On the Product-Determinant-Sum of Central Wishart Matrices and Its Application to Wireless Networks

Abstract-Inspired by multiaccess in dense cellular systems, the paper first considers minimizing the determinant of the sum of a subset of n i.i.d. central Wishart matrices, for any given size. When n goes to infinity, the exact scaling for more general case, selecting a subset to minimize the product-determinant-sum of n i.i.d. central Wishart matrix vectors, is provided. Specifically, suppose each vector has K Wishart matrices of format GG with G of dimension N
_r
× N
_t
. Then for any subset of size n
^α
, the K determinants, each being the sum over one of the vector elements, will have a product no less than exp{K N
_r
(1+ 1/K N
_r
N
_t
)(α - α*) log n}, for all a between α* := 1/(1 + K N
_r
N
_t
) and 1. The paper then applies the results to study multiaccess with cross-cell collaborations in dense environment. When each cell allows multi-users to be on and decodes by treating signals from neighboring cells as noise, the maximum throughput is characterized and achieved by selecting users based on their channels. Specifically, if every cell allows same number of nodes to be on and decodes by successive interference cancellation for in-cell nodes, the maximum throughput is N
_r
/1 + K N
_r
N
_t
log n bit/s/Hz/cell when n goes to infinity, where n,K, N
_r
, N
_t
are numbers of nodes per cell, neighbors per cell, receive antennas per user and transmit antennas per base station, respectively. It is also shown that time-sharing across cells achieves higher throughput, determined by the chromatic number of the interference graph.