Lattice Coding I: From Theory To Application Amin Sakzad Dept of - - PowerPoint PPT Presentation

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Lattice Coding I: From Theory To Application Amin Sakzad Dept of - - PowerPoint PPT Presentation

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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

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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

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Motivation Preliminaries Problems Relation

1

Motivation

2

Preliminaries Definitions Three examples

3

Problems Sphere Packing Problem Covering Problem Quantization Problem Channel Coding Problem

4

Relation Probability of Error versus VNR

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation

Motivation I: Geometry of Numbers

Initiated by Minkowski and studies convex bodies and integer points in Rn.

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

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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

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Motivation Preliminaries Problems Relation Definitions

Definition A set Λ ⊆ Rn 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

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Motivation Preliminaries Problems Relation Definitions

Definition A set Λ ⊆ Rn 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 ⊆ Rn is called a lattice if Λ is a group under addition in Rn.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Every lattice is generated by the integer combination of some linearly independent vectors g1, . . . , gm ∈ Rn, i.e., Λ = {u1g1 + · · · + umgm : u1, . . . , um ∈ Z} .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Every lattice is generated by the integer combination of some linearly independent vectors g1, . . . , gm ∈ Rn, i.e., Λ = {u1g1 + · · · + umgm : u1, . . . , um ∈ Z} . Definition The m × n matrix G = (g1, . . . , gm) 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

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Motivation Preliminaries Problems Relation Definitions

Every lattice is generated by the integer combination of some linearly independent vectors g1, . . . , gm ∈ Rn, i.e., Λ = {u1g1 + · · · + umgm : u1, . . . , um ∈ Z} . Definition The m × n matrix G = (g1, . . . , gm) 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 ∈ Zn} .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The Gram matrix of Λ is M = GGT .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The Gram matrix of Λ is M = GGT . Definition The minimum distance of Λ is defined by dmin(Λ) = min{x: x ∈ Λ \ {0}}, where · stands for Euclidean norm.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The determinate (volume) of an n-dimensional lattice Λ, det(Λ), is defined as det[GGT ]

1 2 . Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The coding gain of a lattice Λ is defined as: γ(Λ) = d2

min(Λ)

det(Λ)

2 n

. Geometrically, γ(Λ) measures the increase in the density of Λ over the lattice Zn.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The set of all vectors in Rn whose inner product with all elements

  • f Λ is an integer form the dual lattice Λ∗.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Definitions

Definition The set of all vectors in Rn whose inner product with all elements

  • f 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

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Motivation Preliminaries Problems Relation Three examples Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Barens-Wall Lattices

Let G = 1 1 1

  • .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Barens-Wall Lattices

Let G = 1 1 1

  • .

Let G⊗m denote the m-fold Kronecker (tensor) product of G.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Barens-Wall Lattices

Let G = 1 1 1

  • .

Let G⊗m denote the m-fold Kronecker (tensor) product of G. A basis matrix for Barnes-Wall lattice BWn, n = 2m, can be formed by selecting the rows of matrices G⊗m, . . . , 2⌊ m

2 ⌋G⊗m

which have a square norm equal to 2m−1 or 2m.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Barens-Wall Lattices

Let G = 1 1 1

  • .

Let G⊗m denote the m-fold Kronecker (tensor) product of G. A basis matrix for Barnes-Wall lattice BWn, n = 2m, can be formed by selecting the rows of matrices G⊗m, . . . , 2⌊ m

2 ⌋G⊗m

which have a square norm equal to 2m−1 or 2m. dmin(BWn) = n

2 and det(BWn) = ( n 2 )

n 4 , which confirms

that γ(BWn) = n

2 .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Dn Lattices

For n ≥ 3, Dn can be represented by the following basis matrix: G =        −1 −1 · · · 1 −1 · · · 1 −1 · · · . . . . . . . . . . . . . . . · · · −1        .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Three examples

