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 $i=1,\mathrm{\dots },n$, 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

$${\mathrm{\Gamma }}_i=\frac{G_{ii}{P}_i}{{\sum }_{j\ne i}{G}_{ij}{P}_j+n_i}$$

To guarantee the quality of received signal, it is required that the signal to interference plus noise ratio ${\mathrm{\Gamma }}_i$ is equal to a predefined constant ${\gamma }_i$ for all $i$. Given ${\gamma }_i$, $i=1,\mathrm{\dots },n$, we want to find $P_1,\mathrm{\dots },P_n$ such that the above equation holds for $i=1,\mathrm{\dots },n$. Let $A$ be the $n\times n$ matrix with zero diagonal and $(i,j)$-entry $(G_{ij}{\gamma }_i)/G_{ii}$ for $i\ne j$. We want to solve

$$(I-A)𝐩=𝐧$$

where $𝐩={(P_1,\mathrm{\dots },P_n)}^T$ is the power vector and $𝐧={(n_i{\gamma }_i/G_{ii})}_{i=1}^n$. The matrix $I-A$ is a Z-matrix, since all $G_{ij}$ and ${\gamma }_i$ are positive constants. The required power vector is ${(I-A)}^{-1}𝐧$ if $I-A$ is invertible. We also required that the components of $𝐩$ to be positive as power cannot be negative. The resulting power vector ${(I-A)}^{-1}𝐧$ 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
Canonical name	AnApplicationOfZmatrixInAMobileRadioSystem
Date of creation	2013-03-22 16:14:16
Last modified on	2013-03-22 16:14:16
Owner	kshum (5987)
Last modified by	kshum (5987)
Numerical id	6
Author	kshum (5987)
Entry type	Application
Classification	msc 15A99