A physics-based perspective for understanding and utilizing spatial resources of wireless channels

Abstract

To satisfy the increasing demands for transmission rates of wireless communications, it is necessary to use spatial resources of electromagnetic (EM) waves. In this context, EM information theory (EIT) has become a hot topic by integrating the theoretical framework of deterministic mathematics and stochastic statistics to explore the transmission mechanisms of continuous EM waves. However, the previous studies were primarily focused on frame analysis, with limited exploration of practical applications and a comprehensive understanding of its essential physical characteristics. In this paper, we present a three-dimensional (3-D) line-of-sight channel capacity formula that captures the vector EM physics and accommodates both near- and far-field scenes. Based on the rigorous mathematical equation and the physical mechanism of fast multipole expansion, a channel model is established, and the finite angular spectral bandwidth feature of scattered waves is revealed. To adapt to the feature of the channel, an optimization problem is formulated for determining the mode currents on the transmitter, aiming to obtain the optimal design of the precoder and combiner. We make comprehensive analyses to investigate the relationship among the spatial degree of freedom, noise, and transmitted power, thereby establishing a rigorous upper bound of channel capacity. A series of simulations are conducted to validate the theoretical model and numerical method. This work offers a novel perspective and methodology for understanding and leveraging EIT, and provides a theoretical foundation for the design and optimization of future wireless communications.

Figures (8)

• Figure 1: Three EM channel analysis models for understanding the mechanism of wireless information transmission. (a) Physical representation of the EM channel. (b) Geometric interpretation of the transmission based on KIT. The black curves on the left represent the transmitted signals, while the blue curves on the right denote the received signals distorted by noises. (c) Block diagram of the system structure for illustrating the mechanism of the spatial information transmission.
• Figure 2: Diagrams for understanding \ref{['eq4']} and \ref{['eq6']}, and the propagation operator. (a) Block diagram for illustrating the propagation operator. (b) Geometric coordinates for depicting the additional theorem expansion of SGF, namely, the equation \ref{['eq4']}. (c) The classic 4-f system as a physical mapping of the propagation operator.
• Figure 3: Illustration for understanding the finite-bandwidth property of the translator and \ref{['eq4']}. (a) The normalized translators with and without the window function. The aperture is $10\lambda$ and the distance between the transmitter and the receiver is $20\lambda$. The source position is $\mathbf{s}=(-5,\,1,\,1)\lambda$, the field is $\mathbf{r}=(-3.5,\,5,\,20)\lambda$, and their centers are $(0,\,0,\,0)\lambda$ and $(0,\,0,\,20)\lambda$, respectively. (b) The relative error as a function of the sampled angular width $\theta_e$. $G_a$ represents the integral on the conical surface $S_{\theta_e}$ defined by \ref{['eq4']}, and $G$ is the value calculated by $e^{-jk|\mathbf{r}-\mathbf{s}|}/(4\pi|\mathbf{r}-\mathbf{s}|)$, which is accurate.
• Figure 4: Illustration for understanding the relationship between the angular bandwidth and the distance between the transmitter and receiver. The configurations of the source and field regions are the same, we just change the distance.
• Figure 5: Illustration of the relative magnitudes of the eigenvalues $\beta_n$ with respect to the mode index. The abscissa of the vertical red dashed line is the value of the spatial DoF $N$ calculated by \ref{['eq20']}, and the blue “+” marker represents the eigenvalues. The red solid line represents a piecewise function used for approximating the calculated eigenvalues.