Figure 4.4 shows an electromagnetic wave incident at an angle θ_i with the plane of the boundary between two dielectric media. As shown in the figure, part of the energy is reflected back to the first media at an angle θ_r, and part of the energy is transmitted (refracted) into the second media at an angle θ_t. The nature of reflection varies with the direction of polarization of the E-field. The behavior for arbitrary directions of polarization can be studied by considering the two distinct cases shown in Figure 4.4. The plane of incidence is defined as the plane containing the incident, reflected, and transmitted rays [Ram65]. In Figure 4.4a, the E-field polarization is parallel with the plane of incidence (that is, the E-field has a vertical polarization, or normal component, with respect to the reflecting surface) and in Figure 4.4b, the E-field polarization is perpendicular to the plane of incidence (that is, the incident E-field is pointing out of the page toward the reader, and is perpendicular to the page and parallel to the reflecting surface).

Figure 4.4. Geometry for calculating the reflection coefficients between two dielectrics.

In Figure 4.4, the subscripts i, r, t refer to the incident, reflected, and transmitted fields, respectively. Parameters ε₁, μ₁, σ₁, and ε₂, μ₂, σ₂ represent the permittivity, permeability, and conductance of the two media, respectively. Often, the dielectric constant of a perfect (lossless) dielectric is related to a relative value of permittivity, ε_r, such that ε = ε₀ε_r, where ε₀ is a constant given by 8.85 × 10⁻¹² F/m. If a dielectric material is lossy, it will absorb power and may be described by a complex dielectric constant given by

Equation 4.17. 

where

Equation 4.18. 

and σ is the conductivity of the material measured in Siemens/meter. The terms ε_r and σ are generally insensitive to operating frequency when the material is a good conductor (f < σ/(ε₀ε_r)). For lossy dielectrics, ε₀ and ε_r are generally constant with frequency, but σ may be sensitive to the operating frequency, as shown in Table 4.1. Electrical properties of a wide range of materials were characterized over a large frequency range by Von Hipple [Von54].

Because of superposition, only two orthogonal polarizations need be considered to solve general reflection problems. The reflection coefficients for the two cases of parallel and perpendicular E-field polarization at the boundary of two dielectrics are given by

Equation 4.19. 

Equation 4.20. 

where η_i is the intrinsic impedance of the i th medium (i = 1, 2), and is given by , the ratio of electric to magnetic field for a uniform plane wave in the particular medium. The velocity of an electromagnetic wave is given by , and the boundary conditions at the surface of incidence obey Snell’s Law which, referring to Figure 4.4, is given by

Equation 4.21. 

The boundary conditions from Maxwell’s equations are used to derive Equations (4.19) and (4.20) as well as Equations (4.22), (4.23.a), and (4.23.b).

Equation 4.22. 

and

Equation 4.23.a. 

Equation 4.23.b. 

where Γ is either Γ_‖ or Γ_┴, depending on whether the E-field is in (vertical) or normal (horizontal) to the plane of incidence.

For the case when the first medium is free space and μ₁ = μ₂, the reflection coefficients for the two cases of vertical and horizontal polarization can be simplified to

Equation 4.24. 

and

Equation 4.25. 

For the case of elliptical polarized waves, the wave may be broken down (depolarized) into its vertical and horizontal E-field components, and superposition may be applied to determine transmitted and reflected waves. In the general case of reflection or transmission, the horizontal and vertical axes of the spatial coordinates may not coincide with the perpendicular and parallel axes of the propagating waves. An angle θ measured counter-clockwise from the horizontal axis is defined as shown in Figure 4.5 for a propagating wave out of the page (toward the reader) [Stu93]. The vertical and horizontal field components at a dielectric boundary may be related by

Equation 4.26. 

Figure 4.5. Axes for orthogonally polarized components. Parallel and perpendicular components are related to the horizontal and vertical spatial coordinates. Wave is shown propagating out of the page toward the reader.

where and are the depolarized field components in the horizontal and vertical directions, respectively, and are the horizontally and vertically polarized components of the incident wave, respectively, and , , , and are time varying components of the E-field which may be represented as phasors. R is a transformation matrix which maps vertical and horizontal polarized components to components which are perpendicular and parallel to the plane of incidence. The matrix R is given by

where θ is the angle between the two sets of axes, as shown in Figure 4.5. The depolarization matrix D_C is given by

where D_xx = Γ_x for the case of reflection and D_xx = T_x = 1 + Γ_x for the case of transmission [Stu93].

Figure 4.6 shows a plot of the reflection coefficient for both horizontal and vertical polarization as a function of the incident angle for the case when a wave propagates in free space (ε_r = 1) and the reflection surface has (a) ε_r = 4, and (b) ε_r = 12.

Figure 4.6. Magnitude of reflection coefficients as a function of angle of incidence for ε_r = 4, ε_r = 12, using geometry in Figure 4.4.

Example 4.4. 

Demonstrate that if medium 1 is free space and medium 2 is a dielectric, both |Γ_‖| and |Γ_┴| approach 1 as θ_i approaches 0° regardless of ε_r.

Solution

Substituting θ_i = 0° in Equation (4.24)

Substituting θ_i = 0° in Equation (4.25)

This example illustrates that ground may be modeled as a perfect reflector with a reflection coefficient of unit magnitude when an incident wave grazes the earth, regardless of polarization or ground dielectric properties (some texts define the direction of E_r to be opposite to that shown in Figure 4.4a, resulting in Γ = −1 for both parallel and perpendicular polarization).

The Brewster angle is the angle at which no reflection occurs in the medium of origin. It occurs when the incident angle θ_B is such that the reflection coefficient Γ_‖ is equal to zero (see Figure 4.6). The Brewster angle is given by the value of θ_B which satisfies

Equation 4.27. 

For the case when the first medium is free space and the second medium has a relative permittivity ε_r, Equation (4.27) can be expressed as

Equation 4.28. 

Note that the Brewster angle occurs only for vertical (i.e. parallel) polarization.

Example 4.5. 

