generated by the integer combination of some linearly independent vectors g 1 , . . . , g m ∈ R n , i.e., Λ = { u 1 g 1 + · · · + u m g m : u 1 , . . . , u m ∈ Z } . Definition The m × n matrix G = ( g 1 , . . . , g m ) which has the generator vectors as its rows is called a generator matrix of Λ . A lattice is called full rank if m = n . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Every lattice is generated by the integer combination of some linearly independent vectors g 1 , . . . , g m ∈ R n , i.e., Λ = { u 1 g 1 + · · · + u m g m : u 1 , . . . , u m ∈ Z } . Definition The m × n matrix G = ( g 1 , . . . , g m ) which has the generator vectors as its rows is called a generator matrix of Λ . A lattice is called full rank if m = n . Note that Λ = { x = uG : u ∈ Z n } . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The Gram matrix of Λ is M = GG T . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The Gram matrix of Λ is M = GG T . Definition The minimum distance of Λ is defined by d min (Λ) = min {� x � : x ∈ Λ \ { 0 }} , where � · � stands for Euclidean norm. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The determinate (volume) of an n -dimensional lattice Λ , det(Λ) , is defined as det[ GG T ] 1 2 . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The coding gain of a lattice Λ is defined as: γ (Λ) = d 2 min (Λ) . 2 det(Λ) n Geometrically, γ (Λ) measures the increase in the density of Λ over the lattice Z n . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The set of all vectors in R n whose inner product with all elements of Λ is an integer form the dual lattice Λ ∗ . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Definitions Definition The set of all vectors in R n whose inner product with all elements of Λ is an integer form the dual lattice Λ ∗ . For a lattice Λ , with generator matrix G , the matrix G − T forms a basis matrix for Λ ∗ . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples Barens-Wall Lattices Let � 1 � 0 G = . 1 1 Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples Barens-Wall Lattices Let � 1 � 0 G = . 1 1 Let G ⊗ m denote the m -fold Kronecker (tensor) product of G . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples Barens-Wall Lattices Let � 1 � 0 G = . 1 1 Let G ⊗ m denote the m -fold Kronecker (tensor) product of G . A basis matrix for Barnes-Wall lattice BW n , n = 2 m , can be formed by selecting the rows of matrices G ⊗ m , . . . , 2 ⌊ m 2 ⌋ G ⊗ m which have a square norm equal to 2 m − 1 or 2 m . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples Barens-Wall Lattices Let � 1 � 0 G = . 1 1 Let G ⊗ m denote the m -fold Kronecker (tensor) product of G . A basis matrix for Barnes-Wall lattice BW n , n = 2 m , can be formed by selecting the rows of matrices G ⊗ m , . . . , 2 ⌊ m 2 ⌋ G ⊗ m which have a square norm equal to 2 m − 1 or 2 m . d min ( BW n ) = � n 2 and det( BW n ) = ( n n 4 , which confirms 2 ) that γ ( BW n ) = � n 2 . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples D n Lattices For n ≥ 3 , D n can be represented by the following basis matrix: − 1 − 1 · · · 0 0 1 − 1 0 · · · 0 − 1 · · · 0 1 0 G = . . . . . . . . . . . . . . . . 0 0 0 · · · − 1 Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Three examples D n Lattices For n ≥ 3 , D n can be represented by the following basis matrix: − 1 − 1 · · · 0 0 1 − 1 0 · · · 0 − 1 · · · 0 1 0 G = . . . . . . . . . . . . . . . . 0 0 0 · · · − 1 √ We have det( D n ) = 2 and d min ( D n ) = 2 , which result in n − 2 n . γ ( D n ) = 2 Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem, Covering Problem, Quantization, Channel Coding Problem. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Let us put a sphere of radius ρ = d min (Λ) / 2 at each lattice point Λ . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Let us put a sphere of radius ρ = d min (Λ) / 2 at each lattice point Λ . Definition The density of Λ is defined as ∆(Λ) = ρ n V n det(Λ) , where V n is the volume of an n -dimensional sphere with radius 1 . Note that V n = π n/ 2 ( n/ 2)! . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Definition The kissing number τ (Λ) is the number of spheres that touches one sphere. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Definition The kissing number τ (Λ) is the number of spheres that touches one sphere. Definition The center density of Λ is then δ = ∆ V n . Note that 4 δ (Λ) 2 /n = γ (Λ) . Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Definition The kissing number τ (Λ) is the number of spheres that touches one sphere. Definition The center density of Λ is then δ = ∆ V n . Note that 4 δ (Λ) 2 /n = γ (Λ) . Definition The Hermite’s constant γ n is the highest attainable coding gain of an n -dimensional lattice. Lattice Coding I: From Theory To Application Amin Sakzad
Motivation Preliminaries Problems Relation Sphere Packing Problem Lattice Sphere Packing Problem Find the densest lattice packing of equal nonoverlapping, solid spheres (or balls) in n -dimensional space. Lattice Coding I: From Theory To Application Amin Sakzad
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