Model for urban and indoor cellular propagation using percolation theory G. Franceschetti, 1 S. Marano, 2 N. Pasquino, 1 and I. M. Pinto 3 1 Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Universita ` degli Studi di Napoli ‘‘Federico II,’’ Napoli I-80125, Italy 2 Dipartimento di Ingegneria dell’Informazione e Ingegneria Elettrica, Universita ` degli Studi di Salerno, Fisciano (SA) I-84084, Italy 3 Universita ` degli Studi del Sannio a Benevento, Benevento I-82100, Italy Received 30 September 1999 A method for the analysis and statistical characterization of wave propagation in indoor and urban cellular radio channels is presented, based on a percolation model. Pertinent principles of the theory are briefly reviewed, and applied to the problem of interest. Relevant quantities, such as pulsed-signal arrival rate, number of reflections against obstacles, and path lengths are deduced and related to basic environment parameters such as obstacle density and transmitter-receiver separation. Results are found to be in good agreement with alter- native simulations and measurements. PACS numbers: 42.25.Dd,41.20.Jb I. INTRODUCTION Telecommunications have experienced, in recent years, unprecedented development, mainly stimulated by wide- spread demand for fast and reliable access to information with steadily improving quality of service. When terrain pro- files, customer distribution, absence of existing facilities, and allowance for user mobility make wired systems unafford- able and not worth the cost, wireless cellular systems are the choice. However, in order to get reliable estimates of the coverage and therefore mean extension of the cells, we need a working knowledge of the propagation properties of the indoor and outdoor radio channels involved. Most widely used techniques devised for this goal adopt an empirical approach whereby measured electromagnetic fields are used to build some statistical characterization of the channel under observation 1–6. On the one hand, these methods are time and cost demanding, since they need ex- tensive measurement campaigns at every location to be char- acterized. On the other hand, they are quite reliable in terms of accuracy. An alternative procedure which is viable when dealing with microcellular environments is referred to as the ‘‘ray tracing’’ 7 method. This basically consists of tracing the trajectories of the electromagnetic energy flux from the trans- mitter to the receiver, making some judicious use of the dif- fraction theory, whenever needed, to compute the principal part of the field in an asymptotic short-wave expansion. To accomplish this, some a priori knowledge of electromagnetic and physical properties of the propagation environment and an adequate amount of processing power is required. The method proposed here is based on a model of the propagation channel borrowed from the percolation theory. This will allow for a remarkably simple description of the channel itself and the propagation phenomena in it. This Rapid Communication is organized as follows. In Sec. II the proposed model is outlined, and the relevant as- sumptions and simplifications are discussed. In Sec. III simu- lations are presented for the propagation of an ideal pulse through a channel within the framework of the proposed model. II. PERCOLATIVE APPROACH Percolation theory PT describes diffusion phenomena in a random lattice, as a function of the interconnection density. It has been widely applied to such diverse fields as biophys- ics, polymer science, and electrical engineering, to name a few 8. By properly tailoring PT concepts and analysis tools, one could study the propagation of an electromagnetic EM wave, e.g., in urban environments, where open paths take the role of the percolative clusters a cluster is defined as a sequence of neighboring empty cells, and EM energy flux plays the role of the diffusing fluid 9,10. As a key feature, PT allows us to describe relatively com- plex phenomena using a relatively small number of simple parameters. For urban channel propagation, the relevant quantities are the scatterer density, the wavelength-scaled cell-side length, and the transmitter-receiver distance. We shall accordingly model a city as a two-dimensional square- cell lattice, where occupied cells represent buildings whose known density is q. Cell-side length will be assumed as fixed and denoted by a. Without loss of generality we can imagine a uniform dis- tribution of buildings, and consider the status of each cell as being totally independent of that of the surrounding ones. An EM wave radiator in the channel is modeled as the source of an isotropic set of rays EM energy-flux lines, and can al- ways be assumed to be located at the center of the lattice if this latter has infinite extent. The ray model of EM wave propagation holds in the short-wave limit, where the EM field characteristic wavelength  =c / f is much smaller than the scatterer size (  / a 1). Electromagnetic energy-flux lines rays percolate through the empty cells from transmitter to receiver, obeying the laws of geometrical optics. Obstacles are taken as opaque, with an average e.g., at normal incidence reflection coefficient R. Refracted waves are supposed to be completely absorbed, so that no ray can traverse an obstacle and re- emerge from it. We ignore diffraction effects contributing terms of higher order in powers of  / a ), and assume locally plane-wave fronts and scatterer surfaces. It is clear that if the density of occupied cells is too high, RAPID COMMUNICATIONS PHYSICAL REVIEW E MARCH 2000 VOLUME 61, NUMBER 3 PRE 61 1063-651X/2000/613/22284/$15.00 R2228 ©2000 The American Physical Society