Calculate the Brewster angle for a wave impinging on ground having a permittivity of ε_r = 4.

Solution

The Brewster angle can be found by substituting the values for ε_r in Equation (4.28).

Thus Brewster angle for ε_r = 4 is equal to 26.56°.

Since electromagnetic energy cannot pass through a perfect conductor a plane wave incident on a conductor has all of its energy reflected. As the electric field at the surface of the conductor must be equal to zero at all times in order to obey Maxwell’s equations, the reflected wave must be equal in magnitude to the incident wave. For the case when E-field polarization is in the plane of incidence, the boundary conditions require that [Ram65]

Equation 4.29. 

and

Equation 4.30. 

Similarly, for the case when the E-field is horizontally polarized, the boundary conditions require that

Equation 4.31. 

and

Equation 4.32. 

Referring to Equations (4.29) to (4.32), we see that for a perfect conductor, Γ_‖ = 1, and Γ_┴ = −1, regardless of incident angle. Elliptical polarized waves may be analyzed by using superposition, as shown in Figure 4.5 and Equation (4.26).

Consider a transmitter and receiver separated in free space as shown in Figure 4.10a. Let an obstructing screen of effective height h with infinite width (going into and out of the paper) be placed between them at a distance d₁ from the transmitter and d₂ from the receiver. It is apparent that the wave propagating from the transmitter to the receiver via the top of the screen travels a longer distance than if a direct line-of-sight path (through the screen) existed. Assuming h « d₁, d₂ and h » λ, then the difference between the direct path and the diffracted path, called the excess path length (Δ), can be obtained from the geometry of Figure 4.10b as

Equation 4.54. 

Figure 4.10. Diagrams of knife-edge geometry.

The corresponding phase difference is given by

Equation 4.55. 

and when tanx ≈ x, then α = β + γ from Figure 4.10c and

(a proof of Equations (4.54) and (4.55) is left as an exercise for the reader).

Equation (4.55) is often normalized using the dimensionless Fresnel–Kirchoff diffraction parameter v which is given by

Equation 4.56. 

where α has units of radians and is shown in Figure 4.10b and Figure 4.10c. From Equation (4.56), ф can be expressed as

Equation 4.57. 

From the above equations it is clear that the phase difference between a direct line-of-sight path and diffracted path is a function of height and position of the obstruction, as well as the transmitter and receiver location.

In practical diffraction problems, it is advantageous to reduce all heights by a constant, so that the geometry is simplified without changing the values of the angles. This procedure is shown in Figure 4.10c.

The concept of diffraction loss as a function of the path difference around an obstruction is explained by Fresnel zones. Fresnel zones represent successive regions where secondary waves have a path length from the transmitter to receiver which are nλ/2 greater than the total path length of a line-of-sight path. Figure 4.11 demonstrates a transparent plane located between a transmitter and receiver. The concentric circles on the plane represent the loci of the origins of secondary wavelets which propagate to the receiver such that the total path length increases by λ/2 for successive circles. These circles are called Fresnel zones. The successive Fresnel zones have the effect of alternately providing constructive and destructive interference to the total received signal. The radius of the nth Fresnel zone circle is denoted by r_n and can be expressed in terms of n, λ, d₁, and d₂ by

Equation 4.58. 

Figure 4.11. Concentric circles which define the boundaries of successive Fresnel zones.

This approximation is valid for d₁, d₂ » r_n.

The excess total path length traversed by a ray passing through each circle is nλ/2, where n is an integer. Thus, the path traveling through the smallest circle corresponding to n = 1 in Figure 4.11 will have an excess path length of λ/2 as compared to a line-of-sight path, and circles corresponding to n = 2, 3, etc. will have an excess path length of λ, 3λ/2, etc. The radii of the concentric circles depend on the location of the plane. The Fresnel zones of Figure 4.11 will have maximum radii if the plane is midway between the transmitter and receiver, and the radii become smaller when the plane is moved toward either the transmitter or the receiver. This effect illustrates how shadowing is sensitive to the frequency as well as the location of obstructions with relation to the transmitter or receiver.

In mobile communication systems, diffraction loss occurs from the blockage of secondary waves such that only a portion of the energy is diffracted around an obstacle. That is, an obstruction causes a blockage of energy from some of the Fresnel zones, thus allowing only some of the transmitted energy to reach the receiver. Depending on the geometry of the obstruction, the received energy will be a vector sum of the energy contributions from all unobstructed Fresnel zones.

As shown in Figure 4.12, an obstacle may block the transmission path, and a family of ellipsoids can be constructed between a transmitter and receiver by joining all the points for which the excess path delay is an integer multiple of half wavelengths. The ellipsoids represent Fresnel zones. Note that the Fresnel zones are elliptical in shape with the transmitter and receiver antenna at their foci. In Figure 4.12, different knife edge diffraction scenarios are shown. In general, if an obstruction does not block the volume contained within the first Fresnel zone, then the diffraction loss will be minimal, and diffraction effects may be neglected. In fact, a rule of thumb used for design of line-of-sight microwave links is that as long as 55% of the first Fresnel zone is kept clear, then further Fresnel zone clearance does not significantly alter the diffraction loss.

Figure 4.12. Illustration of Fresnel zones for different knife-edge diffraction scenarios.

Estimating the signal attenuation caused by diffraction of radio waves over hills and buildings is essential in predicting the field strength in a given service area. Generally, it is impossible to make very precise estimates of the diffraction losses, and in practice prediction is a process of theoretical approximation modified by necessary empirical corrections. Though the calculation of diffraction losses over complex and irregular terrain is a mathematically difficult problem, expressions for diffraction losses for many simple cases have been derived. As a starting point, the limiting case of propagation over a knife-edge gives good insight into the order of magnitude of diffraction loss.

When shadowing is caused by a single object such as a hill or mountain, the attenuation caused by diffraction can be estimated by treating the obstruction as a diffracting knife edge. This is the simplest of diffraction models, and the diffraction loss in this case can be readily estimated using the classical Fresnel solution for the field behind a knife edge (also called a half-plane). Figure 4.13 illustrates this approach.

