Gaussian Multiple and Random Access in the Finite Blocklength Regime

Abstract

This paper presents finite-blocklength achievability bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a sphere and a maximum likelihood decoder, the derived MAC bound on each transmitter's rate matches the MolavianJazi-Laneman bound (2015) in its first-and secondorder terms, improving the remaining terms to 1 2 log n n + O 1 n bits per channel use. The result then extends to a RAC model in which neither the encoders nor the decoder knows which of K possible transmitters are active. In the proposed rateless coding strategy, decoding occurs at a time nt that depends on the decoder's estimate t of the number of active transmitters k. Single-bit feedback from the decoder to all encoders at each potential decoding time ni, i ≤ t, informs the encoders when to stop transmitting. For this RAC model, the proposed code achieves the same first-, second-, and third-order performance as the best known result for the Gaussian MAC in operation. Index Terms-Gaussian multiple access channel, Gaussian random access channel, third-order asymptotics, finite blocklength, maximum likelihood decoder, spherical distribution.

Finite-Blocklength and Error-Exponent Analyses for LDPC Codes in Point-to-Point and Multiple Access Communication

“…Our argument follows the multiple access source coding proof in [27,Th. 11] and is similar to [24,Th. 1].…”

“…Remark 4: As noted in [24], using the same codebook from the LDPC(λ, ρ; n) ensemble for all transmitters has practical advantages. In our case, each device is the same except for its unique random coset vector v j .…”

“…Remark 12: The authors in [24] achieve lower decoder complexity in the symmetrical rate case by replacing the three events in (124) by one. While we do not assume the symmetrical rate point, we note that only events corresponding to constraints that are active at a given rate point have a nonnegligible impact in (128).…”

Variable-Length Sparse Feedback Codes for Point-to-Point, Multiple Access, and Random Access Channels

“…where J(P ) is the constant given in (49), and (63) follows from the fact that P Y n k is product of k output distributions with dimensions n j − n j−1 , j ∈ [K], each induced by a uniform distribution over a sphere with the corresponding radius. As argued in [7], [17], [22], [23], by spherical symmetry, the distribution of the random variable…”

“…. , n K ; our analysis in [23] shows that for any finite K, and sufficiently large increments n i − n i−1 for all i ∈ [K], using this restricted subset instead of the entire n K -dimensional power sphere results in no change in the asymptotic expansion (14) up to the third-order term.…”