Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
7.5 Deterministic Channel-Modeling Methods
In principle, a wireless propagation channel can be viewed as a deterministic channel. Maxwell’s equations, together with electromagnetic boundary conditions (location, shape, and dielectric and conductive properties of all objects in the environment), allow determination of the field strength at all points and times. For outdoor environments, such purely deterministic channel models have to take into account all the geographical and morphological features of a propagation environment;
for indoor environments, the building structure, wall properties, and even furniture should be taken into consideration. In this section, we outline the basic principles of channel models based on such a deterministic point of view.
To make deterministic modeling a viable option, two major challenges had to be overcome:
(i) the high amount of required computer time and (ii) the need for exact knowledge of boundary conditions.
• Computer time and storagewere prohibitive up to about 1990. However, this has changed since then. On one hand, computers have become so much faster that tasks that seemed unfeasible even with supercomputers in the 1990s are realistic options on a personal computer nowadays.
On the other hand, the development of more efficient deterministic algorithms has improved the situation as well.
• Exact knowledge of boundary conditionsis required for the successful application of deterministic models. This implies that the position and the electromagnetic properties of the whole “relevant”
environment have to be known (we will discuss later what “relevant” means in this context).
The creation of digital terrain maps and city plans, based on satellite images or building plans, has also made considerable progress in the last few years.
The most accurate solution (given an environment database) is a “brute force” solution of Maxwell’s equations, employing either integral or differential equation formulations. Integral equations are most often variations of the well-known Method of Moments, where the unknown currents induced in the IOs are represented by a set of basis functions. In their most simple form, basis functions are rectangular functions, extending over a fraction of a wavelength. Differential equation formulations include theFinite Element Method(FEM) or the increasingly popularFinite Difference Time Domain (FDTD) method.
All these methods are highly accurate, but the computational requirements are prohibitive in most environments. It is thus much more common to use approximations to Maxwell’s equations as a basis for solution. The most widespread approximation is the high-frequency approximation (also known asray approximation).4 In this approximation, electromagnetic waves are modeled as rays that follow the laws of geometrical optics (Snell’s laws for reflection and transmission);
further refinements allow inclusion of diffraction and diffuse scattering in an approximate way. In the remainder of this section, we will concentrate on various implementations of ray-based schemes.
4In the literature, such methods are often generally known asray tracing. However, the expression “ray tracing”
is also used for one specific implementation method (described below). We will thus stick with the name “high- frequency approximation” for the general class of algorithms.
Channel Models 139
7.5.1 Ray Launching
In theray-launching approach, the transmit antenna sends out (launches) rays into different direc- tions. Typically, the total spatial angle 4π is divided intoN units of equal magnitude, and each ray is sent in the direction of the center of one such unit (i.e., uniform sampling of the spatial angle) (see Figure 7.8). The number of launched rays is a tradeoff between accuracy of the method and computation time.
TX
Δθ θ
Δφ
φ
zs
Figure 7.8 Principle of ray launching.
Reproduced with permission from Damosso and Correia [1999]©European Union.
The algorithm follows the propagation of each ray until it either hits the RX or becomes too weak to be significant (e.g., drops below the noise level). When following a ray, a number of effects have to be taken into account:
• Free space attenuation: as each ray represents a certain spatial angle, the energyper unit area decreasesd−2 along the path of the ray.
• Reflections change the direction of a ray and cause an additional attenuation. Reflection coeffi- cients can be computed from Snell’s laws (see Chapter 4) depending on the angle of incidence and possibly the polarization of the incident ray.
• Diffraction and diffuse scatteringare included in more advanced models. In those cases, a ray that is incident on an IO gives rise to several new rays. The amplitudes of diffracted rays are usually computed from the geometrical or uniform theory of diffraction, as discussed in Chapter 4.
Theray-splitting algorithm is an important improvement in the accuracy of the method. The algorithm is based on the premise that the effective cross-section of the ray should never exceed a certain size (e.g., the size of a typical IO). Thus, if a ray has propagated too far from the TX, it is subdivided into two rays. Let us explain that principle in more detail using Figure 7.9. To simplify the discussion, we consider only the two-dimensional case. Each ray represents not only a certain angle but rather an angularrange of widthφ – corresponding to the angle between two launched rays. The intersection of such an angular range with a circle of radiusd has a length of approximatelyφd(for the three-dimensional case, think “cross-section” instead of “length”). Thus, the farther we get away from the TX, the larger the length that is covered by the ray. In order to maintain high accuracy of the simulation, this length should not become too large. As soon as it reaches a lengthL, the ray is split (thus reducing the length toL/2).The resulting subrays (which again represent a whole angular range of widthφ) then propagate until they reach a lengthL, etc.
TX
L f d
f f
Figure 7.9 Principle of ray splitting.
