Lattice Strategies for the Dirty Multiple Access Channel

Abstract

In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver.The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies ("lattice precoding") can achieve positive rates, independent of the interference power.Furthermore, in some cases -which depend on the noise variance and power constraints -high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR -where the capacity region of the dirty MAC approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1 2 log 2 (πe/6) ≈ 0.254 bit. Thus, the doubly-dirty MAC is another instance of a network setting, like the Körner-Marton problem, where (linear) structured coding is potentially better than random binning. Finally, it is shown that lattice strategies are at most 0.167 bit from the capacity region for all SNR. The results are also compared and contrasted to the single dirt multiple access channel case (considered by other researchers), where lattice strategies and Gaussian random binning have similar performance.

Outer bounds for Gaussian multiple access channels with state known at one encoder

“…A single-letter characterization for C(P 1 , P 2 , Q) is not known. The best known achievable rate region was derived by Kotagiri and Laneman [3], and is achieved by 1 Note that, the authors of [2] and [5] assumed per-codeword power constraints, i.e., for all messages m 1 and m 2 , the codewords x n 1 and…”

“…The state S ∼ N (0, Q) is known noncausally at encoder 1 (state-cognitive user), but is not known at encoder 2 (non-cognitive user) nor at the decoder. This channel model generalizes Costa's dirty-paper channel [1] to the multipleaccess setting, and is also known as "dirty MAC" or "MAC with a single dirty user" [2]. Although the capacity region of the dirty MAC described in (1) has been studied extensively in the literature [2]- [4], no single-letter expression for the capacity region is available to date.…”

“…This channel model generalizes Costa's dirty-paper channel [1] to the multipleaccess setting, and is also known as "dirty MAC" or "MAC with a single dirty user" [2]. Although the capacity region of the dirty MAC described in (1) has been studied extensively in the literature [2]- [4], no single-letter expression for the capacity region is available to date. Kotagiri and Laneman [3] derived an inner bound on the capacity region using a generalized dirty paper coding scheme at the cognitive encoder, which allows arbitrary correlation between the input X 1 and the state S. Philosof et al [2] showed that the same rate region can be achieved using lattice-based transmission.…”

“…Although the capacity region of the dirty MAC described in (1) has been studied extensively in the literature [2]- [4], no single-letter expression for the capacity region is available to date. Kotagiri and Laneman [3] derived an inner bound on the capacity region using a generalized dirty paper coding scheme at the cognitive encoder, which allows arbitrary correlation between the input X 1 and the state S. Philosof et al [2] showed that the same rate region can be achieved using lattice-based transmission. In general, it is not clear whether a single-letter solution (i.e., random coding/random binning using independent and identically distributed (i.i.d.)…”

“…copies of some scalar distribution) is optimal for the dirty MAC (1). However, as [2] and [4] demonstrated, a singleletter solution is suboptimal for the doubly-dirty MAC, in which the output is corrupted by two states, each known at one encoder noncausally. In this case, (linear) structured lattice coding outperforms the best known single-letter solution.…”