Gaussian Channels: I-MMSE at Every SNR

Séminaire le 20 Octobre 2016, 14h00 à CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40
Prof. Shlomo Shamai, The Andrew and Erna Viterbi Faculty of Electrical Engineering at the Technion-Israel Institute of Technology

Multi-user information theory presents many open problems, even in the simple Gaussian regime. One such prominent problem is the two-user Gaussian interference channel which has been a long standing open problem for over 30 years. We distinguish between two families of multi-user scalar Gaussian settings; a single transmitter (one dimension) and two transmitters (two dimensions), not restricting the number and nature of the receivers. Our first goal is to fully depict the behavior of asymptotically optimal, capacity
achieving, codes in one dimensional settings for every SNR. Such an understanding provides important insight to capacity achieving schemes and also gives an exact measure of the disturbance such codes have on unintended receivers.

We first discuss the Gaussian point-to-point channel and enhance some known results. We then consider the Gaussian wiretap channel and the Gaussian Broadcast channel (with and without secrecy demands) and reveal MMSE properties that confirm "rules of thumb" used in the achievability proofs of the capacity region of these channels and provide insights to the design of such codes.
We also include some recent observations that give a graphical interpretation to rate and equivocation in this one dimensional setting.
Our second goal is to employ these observations to the analysis of the two dimensional setting. Specifically, we analyze the two-user Gaussian interference channel, where simultaneous transmissions from two users interfere with each other. We employ our understanding of asymptotically point-to-point optimal code sequences to the analysis of this channel. Our results also resolve the "Costa Conjecture"
(a.k.a the "missing corner points" conjecture), as has been recently proved by Polyanskiy-Wu, applying Wasserstein Continuity of Entopy.

The talk is based on joint studies with R. Bustin, H. V. Poor and R. F. Schaefer.