Dn Lattices

For n ≥ 3, Dn can be represented by the following basis matrix: G =        −1 −1 · · · 1 −1 · · · 1 −1 · · · . . . . . . . . . . . . . . . · · · −1        . We have det(Dn) = 2 and dmin(Dn) = √ 2, which result in γ(Dn) = 2

n−2 n . Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation

Sphere Packing Problem, Covering Problem, Quantization, Channel Coding Problem.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Let us put a sphere of radius ρ = dmin(Λ)/2 at each lattice point Λ.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Let us put a sphere of radius ρ = dmin(Λ)/2 at each lattice point Λ. Definition The density of Λ is defined as ∆(Λ) = ρnVn det(Λ), where Vn is the volume of an n-dimensional sphere with radius 1. Note that Vn = πn/2 (n/2)!.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Definition The kissing number τ(Λ) is the number of spheres that touches

  • ne sphere.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Definition The kissing number τ(Λ) is the number of spheres that touches

  • ne sphere.

Definition The center density of Λ is then δ = ∆

Vn .

Note that 4δ(Λ)2/n = γ(Λ).

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Definition The kissing number τ(Λ) is the number of spheres that touches

  • ne sphere.

Definition The center density of Λ is then δ = ∆

Vn .

Note that 4δ(Λ)2/n = γ(Λ). Definition The Hermite’s constant γn is the highest attainable coding gain of an n-dimensional lattice.

Lattice Coding I: From Theory To Application Amin Sakzad

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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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Motivation Preliminaries Problems Relation Sphere Packing Problem

Summary of Well-Known Results

Theorem For large n’s we have 1 2πe ≤ γn n ≤ 1.744 2πe ,

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Sphere Packing Problem

Summary of Well-Known Results

Theorem For large n’s we have 1 2πe ≤ γn n ≤ 1.744 2πe , The densest lattice packings are known for dimensions 1 to 8 and 12, 16, and 24.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Covering Problem

Let us supose a set of spheres of radius R covers Rn

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Covering Problem

Let us supose a set of spheres of radius R covers Rn Definition The thickness of Λ is defined as Θ(Λ) = RnVn det(Λ)

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Covering Problem

Let us supose a set of spheres of radius R covers Rn Definition The thickness of Λ is defined as Θ(Λ) = RnVn det(Λ) Definition The normalized thickness of Λ is then θ(Λ) = Θ

Vn .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Covering Problem

Lattice Covering Problem

Ask for the thinnest lattice covering of equal overlapping, solid spheres (or balls) in n-dimensional space.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Covering Problem

Summary of Well-Known Results

Theorem The thinnest lattice coverings are known for dimensions 1 to 5, (all A∗

n).

Davenport’s Construction of thin lattice coverings, (thinner than A∗

n for n ≤ 200).

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Definition For any point x in a constellation A the Voroni cell ν(x) is defined by the set of points that are at least as close to x as to any other point y ∈ A, i.e., ν(x) = {v ∈ Rn : v − x ≤ v − y, ∀ y ∈ A}.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Definition For any point x in a constellation A the Voroni cell ν(x) is defined by the set of points that are at least as close to x as to any other point y ∈ A, i.e., ν(x) = {v ∈ Rn : v − x ≤ v − y, ∀ y ∈ A}. We simply denote ν(0) by ν.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Definition An n-dimensional quantizer is a set of points chosen in Rn. The input x is an arbitrary point of Rn ; the output is the closest point to x.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Definition An n-dimensional quantizer is a set of points chosen in Rn. The input x is an arbitrary point of Rn ; the output is the closest point to x. A good quantizer attempts to minimize the mean squared error of quantization.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Lattice Quantizer Problem

finds and n-dimentional lattice Λ for which G(ν) =

1 n

  • ν x · xdx

det(ν)1+ 2

n

, is a minimum.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Summary of Well-Known Results

Theorem The optimum lattice quantizers are only known for dimensions 1 to 3. As n → ∞, we have Gn → 1 2πe.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Quantization Problem

