A General DoF and Pattern Analyzing Scheme for Electromagnetic Information Theory

TL;DR

This work addresses fundamental DoF limits for electromagnetic information transmission in realistic finite regions by extending the Slepian concentration framework to 3D space and 4D space_time under electromagnetic constraints. It defines functional DoF fDoF_epsilon(H) and channel DoF cDoF_epsilon(H_t,H_r,P), proves a bound that the channel DoF is no larger than the larger of the transmitter or receiver functional DoFs, and derives asymptotic expressions such as N3D = V/(6 pi^2) k0^3 for ball constraints and N3D = V/(6 pi^2) (k0^3 - k1^3) for shell constraints. The paper also provides non_asymptotic DoF schemes and extensive numerical simulations to guide spatial sampling and space_time pattern design for 3D antenna arrays, illustrating that half_wavelength sampling is not always required in narrow_band settings and that Gauss_Legendre spacing can improve DoF efficiency. These results offer practical guidance for EIT based 6G system design, including sampling strategies, orthogonal space_time patterns, and pattern_based information transmission.

Abstract

Electromagnetic information theory (EIT) is one of the emerging topics for 6G communication due to its potential to reveal the performance limit of wireless communication systems. For EIT, one of the most important research directions is degree of freedom (DoF) analysis. Existing research works on DoF analysis for EIT focus on asymptotic conclusions of DoF, which do not well fit the practical wireless communication systems with finite spatial regions and finite frequency bandwidth. In this paper, we use the theoretical analyzing tools from Slepian concentration problem and extend them to three-dimensional space domain and four-dimensional space-time domain under electromagnetic constraints. Then we provide asymptotic DoF conclusions and non-asymptotic DoF analyzing scheme, which suits practical scenarios better, under different scenarios like three-dimensional antenna array. Moreover, we theoretically prove that the channel DoF is upper bounded by the proposed DoF of electromagnetic fields. Finally, we use numerical analysis to provide some insights about the optimal spatial sampling interval of the antenna array, the DoF of three-dimensional antenna array, the impact of unequal antenna spacing, the orthogonal space-time patterns, etc.

Figures (10)

• Figure 1: The eigenvalues of one-dimensional Slepian concentration problem when $\Omega T$ increases.
• Figure 2: The two functional spaces $\mathcal{H}_1$ and $\mathcal{H}_2$ include all fields with space-time constraints and all fields with wavenumber-frequency constraints separately. $\mathcal{H}_1$ and $\mathcal{H}_2$ are subspaces of $\mathcal{H}$. Through projection operators $\Pi_1$ and $\Pi_2$ we can project arbitrary $x \in \mathcal{H}$ onto $\mathcal{H}_1$ and $\mathcal{H}_2$.
• Figure 3: Description of the definition of functional DoF and channel DoF.
• Figure 4: The DoF gain with different frequency band. The three-dimensional antenna array is $0.5\,{\rm m} \times 0.5\,{\rm m} \times 0.5\,{\rm m}$, sampled with $\lambda/2$ antenna spacing. The maximum frequency is $3\,{\rm GHz}$. The thick shell has $2\,{\rm GHz}$ bandwidth. The thin shell has $150\,{\rm MHz}$ bandwidth. The sphere surface in the wavenumber domain corresponds to single frequency point at $3\,{\rm GHz}$.
• Figure 5: The DoF gain with different heights. The three-dimensional antenna array is $0.5\,{\rm m} \times 0.5\,{\rm m} \times 0.05x\,{\rm m}$, sampled with $\lambda/2$ antenna spacing. $x$ is chosen to be $1$, $3$, $7$ and $10$ to represent three-dimensional antenna array with different heights (number of layers).

Theorems & Definitions (13)

• Remark 1
• Theorem 1
• Remark 2
• Definition 1
• Remark 3
• Definition 2
• Remark 4
• Lemma 1
• proof
• Theorem 2