User talk:William C. Y. Lee/Proposed/Lee microcell prediction model

This paper focuses on theoretical and technical aspects of the algorithm of Lee Point-to-point prediction model also called Lee Macrocell Prediction Model [1]. It was created in 1977 at Bell Labs and used in deploying cellular systems in AT&T markets in 1983. Once the locations of the base station and the mobile are determined, based on the parameters of components at base station and mobile unit together with terrain contour data, Lee model provides a fairly accurate prediction on the received signal at both the mobile and the base station.

Introduction

The calculation of macrocell coverage is based on the signal propagation prediction techniques. The Lee “point–to–point” model predicts each local mean of the received signal strength at the mobile terminal based on the radio path between the base station and the mobile unit, and plots all the predicted local means along a path in a mobile cellular environment. The point-to-point model is a very cost-saving tool for planning a system in a large coverage area. Otherwise, selecting the proper sites of base stations based on the measured data in a large area is very costly and labor intensive.

The predicted signal strengths also can be calculated on a sector–by–sector basis. It is used as an area-to-area model or point-to-area model. A recommendation is to use a sector of 60° or 120°. In each sector, we may predict the local means along every 0.25° or 0.5° radial increment within a sector; also use a small distance (radial length) increment along each radial line. The prediction of local means for each sector is stored in a data file. These files can be used to support most analyses, such as to generate a neighboring cell list for Hand-off, frequency planning, traffic demand distribution, hardware dimensioning, etc.

The Lee model uses a reference frequency of 850 MHz, however, the model has been validated with measured data within the frequency range from 150 MHz to 2400 MHz.

Measurements taken from over 500 drive tests conducted in Europe, Asia and the United States have been used to test the Lee model’s implementation. These data cover many different cities and many different characteristics of morphologies – such as dense urban, urban, suburban and rural. All the measured data were recorded based on the local mean [2]. The terrain contour maps [3][4] corresponding to real roads were used. Those drive tests were compared with the predictions from the model to verify and improve it.

The Lee Macrocell Model – A Point-to-Point Model [5][6][7]

The Lee Point-to-Point Model is given by the following equation: \[ \begin{align} P_r &= P_{r_0} - \underbrace{ \overbrace{ \gamma log\left(\frac{r}{r_0}\right) }^1 + \overbrace{ G_{effh}(h_e) }^2 - \overbrace{ L }^3 - \overbrace{ A_f }^4 + \overbrace{ \alpha }^5 }_\text{MODEL COMPONENTS} \tag{1} \end{align} \] where \[ \begin{align} P_r &= \text{the received power in dBm} \\ P_{r_0} &= \text{received power at the intercept point } r_0 \text{ in dBm} \\ \gamma &= \text{slope of path loss in dB per decade} \\ r &= \text{distance between the base station and the mobile unit in miles or kilometers} \\ r_0 &= \text{distance between the base station and the intercept point in miles or kilometers} \\ h_e &= \text{effective base station antenna height in feet or meters} \\ h_2 &= \text{mobile antenna height in feet or meters} \\ h_1' &= \text{actual antenna height at the base station} \\ h_e' &= \text{maximum effective antenna height} \\ A_f &= \text{frequency offset adjustment in dB from the default frequency } f_0 \\ &= 20log\left(\frac{f}{f_0}\right) \tag{2} \\ G_{effh}(h_e) &= \text{gain from effective antenna heights } h_e \text{ due to the terrain contour} \\ &= 20log\left(\frac{h_e}{h_1'}\right) \quad \quad \quad \quad \quad \quad \text{(under a non-shadow condition)} \tag{3} \\ Max. G_{effh}(h_e') &= \text{maximum effective antenna height gain} \\ &= 20log\left(\frac{h_e'}{h_1'}\right) \tag{4} \\ L &= \text{(knife-edge diffraction loss) } - \text{(max effective antenna height gain)} \\ &= L_D - 20log\left(\frac{h_e'}{h_1'}\right) \geq 0 \quad \quad \quad \quad \quad \quad \text{(under a non-shadow condition)} \tag{5} \end{align} \]

The maximum effective antenna height h'_e is measured as the height at the base station from the intersect point of a line that is drawn from the tip of the hill along the slope of the hillside to the base station. \[ \begin{align} \alpha &= \text{the signal adjustment factor in dB, such as an additional gains if the actual} \\ &\phantom{{}={}} \text{antenna gains } g_b' and g_m', \text{ and antenna heights } h_1’ and h_2’ \text{ at two terminals} \\ &\phantom{{}={}} \text{are different from the standard conditions} \\ &= (g_b' - g_b) + (g_m - g_m') + 20log\left(\frac{h_1'}{h_1}\right) + 10log\left(\frac{h_2'}{h_2}\right) \\ &= \Delta g_b + \Delta g_m + \Delta g_{h_1} + \Delta g_{h_2} \tag{6} \end{align} \]

Model Components – For a Point-to-Point Model

The signal strength value given by the Lee point-to-point model is composed of five components. These are briefly described as follows.

