This study deals with theoretical and experimental prediction of types of fading, slow and fast, and the total path loss that occur in the urban land radio channel for various elevations of base station and moving subscriber antenna with respect to building rooftops. A new distribution function, which covers the well-known classical Rice and Rayleigh statistical distributions, is derived on the basis of experimental data obtained for various situations in the urban scene and of the stochastic multiparametric model that describes multiray effects that occur in the urban communication link. The total path loss is derived for various urban propagation scenarios, taking into account the corresponding fading caused by shadowing and multipath effects. Radio maps of two experimental urban sites are designed. On the basis of the topographic features of one of these sites, the optimal number of antennas and their positions are predicted to achieve maximum radio coverage of area of service. The obtained results allow us to use the proposed unified stochastic model for slow and fast fading effects estimation in link budget performance and radio map design only on the basis of the topographic map of the corresponding terrain under investigations, the buildings' overlay profile, and the parameters of the base station antennas.
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 As is well known [Bramley and Cherry, 1973; Cox, 1973; Ho et al., 1979; Ikegami and Yoshida, 1980; Okumura et al., 1968; Ponomarev et al., 1984; Turin et al., 1972], to obtain the precise radio coverage of areas of service and to strictly predict the link budget for various land communication links, rural, mixed, suburban and urban, designers of wireless networks need a precise information about the parameters of the channel. Among these parameters are the path loss and long-term (slow) and short-term (fast) fading, as well as the characteristic functions, correlation or structure, of fading in the space and time domains. These parameters of radio channel are strongly correlated with the topography of the built-up terrain. The topography includes buildings' distribution and their density, street orientation inside the grid plan pattern, buildings' overlay profile, the existence of vegetation, trees, and hills, and so on. The features of the terrain cause multipath fading of radio waves arriving at the receiver from different directions and creation of a complicated interference picture with random fluctuations of signal envelope.
 In this study, in section 2, we analyze experiments carried out in different urban environments to show the existence of two kinds of fading, slow and fast, in urban radio communication links. Here, using the experimental data, we investigate both slow and fast fading. A general description of the signal envelope distribution is obtained on the basis of the multiray concept according to a multiparametric model developed and described by Blaunstein [1999, 2005], Blaunstein et al. [2001, 2002], Blaunstein and Christodoulou , and Blaunstein and Levin . We show that the Rayleigh or Rice distributions of the signal envelope are not valid for more than 30% of investigated data, whereas the proposed general description is valid for all measured data covering both of these canonical distributions for specific ranges of the urban radio links. In section 3, we present several more actual scenarios that occur in the urban scene. All equations are obtained taking into account multipath and shadowing effects and allowed us to obtain radio coverage (i.e., radio map) for two different cities on the basis of experimental data and topographic maps of the tested areas [Blaunstein, 1999, 2005; Blaunstein et al., 2001, 2002; Blaunstein and Christodoulou, 2006; Blaunstein and Levin, 1996]. A good agreement is obtained between theoretical prediction of link budget and radio coverage and between the experimental data. The proposed stochastic approach allows us to minimize the number of terminal antennas and to predict their operational characteristics for the total radio coverage of areas of service.
2. Slow and Fast Fading Phenomena Experimental Verification
 Following Blaunstein and Christodoulou , we collected about 3800 samples of signal strength for analyzing the cross-correlation function of the signal amplitude for three components, x, y, z, of the total vector of the wavefield. Each component was measured along the radio path between the terminal located at point r2 and that at point r1, i.e., along the radio path d = ∣r2 − r1∣. These components were averaged using scales of 30–50 wavelengths at the frequency band from 900 to 2400 MHz. This allows us to exclude the effects of fast fading due to random interference between multipath field components. We have obtained the close results, as was shown experimentally by Blaunstein [1999, 2005], Blaunstein et al. [2001, 2002], and Blaunstein and Christodoulou , namely, the existence of two sharp maxima, which correspond to two scales of signal spatial variations, L1 = 20–30 m and L2 = 80–120 m. As was mentioned by Blaunstein [1999, 2005], Blaunstein et al. [2001, 2002], and Blaunstein and Christodoulou , the first scale can be related to the average length of gaps between buildings. It determines the so-called “light zones,” observed within these gaps. The second scale, according to Blaunstein [1999, 2005], Blaunstein et al. [2001, 2002], and Blaunstein and Christodoulou , can be related to the large “dark zones,” which follow the “light zones” and can be explained by the effect of shadowing caused by buildings surrounding both antennas.
