Lattice Strategies for the Dirty Multiple Access Channel

A generalization of the Gaussian dirty-paper problem to a multiple access setup is considered. There are two additive interference signals, one known to each transmitter but none to the receiver. The rates achievable using Costa’s strategies (i.e. by a random binning scheme induced by Costa’s auxiliary random variables) vanish in the limit when the interference signals are strong. In contrast, it is shown that lattice strategies (“lattice precoding”) can achieve positive rates independent of the interferences, and in fact in some cases - which depend on the noise variance and power constraints - they are optimal. In particular, lattice strategies are optimal in the limit of high SNR. It is also shown that the gap between the achievable rate region and the capacity region is at most 0.167 bit. Thus, the dirty MAC is another instance of a network setup, like the Korner-Marton modulo-two sum problem, where linear coding is potentially better than random binning. Lattice transmission schemes and conditions for optimality for the asymmetric case, where there is only one interference which is known to one of the users (who serves as a “helper” to the other user), and for the “common interference” case are also derived. In the former case the gap between the helper achievable rate and its capacity is at most 0.085 bit.

💡 Research Summary

This paper studies a Gaussian multiple‑access channel (MAC) in which each transmitter knows a non‑causal additive interference that is completely unknown to the receiver. The model, called the “doubly dirty MAC,” is given by
 Y = X₁ + X₂ + S₁ + S₂ + Z,
where X₁ and X₂ satisfy individual power constraints P₁ and P₂, S₁ and S₂ are the interference sequences known only to user 1 and user 2 respectively, and Z ∼ N(0,N) is AWGN. The interferences are assumed to be “strong,” i.e., either arbitrary sequences or Gaussian with variances tending to infinity.

The authors first examine the straightforward extension of Costa’s dirty‑paper coding (DPC) to this multi‑user setting. Using the auxiliary random variables Uᵢ = Xᵢ + αᵢSᵢ (αᵢ = Pᵢ/(Pᵢ+N)) leads to the achievable region (5). However, when both users employ such a scheme, the mutual‑information term I(U₁,U₂;Y) – I(U₁;S₁) – I(U₂;S₂) collapses to zero as the interference power grows, so random‑binning based DPC yields vanishing rates for strong interference.

To overcome this limitation, the paper proposes lattice‑based precoding (often called “lattice precoding”). A lattice Λ ⊂ ℝⁿ is a discrete additive subgroup generated by a full‑rank matrix G. The nearest‑neighbor quantizer Q_Λ(·) and the modulo operation x mod Λ = x – Q_Λ(x) are used to shape the transmitted signals. Each user maps its message wᵢ to a lattice point tᵢ ∈ Λ and transmits
 Xᵢ =