Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity †
Abstract
:1. Introduction
1.1. Contributions and Paper Organization
1.2. Notation
2. Problem Statement
- (i)
- per-antenna amplitude constraints, i.e., for some given ; and
- (ii)
- -dimensional amplitude constraint, i.e., for some given .
3. Properties of an Optimal Input Distribution
3.1. Necessary and Sufficient Conditions for Optimality
- is unique and symmetric if is left invertible [18].
3.2. General Structure of Capacity-Achieving Input Distributions
- Towards a contradiction, it is assumed that the set of points of increase is infinite.
- The assumption in Step 1 is then used to establish a certain property of the function on the input space . For example, by showing that has an analytic continuation to . Then, by means of the Identity Theorem of complex analysis and the Bolzano–Weierstrass Theorem [25], Smith was able to show that must be constant.
- By using either the Fourier or Laplace transform of together with the property of established in Step 2, a new a property of the channel output distribution is established. For example, Smith was able to show that must be constant.
- A conclusion out of Step 3 is used to reach a contradiction. The contradiction implies that must be finite. For example, to reach a contradiction, Smith was using the fact that the channel output distribution results from a convolution with a Gaussian probability density, which cannot be constant.
- (i)
- is an open set.
- (ii)
- is a set of positive Lebesgue measure.
- is unique.
- is symmetric.
- is discrete with the number of mass points being of the order .
- contains probability mass points at .
3.3. Properties of Capacity-Achieving Input Distributions in the Small (But Not Vanishing) Amplitude Regime
4. Upper and Lower Bounds on the Capacity
4.1. Upper Bounds
4.2. Lower Bounds
5. Invertible Channel Matrices
5.1. Diagonal Channel Matrices
5.2. Gap to the Capacity
6. Arbitrary Channel Matrices
7. The SISO Case
7.1. Upper and Lower Bounds
7.2. High and Low Amplitude Asymptotics
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Dytso, A.; Goldenbaum, M.; Poor, H.V.; Shamai, S. Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity. Entropy 2019, 21, 200. https://doi.org/10.3390/e21020200
Dytso A, Goldenbaum M, Poor HV, Shamai S. Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity. Entropy. 2019; 21(2):200. https://doi.org/10.3390/e21020200
Chicago/Turabian StyleDytso, Alex, Mario Goldenbaum, H. Vincent Poor, and Shlomo Shamai (Shitz). 2019. "Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity" Entropy 21, no. 2: 200. https://doi.org/10.3390/e21020200
APA StyleDytso, A., Goldenbaum, M., Poor, H. V., & Shamai, S. (2019). Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity. Entropy, 21(2), 200. https://doi.org/10.3390/e21020200