 For the purpose of investigating fast fading phenomena, we have analyzed the 3800 samples of the signal strength, which we divided into three separate groups, following the approach described by Blaunstein and Christodoulou . These groups were set according to the type of radio path and antenna elevations with respect to buildings' roofs. About 30 different radio paths were chosen; at each radio path, the measurements of the signal envelope were taken each 1 m for frequencies of 1800 and 2400 MHz and each 3 m for 900 MHz. Therefore the gaps between each point of measurement are much smaller than the scale of slow fading mentioned above. The analysis results are presented in the form of histograms (see Figure 1).
 The first group of signal outputs was measured at radio links with obstructive conditions for low elevated terminal antennas with no line of sight (NLOS) between them. The second group of signal outputs was obtained for radio links where both LOS and NLOS conditions were observed between the terminal antennas. In this case, one of the antennas (usually the base station antenna) was higher than the rooftops. The third group of signal outputs was assembled from the same measurements as the last ones but for radio links with LOS conditions.
 The measured output samples for the second and third groups were combined together (approximately 1000 samples for the second group and 800–900 samples for the third group). The corresponding statistical analysis are presented as a histogram in Figure 1a versus the same nondimensional parameter x(t) = r(t)/A, where A is the amplitude of the LOS component. The solid curve in Figure 1a describes the calculations of the Ricean distribution of fast signal variations, obtained by using the information about σ from the statistical analysis of the experimental data r(t) and the information about the regular signal component A [Blaunstein and Christodoulou, 2006]. A correlation is evident between the distribution of the signal strength, which was obtained experimentally and is presented in Figure 1a by a histogram, and between the theoretical predictions (the continuous curve) that is described by Rice distribution function. The agreement between the experimentally obtained histograms and theoretical prediction was observed for about 76% of the samples of the second and third groups of tested measurements of the signal strength envelope.
 Statistical analysis of the first group of outputs (approximately 1200 samples of the magnitudes of the signal strength) is presented as a histogram in Figure 1b versus the normalized parameter x = r(t)/σ, where r(t) is the magnitude of the signal envelope and σ is the standard deviation of the signal envelope with respect to its mean value. The corresponding solid curve presented in Figure 1b was derived from the Rayleigh distribution function where the standard deviation of the signal envelope was found experimentally from measured expected values of the random signal strength r〉 as [Blaunstein and Christodoulou, 2006].
 A good agreement is evident between the distribution of the signal strength, which was obtained experimentally and is presented in Figure 1b by a histogram, and between the theoretical predictions (the solid curve) that is described by Rayleigh law. The agreement between the experimentally obtained histograms and theoretical prediction was observed for about 82% samples of the first group of tested measurements of the signal strength envelope.
 However, as was found experimentally, 30%–35% of about 3800 measurement samples cannot be described by the Rayleigh-Rice statistics [Blaunstein, 1999, 2005; Blaunstein et al., 2001, 2002; Blaunstein and Christodoulou, 2006; Blaunstein and Levin, 1996]. Thus, in the first group, only about 800 of 1200 measured samples fit the Rayleigh statistics description and were analyzed. The same situation occurs with the second group, where only about 1000 samples of 1200 satisfy the Rice law. Finally, from the third group, only about 800–900 samples of 1300 could be analyzed using Rice statistics. In order to explain the mechanism of deep signal strength variations, we have used the unified stochastic approach [Blaunstein, 1999, 2005; Blaunstein et al., 2001, 2002; Blaunstein and Christodoulou, 2006; Blaunstein and Levin, 1996], taking into account multiple reflection and scattering effects from buildings and other obstructions, natural or man-made, randomly distributed, at the terrain, according to Poisson law. As was obtained by Blaunstein  and Blaunstein et al. , only single and double scattered waves must be taken into account for microcell communication channels with ranges less than 2–3 km. Moreover, the magnitudes of wavefields, as complex values, are normally distributed with zero-mean value and dispersion σ12 (for single scattered waves) and σ22 or two-time scattered waves). The average number of waves depends on the characteristic features of the built-up terrain: buildings density ν per km2, average buildings' length , and buildings' contour density γ0 = 2ν/π per km. The average wave number also depends on the distance from BS antenna, d. Thus, according to Blaunstein  and Blaunstein et al.  we getfor average number of single scattered waves:
for average number of double scattered waves:
where Kn (γ0d) is the MacDonald's function of n order.
