# 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

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

where $\mathbf{p}={({P}_{1},\mathrm{\dots},{P}_{n})}^{T}$ 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 |
---|---|

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 |