English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper studies an $n$-dimensional additive Gaussian noise channel with apeak-power-constrained input.  It is well known that, in this case, when $n=1$ the capacity-achieving inputdistribution is discrete with finitely many mass points, and when $n>1$ thecapacity-achieving input distribution is supported on finitely many concentricshells. However, due to the previous proof technique, neither the exact numberof mass points/shells of the optimal input distribution nor a bound on it wasavailable. This paper provides an alternative proof of the finiteness of thenumber mass points/shells of the capacity-achieving input distribution andproduces the first firm bounds on the number of mass points and shells, pavingan alternative way for approaching many such problems.  Roughly, the paper consists of three parts. The first part considers the caseof $n=1$. The first result, in this part, shows that the number of mass pointsin the capacity-achieving input distribution is within a factor of two from thedownward shifted capacity-achieving output probability density function (pdf).The second result, by showing a bound on the number of zeros of the downwardshifted capacity-achieving output pdf, provides a first firm upper on thenumber of mass points. Specifically, it is shown that the number of mass pointsis given by $O(\mathsf{A}^2)$ where $\mathsf{A}$ is the constraint on the inputamplitude.  The second part generalizes the results of the first part to the case of$n>1$. In particular, for every dimension $n>1$, it is shown that the number ofshells is given by $O(\mathsf{A}^2)$ where $\mathsf{A}$ is the constraint onthe input amplitude.  Finally, the third part provides bounds on the number of points for the caseof $n=1$ with an additional power constraint.

The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points