Figure 4.13. Illustration of knife-edge diffraction geometry. The receiver R is located in the shadow region.

Consider a receiver at point R, located in the shadowed region (also called the diffraction zone). The field strength at point R in Figure 4.13 is a vector sum of the fields due to all of the secondary Huygen’s sources in the plane above the knife edge. The electric field strength, E_d, of a knife-edge diffracted wave is given by

Equation 4.59. 

where E_o is the free space field strength in the absence of both the ground and the knife edge, and F(v) is the complex Fresnel integral. The Fresnel integral, F(v), is a function of the Fresnel–Kirchoff diffraction parameter v, defined in Equation (4.56), and is commonly evaluated using tables or graphs for given values of v. The diffraction gain due to the presence of a knife edge, as compared to the free space E-field, is given by

Equation 4.60. 

In practice, graphical or numerical solutions are relied upon to compute diffraction gain. A graphical representation of G_d(dB) as a function of v is given in Figure 4.14. An approximate solution for Equation (4.60) provided by Lee [Lee85] is

Equation 4.61.a. 

Equation 4.61.b. 

Equation 4.61.c. 

Equation 4.61.d. 

Equation 4.61.e. 

Figure 4.14. Knife-edge diffraction gain as a function of Fresnel diffraction parameter v.

Example 4.8. 

Given the following geometry, determine (a) the loss due to knife-edge diffraction, and (b) the height of the obstacle required to induce 6 dB diffraction loss. Assume f = 900 MHz.

Solution

1. The wavelength

   Redraw the geometry by subtracting the height of the smallest structure.

   
   

   and

   α = β + γ = 2.434° = 0.0424 rad

   Then using Equation (4.56)

   

   From Figure 4.14 or (4.61.e), the diffraction loss is 25.5 dB.

2. For 6 dB diffraction loss, ν = 0. The obstruction height h may be found using similar triangles (β = γ), as shown below.

   

   It follows that , thus h = 4.16 m.

In many practical situations, especially in hilly terrain, the propagation path may consist of more than one obstruction, in which case the total diffraction loss due to all of the obstacles must be computed. Bullington [Bul47] suggested that the series of obstacles be replaced by a single equivalent obstacle so that the path loss can be obtained using single knife-edge diffraction models. This method, illustrated in Figure 4.15, oversimplifies the calculations and often provides very optimistic estimates of the received signal strength. In a more rigorous treatment, Millington et al. [Mil62] gave a wave-theory solution for the field behind two knife edges in series. This solution is very useful and can be applied easily for predicting diffraction losses due to two knife edges. However, extending this to more than two knife edges becomes a formidable mathematical problem. Many models that are mathematically less complicated have been developed to estimate the diffraction losses due to multiple obstructions [Eps53], [Dey66].

Figure 4.15. Bullington’s construction of an equivalent knife edge.

In radio channels where large, distant objects induce scattering, knowledge of the physical location of such objects can be used to accurately predict scattered signal strengths. The radar cross section (RCS) of a scattering object is defined as the ratio of the power density of the signal scattered in the direction of the receiver to the power density of the radio wave incident upon the scattering object, and has units of square meters. Analysis based on the geometric theory of diffraction and physical optics may be used to determine the scattered field strength.

For urban mobile radio systems, models based on the bistatic radar equation may be used to compute the received power due to scattering in the far field. The bistatic radar equation describes the propagation of a wave traveling in free space which impinges on a distant scattering object, and is then reradiated in the direction of the receiver, given by

Equation 4.66. 

where d_T and d_R are the distance from the scattering object to the transmitter and receiver, respectively. In Equation (4.66), the scattering object is assumed to be in the far field (Fraunhofer region) of both the transmitter and receiver. The variable RCS is given in units of dB · m², and can be approximated by the surface area (in square meters) of the scattering object, measured in dB with respect to a one square meter reference [Sei91]. Equation (4.66) may be applied to scatterers in the far-field of both the transmitter and receiver (as illustrated in [Van87], [Zog87], [Sei91]) and is useful for predicting receiver power which scatters off large objects, such as buildings, which are for both the transmitter and receiver.

Several European cities were measured from the perimeter [Sei91], and RCS values for several buildings were determined from measured power delay profiles. For medium and large size buildings located 5–10 km away, RCS values were found to be in the range of 14.1 to 55.7 dB · m².

Both theoretical and measurement-based propagation models indicate that average received signal power decreases logarithmically with distance, whether in outdoor or indoor radio channels. Such models have been used extensively in the literature. The average large-scale path loss for an arbitrary T–R separation is expressed as a function of distance by using a path loss exponent, n

Equation 4.67. 

or

Equation 4.68. 

where n is the path loss exponent which indicates the rate at which the path loss increases with distance, d₀ is the close-in reference distance which is determined from measurements close to the transmitter, and d is the T–R separation distance. The bars in Equations (4.67) and (4.68) denote the ensemble average of all possible path loss values for a given value of d. When plotted on a log–log scale, the modeled path loss is a straight line with a slope equal to 10n dB per decade. The value of n depends on the specific propagation environment. For example, in free space, n is equal to 2, and when obstructions are present, n will have a larger value.

It is important to select a free space reference distance that is appropriate for the propagation environment. In large coverage cellular systems, 1 km reference distances are commonly used [Lee85], whereas in microcellular systems, much smaller distances (such as 100 m or 1 m) are used. The reference distance should always be in the far field of the antenna so that near-field effects do not alter the reference path loss. The reference path loss is calculated using the free space path loss formula given by Equation (4.5) or through field measurements at distance d₀. Table 4.2 lists typical path loss exponents obtained in various mobile radio environments.

The model in Equation (4.68) does not consider the fact that the surrounding environmental clutter may be vastly different at two different locations having the same T–R separation. This leads to measured signals which are vastly different than the average value predicted by Equation (4.68). Measurements have shown that at any value of d, the path loss PL(d) at a particular location is random and distributed log-normally (normal in dB) about the mean distance-dependent value [Cox84], [Ber87]. That is

Equation 4.69.a. 

and

Equation 4.69.b. 

where X_σ is a zero-mean Gaussian distributed random variable (in dB) with standard deviation σ (also in dB).

