Spatial Degrees of Freedom and Channel Strength for Antenna Systems

Abstract

The number of spatial degrees of freedom (NDoF) and channel strength in antenna systems are examined within a geometric framework. Starting from a correlation-operator representation of the channel between transmitter and receiver regions, we analyze the associated eigenspectrum and relate the NDoF to its spectral transition (corner). We compare the spectrum-based effective NDoF and effective rank metrics, clarifying their behavior for both idealized and realistic eigenvalue distributions. In parallel, we develop geometry-based asymptotic estimates in terms of mutual shadow (view) measures and coupling strength. Specifically, we show that while the projected length or area predicts the number of usable modes in two- and three-dimensional settings, the coupling strength determines the average eigenvalue level. Canonical configurations of parallel lines and regions are used to derive closed-form asymptotic expressions for the effective NDoF, revealing significant deviations from the spectral corner in closely spaced configurations. The results illustrate that these are physically grounded. The proposed theory and techniques are computationally efficient and form a toolbox for estimating the modal richness in near-field channels, with implications for array design, inverse problems, and high-capacity communication systems.

Figures (9)

• Figure 1: Eigenspectra for the channels between a transmitting disc region $\varOmega_\mathrm{T}$ with radius $r\in\{1,8\}a$ and receiving disc region $\varOmega_\mathrm{R}$ with radius $a=\{10,50\}\lambda$ separated by a distance $d=a$. The modes up to and after the corner are termed here "propagating" and "reactive", respectively.
• Figure 2: NDoF estimates $N_\mathrm{a},N_\mathrm{e}$, and $N_\mathrm{r}$ for spherical shells and balls versus the electrical size $ka$. Insets depict the corresponding eigenvalues $\zeta_n$ at $ka\in\{3,100\}$ with markers indicating the estimated NDoF.
• Figure 3: NDoF estimates $N_\mathrm{a},N_\mathrm{e},N_\mathrm{r},N_\mathrm{c},N_{1/2}$ for the channels between two discs (solid curves) in $\mathbb{R}{}^3$ and two lines (dashed curves) in $\mathbb{R}{}^2$.
• Figure 4: NDoFs per wavelength for the channel between two lines with length $\ell$ and separation distance $d$. The shadow-length estimate \ref{['eq:ShadowLength2lines']} is shown by the solid curve, the asymptotic effective NDoF in 3D and 2D from \ref{['eq:NeNDoF2lines']} and \ref{['eq:eDoF2lines2D']}, respectively, by dashed curves, and the paraxial approximation by a dash-dotted curve. Analytical results are compared with numerical estimates of the effective NDoF \ref{['eq:eNDoF']}, effective rank \ref{['eq:rNDoF']}, and corner position for wavelengths $\lambda\in\{1,2,4\}0.01\ell$, indicated by small to large markers.
• Figure 5: Normalized eigenspectra for two lines with length $\ell$ separated a distance $d$ at wavelength $\lambda/\ell=10^{-3}$, cf., inset in Fig. \ref{['fig:eDoF2lines']}. NDoFs $N_\mathrm{e}$ and $N_\mathrm{r}$ are indicated by the markers and $N_\mathrm{a}$ is evaluated from \ref{['eq:ShadowLength2lines']}.