Classification of solutions to the minimum energy problem in one dimensional sensor networks

Abstract

We classify solutions of the minimum energy problem in one dimensional wireless sensor networks for the data transmission cost matrix which is a power function of the distance between transmitter and receiver with any real exponent. We show, how these solutions can be utilized to solve the minimum energy problem for the data transmission cost matrix which is a linear combination of two power functions. We define the minimum energy problem in terms of the sensors signal power, transmission time and capacities of transmission channels. We prove, that for the point-to-point data transmission method utilized by the sensors in the physical layer, when the transmitter adjust the power of its radio signal to the distance to the receiver, the optimal transmission is without interference. We also show, that the solutions of the minimum energy problem written in terms of data transmission cost matrix and in terms of the sensor signal power coincide.