In this paper, we shall show that for a nT-element base station (BS) communicating with M(≤ nT) single-element mobile stations (MS) (or multi-user MISO) orthogonally in the spatial domain, the optimization problem is equivalent to the least squares (LS) problem for an underdetermined linear system. We then prove that the optimal BS antenna weights can be expressed as the pseudo-inverse of the multi-user channel matrix. This solution decomposes the multi-user system into many single-user systems with maximal resultant channel responses. The average of the squared channel response (defined as channel gain) and the inverse of the normalized variance of the squared channel response (defined as diversity order) are derived for performance analysis. It is found that every individual user of the resulting system behaves like a single-user system with nT-M+1 reception diversity. Finally, by applying the solution on a multi-user MIMO antenna system (i.e., with multiple antennas at the MS as well), an iterative approach is proposed to perform multi-user orthogonal space division multiplexing (OSDM) in the downlink.
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