Degrees of Freedom of Spatial Multiplexing in Distance Domain of Arbitrary Continuous-Aperture Array in Near-Field Region

Abstract: Extremely large aperture array operating in the near-field regime unlocks additional spatial resources that can be exploited to simultaneously serve multiple users even when they share the same angular direction, a capability not achievable in conventional far-field systems. A fundamental question, however, remains: What is the maximum spatial degree of freedom (DoF) of spatial multiplexing in the distance domain? In this paper, we address this open problem by investigating the spatial DoF of a line-of-sight (LoS) channel between a large two-dimensional transmit aperture and a linear receive array with collinearly-aligned elements (i.e., at the same angular direction) but located at different distances from the transmit aperture. We assume that both the aperture and linear array are continuous-aperture (CAP) arrays with an infinite number of elements and infinitesimal spacing, which establishes an upper bound for the spatial degrees of freedom (DoF) in the case of finite elements. First, we assume an ideal case where the transmit array is a single piece and the linear array is on the broad side of the transmit array. By reformulating the channel as an integral operator with a Hermitian convolution kernel, we derive a closed-form expression for the spatial DoF via the Fourier transform. Our analysis shows that the spatial DoF in the distance domain is predominantly determined by the extreme boundaries of the array rather than its detailed interior structure. We further extend the framework to non-broadside configurations by employing a projection method, which effectively converts the spatial DoF to an equivalent broadside case. Finally, we extend our analytical framework to the modular array, which shows the spatial DoF gain over the single-piece array given the constraint of the physical length of the array.