TY - JOUR

T1 - The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel

T2 - An Upper Bound on the Number of Mass Points

AU - Dytso, Alex

AU - Yagli, Semih

AU - Poor, H. Vincent

AU - Shamai, Shlomo

N1 - Funding Information:
Manuscript received February 13, 2019; revised August 27, 2019; accepted September 30, 2019. Date of publication October 21, 2019; date of current version March 17, 2020. The work of A. Dytso, S. Yagli, and H. V. Poor was supported in part by the U.S. National Science Foundation under Grant CCF-1908308 and in part by the United States–Israel Binational Science Foundation under Grant BSF-2018710. The work of S. Shamai was supported in part by the United States–Israel Binational Science Foundation under Grant BSF-2018710 and in part by the European Union’s Horizon 2020 Research and Innovation Programme under Grant 694630. This article was presented in part at the 2019 IEEE International Symposium on Information Theory.
Publisher Copyright:
© 2019 IEEE.

PY - 2020/4

Y1 - 2020/4

N2 - This paper studies an n-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when n = 1 the capacity-achieving input distribution is discrete with finitely many mass points, and when n > 1 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 O(A2) upper bound where A denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when n > 1, showing that, for each and every dimension n ≥ 1, the number of shells that the optimal input distribution contains is O(A2). Finally, the third main result of this paper reconsiders the case n = 1 with an additional average power constraint, demonstrating a similar O(A2) bound.

AB - This paper studies an n-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when n = 1 the capacity-achieving input distribution is discrete with finitely many mass points, and when n > 1 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 O(A2) upper bound where A denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when n > 1, showing that, for each and every dimension n ≥ 1, the number of shells that the optimal input distribution contains is O(A2). Finally, the third main result of this paper reconsiders the case n = 1 with an additional average power constraint, demonstrating a similar O(A2) bound.

KW - Amplitude constraint

KW - additive vector Gaussian noise channel

KW - capacity

KW - discrete distributions

KW - power constraint

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U2 - 10.1109/TIT.2019.2948636

DO - 10.1109/TIT.2019.2948636

M3 - Article

AN - SCOPUS:85082172622

SN - 0018-9448

VL - 66

SP - 2006

EP - 2022

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 4

M1 - 8878162

ER -