The log-normal distribution describes the random shadowing effects which occur over a large number of measurement locations which have the same T–R separation, but have different levels of clutter on the propagation path. This phenomenon is referred to as log-normal shadowing. Simply put, log-normal shadowing implies that measured signal levels at a specific T–R separation have a Gaussian (normal) distribution about the distance-dependent mean of (4.68), where the measured signal levels have values in dB units. The standard deviation of the Gaussian distribution that describes the shadowing also has units in dB. Thus, the random effects of shadowing are accounted for using the Gaussian distribution which lends itself readily to evaluation (see Appendix F).

The close-in reference distance d₀, the path loss exponent n, and the standard deviation σ, statistically describe the path loss model for an arbitrary location having a specific T–R separation, and this model may be used in computer simulation to provide received power levels for random locations in communication system design and analysis.

In practice, the values of n and σ are computed from measured data, using linear regression such that the difference between the measured and estimated path losses is minimized in a mean square error sense over a wide range of measurement locations and T–R separations. The value of in (4.69.a) is based on either close-in measurements or on a free space assumption from the transmitter to d₀. An example of how the path loss exponent is determined from measured data follows. Figure 4.17 illustrates actual measured data in several cellular radio systems and demonstrates the random variations about the mean path loss (in dB) due to shadowing at specific T–R separations.

Figure 4.17. Scatter plot of measured data and corresponding MMSE path loss model for many cities in Germany. For this data, n = 2.7 and σ = 11.8 dB.

Since PL(d) is a random variable with a normal distribution in dB about the distance-dependent mean, so is P_r(d), and the Q-function or error function (erf) may be used to determine the probability that the received signal level will exceed (or fall below) a particular level. The Q-function is defined as

Equation 4.70.a. 

where

Equation 4.70.b. 

The probability that the received signal level (in dB power units) will exceed a certain value γ can be calculated from the cumulative density function as

Equation 4.71. 

Similarly, the probability that the received signal level will be below γ is given by

Equation 4.72. 

Appendix F provides tables for evaluating the Q and erf functions.

It is clear that due to random effects of shadowing, some locations within a coverage area will be below a particular desired received signal threshold. It is often useful to compute how the boundary coverage relates to the percent of area covered within the boundary. For a circular coverage area having radius R from a base station, let there be some desired received signal threshold γ. We are interested in computing U(γ), the percentage of useful service area (i.e. the percentage of area with a received signal that is equal or greater than γ), given a known likelihood of coverage at the cell boundary. Letting d = r represent the radial distance from the transmitter, it can be shown that if P_r[P_r(r)>γ] is the probability that the random received signal at d = r exceeds the threshold γ within an incremental area dA, then U(γ) can be found by [Jak74]

Equation 4.73. 

Using (4.71), Pr[P_r(r) > γ] is given by

Equation 4.74. 

In order to determine the path loss as referenced to the cell boundary (r = R), it is clear that

Equation 4.75. 

and Equation (4.74) may be expressed as

Equation 4.76. 

If we let and , then

Equation 4.77. 

By substituting t = a + blog(r/R) in Equation (4.77), it can be shown that

Equation 4.78. 

By choosing the signal level such that (i.e. a = 0), U(γ) can be shown to be

Equation 4.79. 

Equation (4.78) may be evaluated for a large number of values of σ and n, as shown in Figure 4.18 [Reu74]. For example, if n = 4 and σ = 8 dB, and if the boundary is to have 75% boundary coverage (75% of the time the signal is to exceed the threshold at the boundary), then the area coverage is equal to 94%. If n = 2 and σ = 8 dB, a 75% boundary coverage provides 91% area coverage. If n = 3 and σ = 9 dB, then 50% boundary coverage provides 71% area coverage.

Figure 4.18. Family of curves relating fraction of total area with signal above threshold, U(γ) as a function of probability of signal above threshold on the cell boundary.

Example 4.9. 

Four received power measurements were taken at distances of 100 m, 200 m, 1 km, and 3 km from a transmitter. These measured values are given in the following table. It is assumed that the path loss for these measurements follows the model in Equation (4.69.a), where d₀ = 100 m: (a) find the minimum mean square error (MMSE) estimate for the path loss exponent, n; (b) calculate the standard deviation about the mean value; (c) estimate the received power at d = 2 km using the resulting model; (d) predict the likelihood that the received signal level at 2 km will be greater than −60 dBm; and (e) predict the percentage of area within a 2 km radius cell that receives signals greater than −60 dBm, given the result in (d).

Distance from Transmitter	Received Power
100 m	0 dBm
200 m	−20 dBm
1000 m	−35 dBm
3000 m	−70 dBm

Solution

The MMSE estimate may be found using the following method. Let p_i be the received power at a distance d_i and let _i be the estimate for p_i using the (d/d₀)ⁿ path loss model of Equation (4.67). The sum of squared errors between the measured and estimated values is given by

The value of n which minimizes the mean square error can be obtained by equating the derivative of J(n) to zero, and then solving for n.

1. Using Equation (4.68), we find _i = p_i(d₀)−10nlog(d_i § 100 m). Recognizing that P(d₀) = 0 dBm, we find the following estimates for _i in dBm:

   1 = 0, 2 = −3n, 3 = −10n, 4 = −14.77n.

   The sum of squared errors is then given by

   

   Setting this equal to zero, the value of n is obtained as n = 4.4.

2. The sample variance σ² = J(n)/4 at n = 4.4 can be obtained as follows.

   J(n)	=	(0 + 0) + (−20 + 13.2)2 + (−35 + 44)2 + (−70 + 64.988)2
   	=	152.36.
   σ2	=	152.36/4 = 38.09

   therefore

   σ = 617 dB, which is a biased estimate. In general, a greater number of measurements are needed to reduce σ².

