On the dimensionality of spatial fields with restricted angle of arrival

Wireless communication occurs through continuous fields, over regions of space. Although the communication channel may be modelled in terms of infinite dimensional vector spaces, it is paramount to develop finite dimensional approximations for this channel. We extend previous work which examined the fundamental finite dimensionality of fields over a finite region of space to incorporate restricted angles of signal arrival. By considering a subspace of fields where direction of arrival of the field components, or source spatial distribution, is restricted we show the received field dimensionality is linearly related to the product of the radius of the region of interest and the angular restriction. This result provides a rigourous foundation for analysing the performance and capacity of MIMO systems in non isotropic environments. The proof is presented for the two dimensional case. It is apparent that this work can be extended to the case of three dimensions for a similar result