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
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 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 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(loglogMMSE1) (where
MMSE is the disturbance constraint), is shown by means of the mixed
inputs technique recently introduced by Dytso et al.Comment: Submitted to IEEE Transactions on Information Theor