3. The estimate of the received power at d = 2 km is given by

   (d = 2km) = 0 − 10(4.4)log(2000 § 100) = −57.24 dBm.

   A Gaussian random variable having zero mean and σ = 617 could be added to this value to simulate random shadowing effects at d = 2 km.

4. The probability that the received signal level will be greater than −60 dBm is given by

   

5. If 67.4% of the users on the boundary receive signals greater than −60 dBm, then Equation (4.78) or Figure 4.18 may be used to determine that 92% of the cell area receives coverage above −60 dBm.

The Longley–Rice model [Ric67], [Lon68] is applicable to point-to-point communication systems in the frequency range from 40 MHz to 100 GHz, over different kinds of terrain. The median transmission loss is predicted using the path geometry of the terrain profile and the refractivity of the troposphere. Geometric optics techniques (primarily the two-ray ground reflection model) are used to predict signal strengths within the radio horizon. Diffraction losses over isolated obstacles are estimated using the Fresnel–Kirchoff knife-edge models. Forward scatter theory is used to make troposcatter predictions over long distances, and far field diffraction losses in double horizon paths are predicted using a modified Van der Pol-Bremmer method. The Longley–Rice propagation prediction model is also referred to as the ITS irregular terrain model.

The Longley–Rice model is also available as a computer program [Lon78] to calculate large-scale median transmission loss relative to free space loss over irregular terrain for frequencies between 20 MHz and 10 GHz. For a given transmission path, the program takes as its input the transmission frequency, path length, polarization, antenna heights, surface refractivity, effective radius of earth, ground conductivity, ground dielectric constant, and climate. The program also operates on path-specific parameters such as horizon distance of the antennas, horizon elevation angle, angular trans-horizon distance, terrain irregularity, and other specific inputs.

The Longley–Rice method operates in two modes. When a detailed terrain path profile is available, the path-specific parameters can be easily determined and the prediction is called a point-to-point mode prediction. On the other hand, if the terrain path profile is not available, the Longley–Rice method provides techniques to estimate the path-specific parameters, and such a prediction is called an area mode prediction.

There have been many modifications and corrections to the Longley–Rice model since its original publication. One important modification [Lon78] deals with radio propagation in urban areas, and this is particularly relevant to mobile radio. This modification introduces an excess term as an allowance for the additional attenuation due to urban clutter near the receiving antenna. This extra term, called the urban factor (UF), has been derived by comparing the predictions by the original Longley–Rice model with those obtained by Okumura [Oku68].

One shortcoming of the Longley–Rice model is that it does not provide a way of determining corrections due to environmental factors in the immediate vicinity of the mobile receiver, or consider correction factors to account for the effects of buildings and foliage. Further, multipath is not considered.

A classical propagation prediction approach similar to that used by Longley–Rice is discussed by Edwards and Durkin [Edw69], as well as Dadson [Dad75]. These papers describe a computer simulator, for predicting field strength contours over irregular terrain, that was adopted by the Joint Radio Committee (JRC) in the U.K. for the estimation of effective mobile radio coverage areas. Although this simulator only predicts large-scale phenomena (i.e. path loss), it provides an interesting perspective into the nature of propagation over irregular terrain and the losses caused by obstacles in a radio path. An explanation of the Edwards and Durkin method is presented here in order to demonstrate how all of the concepts described in this chapter are used in a single model.

The execution of the Durkin path loss simulator consists of two parts. The first part accesses a topographic data base of a proposed service area and reconstructs the ground profile information along the radial joining the transmitter to the receiver. The assumption is that the receiving antenna receives all of its energy along that radial and, therefore, experiences no multipath propagation. In other words, the propagation phenomena that is modeled is simply LOS and diffraction from obstacles along the radial, and excludes reflections from other surrounding objects and local scatterers. The effect of this assumption is that the model is somewhat pessimistic in narrow valleys, although it identifies isolated weak reception areas rather well. The second part of the simulation algorithm calculates the expected path loss along that radial. After this is done, the simulated receiver location can be iteratively moved to different locations in the service area to deduce the signal strength contour.

The topographical data base can be thought of as a two-dimensional array. Each array element corresponds to a point on a service area map while the actual contents of each array element contain the elevation above sea level data as shown in Figure 4.19. These types of digital elevation models (DEM) are readily available from the United States Geological Survey (USGS). Using this quantized map of service area heights, the program reconstructs the ground profile along the radial that joins the transmitter and the receiver. Since the radial may not always pass through discrete data points, interpolation methods are used to determine the approximate heights that are observed when looking along that radial. Figure 4.20a shows the topographic grid with arbitrary transmitter and receiver locations, the radial between the transmitter and receiver, and the points with which to use diagonal linear interpolation. Figure 4.20b also shows what a typical reconstructed radial terrain profile might look like. In actuality, the values are not simply determined by one interpolation routine, but by a combination of three for increased accuracy. Therefore, each point of the reconstructed profile consists of an average of the heights obtained by diagonal, vertical (row), and horizontal (column) interpolation methods. From these interpolation routines, a matrix of distances from the receiver and corresponding heights along the radial is generated. Now the problem is reduced to a one-dimensional point-to-point link calculation. These types of problems are well-established and procedures for calculating path loss using knife-edge diffraction techniques described previously are used.

Figure 4.19. Illustration of a two-dimensional array of elevation information.

Figure 4.20. Illustration of terrain profile reconstruction using diagonal interpolation.

At this point, the algorithm must make decisions as to what the expected transmission loss should be. The first step is to decide whether a line-of-sight (LOS) path exists between the transmitter and the receiver. To do this, the program computes the difference, δ_j, between the height of the line joining the transmitter and receiver antennas from the height of the ground profile for each point along the radial (see Figure 4.21).

Figure 4.21. Illustration of line-of-sight (LOS) decision making process.

