When Does UPW Model Become Invalid for XL-MIMO With Directional Array Elements?

For multi-antenna wireless communication systems, uniform plane wave (UPW) model is justified when the link distance is much larger than the array aperture, so that the array elements are assumed to be isotropic when modelling the array response vector. This letter attempts to answer the following question: When does UPW model become invalid for extremely large-scale multiple-input multiple-output (XL-MIMO) communications with directional array elements? By taking into account both the phase and amplitude variations, the normalized difference between the generic near-field non-uniform spherical wave (NUSW) model and the conventional far-field UPW model is introduced. Based on this, we define the generalized far-field region and obtain closed-form expressions of its boundary, which can be treated as an upper-bound to separate the near-field and far-field in general. Simulation and numerical results are provided to verify our theoretical analysis.