The macrodiversity multi-user
MIMO uplink communication system considered here consists of \scriptstyle N distributed single antenna users and \scriptstyle n_{R} distributed single antenna base stations (BS). Following the well established narrow band flat fading
MIMO system model, input-output relationship can be given as :\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n} where \scriptstyle\mathbf{y} and \scriptstyle\mathbf{x} are the receive and transmit vectors, respectively, and \scriptstyle\mathbf{H} and \scriptstyle\mathbf{n} are the macrodiversity channel matrix and the spatially uncorrelated
AWGN noise vector, respectively. The power spectral density of
AWGN noise is assumed to be \scriptstyle N_0. The \scriptstyle i,jth element of \scriptstyle\mathbf{H}, h_{ij} represents the fading coefficient (see
Fading) of the \scriptstyle i,jth constituent link which in this particular case, is the link between \scriptstyle jth user and the \scriptstyle ith base station. In macrodiversity scenario, :E \left \{ \left| h_{ij} \right |^2 \right \} = g_{ij} \quad \forall i,j, where \scriptstyle g_{i,j} is called the average link gain giving average link
SNR of \scriptstyle \frac{g_{ij}}{N_0}. The macrodiversity power profile matrix can thus be defined as : \mathbf{G} = \begin{pmatrix} g_{11} & \dots & g_{1N} \\ g_{21} & \dots & g_{2N} \\ \dots & \dots & \dots \\ g_{n_R1} & \dots & g_{n_RN} \\ \end{pmatrix}. The original input-output relationship may be rewritten in terms of the macrodiversity power profile and so-called normalized channel matrix, \mathbf{H}_w, as :\mathbf{y} = \left( \left( \mathbf{G}^{\circ\frac{1}{2}} \right) \circ \mathbf{H}_w \right) \mathbf{x} + \mathbf{n}. where \mathbf{G}^{\circ \frac{1}{2}} is the element-wise square root of \mathbf{G}, and the operator, \circ, represents Hadamard multiplication (see
Hadamard product). The \scriptstyle i,jth element of \mathbf{H}_w, h_{w,ij}, satisfies the condition given by :E \left \{ \left| h_{w,ij} \right |^2 \right \} = 1 \quad \forall i,j . It has been shown that there exists a functional link between the
permanent of macrodiversity power profile matrix, \mathbf{G} and the performance of multi-user macrodiversity systems in fading. Although it appears as if the macrodiversity only manifests itself in the power profile, systems that rely on macrodiversity will typically have other types of transmit power constraints (e.g., each element of \mathbf{x} has a limited average power) and different sets of coordinating transmitters/receivers when communicating with different users. Note that the input-output relationship above can be easily extended to the case when each transmitter and/or receiver have multiple antennas. ==See also==