2708 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014 Simulation of Long Distance Wave Propagation in 2-D Sparse Random Media: A Statistical S-Matrix Approach in Spectral Domain Amr A. Ibrahim, Student Member, IEEE, and Kamal Sarabandi, Fellow, IEEE Abstract—The problem of long distance wave propagation in a sparse random medium is considered in this paper. A mathemat- ical technique for modeling the behavior of electromagnetic wave propagation as a function of distance in a 2-D sparse random media at millimeter wave (MMW) regime is presented. The pro- posed model is a field method based on Maxwell’s equation and, thus, the phase and magnitude of the field in the random medium can be tracked accurately. The random media is characterized by low volume fraction but electrically large scatterers having rela- tively high dielectric constant. The technique relies on discretizing the random media into thin slabs and relating the forward and backward scattered plane wave spectra from the individual slabs by an equivalent spectral bistatic scattering matrix. By cascading the scattering matrices of the individual slabs, the statistics of the overall scattered wave in both forward and backward directions are obtained. This technique will be referred to as statistical S-matrix approach in spectral domain, or SSWaP-SD. Using this method, it is shown that after propagation to a critical range in- side the random media, the incoherent component of the forward propagating wave overcomes the mean-field component resulting in a dual-slope attenuation curve as a function of distance. The accuracy of the model is examined against a full wave Monte Carlo simulation and a very good agreement is observed. Finally, based on the proposed model, analytical expressions for predicting the forward path-loss as well as the back scattered power from a sparse, translational invariant discrete random media are derived. The analytical expressions are tested against Monte Carlo simu- lation for a very long random medium and very good agreements are demonstrated. Index Terms—Propagation, random media, spectral domain analysis, stochastic fields. I. INTRODUCTION R ECENTLY, the increasing demand on high data rates in modern communication systems has pushed the oper- ating carrier frequencies to the millimeter wave (MMW) regime [1]–[4]. Also MMW and sub-MMW frequencies are being con- sidered for short range radars envisioned for autonomous robotic applications [5], [6]. With the advent of solid state electronics, more power and gain is becoming available at high MMW and sub-MMW bands [7]–[9]. Such technologies will Manuscript received September 18, 2013; revised January 10, 2014; accepted February 07, 2014. Date of publication February 21, 2014; date of current ver- sion May 01, 2014. The authors are with the Department of Electrical Engineering and Com- puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: amralaa@umich.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2014.2307580 enable long range imaging radars and wireless communication systems with unprecedented imaging capabilities and data rates, respectively. In order to build reliable radars or communication systems at these frequencies ( m m), it is important to understand and quantify the phenomenology of wave propagation through typical communication channels such as atmospheric (rain, snow, etc.), foliage, and urban environments. Such environments are statistical in nature and accurate prediction of wave propagation often times entails complex mathematical models. When modeling the scattering from random media, it is common to represent the field as summation of two compo- nents, namely, the coherent or the mean field component, and the incoherent or fluctuating component, i.e., (1) where the symbol refers to the ensemble average. Depending on the communication range and the degree of scattering in the medium, the incoherent part may be smaller, comparable, or even larger than the coherent part. Over the past century, many theories have been proposed in order to estimate both the coherent and incoherent part of the wave propagating in random media [10]–[17]. Such theories involve many approximations and are usually applicable under specific conditions pertaining to: frequency of operation, concentration of the scatterers (sparse versus dense), and permittivity fluctu- ations of the scatterers against the background (strong versus tenuous). For example, at low frequency, dielectric mixing for- mulas, for low volume fractions, can be used to treat the random medium as an effective homogenous medium for which the mean-field can be calculated. However, this approach cannot predict the fluctuating part of the field. For tenuous random media, approximations based on the perturbation techniques (like the Born approximation) can be usually incorporated to simplify the analysis [10]. Such methods are not appropriate when dealing with strong permittivity fluctuations. The single scattering approximation is usually used for scattering from sparse discrete random media [10]–[12]. For dense random media, where multiple scattering takes place, higher order approximations such as the quasi-crystalline approximation (QCA) and the quasi crystalline approximation with coherent potential (QCA-CP) may be incorporated for estimation of the mean-field [13]. For estimation of the fluctuating fields, methods based on radiative transfer model [10], [13], [14], or single scattering theory [15] can be utilized. In [16], many 0018-926X © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.