BASiCS Group University of California at Berkeley Generalized Coset Codes for Symmetric/Asymmetric...
-
date post
21-Dec-2015 -
Category
Documents
-
view
219 -
download
2
Transcript of 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
University of California, Berkeley
Application: Sensor Networks
Scene
Sensor 1
Encoder
Sensor 2
Encoder
Sensor 3
Encoder
Channels are bandwidth or rate-constrained
Joint Decoding
University of California, Berkeley
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)
University of California, Berkeley
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.
University of California, Berkeley
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
University of California, Berkeley
•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
University of California, Berkeley
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
University of California, Berkeley
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
University of California, Berkeley
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.
University of California, Berkeley
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)
University of California, Berkeley
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
University of California, Berkeley
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
University of California, Berkeley
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
University of California, Berkeley
Example 2:
University of California, Berkeley
'
University of California, Berkeley
1
1 is a sublattice of
University of California, Berkeley
2
2 is the set of coset representatives of in 1
University of California, Berkeley
Encoders -> index of subsets in dense lattice containing quantized codewords
1
2
1
1
2
3
4
1
2
3
University of California, Berkeley
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
University of California, Berkeley
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
University of California, Berkeley
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
University of California, Berkeley
Prob. of decoding errorResults
Same prob. of decoding error for all the rate pairs
University of California, Berkeley
Distortion Performance:
Attainable Bound: C-SNR=22 dB, Normalized distortion: -15.5 dB
University of California, Berkeley
Special cases: 2. Lattice codes
Encoder-1
Encoder-2
Hexagonal Lattice
University of California, Berkeley
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