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Register a new userOn Communication through a Gaussian Channel with an MMSE Disturbance Constraint
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Added by
Alex Dytso
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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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 of vector input tends to infinity) expression from the work of Bustin $textit{et al.}$ The key technical novelty is a new upper bound on the MMSE. This bound allows one to bound the MMSE for all signal-to-noise ratio (SNR) values $textit{below}$ a certain SNR at which the MMSE is known (which corresponds to the disturbance constraint). This bound complements the `single-crossing point property of the MMSE that upper bounds the MMSE for all SNR values $textit{above}$ a certain value at which the MMSE value is known. The 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. For vector inputs of size $n=1$, a matching lower bound, to within an additive gap of order $O left( log log frac{1}{sf MMSE} right)$ (where ${sf MMSE}$ is the disturbance constraint), is shown by means of the mixed inputs technique recently introduced by Dytso $textit{et al.}$
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This paper extends the single crossing point property of the scalar MMSE function, derived by Guo, Shamai and Verdu (first presented in ISIT 2008), to the parallel degraded MIMO scenario. It is shown that the matrix Q(t), which is the difference between the MMSE assuming a Gaussian input and the MMSE assuming an arbitrary input, has, at most, a single crossing point for each of its eigenvalues. Together with the I-MMSE relationship, a fundamental connection between Information Theory and Estimation Theory, this new property is employed to derive results in Information Theory. As a simple application of this property we provide an alternative converse proof for the broadcast channel (BC) capacity region under covariance constraint in this specific setting.
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