In recent years, with the increasing interest in indoor wireless communications systems, the development of appropriate tools for modeling the propagation within an indoor environment is becoming of utmost importance. A versatile technique for studying propagation in such a complex scenario is ray tracing [1-2]. Through this approach, a number of paths, stemming from the transmitter, are traced along their way to the receiver, accounting for reflection over the obstacles within the scenario. Other mechanisms of interaction between the wave and the environment, such as diffraction, can be accommodated in ray tracing procedures by appropriate generalization of the basic theory .
While this method is purely deterministic, in actual environments with many randomly placed scatterers of size comparable to the wavelength (Mie scattering), statistical characterization of the multipath channel [3-6] may be the only viable approach in order to have an accurate model of the propagation [7-8]. Statistical modeling built on iteration of ray tracing results suffers of its inner computational intensity. As for the radio channel design what it is of interest are the fluctuations about a mean value of the received power, one can use a simpler and efficient method to take into account variations due to randomly placed obstacles in the propagation environment. Radiative transfer theory seems to be appropriate, as it deals with the wave propagation within a random medium characterized by randomly placed scatterers. Based on a phenomenological description of the transfer of energy, the basic equation (referred to as radiative transfer equation, RTE) simply states the conservation of energy in terms of the specific intensity , i. e. the power per unit area and per unit solid angle propagating along , and which is a function of position
In this chapter, the author reports the RTE results for the evaluation of the power fluctuations in an indoor environment described as a homogeneous medium filled with scatterers arbitrarily placed. The radiative transfer outcome is compared with the ray tracing predictions to assess its limits of applicability.
For the sake of simplicity and without introducing inessential complications in our analysis, the investigation is limited to a propagation environment that can be modeled as a layered parallel plane medium. This modeling is a first approximation of an indoor environment including people, benches or, for instance, a row of chairs in an auditorium, whose location is not fixed. The random medium layer accounts for the average condition out of many possible spatial configurations, given or assumed the number density of scatterers of a succession of layers where each layer is modeled as random medium containing randomly placed scatterers. Moreover, we consider infinite-length circular cylinder as scatterers so as to simplify the solution of the RTE. It should be clear that while the ray tracing approach can be in principle used for any geometry and provide information about the phase of the wave, RT is in practice only applicable to simple geometries and can only yield information about the second order statistics of the wave. The scatterer’s number density is chosen so as interference and interaction between scatterers could be neglected. RT reliability under this condition has been thoroughly investigated [15-17]; here, it is shown how it can be a useful and simple tool for indoor propagation analysis regarding the spatial correlation as defined in Section 5.
The chapter is organized as follows. A review of the radiative transfer is presented in Section 2, together with the definition of main quantities. Section 3 and 4 are devoted to the description of the RTE and of the numerical techniques for its solution. Iterative procedure and its limits of applicability are discussed. Section 5 reports the numerical results in two study cases in a 2D geometry with focus on the impact of the system parameters on the specific intensity and spatial correlation. The results of the comparison between the radiative transfer results and the predictions obtained through ray tracing are also reported. Section 6 is devoted to comments and conclusions.
2. Radiative Transfer Theory: physical background
Two basic theories have been developed in order to approach the study of wave propagation within a random medium characterized by randomly placed scatterers. The first is the analytical theory, where taking into account the scattering and absorption characteristic of the particles solves the Maxwell equations. This approach is mathematically and physically rigorous since in principle the effects of the mechanisms involved in multiple scattering, diffraction and interference can be appropriately modeled. However, in practice, various approximations have to be made in order to obtain feasible solutions (see  or  for an overview). Recent developments  adopt the random medium as paradigm to describe the propagation channel. Stochastic Green’s functions are computed to obtain the channel transfer matrix T in MIMO applications. On the other hand, the transport theory is based on a phenomenological description of the transfer of energy. The basic equation simply states the conservation of energy expressed in terms of the
In this chapter, the basic concepts and the quantities of interest when dealing with radiative transfer theory are reviewed, while the reader is referred to literature for a deeper insight. As the problem is defined in a two-dimensional domain, the classic theoretical formulation is adapted to this framework. Accordingly, the chosen reference system is in cylindrical coordinates, as shown in Figure 1. The analysis deals with monochromatic signals with frequency
To study the propagation of a wave in presence of randomly distributed particles, the main results related to scattering and absorption of a single particle in vacuum are reviewed. Let us consider an elliptically polarized incident plane wave E
and (μ0, ε0) represent the vacuum dielectric constant and its permeability. As the propagation takes place in the
and it is assumed for simplicity to be homogeneous with:
In far field, the scattered field behaves like a cylindrical wave:
Therefore, for each polarization
In the following, where not stated otherwise the subscript
2.1. Scattering and absorption cross widths
and integrating over all angles:
where the scattering cross width σs is defined as:
This quantity represents the equivalent width that would produce the amount
The geometric cross width
According to Rayleigh scattering theory , it means meaning that in this regime the power scattered by the particle is much smaller than the product of geometric cross width and the amplitude of the Poynting vector. Besides, in the high frequency regime
which is known as geometric optics limit.
