The I-MMSE approach on the weak Gaussian Z-interference channel and the type I Gaussian Broadcast-Z-interference channel

A fundamental relationship between Estimation Theory and Information Theory for Gaussian channels was derived by Guo, Shamai and Verdú (first presented in ISIT 2008); in particular, it was shown that for the MIMO standard Gaussian channel, the mutual information and the minimum mean-square error (MMSE) are related. This fundamental relationship and its generalizations, referred to as the I-MMSE relationships, have already been shown to be useful in several aspects of Information Theory. An inherent property of the MMSE, is the “single crossing point” property: as a function of snr, the MMSE of the Gaussian input distribution and the MMSE of an arbitrary input distribution intersect at most once. In this work we will use this property and its recent extensions to the MIMO scenario. In this paper we take a look at two variations of the interference channel: the Gaussian Z-interference channel and the type I Gaussian Broadcast-Z-interference channel. We use the I-MMSE approach to derive outer bounds on the capacity region of these channels. The Z-interference problem is a simplified case of the more general interference channel, for which the capacity region is unknown, in general. The Gaussian Z-interference channel in its standard form is given by: Y₁ = X₁ + √aX₂ + N₁ Y₂ = X₂ + N₂ where Ni and N2 are standard Gaussian and may be considered independent. We consider a power constraints P_i, i ϵ {1, 2}, on both inputs. For a ϵ (0, 1), that is, weak interference, there is no known single letter expression for the capacity region. The best known general outer bound is due to Sato. Using Ahlswede´s limiting expression for the capacity region, we re-derive Sato´s outer bound directly from the limiting expression using the I-MMSE approach and show why this outer bound can not be tight in general. As an additional example of the usage of t- e I-MMSE approach, we examine the type I Gaussian Broadcast-Z-interference channel, recently presented by Shang and Poor. For this channel we show that the I-MMSE approach can derive equivalent, more natural and insightful, outer bound for the weak interference case.