basics group university of california at berkeley generalized coset codes for symmetric/asymmetric...

BASiCS Group

University of California at Berkeley

Generalized Coset Codes for Symmetric/Asymmetric Distributed Source Coding

S. Sandeep Pradhan

Kannan Ramchandran{pradhan5, kannanr}@eecs.berkeley.edu

University of California, Berkeley

Outline

Introduction and motivation Preliminaries Generalized coset codes for distributed

source coding Simulation results Conclusions


Application: Sensor Networks

Scene

Sensor 1

Encoder

Sensor 2

Encoder

Sensor 3

Encoder

Channels are bandwidth or rate-constrained

Joint Decoding


Introduction and motivation

Distributed source coding Information theoretic results (Slepian-Wolf ‘73,

Wyner-Ziv, ‘76) Little is known about practical systems based

on these elegant concepts Applications: Distributed sensor networks/web

caching, ad-hoc networks, interactive comm.

Goal: Propose a constructive approach (DISCUS) (Pradhan & Ramchandran, 1999)


Source Coding with Side Information at Receiver (illustration) X and Y => length-3 binary data (equally likely), Correlation: Hamming distance between X and Y is at most 1.

Example: When X=[0 1 0], Y => [0 1 0], [0 1 1], [0 0 0], [1 1 0].

Encoder DecoderX

Y

XX ˆ)|( YXHR

•X and Y correlated•Y at encoder and decoder

System 1

X+Y=

0 0 00 0 10 1 01 0 0

Need 2 bits to index this.


What is the best that one can do?

XEncoder Decoder

Y

XX ˆ)|( YXHR

•X and Y correlated•Y at decoder

System 2

The answer is still 2 bits!

How?0 0 0 1 1 1Coset-1

000001010100

111110101011

X

Y


•Encoder -> index of the coset containing X.

•Decoder -> X in given coset.

Note:•Coset-1 -> repetition code.•Each coset -> unique “syndrome” •DIstributed Source Coding Using Syndromes

111

000Coset-1

110

001Coset-4

101

010Coset-3

011

100Coset-2


Symmetric CodingX and Y both encode partial information

Example:• X and Y -> length-7 equally likely binary data.• Hamming distance between X and Y is at most 1.• 1024 valid X,Y pairs

Solution 1:• Y sends its data with 7 bits.• X sends syndromes with 3 bits.• { (7,4) Hamming code } -> Total of 10 bits

Can correct decoding be done if X and Y send 5 bits each ?

Encoder Decoder

Y

X̂X


Solution 2: Map valid (X,Y) pairs into a coset matrix

1 2 3 . . . 32

32

.

.

.

21

Coset Matrix

Y

X •Construct 2 codes, assign them to encoders •Encoders -> index of coset of codes containing the outcome


1 0 1 1 0 1 0 0 1 0 0 1 0 10 1 1 0 0 1 01 1 1 0 0 0 1

G =

1 0 1 1 0 1 00 1 0 0 1 0 1

0 1 1 0 0 1 01 1 1 0 0 0 1

G1 =

G2 =

Example

Theorem 1: With (n,k,2t+1) code, X and Y -> rate pairs (R1,R2) :

,),(,}1,0{, tYXdYX Hn

.,,2 2121 knRRknRR

This concept can be generalized to Euclidean-space codes.


Achievable Rate Region for the Problem

3,3:, yxyx RRRR

10 yx RRThe rate region is:

xR

yR

3 4 5 6 7

76543

• All 5 optimal points can be constructively achieved with the same complexity.• An alternative to source-splitting approach (Rimoldi-97)


Generalized coset codes: (Forney, ’88) S = lattice S’=sublattice Construct sequences of cosets of S’ in S in

n-dimensions

S’

-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5

Example:

S

-4.5 -2.5 -0.5 1.5 3.5 5.5


C =0 0 0 0 1 0 1 1

Example: Let n=4

c=1011

1

0

1

1

sequence coming from the above sets -> valid codeword sequence-2.5 2.5 -0.5 -4.5

4-d Euclidean space code

-4.5 -2.5 -0.5 1.5 3.5 5.5

-5.5 -3.5 -1.5 0.5 2.5 4.5

-4.5 -2.5 -0.5 1.5 3.5 5.5

-4.5 -2.5 -0.5 1.5 3.5 5.5


Generalized coset codes for distributed source coding

x x x x x x x x x x x x x x x x x x x x x1 3 5 7 9 13 19

x x25-5-17-23 -11

xx

1 7 13 19 25-5-17 -11

1 19-17

1 7 13

'

1

2

6Two-level hierarchy of subcode construction:

Subset -> encoder 1

Subset -> encoder 2


Example 2:


'


1

1 is a sublattice of


2

2 is the set of coset representatives of in 1


Encoders -> index of subsets in dense lattice containing quantized codewords

1

2

1

1

2

3

4

1

2

3


Encoding: •Encoders quantize with main lattice•Index of the coset of subsets in the main lattice is sent

Decoding: •Decoder -> pair of codewords in the given coset pairs•Estimate the source

Similar subcode construction for generalized coset codeComputationally efficient encoding and decoding

Theorem 2: Decoding complexity = decoding a codeword in );'/( CSS


Correlation distance

• dc => second minimum distance between 2 codevectors in coset pairs i,j • Decoding error => distance between quantized codewords > dc.

Theorem 3: 2/minddc dmin => min. distance of the code

1

2

1

1

2

34

1

2

3


Simulation Results:Trellis codes

Model: Source = X~ i.i.d. Gaussian , Observation= Y i= X+Ni, where Ni ~ i.i.d. Gaussian. Correlation SNR= ratio of variances of X and N. Effective Source Coding Rate = 2bit / sample/encoder.

Quantizers: Fixed-length scalar quantizers with 8 levels.

Trellis codes with 16- states based on 8 level root scalar quantizer


Prob. of decoding errorResults

Same prob. of decoding error for all the rate pairs


Distortion Performance:

Attainable Bound: C-SNR=22 dB, Normalized distortion: -15.5 dB


Special cases: 2. Lattice codes

Encoder-1

Encoder-2

Hexagonal Lattice


Conclusions

Proposed constructive framework for distributed source coding

-> arbitrary achievable rates

Generalized coset codes for framework

Distance properties & complexity -> same for

all achievable rate points

Trellis & lattice codes -> special cases

Simulations

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