1. The area–to–area path loss, used as a baseline for the model, is derived from a propagation slope ($\gamma$) and 1–mile (or extrapolate to 1-km if not measured at 1-km) intercept value (P_r0). P_r0 may be obtained from the measured data. Because P_r0 and $\gamma$ vary from city to city due to the man- made structures and only measured data can give the answer. The effect due to the man-made structures is accounted for in the model.
2. The effective antenna height gain, G_effh, is determined by the terrain contour between the base station and the mobile where a specular reflection point is located. This component is significant for the case of a non–obstructed direct signal path, including LOS and NLOS paths. It accounts for the terrain-contour effect into the model.
3. Diffraction loss, L, is predicted using the Fresnel–Kirchoff diffraction theory. For multiple knife-edges, Lee uses both a modified Epstein–Petersen method and a separated knife-edge check to evaluate diffraction loss. This component is significant for the condition of an obstructed direct signal path or called diffraction path. It is also the effect due to the terrain-contour and is accounted into the model.
4. The area–to–area component includes a frequency-offset adjustment, A_f, which is used to adjust the actual center frequency of an actual system to the model's reference frequency of 850 MHz. (See Table 3)
5. An adjustment factor compensates for the difference between a set of default conditions for a base station transmitter with mobile parameters (which are used as assumptions for calculating the area–to–area path loss component) and the actual values for these parameters in each sector to be predicted.

Once received signal strength values have been predicted for each point along the signal path (radial), Lee model can use a signal smoothing process to produce the final prediction.

Signal Path Conditions

Figure 1: Direct path and shadow region

A point–to–point model constantly calculates along the signal path for path loss and checks for signal obstructions. There are three conditions for the direct path. When the direct path is not obstructed by the terrain, it has two conditions—i.e., the line–of–sight (LOS) path and the non-line-of-sight (NLOS) path. Also there is a diffraction path that is obstructed by the terrain.

1. Non–obstructed direct path, also called Line-of-Sight (LOS) path— When a direct path is not obstructed by any objects between the transmitter and the mobile, and the reflected path from the ground is weak, the free space path-loss formula will be used. If the mobile is very close to the ground, the reflected path from the ground is strong, then the open area case is considered.
2. Non-Line-of Sight (NLOS) path – When the direct path is obstructed from the man-made structures and trees— This is a very common condition in mobile communications. Under this condition, there is a gain or loss according to the terrain contour along the radial path. The effective antenna height gain is a significant contributor to the received signal. This gain varies according to the terrain contour along the radial path. The effective antenna height gain is represented by G_effh(h_e) in Eq.(3).
3. Diffraction path — When the direct path is obstructed by the terrain contour between the transmitter and the mobile, the signal actually incurs a loss due to terrain–related diffraction. This diffraction loss is designated by symbol L in Eq.(5). The direct path and a shadow region are illustrated in Figure 1. In the area of transition from the shadow region to the clear region, the non-obstructed direct path only appears in the clear region, and the signal received at the mobile may have two wave components, a direct path and a reflected path. The reflected path occurs if the effective reflective point exits clear from the shadow region and an effective antenna height gain is yielded from the reflected wave.

Both the effective antenna height gain and diffraction loss are also known as "attenuation adjustments". The calculation of these adjustments depends on terrain elevation effects.

Engineering Five Components

First Component of the Model - Area-to-Area Path Loss (Due to Man-Made Effect on Structures)

There is a number of possible path-loss curves measured in different areas used as references in predicting the area-to-area path loss in the modeling of a macrocell. The Lee Single Breakpoint Model uses the suburban area-to-area path-loss curve as its starting reference curve.

The path-loss curve is derived from the slope $\gamma$ and 1-mile intercept values (P_r0), which are used to predict the received signal strength (P_r) at each mobile point along the radio path from the base station to the mobile. The one-mile intercept value is based on the given ERP at the base station, the antenna heights of both at the base station and at the mobile.

Figure 2: Path loss reference curve (suburban)

The slope and intercept values, which can be adjusted by the user, are defined as follows:

• The slope indicates signal path-loss along the radio path in decibels per decade (dB/dec). The suburban curve uses a default value of 38.4 dB/dec for the path-loss slope over land shown in Figure 2.
• The 1–mile intercept indicates the signal level (dB m) received, under the standard conditions as shown below, at a distance of one mile from the transmitter. The reference value for this 1-mile intercept is –61.7 dB m shown in Figure 2.

