A Concise Tutorial for Analyzing Electromagnetic Degrees of Freedom for Continuous-Aperture Array (CAPA) Systems

A concise tutorial is provided for analysis of the spatial degrees of freedom (DoFs) in continuous-aperture array (CAPA)-based continuous electromagnetic (EM) channels. First, a simplified spatial model is introduced using the Fresnel approximation. By leveraging this model and Landau’s theorem, a closed-form expression for the spatial DoFs is derived. The results show that the number of DoFs is proportional to the transmit and receive aperture sizes and inversely proportional to the propagation distance. Numerical results are presented to illustrate the properties of EM DoFs in CAPA-based channels.