If any δ_j(j = 1, .....,n) is found to be positive along the profile, it is concluded that a LOS path does not exist, otherwise it can be concluded that a LOS path does exist. Assuming the path has a clear LOS, the algorithm then checks to see whether first Fresnel zone clearance is achieved. As shown earlier, if the first Fresnel zone of the radio path is unobstructed, then the resulting loss mechanism is approximately that of free space. If there is an obstruction that just barely touches the line joining the transmitter and the receiver then the signal strength at the receiver is 6 dB less than the free space value due to energy diffracting off the obstruction and away from the receiver. The method for determining first Fresnel zone clearance is done by first calculating the Fresnel diffraction parameter v, defined in Equation (4.59), for each of the j ground elements.

If v_j≤ −0.8 for all j = 1, ....,n, then free space propagation conditions are dominant. For this case, the received power is calculated using the free space transmission formula given in Equation (4.1). If the terrain profile failed the first Fresnel zone test (i.e. any v_j > −0.8), then there are two possibilities:

For both of these cases, the program calculates the free space power using Equation (4.1) and the received power using the plane earth propagation equation given by Equation (4.52). The algorithm then selects the smaller of the powers calculated with Equations (4.1) and (4.52) as the appropriate received power for the terrain profile. If the profile is LOS with inadequate first Fresnel zone clearance, the next step is to calculate the additional loss due to inadequate Fresnel zone clearance and add it (in dB) to the appropriate received power. This additional diffraction loss is calculated by Equation (4.60).

For the case of non-LOS, the system grades the problem into one of four categories:

The method tests for each case sequentially until it finds the one that fits the given profile. A diffraction edge is detected by computing the angles between the line joining the transmitter and receiver antennas and the lines joining the receiver antenna to each point on the reconstructed terrain profile. The maximum of these angles is located and labeled by the profile point (d_i, h_i). Next, the algorithm steps through the reverse process of calculating the angles between the line joining the transmitter and receiver antennas and the lines joining the transmitter antenna to each point on the reconstructed terrain profile. The maximum of these angles is found, and it occurs at (d_j, h_j) on the terrain profile. If d_i = d_j, then the profile can be modeled as a single diffraction edge. The Fresnel parameter, v_j, associated with this edge can be determined from the length of the obstacle above the line joining the transmitter and receiver antennas. The loss can then be evaluated by calculating PL using the Equation (4.60). This extra loss caused by the obstacle is then added to either the free space or plane earth loss, whichever is greater.

If the condition for a single diffraction edge is not satisfied, then the check for two diffraction edges is executed. The test is similar to that for a single diffraction edge, with the exception that the computer looks for two edges in sight of each other (see Figure 4.22).

Figure 4.22. Illustration of multiple diffraction edges.

The Edwards and Durkin [Edw69] algorithm uses the Epstein and Peterson method [Eps53] to calculate the loss associated with two diffraction edges. In short, it is the sum of two attenuations. The first attenuation is the loss at the second diffraction edge caused by the first diffraction edge with the transmitter as the source. The second attenuation is the loss at the receiver caused by the second diffraction edge with the first diffraction edge as the source. The two attenuations sum to give the additional loss caused by the obstacles that is added to the free space loss or the plane earth loss, whichever is larger.

For three diffraction edges, the outer diffraction edges must contain a single diffraction edge in between. This is detected by calculating the line between the two outer diffraction edges. If an obstacle between the two outer edges passes through the line, then it is concluded that a third diffraction edge exists (see Figure 4.22). Again, the Epstein and Peterson method is used to calculate the shadow loss caused by the obstacles. For all other cases of more than three diffraction edges, the profile between the outer two obstacles is approximated by a single, virtual knife edge. After the approximation, the problem is that of a three edge calculation.

This method is very attractive because it can read in a digital elevation map and perform a site-specific propagation computation on the elevation data. It can produce a signal strength contour that has been reported to be good within a few dB. The disadvantages are that it cannot adequately predict propagation effects due to foliage, buildings, other man-made structures, and it does not account for multipath propagation other than ground reflection, so additional loss factors are often included. Propagation prediction algorithms which use terrain information are typically used for the design of modern wireless systems.

Okumura’s model is one of the most widely used models for signal prediction in urban areas. This model is applicable for frequencies in the range 150 MHz to 1920 MHz (although it is typically extrapolated up to 3000 MHz) and distances of 1 km to 100 km. It can be used for base station antenna heights ranging from 30 m to 1000 m.

Okumura developed a set of curves giving the median attenuation relative to free space (A_mu), in an urban area over a quasi-smooth terrain with a base station effective antenna height (h_te) of 200 m and a mobile antenna height (h_re) of 3 m. These curves were developed from extensive measurements using vertical omnidirectional antennas at both the base and mobile, and are plotted as a function of frequency in the range 100 MHz to 1920 MHz and as a function of distance from the base station in the range 1 km to 100 km. To determine path loss using Okumura’s model, the free space path loss between the points of interest is first determined, and then the value of A_mu(f, d) (as read from the curves) is added to it along with correction factors to account for the type of terrain. The model can be expressed as

Equation 4.80. 

where L₅₀ is the 50th percentile (i.e., median) value of propagation path loss, L_F is the free space propagation loss, A_mu is the median attenuation relative to free space, G(h_te) is the base station antenna height gain factor, G(h_re) is the mobile antenna height gain factor, and G_AREA is the gain due to the type of environment. Note that the antenna height gains are strictly a function of height and have nothing to do with antenna patterns.

Plots of A_mu(f, d) and G_AREA for a wide range of frequencies are shown in Figure 4.23 and Figure 4.24. Furthermore, Okumura found that G(h_te) varies at a rate of 20 dB/decade and G(h_re) varies at a rate of 10 dB/decade for heights less than 3 m

Equation 4.81.a. 

Equation 4.81.b. 

Equation 4.81.c. 

Figure 4.23. Median attenuation relative to free space (A_mu(f,d)), over a quasi-smooth terrain.

Figure 4.24. Correction factor, G_AREA, for different types of terrain.