The standard conditions assumed for the path-loss prediction include the following base station transmit power and mobile parameters: \[ \begin{align} P_r &= \text{base station transmit power} \\ &= \text{10 Watts} \\ h_1 &= \text{base station antenna height} \\ &= \text{100 feet (∼ 30.5 meters)} \\ h_2 &= \text{mobile antenna height} \\ &= \text{10 feet (∼ 3 meters)} \\ g_b &= \text{base station antenna gain} \\ &= \text{6 dBd (dB over dipole)} \\ g_m &= \text{mobile antenna gain} \\ &= \text{0 dBd (dB over dipole)} \end{align} \]

The Lee model uses the standard base station antenna height to determine effective antenna height gain. The model also uses the actual base station antenna height to determine the diffraction loss.

An adjustment factor is also applied to the signal strength prediction to fine tune for the difference between the standard condition values and the values of the actual parameters.

Significance of the 1–Mile Intercept

Figure 3: Area-to-area path loss reference curves

The 1–mile intercept (P_r0) used by the Lee Single Breakpoint model is an initial parameter at which the signal received, under standard conditions, at a distance of one mile (1.609 km) from the base station. There are four main reasons the model uses a 1–mile intercept for predicting propagation for near–in distances, as follows:

1. Within a radius of one mile, the antenna beam width is narrow in the vertical plane; this is especially true of high gain Omni–directional antennas. Thus, the signal reception is reduced at a mobile less than one mile away, because of the large elevation angle. This angle causes the mobile to be in the shadow region outside the main beam. The larger the elevation angle, the weaker the reception level due to the antenna's vertical pattern.
2. There are fewer roads within a radius of one mile around the base station, and the data is insufficient to create a statistical curve. Also the road orientation—both in–line and perpendicular—close to the base station can cause a significant difference (from 10 dB to 20 dB) in signal reception levels on those roads.
3. The nearby surroundings of the base station can bias the reception level—either up or down—when the mobile is within the 1–mile radius. When the mobile is more than one mile away from the base station, the effect due to the nearby surroundings of the base station become negligible.
4. For a land–to–mobile propagation, the antenna height at the base station strongly affects mobile reception in the near field; therefore, the mobile reception at one mile away can be referred to a given base station antenna height.
5. For distances of less than one mile, Lee Macrocell model projects the path-loss curve predicted by the Single Breakpoint model extending backward from one mile to the base station.

Slope and Intercept Reference Values

Slope and intercept values for a specific city can be obtained from the mean value of measured data. There are slopes and intercept values for the major cities in a list. For future predictions in similar areas or cities as in the list, we may copy the slopes and intercept values without further measurement data. The available values from Table 1. shown below can be used.

 

Table 1 Slope and Intercept Reference Values

Environment	Pr0 (1-mile intercept)	(path loss slope)
Free space	-45 dBm	20dB/dec
Open area	-49 dBm	43.5dB/dec
Suburban	-61.7 dBm	38.4dB/dec
Newark	-64 dBm	43.1dB/dec
Philadelphia	-70 dBm	36.8dB/dec
Tokyo, Japan	-84 dBm	30.5dB/dec
New York City	-77 dBm	48.0dB/dec

 

The measured data shown in Table 1 can be illustrated by plotting signal loss against distance on a logarithmic scale, as the area–to–area path loss curves shown in Figure 3. Some cities have similar man-made structures as those cities shown in the chart. They can use the value of that similar city. Therefore, the Area-to-Area Prediction Model is based on the man-made structures in different environments, such as open areas, suburban, urban, cities. In the Area-to-Area Model, the terrain contour is not considered, but it will be considered by the Point-to-Point Prediction Model as shown in the following section.

Second Component of The model - Effective Antenna Height Gain (Due to The Terrain-contour Effect)

When the direct path from the base station to the mobile is not obstructed by terrain, the received signal consists of direct and reflected waves. In this situation, the adjustment of received signal strength due to a change in antenna height at the base station does not depend solely on the actual antenna height above the local ground level. Rather, it depends on the effective antenna height , h_e, as determined by the terrain contour between the base station and the mobile.

Once the effective antenna height is found, we can use it with the standard condition antenna height, h₁, to calculate the effective antenna height gain, G_effh, from the Lee point-to-point model shown in Eq.(3).

