Scaling Laws for Gaussian Random Many-Access Channels

Abstract

An earlier version of this paper was presented in part at the 2019 IEEE International Symposium on Information Theory [DOI:10.1109/ISIT.2019.8849751], in part at the 2020 International Zurich Seminar on Information and Communication [DOI:10.3929/ethz-b-000403243], and in part at the 2020 IEEE International Symposium on Information Theory [DOI:10.1109/ISIT44484.2020.9174091]. ; This paper considers a Gaussian multiple-access channel with random user activity where the total number of users ℓn and the average number of active users kn may grow with the blocklength n . For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a function of ℓn and kn . When all users are active with probability one, i.e., ℓn=kn , it is demonstrated that, if kn is of an order strictly below n/logn , then each user can achieve the single-user capacity per unit-energy (loge)/N0 (where N0/2 is the noise power) by using an orthogonal-access scheme. In contrast, if kn is of an order strictly above n/logn , then the users cannot achieve any positive rate per unit-energy. Consequently, there is a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate per unit-energy is infeasible. It is further demonstrated that orthogonal-access schemes in combination with orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal. When the user activity is random, i.e., when ℓn and kn are different, it is demonstrated that, if knlogℓn is sublinear in n , then each user can achieve the single-user capacity per unit-energy (loge)/N0 . Conversely, if knlogℓn is superlinear in n , then the users cannot achieve any positive rate per unit-energy. Consequently, there is again a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate is infeasible that depends on the asymptotic behaviors of both ℓn and kn . It is further demonstrated that orthogonal-ac. ; The work of Jithin Ravi was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program under Grant 714161. The work of Tobias Koch was supported in part by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program under Grant 714161; and in part by the Spanish Ministerio de Ciencia e Innovación under Grant RYC-2014-16332, Grant TEC2016-78434-C3-3-R (AEI/FEDER, EU), and Grant PID2020-116683GB-C21 / AEI / 10.13039/501100011033.