Other corrections may also be applied to Okumura’s model. Some of the important terrain related parameters are the terrain undulation height (Δh), isolated ridge height, average slope of the terrain and the mixed land–sea parameter. Once the terrain related parameters are calculated, the necessary correction factors can be added or subtracted as required. All these correction factors are also available as Okumura curves [Oku68].

Okumura’s model is wholly based on measured data and does not provide any analytical explanation. For many situations, extrapolations of the derived curves can be made to obtain values outside the measurement range, although the validity of such extrapolations depends on the circumstances and the smoothness of the curve in question.

Okumura’s model is considered to be among the simplest and best in terms of accuracy in path loss prediction for mature cellular and land mobile radio systems in cluttered environments. It is very practical and has become a standard for system planning in modern land mobile radio systems in Japan. The major disadvantage with the model is its slow response to rapid changes in terrain, therefore the model is fairly good in urban and suburban areas, but not as good in rural areas. Common standard deviations between predicted and measured path loss values are around 10 dB to 14 dB.

Example 4.10. 

Find the median path loss using Okumura’s model for d = 50 km, h_te = 100 m, h_re = 10 m in a suburban environment. If the base station transmitter radiates an EIRP of 1 kW at a carrier frequency of 900 MHz, find the power at the receiver (assume a unity gain receiving antenna).

Solution

The free space path loss L_F can be calculated using Equation (4.6) as

From the Okumura curves

Amu(900 MHz(50 km)) = 43 dB

and

GAREA = 9 dB.

Using Equation (4.81.a) and (4.81.c), we have

Using Equation (4.80), the total mean path loss is

L50(dB)	=	LF + Amu(f, d) − G(hte) − G(hre) − GAREA
	=	125.5 dB + 43 dB −(−6) dB − 10.46 dB −9 dB
	=	155.04 dB.

Therefore, the median received power is

Pr(d)	=	EIRP(dBm) − L50(dB) + Gr(dB)
	=	60 dBm − 155.04 dB + 0 dB = −95.04 dBm.

The Hata model [Hat90] is an empirical formulation of the graphical path loss data provided by Okumura, and is valid from 150 MHz to 1500 MHz. Hata presented the urban area propagation loss as a standard formula and supplied correction Equations for application to other situations. The standard formula for median path loss in urban areas is given by

Equation 4.82. 

where f_c is the frequency (in MHz) from 150 MHz to 1500 MHz, h_te is the effective transmitter (base station) antenna height (in meters) ranging from 30 m to 200 m, h_re is the effective receiver (mobile) antenna height (in meters) ranging from 1 m to 10 m, d is the T–R separation distance (in km), and a(h_re) is the correction factor for effective mobile antenna height which is a function of the size of the coverage area. For a small to medium sized city, the mobile antenna correction factor is given by

Equation 4.83. 

and for a large city, it is given by

Equation 4.84.a. 

Equation 4.84.b. 

To obtain the path loss in a suburban area, the standard Hata formula in Equation (4.82) is modified as

Equation 4.85. 

and for path loss in open rural areas, the formula is modified as

Equation 4.86. 

Although Hata’s model does not have any of the path-specific corrections which are available in Okumura’s model, the above expressions have significant practical value. The predictions of the Hata model compare very closely with the original Okumura model, as long as d exceeds 1 km. This model is well suited for large cell mobile systems, but not personal communications systems (PCS) which have cells on the order of 1 km radius.

The European Cooperative for Scientific and Technical research (EURO-COST) formed the COST-231 working committee to develop an extended version of the Hata model. COST-231 proposed the following formula to extend Hata’s model to 2 GHz. The proposed model for path loss is [EUR91]

Equation 4.87. 

where a(h_re) is defined in Equations (4.83), (4.84.a), and (4.84.b) and

Equation 4.88. 

The COST-231 extension of the Hata model is restricted to the following range of parameters:

A model developed by Walfisch and Bertoni [Wal88] considers the impact of rooftops and building height by using diffraction to predict average signal strength at street level. The model considers the path loss, S, to be a product of three factors.

Equation 4.89. 

where p₀ represents free space path loss between isotropic antennas given by

Equation 4.90. 

The factor Q² gives the reduction in the rooftop signal due to the row of buildings which immediately shadow the receiver at street level. The P₁ term is based upon diffraction and determines the signal loss from the rooftop to the street.

In dB, the path loss is given by

Equation 4.91. 

where L₀ represents free space loss, L_rts represents the “rooftop-to-street diffraction and scatter loss,” and L_ms denotes multiscreen diffraction loss due to the rows of buildings [Xia92]. Figure 4.25 illustrates the geometry used in the Walfisch Bertoni model [Wal88], [Mac93]. This model is being considered for use by ITU-R in the IMT-2000 standards activities.

Figure 4.25. Propagation geometry for model proposed by Walfisch and Bertoni.

Work by Feuerstein et al. in 1991 used a 20 MHz pulsed transmitter at 1900 MHz to measure path loss, outage, and delay spread in typical microcellular systems in San Francisco and Oakland. Using base station antenna heights of 3.7 m, 8.5 m, and 13.3 m, and a mobile receiver with an antenna height of 1.7 m above ground, statistics for path loss, multipath, and coverage area were developed from extensive measurements in line-of-sight (LOS) and obstructed (OBS) environments [Feu94]. This work revealed that a two-ray ground reflection model (shown in Figure 4.7) is a good estimate for path loss in LOS microcells, and a simple log-distance path loss model holds well for OBS microcell environments.

For a flat earth ground reflection model, the distance d_f at which the first Fresnel zone just becomes obstructed by the ground (first Fresnel zone clearance) is given by

Equation 4.92.a. 

For LOS cases, a double regression path loss model that uses a regression breakpoint at the first Fresnel zone clearance was shown to fit well to measurements. The model assumes omnidirectional vertical antennas and predicts average path loss as

Equation 4.92.b. 

where p₁ is equal to (the path loss in decibels at the reference distance of d₀ = 1 m), d is in meters and n₁, n₂ are path loss exponents which are a function of transmitter height, as given in Figure 4.8. It can easily be shown that at 1900 MHz, p₁ = 38.0 dB.

