# A relay-assisted distributed source coding problem

TL;DR: A relay-assisted distributed source coding problem with three statistically correlated sources is formulated and studied and a single unified coding strategy which subsumes all four special classes is derived.

Abstract: A relay-assisted distributed source coding problem with three statistically correlated sources is formulated and studied. Natural lower bounds for the rates are presented. The lower bounds are shown to be achievable for three special classes of joint source distributions. The achievable coding strategies for the three special classes are observed to have markedly different characteristics. A special class for which the lower bounds are not achievable is presented. A single unified coding strategy which subsumes all four special classes is derived.

## Summary (1 min read)

### Remark:

- For any arbitrarily small positive real number and all n sufficiently large, X n can be encoded at location-a at the rate H(X|Y) bits per sample, and decoded at location-b with a probability which can be made smaller than .
- The converse to the Slepian-Wolf coding theorem proves that the lower bound R a ≥ H(X|Y) cannot be improved.
- The cutset bound (3.5) in Section III-A proves that the lower bound R b ≥ H(X|Z) cannot be improved.

### B. Special cases in which the cutset bounds are tight

- Thus for any arbitrarily small positive real number and all n sufficiently large, X n can be encoded at location-a at the rate H(X|Y, Z) bits per sample, and decoded at location-b with a probability which can be made smaller than .
- The authors shall call this relay-assisted distributed source coding strategy as relay decode and re-encode.
- The operation at the intermediate location-b is to simply forward the message from location-a to location-c without any processing.
- The authors shall call this relay-assisted distributed source coding strategy as relay forward.
- The operation at the intermediate location-b is to (i) encode its observations to be reproduced at location-c and (ii) forward the message from location-a to location-c without any processing.

### is, the auxiliary random variables U, V satisfy the following two Markov chains: U-X-(Y, Z) and V-(U, Y)-(X, Z). Then all rate-pairs

- EQUATION are operationally admissible for relay-assisted distributed source coding and cover all the four previously discussed special cases.
- The authors first show that all the four previously discussed special cases are covered by Theorem 1.
- The authors now sketch the proof that all rate-pairs (R a , R b ) satisfying (4.7) and (4.8) are operationally admissible for relayassisted distributed source coding.
- The encoder also randomly partitions the tuple of codewords C u into nonoverlapping bins.

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...F (X1, X2, X3) = X1 was considered in [7]....

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##### References

42,928 citations

### "A relay-assisted distributed source..." refers background or methods in this paper

...P(E3 | E2) ↓ 0 as n ↑ ∞ by the Markov lemma [10] because U–X–Y is a Markov chain....

[...]

...This follows from the coding theorem due to Slepian and Wolf [10] with Xn as the source of information to be encoded at location-a and Yn as the sideinformation available to a decoder at location-b....

[...]

...5h2(p) where h2(α) := −α log2(α) − (1 − α) log2(1 − α), α ∈ [0, 1] is the binary entropy function [10]....

[...]

...The Slepian-Wolf coding theorem with Yn as the source of information available at location-b and Zn as the sideinformation available to a decoder at location-c shows that, for any arbitrarily small positive real number and all n sufficiently large, Yn can be encoded at location-b at the rate H(Y |Z) bits per sample, and decoded at locationc with a probability which can be made smaller than ....

[...]

...The Slepian-Wolf coding theorem applied a second time with Xn as the source of information available at location-a and (Yn,Zn) as the side-information available to a decoder at location-c shows that, for any arbitrarily small positive real number and all n sufficiently large, Xn can be encoded at location-a at the rate H(X|Y,Z) bits per sample, and decoded at location-c with a probability which can be made smaller than ....

[...]

3,247 citations

562 citations

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...Beginning the late 90’s, the construction of practical distributed source codes for emerging wireless sensor network applications [2] received...

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