Gaussian Channels: I-MMSE at Every SNR
Abstract: 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 "good", 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 Gaussian Broadcast channel 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. 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 "good" point-to-point code sequences to the analysis of this channel .Our results resolve the "Costa Conjecture" for bounded variance inputs (a.k.a the "missing corner points" conjecture).
* Joint work with H. V. Poor , R. F. Schaefer and S. Shamai.