On the Redundancy of Slepian–Wolf Coding

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

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