 The probability of receiving single to double scattered waves at the mobile station (MS) antenna is computed according to the following equation [Blaunstein and Christodoulou, 2006]:
Therefore, in our analysis of experimental data of more than 1500 samples, for the first to the third group of measurements, where the fast strong signal strength variations are observed, we constructed a distribution function for each selected group of rays. We took into account the effects of independent single scattering (the first term in (3)) and two-time scattering (the second term in (3)), as well as their mutual influences on each other (the third term in (3)), i.e.,
where P0 = 1 − (1 − P1)(1 − P2) = 1 − exp [−(1 + 2)] is the probability of direct visibility, P1 and P2 are defined by equation (2) combined with (1a) and (1b), respectively.
 From Figure 1 we observe that for d < 1 km, the general distribution (3) and its parameters that were obtained from the measurements (see curves 1 and 2 in Figure 1d) differ from the Rayleigh and Rice distributions for d < 0.5 km. At the same time, for d = 1.5 km, the general distribution calculated using (3) and indicated by curve 1, and the Rayleigh distribution, indicated by curve 2, are very close. Thus at large distances from the BS antenna, the effect of multipath becomes predominant.
3. Link Budget Design for Different Scenarios of Multipath Fading
 In link budget performance, we use the path loss, as an average characteristic of signal energy attenuation (in decibels, dB) and effects of fading caused by shadowing due to diffraction from building rooftops or corners and multiple scattering from buildings' walls. The parameters of antennas, base station (BS), and mobile station (MS) can be easily included in the equations obtained below.
3.1. First Scenario: Quasi-LOS Condition
 In this scenario, depicted in Figure 2, the total path loss (in dB) can be presented in the following manner:
where h1 and h2 are the minimum and maximum heights of built-up terrain in m, r is the range between the terminal antennas in km, and z2 and z1 are the height of the BS and MS antenna, respectively. Here γ0 = 2ν/π is the buildings' contour density in km−1, is the average length of buildings in km, ν is the number of buildings per square kilometer, and GBS and GMS are the antenna gains for the BS and MS, respectively. In urban areas, depending on the building's density, γ0 ranges between 5 × 10−3 m−1 and 10 × 10−3 m−1.
 The function F(z1,z2), in meters, describes the buildings' overlay profile surrounding two terminal antennas. It can be simply evaluated following Blaunstein et al.  for two typical cases:
 For urban areas with an approximately equal number of tall and small buildings, n = 1; when the number of tall buildings is dominant (Manhattan grid plan), n = 0.1−0.5; when the number of small buildings prevails (suburban and residential areas), n = 5−10. In most Israeli cities, n = 1 is a very precise value. In Lisbon, for example, this value varies from 0.89 to 1.17, which is close to n = 1 [Blaunstein et al., 2001]. Therefore we propose to use n = 1 for areas which are not the same as the Manhattan grid scenario. In the case of n = 1, the average building height equals
3.2. Second Scenario: Non-LOS Conditions With Single Diffraction
 In the second scenario, as shown in Figure 3, diffraction from the roof close to the MS antenna is the source of shadowing and the slow fading phenomenon. Here we have
In (9) the parameters units are meters and are the same as above, F2 (z1,z2) = ( − z1)2; lν is the parameters of walls' roughness, usually equal to 1–3 m; ∣Γ∣ is the absolute value of the reflection coefficient, which equals ∣Γ∣ = 0.4 for glass, ∣Γ∣ = 0.5–0.6 for wood, ∣Γ∣ = 0.7–0.8 for stones, and ∣Γ∣ = 0.9 for concrete [Blaunstein and Christodoulou, 2006; Rappaport, 1996; Bertoni, 2000].
 The wavelength of the radio wave has a wide range that varies from λ = 0.05 m to λ = 0.53 m and covers most of the modern wireless networks. Equation (8) with equation (9) can be used for link budget design for various scenarios that occur in the built-up terrain and for high-elevated BS antenna with respect to building profile.