For the OBS case, the path loss was found to fit the standard log-distance path loss law of Equation (4.69.a)

Equation 4.92.c. 

where n is the OBS path loss exponent given in Figure 4.26 as a function of transmitter height. The standard deviation (in dB) of the log-normal shadowing component about the distance-dependent mean was found from measurements using the techniques described in Example 4.9. The log-normal shadowing component is also listed as a function of height for both the LOS and OBS microcell environments. Figure 4.26 indicates that the log-normal shadowing component is between 7 and 9 dB regardless of antenna height. It can be seen that LOS environments provide slightly less path loss than the theoretical two-ray ground reflected model, which would predict n₁ = 2 and n₂ = 4.

Figure 4.26. Parameters for the wideband microcell model at 1900 MHz.

Buildings have a wide variety of partitions and obstacles which form the internal and external structure. Houses typically use a wood frame partition with plaster board to form internal walls and have wood or nonreinforced concrete between floors. Office buildings, on the other hand, often have large open areas (open plan) which are constructed by using moveable office partitions so that the space may be reconfigured easily, and use metal reinforced concrete between floors. Partitions that are formed as part of the building structure are called hard partitions, and partitions that may be moved and which do not span to the ceiling are called soft partitions. Partitions vary widely in their physical and electrical characteristics, making it difficult to apply general models to specific indoor installations. Nevertheless, researchers have formed extensive data bases of losses for a great number of partitions, as shown in Table 4.3.

The losses between floors of a building are determined by the external dimensions and materials of the building, as well as the type of construction used to create the floors and the external surroundings [Sei92a], [Sei92b]. Even the number of windows in a building and the presence of tinting (which attenuates radio energy) can impact the loss between floors. Table 4.4 illustrates values for floor attenuation factors (FAF) in three buildings in San Francisco [Sei92a]. It can be seen that for all three buildings, the attenuation between one floor of the building is greater than the incremental attenuation caused by each additional floor. Table 4.5 illustrates very similar tendencies. After about five or six floor separations, very little additional path loss is experienced.

Indoor path loss has been shown by many researchers to obey the distance power law in Equation (4.93)

Equation 4.93. 

where the value of n depends on the surroundings and building type, and X_σ represents a normal random variable in dB having a standard deviation of σ dB. Notice that Equation (4.93) is identical in form to the log-normal shadowing model of Equation (4.69.a). Typical values for various buildings are provided in Table 4.6 [And94].

The Ericsson radio system model was obtained by measurements in a multiple floor office building [Ake88]. The model has four breakpoints and considers both an upper and lower bound on the path loss. The model also assumes that there is 30 dB attenuation at d₀ = 1 m, which can be shown to be accurate for f = 900 MHz and unity gain antennas. Rather than assuming a log-normal shadowing component, the Ericsson model provides a deterministic limit on the range of path loss at a particular distance. Bernhardt [Ber89] used a uniform distribution to generate path loss values within the maximum and minimum range as a function of distance for in-building simulation. Figure 4.27 shows a plot of in-building path loss based on the Ericsson model as a function of distance.

Figure 4.27. Ericsson in-building path loss model.

An in-building site-specific propagation model that includes the effect of building type as well as the variations caused by obstacles was described by Seidel [Sei92b] and has been used to accurately deploy indoor and campus networks [Mor00, Ski96]. This model provides flexibility and was shown to reduce the standard deviation between measured and predicted path loss to around 4 dB, as compared to 13 dB when only a log-distance model was used in two different buildings. The attenuation factor model is given by

Equation 4.94. 

where n_SF represents the exponent value for the “same floor” measurement, FAF represents a floor attenuation factor for a specified number of building floors, and PAF represents the partition attenuation factor for a specific obstruction encountered by a ray drawn between the transmitter and receiver in 3-D. This technique of drawing a single ray between the transmitter and receiver is called primary ray tracing. Summing the cumulative partition losses along the primary ray has been shown to yield excellent accuracy while offering tremendous computational efficiency [Mor00, Ski96, Rap00]. Thus, if a good estimate for n exists (e.g., selected from Table 4.4 or Table 4.6) on the same floor, then the path loss on a different floor can be predicted by adding an appropriate value of FAF (e.g., selected from Table 4.5), and then summing the partition losses selected from Table 4.3. Alternatively, in Equation (4.94), FAF may be replaced by an exponent which already considers the effects of multiple floor separation

Equation 4.95. 

where n_MF denotes a path loss exponent based on measurements through multiple floors.

Table 4.7 illustrates typical values of n for a wide range of locations in many buildings. This table also illustrates how the standard deviation decreases as the average region becomes smaller and more site specific. Scatter plots illustrating actual measured path loss in two multi-floored office buildings are shown in Figure 4.28 and Figure 4.29.

Figure 4.28. Scatter plot of path loss as a function of distance in Office Building 1.

Figure 4.29. Scatter plot of path loss as a function of distance in Office Building 2.

Devasirvatham et al. [Dev90b] found that in-building path loss obeys free space plus an additional loss factor which increases exponentially with distance, as shown in Table 4.8. Based on this work in multi-floor buildings, it would be possible to modify Equation (4.94) such that

Equation 4.96. 

where α is the attenuation constant for the channel with units of dB per meter (dB/m). Table 4.8 provides typical values of α as a function of frequency as measured in [Dev90b].

Example 4.11. 

This example demonstrates how to use Equations (4.94) and (4.95) to predict the mean path loss 30 m from the transmitter, through three floors of Office Building 1 (see Table 4.5). Assume that two concrete block walls are between the transmitter and receiver on the intermediate floors. From Table 4.5, the mean path loss exponent for same-floor measurements in a building is n = 3.27, the mean path loss exponent for three-floor measurements is n = 5.22, and the average floor attenuation factor is FAF = 24.4 dB for three floors between the transmitter and receiver. Table 4.3 shows a concrete block wall has about 13 dB of attenuation.

Solution

The mean path loss using Equation (4.94) is

The mean path loss using Equation (4.95) is