BASiCS Group

1
Generalized Coset Codes for Symmetric/Asymmetric
Distributed Source Coding

• S. Sandeep Pradhan
• Kannan Ramchandran
• pradhan5, kannanr_at_eecs.berkeley.edu

2
Outline

• Introduction and motivation
• Preliminaries
• Generalized coset codes for distributed source
  coding
• Simulation results
• Conclusions

3
Application Sensor Networks
Joint Decoding
Scene
Channels are bandwidth or rate-constrained
4
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)

5
Source Coding with Side Information at Receiver
(illustration)

• X and Y gt length-3 binary data (equally likely),
  
• Correlation Hamming distance between X and Y is
  at most 1.
• Example When X0 1 0,
• Y gt 0 1 0, 0 1 1, 0 0 0, 1 1 0.

6
System 2
X

• X and Y correlated
• Y at decoder

• What is the best that one can do?

• The answer is still 2 bits!

How?
7

• Encoder -gt index of the coset containing X.
  
• Decoder -gt X in given coset.

• Note
• Coset-1 -gt repetition code.
• Each coset -gt unique syndrome
• DIstributed Source Coding Using Syndromes

8
Symmetric CodingX and Y both encode partial
information

• Example
• X and Y -gt 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 -gt Total of 10 bits
• Can correct decoding be done if X and Y send 5
  bits each ?

Y
9

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

Coset Matrix
Y
X

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

10
Example
This concept can be generalized to
Euclidean-space codes.
11
Achievable Rate Region for the Problem
The rate region is

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

12
Generalized coset codes (Forney, 88)

• S lattice
• Ssublattice
• Construct sequences of cosets of S in S in
• n-dimensions

S
13
Example Let n4
4-d Euclidean space code
c1011
1
0
1
1
-2.5 2.5 -0.5 -4.5
sequence coming from the above sets -gt valid
codeword sequence
14
Generalized coset codes for distributed source
coding
1
3
5
7
9
13
-5
-17
19
25
-23
-11
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
1
7
13
-5
19
25
-17
-11
6
Two-level hierarchy of subcode construction
1
-17
19
Subset -gt encoder 1
1
7
13
Subset -gt encoder 2
15
Example 2
16
(No Transcript)
17

18

is the set of coset representatives of in
19
1
1
1
2
2
3
3
4
Encoders -gt index of subsets in dense lattice
L, containing quantized codewords
20
Encoding

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

Decoding

• Decoder -gt pair of codewords in the given coset
  pairs
• Estimate the source

Similar subcode construction for generalized
coset code Computationally efficient encoding and
decoding
Theorem 2 Decoding complexity decoding a
codeword in
21
Correlation distance

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

Theorem 3
dmin gt min. distance of the code
22
Simulation ResultsTrellis codes
Model Source X i.i.d. Gaussian
, Observation Y i XNi, 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
23
Results
Prob. of decoding error
Same prob. of decoding error for all the rate
pairs
24
Distortion Performance
Attainable Bound C-SNR22 dB, Normalized
distortion -15.5 dB
25
Special cases 2. Lattice codes
Hexagonal Lattice
Encoder-1
26
Conclusions

• Proposed constructive framework for distributed
  source coding
• -gt arbitrary achievable rates
• Generalized coset codes for framework
• Distance properties complexity -gt same for
• all achievable rate points
• Trellis lattice codes -gt special cases
• Simulations