The Capacity of the Amplitude-Constrained Vector Gaussian Channel

Favano, Antonino; Ferrari, Marco; Magarini, Maurizio; Barletta, Luca

doi:10.1109/isit45174.2021.9518071

Cited by 10 publications

(6 citation statements)

References 14 publications

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“…where N ∼ N (0 n , σ 2 I n ). The first problem seeks to characterize the capacity of the point-topoint channel under an amplitude constraint, and the second problem seeks to find the largest minimum mean squared error under the assumption that the signal has bounded amplitude; the interested reader is referred to [27][28][29] for a detailed background on both problems. Similarly to the wiretap channel, we can define the low-amplitude regime for both problems as the largest R such that P X R is optimal and denote these by Rptp n (σ 2 ) and RMMSE n (σ 2 ).…”

Section: Connections To Other Optimization Problemsmentioning

confidence: 99%

“…2 )/ √ n versus n for σ 2 1 = 1 and σ 2 2 = 1.001, 1.5, 10, 1000. In red, we show c(1, σ 2 2 ) defined in (46).…”

Section: Characterizing the Low-amplitude Regimementioning

confidence: 99%

“…We check the optimality of P X by verifying whether the KKT conditions in Lemma 2 are satisfied. Since the algorithm has to verify the KKT conditions numerically, i.e., with finite precision, we find it more convenient to check the negated version of (28), where a tolerance parameter ε is introduced that trades off accuracy with computational burden. Specifically, P X is not an optimal input pmf if any of the following conditions are satisfied: Ξ(t; P X ) − I s ( X ; P X ) > ε, for some t ∈ supp( P X ) (67a)…”

Section: Numerical Algorithmmentioning

confidence: 99%

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Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution

Favano

Barletta

Dytso

2023

Entropy

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This paper studies the secrecy capacity of an n-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint R¯n, such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value of R¯n as n goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy capacity is also characterized in a form amenable to computation. Several numerical examples are provided, such as the example of the secrecy-capacity-achieving distribution beyond the low-amplitude regime. Furthermore, for the scalar case (n=1), we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order of R2σ12, where σ12 is the variance of the Gaussian noise over the legitimate channel.

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Section: Connections To Other Optimization Problemsmentioning

confidence: 99%

“…2 )/ √ n versus n for σ 2 1 = 1 and σ 2 2 = 1.001, 1.5, 10, 1000. In red, we show c(1, σ 2 2 ) defined in (46).…”

Section: Characterizing the Low-amplitude Regimementioning

confidence: 99%

Section: Numerical Algorithmmentioning

confidence: 99%

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Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution

Favano

Barletta

Dytso

2023

Entropy

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“…The authors of [15] and [16] derive capacity bounds for multiple-input multiple-output (MIMO) systems with identity channel matrix and subject to a peak amplitude constraint that limits the norm of the input vector. In [17] for the same MIMO systems and constraint, we provide further insights into the capacity-achieving input distribution and define an iterative procedure to numerically evaluate an arbitrarily accurate estimate of the channel capacity. As for systems characterized by nonidentity channel matrices, the authors of [18] investigate the capacity of 2 × 2 MIMO systems for rectangular input constraint regions.…”

Section: Introductionmentioning

confidence: 99%

A Sphere Packing Bound for Vector Gaussian Fading Channels Under Peak Amplitude Constraints

Favano

Ferrari

Magarini

et al. 2023

IEEE Trans. Inform. Theory

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An upper bound on the capacity of multiple-input multiple-output (MIMO) Gaussian fading channels is derived under peak amplitude constraints. The upper bound is obtained borrowing concepts from convex geometry and it extends to MIMO channels notable results from the geometric analysis on the capacity of scalar Gaussian channels. Relying on a sphere packing argument and on the renowned Steiner's formula, the proposed upper bound depends on the intrinsic volumes of the constraint region, i.e., functionals defining a measure of the geometric features of a convex body. The tightness of the bound is investigated at high signal-to-noise ratio (SNR) for any arbitrary convex amplitude constraint region, for any channel matrix realization, and any dimension of the MIMO system. In addition, two variants of the upper bound are proposed: one is useful to ensure the feasibility in the evaluation of the bound and the other to improve the bound's performance in the low SNR regime. Finally, the upper bound is specialized for two practical transmitter configurations, either employing a single power amplifier for all transmitting antennas or a power amplifier for each antenna.

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“…Furthermore, they improve on McKellips' result and define a more accurate upper bound, which they refer to as refined upper bound. In [9], the present authors define a numerical algorithm to evaluate an arbitrarily precise estimate of the channel capacity and of its capacity-achieving distribution.…”

Section: Introductionmentioning

confidence: 99%

The Capacity of Fading Vector Gaussian Channels Under Amplitude Constraints on Antenna Subsets

Favano¹,

Ferrari²,

Magarini³

et al. 2022

Preprint

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Upper bounds on the capacity of vector Gaussian channels affected by fading are derived under peak amplitude constraints at the input. The focus is on constraint regions that can be decomposed in a Cartesian product of sub-regions. This constraint models a transmitter configuration employing a number of power amplifiers less than or equal to the total number of transmitting antennas. In general, the power amplifiers feed distinct subsets of the transmitting antennas and partition the input in independent subspaces. Two upper bounds are derived: The first one is suitable for high signal-to-noise ratio (SNR) values and, as we prove, it is tight in this regime; The second upper bound is accurate at low SNR. Furthermore, the derived upper bounds are applied to the relevant case of amplitude constraints induced by employing a distinct power amplifier for each transmitting antenna.

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The Capacity of the Amplitude-Constrained Vector Gaussian Channel

Cited by 10 publications

References 14 publications

Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution

Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution

A Sphere Packing Bound for Vector Gaussian Fading Channels Under Peak Amplitude Constraints

The Capacity of Fading Vector Gaussian Channels Under Amplitude Constraints on Antenna Subsets

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