1316 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 6, NOVEMBER 2002 Optimization Methods for Optimal Transmitter Locations in a Mobile Wireless System Fernando Aguado-Agelet, Aurea M. Martínez Varela, Lino J. Alvarez-Vázquez, José M. Hernando, Member, IEEE, and Arno Formella Abstract—A combination of a simple indoor propagation model with different optimization methods enables optimal single and multiple transmitter locations and antenna sectorization in wire- less systems. Index Terms—Antenna sectorization, optimization methods, transmitter location, wireless systems. I. INTRODUCTION O PTIMIZATION algorithms in combination with propaga- tion models allow the development of automatic planning strategies for wireless systems. In [1], a bundle method for opti- mizing a single transmitter location is presented. From such an optimal location, a minimum transmitted power for an isotropic antenna is required for assuring a specific received level in the selected area. In this paper, we extend the planning optimiza- tion presented in [1] to the multiple transmitter problem, where a single antenna is not sufficient to accomplish the coverage re- quirements over all the indoor environment. Simple coverage or mixed coverage–interference cost functions can be consid- ered. Both formulations with a given number of transmitter an- tennas look for the best possible locations where the transmitted powers are minimized. In the mixed cost function, an interfer- ence penalty term is included to minimize the coverage in the overlapping regions, allowing a better channel assignment with a higher frequency reuse. Additionally, an antenna sectorization solution is presented. Given a fixed antenna location, we cannot change to the optimal location to minimize the transmitted power. Nevertheless, we propose an alternative optimization strategy for the single an- tenna case, where the azimuth radiation area of 360 around the transmitting antenna is split into a number of angular sectors using different transmitter powers in each sector. This leads to a homogenous space subdivision in terms of propagation con- ditions minimizing the sum of the overall power. Manuscript received October 17, 2001; revised March 15, 2002 and May 2, 2002. F. Aguado-Agelet is with the Departamento de Tecnologías de las Comu- nicaciones, ETSI Telecomunicación, Universidad de Vigo, 36200 Vigo, Spain (e-mail: faguado@tsc.uvigo.es). A. M. Martínez Varela and L. J. Alvarez-Vázquez are with the Departamento de Matemática Aplicada, ETSI Telecomunicación, Universidad de Vigo, 36200 Vigo, Spain (e-mail: aurea@dma.uvigo.es, lino@dma.uvigo.es). J. M. Hernando is with SSR, ETSI Telecomunicación, Universidad Politéc- nica de Madrid, 28080 Madrid, Spain (e-mail: hernando@grc.ssr.upm.es). A. Formella is with the Departamento de Lenguajes y Sistemas Informáticos, Universidad de Vigo, 32004 Ourense, Spain (e-mail: formella@ei.uvigo.es). Digital Object Identifier 10.1109/TVT.2002.804844 II. PROPAGATION MODEL Several propagation models can be followed in order to com- pute the cost function in the optimization procedure. In this paper, results obtained from computer runs with a simulated in- door office plant model with a simple propagation law will be used [2]. This model computes the propagation loss as a func- tion of an average loss parameter obtained in different mea- surement campaigns [1] and the number of intermediate walls crossed by the ray from the transmitter to the receiver. The fol- lowing equation summarizes the path loss: dB (1) where is the average loss factor ( for free space loss), is the distance between the receiver and transmitter, is the wavelength, is the number of wall crossings, and is the attenuation in dB for each crossing. The recommended values for and are 3 and 2.38 dB, respectively [1]. III. OPTIMIZATION METHODS According to the planning strategy, different cost functions with bound constraints must be optimized, but in all cases the functions are nondifferentiable and nonconvex. Thus, we need to use suitable methods for solving a nonlinear programming problem, which will be formulated as the minimization of a function subject to same constraints. To solve the problem, four different methods are proposed. A. Nelder–Mead Method The Nelder–Mead simplex method [3] is a gradient-free method that merely compares objective function values taken from a set of sample points. The values are used to continue the sampling. A simplex is defined as the convex hull of 1 points in (for instance, a triangle for or a tetrahedron for ). The Nelder–Mead method is a direct search algorithm that takes into account the local geometry of the function and constructs a sequence of simplices as approximations to an optimal point. In the algorithm, the 1 vertices of each simplex are sorted according to the objective function values (2) and the worst vertex is replaced with a new point of the form , where is the centroid of 0018-9545/02$17.00 © 2002 IEEE