Summary of Well-Known Results

Theorem The optimum lattice quantizers are only known for dimensions 1 to 3. As n → ∞, we have Gn → 1 2πe. It is worth remarking that the best n-dimensional quantizers presently known are always the duals of the best packings known.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Definition For two points x and y in Fn

q the Hamming distance is defined as

d(x, y) = {i: xi = yi} .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Definition For two points x and y in Fn

q the Hamming distance is defined as

d(x, y) = {i: xi = yi} . Definition A q-ary (n, M, dmin) code C is a subset of M points in Fn

q , with

minimum distance dmin(C) = min

x=y∈C d(x, y).

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures I

Suppose that x, which is in a constellation A, is sent, y = x + z is received, where the components of z are i.i.d. based on N(0, σ2), The probability of error is defined as Pe(A, σ2) = 1 − 1 ( √ 2πσ)n

  • ν

exp −x2 2σ2

  • dx.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures II

Rate Definition The rate r of an (n, M, dmin) code C is r = log2(M) n .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures II

Rate Definition The rate r of an (n, M, dmin) code C is r = log2(M) n . The power of a transmission has a close relation with the rate of the code.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures II

Rate Definition The rate r of an (n, M, dmin) code C is r = log2(M) n . The power of a transmission has a close relation with the rate of the code. Normalized Logarithmic Density Definition The normalized logarithmic density (NLD) of an n-dimensional lattice Λ is 1 n log

  • 1

det(Λ)

  • .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures III

Capacity Definition The capacity of an AWGN channel with noise variance σ2 is C = 1 2 log

  • 1 + P

σ2

  • ,

where P

σ2 is called the

signal-to-noise ratio.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Performance Measures III

Capacity Definition The capacity of an AWGN channel with noise variance σ2 is C = 1 2 log

  • 1 + P

σ2

  • ,

where P

σ2 is called the

signal-to-noise ratio. Generalized Capacity Definition The capacity of an “unconstrained” AWGN channel with noise variance σ2 is C∞ = 1 2 ln

  • 1

2πeσ2

  • .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Channel Coding Problem

Approaching Capacity

Capacity-Achieving Codes Definition A (n, M, dmin) code C is called capacity-achieving for the AWGN channel with noise variance σ2, if r = C when Pe(C, σ2) ≈ 0. Sphere-Bound- Achieving Lattices Definition An n-dimensional lattice Λ is called capacity-achieving for the unconstrained AWGN channel with noise variance σ2, if

NLD(Λ) = C∞ when

Pe(Λ, σ2) ≈ 0.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR

Definition The volume-to-noise ratio of a lattice Λ over an unconstrained AWGN channel with noise variance σ2 is defined as α2(Λ, σ2) = det(Λ)

2 n

2πeσ2 .

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR

Definition The volume-to-noise ratio of a lattice Λ over an unconstrained AWGN channel with noise variance σ2 is defined as α2(Λ, σ2) = det(Λ)

2 n

2πeσ2 . Note that α2(Λ, σ2) = 1 is equivalent to NLD(Λ) = C∞.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR

Union Bound Estimate

Using the formula of coding gain and α2(Λ, σ2), we obtain an estimate upper bound for the probability of error for a maximum-likelihood decoder: Pe(Λ, σ2) ≤ τ(Λ) 2 erfc πe 4 γ(Λ)α2(Λ, σ2)

  • ,

where erfc(t) = 2 √π ∞

t

exp(−t2)dt.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR −2 2 4 6 8 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 VNR(dB) Normalizeed Error Probability (NEP) Sphere bound Uncoded system Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR

Thanks for your attention! Friday 18 Oct. Building 72, Room 132.

Lattice Coding I: From Theory To Application Amin Sakzad

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Motivation Preliminaries Problems Relation Probability of Error versus VNR

Thanks for your attention! Friday 18 Oct. Building 72, Room 132.

Lattice Coding I: From Theory To Application Amin Sakzad