Fundamental Limits of Rate-Constrained Multi-User Channels and Random Wireless Networks
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This thesis contributes toward understanding fundamental limits of multi-user fading channels and random wireless networks. Specifically, considering different samples of channel gains corresponding to different users/nodes in a multi-user wireless system, the maximum number of channel gains supporting a minimum rate is asymptotically obtained. First, the user capacity of fading multi-user channels with minimum rates is analyzed. Three commonly used fading models, namely, Rayleigh, Rician and Nakagami are considered. For broadcast channels, a power allocation scheme is proposed to maximize the number of active receivers, for each of which, a minimum rate Rmin>0 can be achieved. Under the assumption of independent Rayleigh fading channels for different receivers, as the total number of receivers n goes to infinity, the maximum number of active receivers is shown to be arbitrarily close to ln(P.ln(n))/Rmin with probability approaching one, where P is the total transmit power. The results obtained for Rayleigh fading are extended to the cases of Rician and Nakagami fading models. Under the assumption of independent Rician fading channels for different receivers, as the total number of receivers n goes to infinity, the maximum number of active receivers is shown to be equal to ln(2P.ln(n))/Rmin with probability approaching one. For broadcast channels with Nakagami fading, the maximum number of active receivers is shown to be equal to ln(ω/μ.P.ln(n))/Rmin with probability approaching one, where ω and μ are the Nakagami distribution parameters. A by-product of the results is to also provide a power allocation strategy that maximizes the total throughput subject to the rate constraints. In multiple-access channels, the maximum number of simultaneous active transmitters (i.e. user capacity) is obtained in the many user case in which a minimum rate must be maintained for all active users. The results are presented in the form of scaling laws as the number of transmitters increases. It is shown that for all three fading distributions, the user capacity scales double logarithmically in the number of users and differs only by constants depending on the distributions. We also show that a scheduling policy that maximizes the number of simultaneous active transmitters can be implemented in a distributed fashion. Second, the maximum number of active links supporting a minimum rate is asymptotically obtained in a wireless network with an arbitrary topology. It is assumed that each source-destination pair communicates through a fading channel and destinations receive interference from all other active sources. Two scenarios are considered: 1) Small networks with multi-path fading, 2) Large Random networks with multi-path fading and path loss. In the first case, under the assumption of independent Rayleigh fading channels for different source-destination pairs, it is shown that the optimal number of active links is of the order log(N) with probability approaching one as the total number of nodes, N, tends to infinity. The achievable total throughput also scales logarithmically with the total number of links/nodes in the network. In the second case, a two-dimensional large wireless network is considered and it is assumed that nodes are Poisson distributed with a finite intensity. Under the assumption of independent multi-path fading for different source-destination pairs, it is shown that the optimal number of active links is of the order N with probability approaching one. As a result, the achievable per-node throughput obtained by multi-hop routing scales with Θ(1/√N).