A Finite-Blocklength Perspective on Gaussian Multi-Access Channels

Motivated by the growing application of wireless multi-access networks with stringent delay constraints, we investigate the Gaussian multiple access channel (MAC) in the finite blocklength regime. Building upon information spectrum concepts, we develop several non-asymptotic inner bounds on channel coding rates over the Gaussian MAC with a given finite blocklength, positive average error probability, and maximal power constraints. Employing Central Limit Theorem (CLT) approximations, we also obtain achievable second-order coding rates for the Gaussian MAC based on an explicit expression for its dispersion matrix. We observe that, unlike the pentagon shape of the asymptotic capacity region, the second-order region has a curved shape with no sharp corners. A main emphasis of the paper is to provide a new perspective on the procedure of handling input cost constraints for tight achievability proofs. Contrary to the complicated achievability techniques in the literature, we show that with a proper choice of input distribution, tight bounds can be achieved via the standard random coding argument and a modified typicality decoding. In particular, we prove that codebooks generated randomly according to independent uniform distributions on the respective "power shells" perform far better than both independent and identically distributed (i.i.d.) Gaussian inputs and TDMA with power control. Interestingly, analogous to an error exponent result of Gallager, the resulting achievable region lies roughly halfway between that of the i.i.d. Gaussian inputs and that of a hypothetical "sum-power shell" input. However, dealing with such a non-i.i.d. input requires additional analysis such as a new change of measure technique and application of a Berry-Esseen CLT for functions of random variables.