3.3. Third Scenario: Non-LOS Conditions With Multiple Diffraction
 In this type of situation, depicted in Figure 4, diffraction from roofs close to the MS antenna and BS antenna are the sources of shadowing and the slow fading phenomena. Here we have
where all parameters are the same as above and are measured in meters. In this situation, the profile function can be presented following Blaunstein et al.  as
As above, equation (10) with equations (11)–(12) can be used for link budget design for various scenarios of the built-up terrain, for both BS and MS antennas lower than the rooftops.
3.4. Fourth Scenario: Communication Along the Street With Two Antennas Below the Rooftops
 This scenario describes a situation when both antennas are located in direct visibility along a street at the range of r = 100–1000 m. The width of the street is a = 10–20 m, the wave band of the transmitter/receiver is λ = 0.01 − 0.05 m, and the parameter of brokenness is X = /( + ), which describes the distribution of buildings lining the street according to Blaunstein  and Blaunstein and Levin . The parameter of brokenness is defined by the ratio between the average length of building and the sum of average length of slits plus the average length of buildings , as shown in Figure 5. In the urban scene, such as Manhattan grid plan, X = 0.5 − 0.8.
 Here the heights of antennas are lower or at the same height compared with the average buildings' height. In this situation in the urban scene, the total path loss (in dB) is [Blaunstein, 1999; Blaunstein and Levin, 1996]
where r, a, and λ are in meters.
3.5. Fifth Scenario: Communication Along the Street With One Antenna Above the Rooftops
 In this case, as shown in Figure 6, the base station (BS) antenna, with the height z2, is higher than the buildings' overlay profile maximum height h2, but the moving subscriber (MS) antenna with the height z1 is lower than h2. In such a situation, the waveguide effects become small enough, and the effect of building array randomly distributed at the terrain must be taken into account by using the proposed stochastic approach [Blaunstein, 2005; Blaunstein et al., 2001].
 However, here we must take into account the effects of buildings lining the street and placed within the layer denoted by “1” in Figure 7 in both parts of the street. (Because of the symmetry in Figure 7 we depict only one side of the building layer.)
 For all other buildings located in area “2” (see Figure 7), we can use the stochastic description and equations (4)–(8) for the first scenario or (9)–(10) for the second scenario. The additional term, which accounts for effects of buildings lining the street in area “1,” cannot be used additively when we present all equations in decibels. Therefore we can rewrite Lfading using the following equations.
 For the first scenario, accompanied by the buildings lining the street, we get
where is the average buildings' height defined in meters by (3.4), lν is the parameter of roughness of buildings' walls (usually lν = 1 − 3 m), and k = 2π/λ.
 For the second scenario, accompanied by the buildings lining the street, we get
As above, all parameters are in meters. Using expression (14) or (15), we can finally introduce them in one of the equations (4) or (9) according to the corresponding scenario that occurs in the urban scene.
4. Radio Coverage Design in Multipath Fading Urban Channels
 Let us compare the results of the theoretical prediction of link budget within different urban communication links with experimental data. Here we use as an example the radio map design based on propagation characteristics of a test site area with specific topographic features of built-up terrain and elevations of the base station and mobile subscriber antenna with respect to the buildings surrounding it.
4.1. Ramat-Gan Stock Market Radio Map Design
 As seen in Figure 8, the Ramat-Gan stock market area is a very dense area. It can be easily prototyped as a typical “downtown” area with tall buildings that are very close to each other.
 The first stage of the simulation was to process the terrain data. We have measured the dimensions of each building near the antenna, and using equation (7), we calculated the buildings' average height; this was done for 72 different radio paths. We have generated each beam with 5° separation.
 In Figure 9 the antenna locations are presented in the stock market area of Ramat-Gan; the dotted lines represent the 72 radio paths from BS to any subscriber located at the test site area. Then we have estimated the number of buildings per km2 in the test site area and their average length or width (related to the direction of the antenna main lobe orientation). Then the buildings' density γ0 of the test site area was calculated according to the definition γ0 = 2ν/π and is presented in Figure 10. In Figure 10 the different magnitudes of γ0 are shown on the right; each magnitude is assigned a different shade. The maximum density of the buildings is more than 12 km−1; it is observed at the center of the market site of Ramat-Gan.
