On Capacity-Achieving Input Distributions to Additive Vector Gaussian Noise Channels Under Peak and Even Moment Constraints
Abstract
We investigate the support of the capacity-achieving input distribution to a vector-valued
Gaussian noise channel. The input is subject to a radial even-moment constraint and, in some
cases, is additionally restricted to a given compact subset of R^n. Unlike much of the prior work
in this field, the noise components are permitted to have different variances and the compact
input alphabet is not necessarily a ball. Therefore, the problem considered here is not limited to
being spherically symmetric, which forces the analysis to be done in n dimensions.
In contrast to a commonly held belief, we demonstrate that the n-dimensional (real-analytic)
Identity Theorem can be used to obtain results in a multivariate setting. In particular, it is determined that when the even-moment constraint is greater than n, or when the input alphabet is
compact, the capacity-achieving distribution’s support has Lebesgue measure 0 and is nowhere
dense in R^n. An alternate proof of this result is then given by exploiting the geometry of the zero
set of a real-analytic function. Furthermore, this latter approach is used to show that the support
is composed of a countable union of submanifolds, each with dimension n − 1 or less. In the
compact case, the support is a finite union of submanifolds.
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Cite this version of the work
Jonah Sean Eisen
(2021).
On Capacity-Achieving Input Distributions to Additive Vector Gaussian Noise Channels Under Peak and Even Moment Constraints. UWSpace.
http://hdl.handle.net/10012/17256
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