Variable-rate Slepian-Wolf Research Articles

On the Reliability Function of Variable-Rate Slepian-Wolf Coding

The reliability function of variable-rate Slepian-Wolf coding is linked to the reliability function of channel coding with constant composition codes, through which computable lower and upper bounds are derived. The bounds coincide at rates close to the Slepian-Wolf limit, yielding a complete characterization of the reliability function in that rate region. It is shown that variable-rate Slepian-Wolf codes can significantly outperform fixed-rate Slepian-Wolf codes in terms of rate-error tradeoff. Variable-rate Slepian-Wolf coding with rate below the Slepian-Wolf limit is also analyzed. In sharp contrast with fixed-rate Slepian-Wolf codes for which the correct decoding probability decays to zero exponentially fast if the rate is below the Slepian-Wolf limit, the correct decoding probability of variable-rate Slepian-Wolf codes can be bounded away from zero.

Optimum Tradeoffs Between the Error Exponent and the Excess-Rate Exponent of Variable-Rate Slepian–Wolf Coding

We analyze the optimal tradeoff between the error exponent and the excess-rate exponent for variable-rate Slepian-Wolf codes. In particular, we first derive upper (converse) bounds on the optimal error and excess-rate exponents, and then lower (achievable) bounds, via a simple class of variable-rate codes which assign the same rate to all source blocks of the same type class. Then, using the exponent bounds, we derive bounds on the optimal rate functions, namely, the minimal rate assigned to each type class, needed in order to achieve a given target error exponent. The resulting excess-rate exponent is then evaluated. Iterative algorithms are provided for the computation of both bounds on the optimal rate functions and their excess-rate exponents. The resulting Slepian-Wolf codes bridge between the two extremes of fixed-rate coding, which has minimal error exponent and maximal excess-rate exponent, and average-rate coding, which has maximal error exponent and minimal excess-rate exponent.

On the Redundancy of Slepian–Wolf Coding

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, the redundancy of both variable and fixed rate Slepian–Wolf coding is considered. Given any jointly memoryless source-side information pair <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$\{(X_i, Y_i)\}_{i=1}^{\infty}$</tex></formula></emphasis> with finite alphabet, the redundancy <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$R^n(\epsilon_n)$</tex></formula></emphasis> of variable rate Slepian–Wolf coding of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$X_1^n$</tex></formula></emphasis> with decoder only side information <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$Y_1^n$</tex></formula></emphasis> depends on both the block length <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$n$</tex></formula></emphasis> and the decoding block error probability <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$\epsilon_n$</tex></formula></emphasis>, and is defined as the difference between the minimum average compression rate of order <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$n$</tex> </formula></emphasis> variable rate Slepian–Wolf codes having the decoding block error probability less than or equal to <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$\epsilon_n$</tex></formula></emphasis>, and the conditional entropy <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$H(X\vert Y)$</tex></formula></emphasis>, where <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$H(X\vert Y)$</tex></formula></emphasis> is the conditional entropy rate of the source given the side information. The redundancy of fixed rate Slepian–Wolf coding of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$X_1^n$</tex></formula></emphasis> with decoder only side information <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$Y_1^n$</tex> </formula></emphasis> is defined similarly and denoted by <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$R^n_F(\epsilon_n)$</tex></formula></emphasis>. It is proved that under mild assumptions about <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$\epsilon_n,$</tex></formula></emphasis> <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$R^n(\epsilon_n) = d_v \sqrt{-\log\epsilon_n/n} + o(\sqrt{-\log \epsilon_n/n})$</tex></formula></emphasis> and <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$R^n_{F}(\epsilon_n) = d_f \sqrt{- \log \epsilon_n / n} + o(\sqrt{-\log \epsilon_n/n})$</tex></formula></emphasis>, where <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$d_f$</tex> </formula></emphasis> and <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$d_v$</tex></formula></emphasis> are two constants completely determined by the joint distribution of the source-side information pair. Since <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$d_v$</tex> </formula></emphasis> is generally smaller than <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$d_f$</tex></formula></emphasis>, our results show that variable rate Slepian–Wolf coding is indeed more efficient than fixed rate Slepian–Wolf coding. </para>

On the Duality Between Slepian–Wolf Coding and Channel Coding Under Mismatched Decoding

In this paper, Slepian-Wolf coding with a mismatched decoding metric is studied. Two different dualities between Slepian-Wolf coding and channel coding under mismatched decoding are established. These two dualities provide a systematic framework for comparing linear Slepian-Wolf codes, nonlinear Slepian-Wolf codes, and variable-rate Slepian-Wolf codes. In contrast with the fact that linear codes suffice to achieve the Slepian-Wolf limit under matched decoding, the minimum rate achievable with nonlinear Slepian-Wolf codes under mismatched decoding can be strictly lower than that achievable with linear Slepian-Wolf codes.