Fundamental dependencies in angle-delay-Doppler in wireless channels

The degrees of freedom (DoF) in a wireless channel determine its statistics and capacity. The DoF in turn depend on the number and spatial distribution of physical propagation paths in the environment as well as the signal space dimensions (number of antennas, bandwidth). In this paper, we use a virtual representation of time- and frequency-selective MIMO channels to relate the DoF to the physical propagation paths. The virtual representation is characterized by virtual channel coefficients in angle-delay-Doppler space. It induces a partitioning of paths that reveals the contribution of each path to the DoF as well as fundamental dependencies in angle, delay and Doppler that constrain the DoF. As the signal space dimensions increase, the DoF increase whereas the statistics of the virtual channel coefficients deviate from Gaussian (Rayleigh) to "specular" in which the amplitude of the coefficients is deterministic and the channel variations are manifested solely in the phase. Implications of the angle-delay-Doppler dependencies on channel structure and capacity are discussed, which are particularly relevant to ultra-wideband systems.