On Communications through a Gaussian Noise Channel with an MMSE Disturbance Constraint Alex Dytso, Ronit Bustin, Daniela Tuninetti, Natasha Devroye, H.Vincent Poor, Shlomo Shamai (Shitz) Corresponding Author: Alex Dytso, University of Illinois at Chicago, USA, odytso2@uic.edu Abstract—This paper considers a Gaussian channel with one transmitter and two receivers. The goal is to maximize the communication rate at the intended/primary receiver subject to a disturbance constraint at the unintended/secondary receiver. The disturbance is measured in terms of minimum mean square error (MMSE) of the interference that the transmission to the primary receiver inflicts on the secondary receiver. The paper presents a new upper bound for the problem of maximizing the mutual information subject to an MMSE constraint. The new bound holds for vector inputs of any length and recovers a previously known limiting (when the length for vector input tends to infinity) expression from the work of Bustin et al. The key technical novelty is a new upper bound on MMSE. This new bound allows one to bound the MMSE for all signal-to- noise ratio (SNR) values below a certain SNR at which the MMSE is known (which corresponds to the disturbance constraint). This new bound complements the ‘single-crossing point property’ of the MMSE that upper bounds the MMSE for all SNR values above a certain value at which the MMSE value is known. The new MMSE upper bound provides a refined characterization of the phase-transition phenomenon which manifests, in the limit as the length of the vector input goes to infinity, as a discontinuity of the MMSE for the problem at hand. A matching lower bound, to within an additive gap of order O ( log log 1 MMSE ) (where MMSE is the disturbance constraint), is shown by means of the mixed inputs recently introduced by Dytso et al. I. I NTRODUCTION Consider a Gaussian noise channel with one transmitter and two receivers: Y = √ snrX + Z, (1a) Y snr0 = √ snr 0 X + Z 0 , (1b) where Z, Z 0 , X, Y, Y snr0 ∈ R n , Z, Z 0 ∼N (0, I) and (Z, Z 0 , X) are mutually independent. When it will be nec- essary to stress the SNR at Y we will denote it with Y snr . We denote the mutual information between input X and output Y as I (X; Y)= I (X, snr) := E  log  p Y|X (Y|X) p Y (Y)  . (2) We also denote the mutual information normalized by n as I n (X, snr) := 1 n I (X, snr). (3) We denote the minimum mean squared error (MMSE) of estimating X from Y as mmse(X|Y) = mmse(X, snr) := 1 n Tr (E [Cov(X|Y)]) , (4) where Cov(X|Y) is the conditional covariance matrix of X given Y and is defined as Cov(X|Y) := E h (X - E[X|Y]) (X - E[X|Y]) T |Y i . Moreover, since the distribution of the noise is fixed, the quan- tities I (X; Y) and mmse(X|Y) are completely determined by X and snr and there is no ambiguity in using the notation I (X, snr) and mmse(X, snr). We consider a scenario in which a message, encoded as X, must be decoded at the primary receiver Y snr while it is also seen at the unintended/secondary receiver Y snr0 for which it is an interferer. This scenario is motivated by the two-user Gaussian Interference Channel (G-IC), whose capacity is only known for some special cases. The following strategies are commonly used to manage interference in the G-IC: 1) interference is treated as Gaussian noise: in this ap- proach the interference structure is neglected. It has been shown to be sum-capacity optimal in the so called very- weak interference regime [1]. 2) partial interference cancellation: by using the Han- Kobayashi (HK) achievable scheme [2], part of the interfering message is decoded and subtracted off, and the remaining part is treated as Gaussian noise. This approach has been show to be capacity achieving in strong interference [3] and optimal within 1/2 bit per channel per user otherwise [4]. 3) soft-decoding / estimation: the unintended receiver em- ploys soft-decoding of part of the interference. This is enabled by using non-Gaussian inputs and designing the decoders that treat interference as noise by taking into account the correct (non-Gaussian) distribution of the interference. Such scenarios were considered in [5], [6] and [7], and shown to be optimal to within either a constant or a O(log log(snr)) gap in [8]. In this paper we look at a somewhat simplified scenario as opposed to the G-IC as shown in Fig. 1. We assume that there is only one message for the primary receiver, and the primary user inflicts interference (disturbance) on a secondary receiver. The primary transmitter wishes to maximize its transmission rate, while subject to a constraint on the disturbance it inflicts on the secondary receiver. The disturbance is measured in terms of MMSE. Intuitively, the MMSE disturbance constraint quantifies the remaining interference after partial interference cancellation or soft-decoding have been performed [9], [10]. Formally, we aim to solve the following problem.