Similarly to (10), the absorption cross width can be defined as the ratio between the absorbed power
2.2. Extinction cross width and albedo
The extinction cross-section σ
and it represents the total power loss from the incident wave due to scattering and absorption. The fraction of scattering over the extinction cross width is defined as albedo
The computations of the extinction cross width (12) can be carried out from the knowledge of the scattering matrix, as stated by the forward scattering theorem (also known as optical theorem):
This result can be proved either undertaking the explicit computation of (11)  or computing the received power over a given width and relating this quantity to the geometric dimensions . Besides, it can be shown that for the high frequency regime :
Equation (15) is also known as extinction paradox.
2.3. Example: scattering from a circular cylinder
Figure 2 shows the scattering, the absorption and the extinction cross widths are computed for a circular cylinder of radius
2.4. Specific intensity
Transport theory deals with the propagation of energy in a medium containing randomly placed particles. For a given point
The specific intensity
Hence, the square modulus of the scattering function F(
3. Radiative transfer equation
The radiative transfer equation is an integro-differential equation that governs the propagation of specific intensity within a random medium. Let us assume that the random medium is a made of uniform slabs in the
3.1. Reduced intensity
Let us consider an area with side 1 in the
where < > represents the ensemble average over the distribution of particles orientations. Equation (19) defines the so-called reduced intensity since it only takes into account the extinction of the incident wave .
3.2. Independent scattering and limits of the transfer theory
Scattering of waves impinging on the area
The specific intensity is proportional to the square modulus of the electric field averaged over the distribution of scatterers:
Now, let, we get that:
The phase difference (
From the discussion above, the assumption underlying the radiative transfer equation of independent scattering limits the applicability of the transfer theory to cases where the distance between particles is large enough (see (22)) so as to make negligible the near far interactions between particles. Experimental studies confirm this conclusion: for the radiative transfer to be applicable the spacing between scatterers must be larger than λ/3 and 0.4
Assuming independent scattering, the increase on the specific intensity along direction
whereis the phase function. Notice that if the random medium contains particles of different kinds, the overall phase function is defined as :
3.3. Progressive and regressive intensity
The radiative transfer equation is obtained by combining (19) and (23):
In this formulation, the extinction coefficient and the phase function are considered function of the position
where the azimuth direction
This equivalent formulation of (25) makes it easier to set the boundary conditions as explained in §3.3.1 and §3.3.2. For a uniform distribution of the particles over the random medium and circular cylindrical particles, the phase matrix becomes a function
where and, based on the 2π periodicity of both
3.3.1. Boundary conditions on the specific intensity
The radiative transfer equation has to be solved by imposing appropriate boundary conditions. Here, the boundary conditions that the specific intensity must satisfy on a plane boundary between two media with indices of refraction
As far as the transmitted specific intensity is concerned, we can write the conservation of power on a segment
As by the Snell’s law, it ends:
4. Solution of the radiative transfer equation through numerical quadrature
The radiative transfer equation is an integro-differential equation whose solution in analytical form is very difficult, if not impossible. However, efficient numerical solutions can be devised. A comprehensive treatment of the main techniques can be found in [15, 24].
The case under study concerns a random medium where relevant scattering occurs. An approximate solution can be obtained by computing the integrals in (26) by numerical quadrature as firstly proposed in . The continuum of propagation directions
The two integro-differential equations (27) can be approximate as follows:
where then ×n matrices P+ and P- are defined as:
The n×n K(
Now, defining the 2n×1 vector, one obtains the system of first order linear equation:
4.1. Discrete ordinate Eigen analysis
If the space contains slabs of homogeneous random (and non-random) media, the linear differential system (36) within each slab (say the
being matrix G
As a result, (
by ordering the eigenvalues in increasing order () it follows:
and the corresponding eigenvectors satisfy the condition:
The solution of (36) can be written as the linear combination
where, , and D(y) is a diagonal matrix with elements.
The vector of unknown constant c can be computed from the knowledge of I(
4.1.1. Setting the boundary conditions for a single slab
Given geometry depicted in figure 4, with only one slab of random medium (ranging within
where is the incident specific intensity and AR is the n×n reflection matrix relative to the interface at
Then, using (40) and (41), and partitioning the matrix T as:
in which Tij is n×n, one gets:
The substitution of this finding in (39) is used for the computation of the vector c.
