On the capacity of vector Gaussian channels with bounded inputs

Abstract

Abstract-The capacity of a deterministic multiple-input multiple-output (MIMO) channel under the peak and average power constraints is investigated. For the identity channel matrix, the approach of Shamai et al. is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function. This approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi-dimensional scenarios. It is proved that the support of the capacity-achieving distribution is a finite set of hyper-spheres with mutual independent phases and amplitude in the spherical domain. Subsequently, it is shown that when the average power constraint is relaxed, if the number of antennas is large enough, the capacity has a closed form solution and constant amplitude signaling at the peak power achieves it. Moreover, it will be observed that in a discrete-time memoryless Gaussian channel, the average power constrained capacity, which results from a Gaussian input distribution, can be closely obtained by an input where the support of its magnitude is a discrete finite set. Finally, we investigate some upper and lower bounds for the capacity of the non-identity channel matrix and evaluate their performance as a function of the condition number of the channel.

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

“…Remark 1. Observe that the upper and lower bounds in (4) are not of equal order. We conjecture that the order of the lower bound is tight.…”

“…Given open intervals I 1 and I 2 , let p : I 1 × I 2 → R be a strictly Polyá type-∞ function. 4 For an arbitrary y, suppose p(•, y) : I 1 → R is an n-times differentiable function. Assume that µ is a measure on I 2 , and let ξ : I 2 → R be a function with S (ξ) = n. For x ∈ I 1 , define Ξ(x) = ξ(y)p(x, y)dµ(y).…”

“…As a matter of fact, this phenomena that the optimal input distribution lies on finitely many concentric spheres remains true for any n ≥ 2, cf. [3], [4] and [5].…”

Optimal Signaling Schemes and Capacities of Non-Coherent Correlated MISO Channels Under Per-Antenna Power Constraints

“…For example, the author in [15] studies the optimal inputs and capacity of multi-antenna channels under peak power constraints via lower and upper bounds on the capacity. The work in [27] addresses the discreteness of the optimal input and capacity of multi-antenna channels under both peak and power constraints.…”

On the Capacity of Multiple-Antenna Systems and Parallel Gaussian Channels With Amplitude-Limited Inputs

“…Also, the authors in [6] show (through a different approach) that the capacity-achieving distribution is discrete. The authors in [11] study the capacity of Gaussian vector channel under peak and average power constraints and extend the results of [5] to show that the capacity achieving distribution has a finite number of mass points for its amplitude and the points are uniformly distributed on the hyper-spheres determined by the amplitude mass points. They also, via relaxing the constraints on the problem, derive bounds on the capacity of the channel.…”