Ray launching gives the channel characteristics in the whole environment – i.e., for many differ- ent RX positions and a given TX position. In other words, once we have decided on a BS location, we can compute coverage, delay spread, and other channel characteristics in the whole envisioned cell area. Furthermore, a preprocessing scheme allows the inclusion of multiple TX locations. The environment (the IOs) is subdivided into “tiles” (areas of finite size, typically the same size as the maximum effective area of a ray) and the interaction between all tiles is computed. Then, for each TX position, only the interaction between the TX and the tiles that can act as first IOs has to be computed [Hoppe et al. 2003].
7.5.2 Ray Tracing
Classicalray tracing determines all rays that can go fromone TX location toone RX location.
The method operates in two steps:
1. First, all rays that can transfer energy from the TX location to the RX location are determined.
This is usually done by means of the image principle. Rays that can get to the RX via a reflection show the same behavior as rays from a virtual source that is located where an image of the original source (with respect to the reflecting surface) would be located (see Figure 7.10).
2. In a second step, attenuations (due to free space propagation and finite reflection coefficients) are computed, thus providing the parameters of all MPCs.
Ray tracing allows fast computation of single- and double-reflection processes, and also does not require ray splitting. On the downside, effort increases exponentially with the order of reflections that are included in the simulation. Also, the inclusion of diffuse scattering and diffraction is nontrivial. Finally, the method is less efficient than ray launching for the computation of channel characteristics over a wide area.
7.5.3 Efficiency Considerations
Both for ray launching and ray tracing, it is almost impossible to correctly predict the phases of arriving rays. Such a prediction would require a geographical and building database that is accurate to within a fraction of a wavelength. It is thus preferable to assume that all rays have uniformly distributed random phases. In this case, it is only possible to deterministically predict the small-scalestatisticsof channel characteristics; realizations of the impulse responses are obtained by ascribing random phases to MPCs. This is another form of the mixed deterministic–stochastic approach mentioned at the beginning of this chapter.
Channel Models 141
TX
RX
Figure 7.10 The image principle. Grey circles: virtual sources corresponding to a single reflection. White circles:
virtual sources corresponding to double reflections. Dotted lines: rays from the virtual sources to the RX. Dashed lines: actual reflections. Solid lines: line of sight.
A further method to reduce the computational effort is to perform ray tracing not in all three dimensions but rather only in two dimensions. It depends on the propagation environment whether this simplification is admissible:
• Indoor: indoor environments practically always require three-dimensional considerations. Even when the BS and the MS are on the same floor, reflections at floors and ceilings represent important propagation paths.
• Macrocells: by definition, the BS antenna is considerably above the rooftops. Propagation thus occurs mostly over the rooftops to points that are close to the MS. From these points, they then reach the MS, possibly via a diffraction or a reflection from the wall of the house opposite. Ray tracing in the vertical plane alone can thus be sufficient forsome cases. This is especially true when ray tracing should only predict the received power and delay spread. On the other hand, such a purely vertical ray tracing will not correctly predict the directions of the rays at the MS.
• Microcells, small distance BS–MS: as both BS and MS antennas are below the rooftop, the diffraction loss of over-the-rooftop propagation is large. Propagation in the horizontal plane – i.e., through street canyons – can be a much more efficient process. Under these conditions, ray tracing in just the horizontal plane can be sufficient.
• Microcells, large distance BS–MS: in this case, the relative power of rays propagating in the horizontal plane (compared with over-the-rooftop components) is smaller. Horizontal compo- nents undergo multiple diffraction and reflection processes, while losses from over-the-rooftop components are mostly determined by diffraction losses near the BS and the MS, and thus depend less on distance. In this case, a so-called2.5-dimensional modelcan be used: only propagation in
the horizontal plane, on one hand, and only in the vertical plane, on the other hand, is simulated, and these two contributions are added together.
2.5-dimensional modeling can also be used for macrocells. However, both in macrocells and in microcells, with the BS antenna close to the rooftop height, there are propagation processes that cannot be correctly modeled by 2.5-dimensional ray tracing. For example, reflections at a far IO cluster (high-rise building) are not accounted for in this approach (see Figure 7.11).
RX
Horizontal propagation
plane
TX 3D-ray
Vertical propagation plane
Figure 7.11 Two- and three-dimensional modeling.
7.5.4 Geographical Databases
The foundation of all deterministic methods is the information about the geography and morphology of the environment. The accuracy of that information determines the achievable accuracy of any deterministic channel model.
Forindoor environments, that information can usually be obtained from building plans, which nowadays are often available in digital form.
InRAs, geographical databases are available with a resolution of 10–100 m. These databases are often created by means of satellite observations. In many countries, morphological information (land usage) is also available; however, obtaining this information in an automated and consistent way can be quite challenging.
Inurban areas, digital databases use two different types of data: vector data and pixel data. For vector data, the actual location of building endpoints is stored. For pixel data, a regular grid of points is superimposed on the area, and for each pixel it is stated whether it falls on “free space”
(streets, parks, etc.) or is covered by a building. In both cases, building heights and materials might be included in the database.