4.1.2. Setting the boundary conditions for a multi-layer medium
In the multi-layered geometry of figure 4, each slab has depth
The specific intensity Il,2 - at the interface between the
Therefore, similarly to (42) we can write:
Proceeding as in the previous Section, the specific intensity I(0) is now calculated and used in (39) to obtain the constant vector c1 for the first slab. Then, equation (44) can be iteratively applied so as to compute the specific intensity at the beginning of each slab. The latter allows through (39) the computation of cl for each slab.
5. Numerical results
In this section two examples illustrate how radiative transfer theory could be employed to study the beam broadening and the corresponding spatial correlation for an indoor environment within the context of a communication system.
The main assumption is that the propagation environment can be modeled as a layered parallel plane medium. This situation is a first approximation of an open space office made of a succession of tables where each table is modeled as random medium containing randomly placed scatterers. These are modeled as circular cylinder so as to simplify the solution of the radiative transfer equation.
5.1. Case study I: one table
In this example, the geometry under study is depicted in Figure 5. The whole
5.1.1. Beam broadening
Figure 6 shows the specific intensity
The specific intensities are assumed to be normalized with respect to and thus are shown in dB. Right after the entrance into the table region the progressive beam
In order to get a quantitative insight into the beam broadening discussed above, Figure 7 shows the specific intensity
The effect of an increase in the carrier frequency, envisioned for next generation wireless LAN, is shown in Figure 8. A larger carrier frequency (
5.1.2. Spatial correlation
In a communication link, it is of great interest to assess the degree of correlation between the signals received by different antennas as a function of their inter-spacing
Then, recalling that the ratio between signals received at two points separated by Δ in the direction
The correlation (46) is evaluated for the example at hand under the same assumptions as in Figure 7.
The results are shown in figure 10. The correlation decreases with increasing object density. Thus, from the perspective of communication system performance, increasing the object density is beneficial in terms of degree of diversity. For instance, for Δ = λ the correlation decreases from around 0.9 to around 0.7.
5.2. Case study II: two tables with interface
In this example the environment has a more complicated geometry, where two tables are followed by a semi-infinite dielectric slab (dielectric constant εr,I) as shown in figure 11. The simulation parameters follow the setting described for Case study I, in particular the radius of the scatterers is
5.2.1. Beam broadening
Figure 12 shows the specific intensity
5.2.2. Spatial correlation
The spatial correlation
5.3. Limits and validity of radiative transfer predictions
The validation of numerical results presented in §5.1 and §5.2 relies on a discussion, as there is no empirical evidence (measurements) to compare with. The two chosen cases are not realistic but they represent a possible benchmark to be employed against the numerical results that issue from analytical methods. Among these, the T-matrix approach [26, 27] seems to be the best suited to compute scattering from a random distribution of cylinders and to compare the radiated fields on a realization-by-realization basis with ray tracing. At this stage of development, the only comparison that the author is able to provide to assess the validity of a radiative transfer approach is against a ray tracing technique based upon the beam tracing method . The numerical code used for the simulations was developed at the Politecnico di Milano  and it has already been used for different purposes related to the indoor propagation .
To make the beam tracing procedure suitable for the study of electromagnetic propagation, it is necessary to include the reflection coefficient associated with the interactions of the path with the environment. In particular, the signal received from each path has to be scaled by the product of the reflection coefficients corresponding to the bounces each path goes through when propagating from the transmitter to the receiver. To compare the outcome of the ray tracing simulation with the results of radiative transfer theory some approximations were made. Case I was reproduced in such a way to replicate the same geometrical and radio electrical conditions. The reader is referred to  for further details about the numerical computation and the comparison results, while here the author recalls the general approach and the basic information. The transmitted plane wave was approximated by building a linear antenna array of
The circular scatterers were approximated by polygons of
The specific numerical values for the ray-tracing algorithm were selected after careful empirical investigation to yield negligible approximation errors. The transmitting array has
Figure 17 shows the progressive specific intensity
Figure 18 shows the comparison between the spatial correlation
In this work, the use of radiative transfer theory to study the propagation in an indoor environment was reported. A particular focus was made on two specific aspects related to the channel performances, such as the beam broadening and the spatial correlation, as they are parameters of interest when deploying a network. The pretty good agreement, although under well-controlled conditions, encourages proceeding along two distinct directions. From one side, an improvement in the modeling and in the sketching of actual environments is mandatory if one wants to use the radiative transfer technique as a possible approach to analyze indoor channel performances in practical scenarios. From the other side, validation should be performed either against measurements or - at least - against appropriate analytical solution without limitations or approximations in the frequency domain (or wavelength scale) of interest.
The author would like to gratefully acknowledge the work of Osvaldo Simeone (now Assistant Professor at the New Jersey Institute of Technology), who contributed with ideas, numerical developments and simulations to this research during his doctoral studies in Milano at the Politecnico, Faculty of Telecommunication Engineering.