On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective

Dytso, Alex; Al, Mert; Poor, H. Vincent; Shamai, Shlomo

doi:10.1109/tit.2018.2890208

Cited by 34 publications

(42 citation statements)

References 33 publications

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“…The first 1 The formal proof is incomplete as the argument in [8] uses a conjecture which presumes that the number of points in the optimal distribution increases by 1 as the amplitude constraint is relaxed. Proof of this fact was later shown to be true in [9]. approach uses the maximum entropy principle [11,Chapter 12] and upper bounds the output differential entropy, h(Y ), subject to some moment constraint [12].…”

Section: A Prior Workmentioning

confidence: 99%

“…A suboptimal choice of an output distribution in the dual capacity expression results in an upper bound on the capacity [15], [16], [17], [18]. The third approach uses a characterization of the mutual information as an integral of the minimum mean square error (MMSE) [19], and leads to an upper bound by replacing the optimal estimator in the MMSE term by a suboptimal one [9]. As for the lower bounds on the capacity, the first one relevant to our setting, as mentioned above, was proposed by Shanon in [7] which was based on the entropy power inequality.…”

Section: A Prior Workmentioning

confidence: 99%

“…For example, in the context of a Bayesian estimation problem [28], Oscillation Theorem has been used to show the necessary and sufficient conditions for a binary distribution to be the least favorable. In [9], in a vector version of the optimization in (2), Oscillation Theorem has been used to show the necessary and sufficient conditions for a uniform distribution on a single sphere to be optimal.…”

Section: A Prior Workmentioning

confidence: 99%

“…The following two lemmas find upper and lower bounds on absolute value of the complex analytic extension 10 of f ′ Y over a disc of finite radius centered at the origin. 9 In fact, D R can be any open connected set that contains the interval [−R, R]. For example, a rectangle of width 2(R + ǫ) and arbitrary hight 2ǫ is a typical choice.…”

Section: A Bounds On the Number Of Extreme Points Of A Gaussian Convmentioning

confidence: 99%

See 3 more Smart Citations

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

Dytso

Yagli

Poor

et al. 2020

IEEE Trans. Inform. Theory

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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 capacityachieving 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(A 2 ) 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(A 2 ). Finally, the third main result of this paper reconsiders the case n = 1 with an additional average power constraint, demonstrating a similar O(A 2 ) bound.

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Section: A Prior Workmentioning

confidence: 99%

Section: A Prior Workmentioning

confidence: 99%

Section: A Prior Workmentioning

confidence: 99%

Section: A Bounds On the Number Of Extreme Points Of A Gaussian Convmentioning

confidence: 99%

See 2 more Smart Citations

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

Dytso

Yagli

Poor

et al. 2020

IEEE Trans. Inform. Theory

Self Cite

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show abstract

“…Sharma and Shamai [6] extend the result of Smith, and argue that an equiprobable input on {±A} is optimal if and only if A ≤Ā ≈ 1.665. The proof of the result in [6], which generalizes to vector channels, is shown in [7].…”

Section: Introductionmentioning

confidence: 96%

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

Yagli

Dytso

Poor

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

Self Cite

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This paper studies an n-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, neither the exact number of shells of the optimal input distribution nor a bound on it was available. This paper provides an alternative proof of the finiteness of the number shells of the capacity-achieving input distribution and produces the first firm upper bound on the number of shells, paving an alternative way for approaching many such problems. In particular, for every dimension n, it is shown that the number of shells is given by O(A 2) where A is the constraint on the input amplitude. Moreover, this paper also provides bounds on the number of points for the case of n = 1 with an additional power constraint.

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One-shot achievability and converse bounds of Gaussian random coding in AWGN channels under covert constraint

Wei

Luo

2019

2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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It is well known that finite blocklength plays an important role in the analysis of practical communication. This paper considers the achievability and converse bounds on the maximal channel coding rate (throughput) at a given blocklength and error probability in covert communication over AWGN channels. The covert constraint is in the form of an upper bound of total variation distance (TVD). For the achievability, Gaussian random coding scheme is adopted for convenience in the analysis of TVD. The classical results of finite blocklength regime are not applicable in this case. By revisiting [18], we first present new and more general achievability bounds for random coding schemes under maximal or average probability of error requirements. The general bounds are then applied to covert communication in AWGN channels where codewords are generated from Gaussian distribution while meeting the maximal power constraint. We further show an interesting connection between attaining tight achievability and converse bounds and solving two total variation distance related minimax problems. Further comparison is made between the new achievability bound and existing one with deterministic codebooks. The TVD at the adversary is analyzed under the given coding scheme, and the power level induced by covert constraint is bounded by applications of divergence inequalities. Our thorough analysis thus leads us to a comprehensive characterization of the attainable throughput in covert communication over AWGN channels. The summary of the initial part of this paper was published in [29].

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On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective

Cited by 34 publications

References 33 publications

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

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

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

One-shot achievability and converse bounds of Gaussian random coding in AWGN channels under covert constraint

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