Scaling Laws for Gaussian Random Many-Access Channels
Abstract
This paper considers a Gaussian multiple-access channel with random user activity where the total number of users n 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 n and kn. When all users are active with probability one, i.e., n = kn, it is demonstrated that if kn is of an order strictly below n/ n, then each user can achieve the single-user capacity per unit-energy ( e)/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/ n, then the capacity per unit-energy is zero. 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 n and kn are different, it is demonstrated that if kn n is sublinear in n, then each user can achieve the single-user capacity per unit-energy ( e)/N0. Conversely, if kn n is superlinear in n, then the capacity per unit-energy is zero. 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 behaviours of both n and kn. It is further demonstrated that orthogonal-access schemes, which are optimal when n = kn, can be strictly suboptimal.