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Distributed Source Coding Using Syndromes (DISCUS): Design and ConstructionPowerPoint Presentation

Distributed Source Coding Using Syndromes (DISCUS): Design and Construction

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### Distributed Source Coding Using Syndromes (DISCUS): Design and Construction

S.Sandeep Pradhan, Kannan Ramchandran

IEEE Transactions on Information Theory,

vol. 49, no.3, pp.626-643, Mar 2003

Outline

- Introduction
- Preliminaries
- Encoding with a Fidelity Criterion
- Problem Formulation
- Design Algorithm
- Constructions based on Trellis Codes

- Simulation Results
- Conclusion

Introduction

- Slepian-Wolf theorem:
By knowing joint distribution of X and Y, without explicitly knowing Y, encoder of X can perform as well as encoder who knows Y.

- Both encoder and decoder have access to side information Y
- Only decoder has access to side information Y

Introduction

- Wyner-Ziv Problem:
If decoder knows Y, then the information-theoretic rate-distortion performance for coding X is identical, no matter encoder knows Y or not.(X &Y are Gaussian.)

- Prior work on source quantizer design.
- Contributions:
- Construction of a framework resting on algebraic channel coding principles
- Performance analysis on Gaussian signals.

Source: discrete-alphabet continuous-valued

Compression: lossless lossy

Outline

- Introduction
- Preliminaries
- Encoding with a Fidelity Criterion
- Problem Formulation
- Design Algorithm
- Constructions based on Trellis Codes

- Simulation Results
- Conclusion

Preliminaries

- Example:
X, Y: equiprobable 3-bit binary words

Hamming distance is no more than 1.

Y is available to decoder.

Solution?

Cosets: {000,111},{100,011},{010,101},{001, 110}

Only transmit coset index/syndrome.

-0.5

1

Preliminaries- Quantization:
Digitizes an analog signal.

Two parameters: a partition and a codebook.

Codebook: [-2, 0.4, 2.3, 6]

yi-2

ai

yi

ai-1

yi-1

Preliminaries- Lloyd Max Quantization:
partition: ai are midpoints.

codebook: yiare centroids.

Optimal scalar quantization.

Preliminaries

- Trellis Coded Quantization (TCQ):[24]
- Dual of TCM
- Example:
- Uniformly distributed source in [-A, A]

- Implemented by Viterbi algorithm
[24] M.W. Marcellin and T. R. Fischer, “Trellis coded quantization of memoryless and Gauss-Markov sources,” IEEE Trans. Commun., vol. 38, pp.82–93, Jan. 1990.

Outline

- Introduction
- Preliminaries
- Encoding with a Fidelity Criterion
- Problem Formulation
- Design Algorithm
- Constructions based on Trellis Codes

- Simulation Results
- Conclusion

Encoding with a Fidelity Criterion

- Problem Formulation
- X, Y: correlated, memoryless, i.i.d distributed sequences
- Yi = Xi + Ni
- Xi, Yi, Ni: continuous-valued
- Ni: i.i.d distributed, independent from X
- Xi, Ni: zero-mean Gaussian random variables with known variance
- Decoder alone has access to Y.
- Goal: Form best approximation to X given R bits per sample
- Encoding in blocks of length L
- Distortion measure:
- Min R, s.t. reconstruction fidelity is less than given value D.

Encoding with a Fidelity Criterion

System Model: encoder and decoder.

Interplay of source coding, channel coding and estimation

Encoding with a Fidelity Criterion

- Design Algorithm
- Source Coding (M1, M2):
- Partition source space:
- Defining source codebook (S)
- Characterizing active codeword by W (r.v.)

- Estimation (M3):
Get best estimate of X (minimizing distortion) conditioned on outcome of Y and the element in .

- Channel Coding (M4, M5):
- Transmit over an error-free channel with rate R (less than Rs)
- Doable: I(W;Y) > 0, so H(W|Y) = H(W) – I(W;Y)
- Build channel code with rate Rc on channel P(Y|W)
- R = Rs – Rc.

- Source Coding (M1, M2):

Encoding with a Fidelity Criterion

- Summary of Design Algorithm:
- M1 and M3:
- minimize Rs, s.t. reconstruction distortion within given criterion.

- M2: maximize I(W;Y).
- M4:
- maximize Rc, s.t. error probability meets a desired tolerance level.

- M5: minimize decoding computational complexity.

- M1 and M3:

Encoding with a Fidelity Criterion

- Scalar Quantization and Memoryless Coset Construction (C1):
- Lloyd-Max (memoryless) quantizer
- Memoryless coset partition (M4)
- Example:
L=1, (sample by sample)

Quantization codebook: {r0, r1, …, r7}, (Rs = 3)

Channel coding codebook: {r0, r2, r4, r6}, {r1, r3, r5, r7}. (Rc = 2)

R = Rs – Rc = 1 bit/sample.

Encoding with a Fidelity Criterion

- Scalar Quantization and Trellis-Based Coset Construction (C2):
- Scalar quantizer for {Xi}i=1L
- Coset partition (M4) by trellis code.

Codebook (size of 8L), Rs = 3 bits/sample, two cosets

Encoding with a Fidelity Criterion

- Example:
Computing syndrome (Rs = 3, Rc = 2)

outcome of quantization be 7, 3, 2, 1, 4.

L = 5,

Syndrome is given by 10110 for 5 samples.

Encoding with a Fidelity Criterion

- Trellis-Based Quantization and Memoryless Coset Construction (C3):
- Trellis coded quantizer
- Memoryless coset partition
- Example:
Quantization codebook: Rs = 2

D0={r0, r4}, D1={r1, r5}, D2={r2, r6}, D3={r3, r7}.

Memoryless channel code: Rc = 1

1 coded bit with another 1 uncoded bit (from Y) to recover Di.

Encoding with a Fidelity Criterion

- Trellis-Based Quantization and Trellis-Based Coset Coset Construction (C4):
- Trellis coded quantizer
- Trellis coded coset partition

Comparison between C3 and C4.

Encoding with a Fidelity Criterion

- Distance Property
- Given a uniform partition, four cases of coset constructions have same distance property.
- Non-uniform quantizer, analyze performance by simulations.

Outline

- Introduction
- Preliminaries
- Encoding with a Fidelity Criterion
- Problem Formulation
- Design Algorithm
- Four Constructions

- Simulation Results
- Conclusion

Simulation Results

Quantization levels decrease distortion. (C1)

Correlation

-SNR:

ratio of X’s

variance and

N’s variance.

Simulation Results

Correlation

-SNR:

ratio of X’s

variance and

N’s variance.

Quantization levels increase prob. Of error. (C1)

Simulation Results

Correlation

-SNR:

ratio of X’s

variance and

N’s variance.

Error probability comparison of C1 and C2

(3-4dB gain)

Simulation Results

Correlation

-SNR:

ratio of X’s

variance and

N’s variance.

Error probability of C4 codes.

Conclusions

- Constructive practical framework based on algebraic trellis codes.
- Promising performance.

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