 The antenna characteristics, such as “sector” size, tilt, height (z2), effective power (“Eff”), and gain (“G”), as well as the terrain parameters, such as γ0, the distance d between BS and MS, and the step Δd along the radio path between BS and MS (used for this simulation), are presented in Table 1. Notice that in the first column, the values represent the angle of the sector (in degrees) with respect to 0° (vertical axis).
 The simulation running constants and variables are the following:
Finally, using the multiparametric stochastic model and its corresponding equations (8) and (9), we have created a radio map for the test site area of an 800 m radius (with respect to the BS antenna location). The radio map is depicted in Figure 11, with the path loss (in dBm) presented on the right by colored segments corresponding to the obtained magnitudes of path loss.
Figure 11 was depicted using equations (8) and (9), since the BS antenna is elevated above the built-up profile average building height of = 18 m.3. As seen from Figure 11, in the vicinity of the BS antenna (100–300 m), the path loss varies from −100 dBm to −120 dBm, whereas far from the BS antenna (at 500–800 m) it varies from −150 dBm to −180 dBm.
4.2. Ben-Gurion University Radio Map Simulation
 The same computation process was done to simulate the premises of Ben-Gurion University, with one major difference. Here we did not simulate γ0; rather, we have divided the university into five major parts, as depicted in Figure 12, which are different in their buildings' density. Each part has its own general building density. This decision was done on the basis of the university's topographic plan.
 For section II, γ0 = 6 × 10−3 m−1 , for sections I, III and IV, γ0 = 3 × 10−3 m−1, and for section V, γ0 = 4 × 10−3 m−1. The average building height is = 14.5 m. The antenna characteristics used for the simulation presented in Table 2.
Table 2. Ben-Gurion University Simulation Parametersa
Tilt (α), deg
Effective Power, dB
Read 3E-3 as 3 × 10−3.
3E-3, 4E-3, 6E-3
3E-3, 4E-3, 6E-3
 Two sectorial antennas with sector numbers 52, 400, and 408 cover the area of the university (see Figure 13). In Figure 14, the radio coverage of the university test site area was made using these two sectors (the path loss is described on the right in dBm by the corresponding color).
 The results presented in Figures 13 and 14, show that two sectors cannot fully cover the university campuses. Therefore, according to our simulations we have proposed to assemble the third antenna. The third antenna location and sector 468 are shown in Figure 12. Now using three sectors, according to our theoretical prediction, based only on the topography of the tested area, full radio coverage was obtained (see Figure 15) for each subscriber located in the university area of service.
 The radio coverage improvement for stationary communication for any subscriber located in the area of service is evident. The decrease of loss is about 30–40 dBm compared with the results obtained by using two sectorial antennas and are presented in Figure 14. In order to obtain such results, we also propose to rearrange the antennas in order to receive full radio coverage of the area of service. Optimal arrangements can be done using the proposed theoretical predicting algorithm for performing link budget for various urban areas.
 In this study, we investigated the nature of slow and fast fading of the total received signal for mobile wireless antenna in urban communication links. We relied on the unified multiparametric stochastic approach, which was verified by numerous experiments carried out in different urban and suburban areas [Blaunstein, 1999, 2005; Blaunstein et al., 2001, 2002; Blaunstein and Christodoulou, 2006; Blaunstein and Levin, 1996]. According to this approach, the average intensity and path loss were analyzed in the space domain, taking into account slow and fast fading. In order to understand these phenomena, a new cumulative distribution function (CDF) was created. The CDF explains the complicated spatial variations of the signal strength envelope. On the basis of multipath fading presentation, this function covers Rayleigh distribution in NLOS conditions and covers the Ricean distribution in quasi-LOS conditions and lognormal (or Gaussian) distribution in LOS conditions, which occur in the urban scene for various elevations of terminal antennas with respect to buildings' overlay profile.
 The obtained equations can predict different propagation situations in various urban channels for different heights of the BS and MS terminal antennas. Furthermore, the proposed equations can a priori predict the type of fading that occurs in various scenarios in the urban and suburban areas using the corresponding topographic maps. They also predict the total path loss and, finally, give the radio coverage of each area, as well as the minimum number of BS antennas needed for this purpose and their position, tilt, sector, and height with respect to the overlay profile of the surroundings.