On the capacity of volume limited current distributions

The capacity of MIMO channel is related to the channel statistics and this in turn related to the choice of antennas and the channel propagation environment. Recently, there has been some work on the evaluation of antenna independent spatial capacity. This analysis considers a multiple-antenna system in which unlimited number of antenna array elements are available, but there is a restriction on the volume they can occupy. The problem is viewed in electro-magnetic (EM) theory setting where a spatially continuous current distribution radiates into free space, with a receiver in the surrounding space measuring the radiated EM field. The advantage the aforementioned approach is that it provides an antenna independent analysis and this forms and upper bound on the achievable capacity with finite number of antennas. However, in computing the capacity, simplifying assumptions that reduce the vectorial EM problem to a scalar wave problem is made and heuristic arguments are used to extrapolate the results to the vectorial case. The impact of this on the results is not readily apparent. In this work, we consider the complete vector EM problem for a system comprising of a transmitter restricted to spherical volume and a receiver, which is a concentric spherical surface in the far-field, and derive some bounds on capacity. We first develop some tools essential for calculation the capacity of the system and then we compute the singular values of this channel in closed form. The main results of this paper are as follows: 1) the capacity scaling law in the high signal-to-noise ratio (SNR) regime is given by (c₂ + c₁ log SNR) log SNR + 0(log SNR), where c₁ is linear and c₂ quadratic in the radius of the transmitting volume; and 2) the received power scales as a cubic function of the radius of the transmitting spherical volume