[Paper Review] An Achievability Scheme for the Compound Channel with State Noncausally Available at the Encoder

Chandra Nair, Abbas El Gamal+1 more|arXiv (Cornell University)|Apr 20, 2010

Wireless Communication Security Techniques4 references17 citations

TL;DR

This paper proposes a novel coding scheme for the compound broadcast channel with noncausal state information at the encoder, using superposition coding, Marton coding, joint typicality, and indirect decoding. It achieves a strictly higher rate than the standard Gelfand-Pinsker extension and is optimal for certain classes of channels, including deterministic and compound Gaussian channels where dirty paper coding rates are achieved simultaneously for both receivers.

ABSTRACT

A new achievability scheme for the compound channel with discrete memoryless (DM) state noncausally available at the encoder is established. Achievability is proved using superposition coding, Marton coding, joint typicality encoding, and indirect decoding. The scheme is shown to achieve strictly higher rate than the straightforward extension of the Gelfand-Pinsker coding scheme for a single DMC with DM state, and is optimal for some classes of channels.

Motivation & Objective

• To address the suboptimality of the straightforward Gelfand-Pinsker extension for compound channels with noncausal state.
• To develop a new achievability scheme that outperforms existing approaches in rate for the 2-receiver discrete memoryless broadcast channel with state.
• To establish conditions under which the new scheme achieves capacity, including for deterministic and compound Gaussian channels.
• To demonstrate that the Gelfand-Pinsker approach is not optimal for the compound broadcast setting.

Proposed method

• The scheme uses superposition coding with auxiliary random variables W, U, V, and a state-dependent encoder function x(w,u,v,s).
• Codebook generation involves independent random sequences for W, U, and V, with rates T0, T1, and T2, respectively.
• Encoding uses joint typicality: the encoder finds indices l0, l1, l2 such that (w^n(l0), s^n) and (w^n(l0), u^n(l0,l1), v^n(l0,l2), s^n) are jointly typical.
• Decoding employs indirect decoding: each receiver decodes its own message using typicality decoding based on its channel output.
• The achievable rate is derived from a minimax expression involving mutual information terms: I(W,U;Y1)−I(W,U;S), I(W,V;Y2)−I(W,V;S), and a symmetric sum term.
• The scheme generalizes to continuous channels via Gaussian analogs, achieving dirty paper coding rates in compound Gaussian settings.

Experimental results

Research questions

• RQ1 Can a new coding scheme achieve higher rates than the straightforward extension of Gelfand-Pinsker coding for the compound broadcast channel with noncausal state?
• RQ2 Is the Gelfand-Pinsker approach suboptimal for the 2-receiver broadcast channel with state when state is noncausally known at the encoder?
• RQ3 For which classes of channels does the new scheme achieve capacity?
• RQ4 Can the new scheme achieve the dirty paper coding rate simultaneously for both receivers in compound Gaussian channels?
• RQ5 What role do auxiliary random variables W, U, V play in enabling higher rates through joint typicality and superposition?

Key findings

• The proposed scheme achieves a strictly higher rate than the standard Gelfand-Pinsker extension, with a quantitative gap of approximately 0.09 in a specific example.
• For deterministic channels where Y1 and Y2 are functions of (X,S) and I(Y1;Y2|S)=0, the scheme achieves capacity, with C = max_p(x|s) min{H(Y1|S), H(Y2|S)}.
• In a class of compound Gaussian channels with state variance depending on a Bernoulli state, the scheme achieves the dirty paper coding rate for both receivers simultaneously.
• The scheme is optimal for channels where Y1 and Y2 are functions of (X,S), share common information Z, and I(Y1;Y2|S,Z)=0, with W=Z, U=Y1, V=Y2.
• The symmetric sum term in the rate expression ensures robustness to the worst-case receiver, and the scheme achieves the upper bound in the considered classes.
• The gap between the Gelfand-Pinsker rate and the true capacity is shown to be non-zero and analytically computable, with an exact value of approximately 0.41 in a specific example.