Paper Info
Title
The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
Abstract
This paper studies an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=1$ </tex-math></inline-formula> the capacity-achieving input distribution is discrete with finitely many mass points, and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n&gt;1$ </tex-math></inline-formula> the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity-achieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\mathsf {A}^{2})$ </tex-math></inline-formula> upper bound where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {A}$ </tex-math></inline-formula> denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n&gt;1$ </tex-math></inline-formula> , showing that, for each and every dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\ge 1$ </tex-math></inline-formula> , the number of shells that the optimal input distribution contains is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\mathsf {A}^{2})$ </tex-math></inline-formula> . Finally, the third main result of this paper reconsiders the case <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=1$ </tex-math></inline-formula> with an additional average power constraint, demonstrating a similar <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\mathsf {A}^{2})$ </tex-math></inline-formula> bound.
Year
DOI
Venue
2020
10.1109/TIT.2019.2948636
IEEE Transactions on Information Theory
Keywords
DocType
Volume
Upper bound,Entropy,Probability density function,Additives,Random variables,Oscillators,Gaussian noise
Journal
66
Issue
ISSN
Citations 
4
0018-9448
1
PageRank 
References 
Authors
0.36
0
4
Name
Order
Citations
PageRank
Alex Dytso14520.03
Semih Yagli211.37
H. V. Poor3254111951.66
S. Shamai46400669.53