On Stochastic Orders and Fast Fading Multiuser Channels with Statistical CSIT

In this paper, we investigate the ergodic capacity of fast fading Gaussian multiuser channels when only the statistics of the channel state are known at the transmitter. In general, the characterization of capacity regions of multiuser channels with only statistical channel state information at the transmitter (CSIT) is open. Instead of directly matching achievable rate regions and the corresponding outer bounds, in this work we resort to classifying the random channels through their probability distributions. To be more precise, in order to attain capacity results, we first derive sufficient conditions to attain some information theoretic channel orders such as degraded and strong/very strong interference by applying the usual stochastic order and exploiting the same marginal property such that the capacity regions of the memoryless Gaussian multiuser channels can be characterized. These include Gaussian interference channels, Gaussian broadcast channels, and Gaussian wiretap channels/secret key generation. We also extend the framework to channels with a specific memory structure, namely, channels with finite-state, wherein the Markov fading channel is discussed as a special case. Several practical examples such as Rayleigh fading and Nakagami-m fading, etc., illustrate the application of the derived results. Index Terms Stochastic orders, Same marginal property, Imperfect CSIT, Coupling, Maximal coupling, Copula, Multi-user channels, Capacity regions, Channel with memory Parts of the work were presented in ITW 2016, Cambridge, UK and SCC 2017, Hamburg, Germany. Pin-Hsun Lin, Eduard A. Jorswieck, Martin Mittelbach and Carsten R. Janda are with the Communications Laboratory, Department of Electrical Engineering and Information Technology, Technische Universität Dresden, Germany. Rafael F. Schaefer is with the Information Theory and Applications Chair, Technische Universität Berlin, Germany Emails: {pin-hsun.lin, eduard.jorswieck, martin.mittelbach, carsten.janda}@tu-dresden.de, [email protected]. Part of this work is funded by FastCloud 03ZZ0517A and FastSecure 03ZZ0522A. ar X iv :1 71 2. 03 69 2v 1 [ cs .I T ] 1 1 D ec 2 01 7