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,,n, mobile user i transmits radio signal to antenna i. Mobile user i transmits at power Pi. The radio channel attenuate the signal and user i’s signal is received at antenna i with power GiiPi, where Gii denote the channel gain. The radio signals also interfere each other. At antenna i, the interference due to user j has power GijPj. The receiver noise power at antenna i is denoted by ni. The signal to interference plus noise at receiver i is


To guarantee the quality of received signal, it is required that the signal to interference plus noise ratio Γi is equal to a predefined constant γi for all i. Given γi, i=1,,n, we want to find P1,,Pn such that the above equation holds for i=1,,n. Let A be the n×n matrix with zero diagonal and (i,j)-entry (Gijγi)/Gii for ij. We want to solve


where 𝐩=(P1,,Pn)T is the power vector and 𝐧=(niγi/Gii)i=1n. The matrix I-A is a Z-matrix, since all Gij and γi are positive constants. The required power vector is (I-A)-1𝐧 if I-A is invertiblePlanetmathPlanetmathPlanetmath. We also required that the componentsPlanetmathPlanetmathPlanetmath of 𝐩 to be positive as power cannot be negative. The resulting power vector (I-A)-1𝐧 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