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A Brief Introduction to Lattice Coding Theory (in two parts)
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Introduction
- What is a lattice?
- Why study lattices?
- Real-world applications of lattices
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A Brief Introduction to Lattice Coding Theory (in two parts) Brian - - PowerPoint PPT Presentation
A Brief Introduction to Lattice Coding Theory (in two parts) Brian - - PowerPoint PPT Presentation
A Brief Introduction to Lattice Coding Theory (in two parts) Brian Kurkoski 1 Introduction What is a lattice? Why study lattices? Real-world applications of lattices 2 Lattice Definition Definition 1 An n -dimensional lattice
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Lattice Definition
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Definition 1 An n-dimensional lattice Λ is a discrete additive subgroup of Rn.
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Lattice Definition
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Definition 1 An n-dimensional lattice Λ is a discrete additive subgroup of Rn.
Vector addition in Rn: x = [x1 , . . . , xn ] y = [y1 , . . . , yn ] x + y = [x1 + y1, . . . , xn + yn] Group properties:
- has identity
- has inverse
- associative
- closure
- (commutative)
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Lattices in R2
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Lattices in R2
a
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Lattices in R2
a
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Lattices in R2
a b
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Lattices in R2
a b
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Lattices Beyond n=2 dimensions
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n = 2
Warning
image not available
n = 3 n = 4, 5, 6, ...
−3 −1 1 3 −3 −1 1 3 SLIDE 11
From Such A Simple Definition
Properties of lattices:
- Fundamental regions
- Minimum distance and coding gain
- Lattice transformations — scaling, rotation, reflections
- Lattice cosets
- Generator matrix, check matrix
- Quantization
- Lattice codes
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Why Study Lattices?
- Lattices are error-correcting codes defined over the real numbers
- Lattices use the same real-number algebra as the physical world
- physical-layer network coding or compute-forward
- Near-ideal codes for the AWGN channel
- Lossy source coding
- Lattices are fun
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Communications on AWGN Channel
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z y x
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x
SLIDE 15 4 3 2 1 1 2 3 4
x y
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2 3 5
User 1 User 2
Electromagnetic signals have linear superposition
- Codes over the real numbers are natural
- Lattices have a group structure — physical layer network coding
Physical World is Real-Valued
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Destination
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2 Users Signals Superimpose Linearly
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s1(t) = A1 cos
- ωt + φ1)
s2(t) = A2 cos
- ωt + φ2)
s1(t) + s2(t)
s1(t) + s2(t) s2(t) s1(t) Transmitted signals add Signal real/imaginary Ai exp
- φ√−1
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Lattice Compute and Forward
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x 1 h 1x 1 + h 2x 2
relay
x 2 h 1 h 2 u 1 = h 1x 1 + h 2x 2
Fading coefficients relay only wants a linear combination transmitters do not need to know h1, h2
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Lattice Compute and Forward
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x 1 h 1x 1 + h 2x 2
relay
x 2 h 3x 1 + h 4x 2
relay
h 1 h 2 h 3 h 4 u 1 u 2
Invert this matrix, recover x1, x2 u1 u2
- =
h1 h2 h3 h4
- ·
x1 x2
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AWGN Channel, Power Constraint P
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A Gaussian input distribution will achieve channel capacity
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Input Distribution of a Codebook
21 Spherical codebook Gaussian input distribution SHANNON Nested Lattice Code LATTICE
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Practical Applications of Lattices
- V.34 telephone cmodems at 33.6 kbits uses D4 lattice.
- Lossy source coding ITU-T 729.1 speech-coding Gosset lattice E8
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Input y = (y1, y2, … , yn) speech, image, etc. Output
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Lattices Are Fun
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Study of Regular Division of the Plane with Reptiles (1939) by M. C. Escher
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Systematic Way to Form Tessellations
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- J. Croft, http://ssrsbstaff.ednet.ns.ca/jcroft2
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1 1 2 1 1 2
In two dimensions
Rectangular Hexagonal is most efficient
Efficient Arrangement of Spheres
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Efficient Arrangement in Space
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Rectangular Hexagonal is most efficient
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John Conway and Neil Sloane, Sphere Packings, Lattices and Groups, Springer, Third Edition, 1999
All the Elegance of Lattices
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Forney’s Lecture Notes
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- G. David Forney,
Lecture notes for Principles of Digital Communications II Course at MIT Google “David Forney lecture notes”
http://dspace.mit.edu/
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Ram Zamir, Lattice Coding for Signals and Networks Cambridge University Press September 2014
The Latest In Lattice Coding
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