an application of Z-matrix in a mobile radio system

The following is an application of Z-matrix in wireless communication called power balancing problem.

Consider $n$ pairs of mobile users and receiving antennae. For $i=1,\ldots,n$, mobile user $i$ transmits radio signal to antenna $i$. Mobile user $i$ transmits at power $P_{i}$. The radio channel attenuate the signal and user $i$’s signal is received at antenna $i$ with power $G_{ii}P_{i}$, where $G_{ii}$ denote the channel gain. The radio signals also interfere each other. At antenna $i$, the interference due to user $j$ has power $G_{ij}P_{j}$. The receiver noise power at antenna $i$ is denoted by $n_{i}$. The signal to interference plus noise at receiver $i$ is

 $\Gamma_{i}=\frac{G_{ii}P_{i}}{\sum_{j\neq i}G_{ij}P_{j}+n_{i}}$

To guarantee the quality of received signal, it is required that the signal to interference plus noise ratio $\Gamma_{i}$ is equal to a predefined constant $\gamma_{i}$ for all $i$. Given $\gamma_{i}$, $i=1,\ldots,n$, we want to find $P_{1},\ldots,P_{n}$ such that the above equation holds for $i=1,\ldots,n$. Let $A$ be the $n\times n$ matrix with zero diagonal and $(i,j)$-entry $(G_{ij}\gamma_{i})/G_{ii}$ for $i\neq j$. We want to solve

 $(I-A)\mathbf{p}=\mathbf{n}$

where $\mathbf{p}=(P_{1},\ldots,P_{n})^{T}$ is the power vector and $\mathbf{n}=(n_{i}\gamma_{i}/G_{ii})_{i=1}^{n}$. The matrix $I-A$ is a Z-matrix, since all $G_{ij}$ and $\gamma_{i}$ are positive constants. The required power vector is $(I-A)^{-1}\mathbf{n}$ if $I-A$ is invertible. We also required that the components of $\mathbf{p}$ to be positive as power cannot be negative. The resulting power vector $(I-A)^{-1}\mathbf{n}$ has positive components if $(I-A)^{-1}$ is a non-negative matrix. In such case, $I-A$ is an M-matrix.

Title an application of Z-matrix in a mobile radio system AnApplicationOfZmatrixInAMobileRadioSystem 2013-03-22 16:14:16 2013-03-22 16:14:16 kshum (5987) kshum (5987) 6 kshum (5987) Application msc 15A99