Determining Effective Antenna Height

Figure 4: Effectual antenna height gain G_effh - positive gain

Figure 5: Effectual antenna height gain G_effh - negative gain

Effective antenna height is determined by deriving from a specular reflection point, which can be found using the following parameters:

• Base station antenna height, h₁
• Mobile antenna height, h₂
• Distance from the base station to the mobile, d
• Terrain elevations between the base station and the mobile

Note that both the base station antenna height and mobile antenna heights are considered. Both antennas are vertically lined up with y-axis, not perpendicular to the ground slope, as drawing on the figure. It is because for illustration that the scales of antenna heights and the scale of the ground are not the same. If the scales of x-axis and y-axis are the same, then both antennas are almost perpendicular to the ground.

Effective antenna height is determined using the following steps:

1. Find the specular reflection point.
   1. Connect the negative image of the transmitter antenna to the positive image of the mobile antenna; the intercept point at the ground level is considered as a reflection point, R₁.
   2. Connect the negative image of the mobile antenna to the positive image of the transmitter antenna; the intercept point at the ground level is also considered as a reflection point, R₂.

   Lee Model uses the reflection point found closest to the mobile as the specular reflection point.

2. Extend a ground plane—tangent to the average elevation along the terrain contour at the specular reflection point—from the specular reflection point back to the location of the base station (transmitter) antenna.
3. Measure the effective antenna height, h_e, from the intersection of the extended ground plane and the y-axis on which the vertical mast of the base station antenna is located.

Calculating Effective Antenna Height Gain

The effective antenna height gain calculation considers the terrain contour and the relation of effective antenna height, h_e, to the standard condition antenna height, h₁, of 100 feet (∼30.5 meters). Lee model considers four different cases—i.e., 1) terrain sloping is upward when h_e > h₁; 2) over a flat terrain when h_e > h₁; 3) terrain sloping is downward when h_e < h₁; 4) over a flat terrain when h_e < h₁. These cases are illustrated in Figure 4 and Figure 5.

The formula of the effective antenna height gain from Lee model is shown in Eq.(3) when the actual antenna height is the same as the standard antenna height ( h'₁ = h₁) as follows: \[ \begin{align} G_{effh} &= 20log\left(\frac{h_e}{h_1}\right) \tag{7} \end{align} \]

Figure 6: Influence of terrain contour on G_effh and signal strength

1. For a terrain sloping upward or for a flat terrain when h_e > h₁,

   Eq.(7) results in a positive gain (G_effh > 0 dB).

2. For a terrain sloping downward or for a flat terrain when

   h_e < h₁, Eq.(7) result in a negative gain (G_effh < 0 dB). If h_e < h₁/10, then

   h_e is forced to cap at h₁/10.

Figure 6 illustrates the path loss prediction which is obtained based on the area-to area path loss curve adding or subtracting the effective antenna height gain at each local point due to the influence of the local terrain contour, and as a result, the overall point–to–point signal strength prediction is plotted for each local point along the mobile path. Note that the variation in prediction can be significant, as shown at points D through G in the figure.

Third Component of The Model - Diffraction Loss From Diffraction Path

Single knife-edge Case

Lee Model uses Fresnel–Kirchoff diffraction theory [8] to predict the diffraction loss component, L , in Eq.(5) for the Lee Single Breakpoint model. The diffraction loss L consists of two parts. One is based on the knife-edge diffraction loss L_D and the other is the correction factor due to the loss from a real obstacle, which shape is not a knife-edge. The correction factor is obtained from the effective antenna height will be described in the next section. The knife-edge diffraction loss LD is obtained based on a dimensionless parameter v, a diffraction factor, given by \[ \begin{align} v &= (-h_p)\sqrt{\left(\frac{2}{\lambda}\right) \left(\frac{1}{r_1} + \frac{1}{r_2}\right)} \tag{8} \end{align} \] where (parameters expressed in Figure 7) \[ \begin{align} \lambda &= \text{wavelength} \\ r_1 &= \text{distance from the base station to the knife–edge } (r_1' \approx r_1) \\ r_2 &= \text{distance from the knife-edge to the mobile } (r_2' \approx r_2) \\ h_p &= \text{height of the knife–edge which can be above or below the line} \\ &\phantom{{}={}} \text{that connects the base station and mobile antennas.} \\ \end{align} \]

Figure 7: Determining for the diffraction loss calculation

Once the value of diffraction factor v is obtained, the knife-edge diffraction loss L_D(v) can be found from Table 2 below. It gives an approximation of this curve for different values of the diffraction factor, v.

The method for calculating the height of a knife–edge obstruction is discussed earlier. When a signal is blocked by multiple knife–edges, Lee model evaluates the parameters for all knife–edges together and treats each one separately.

 

Table 2 Diffraction loss (L_D) values [9]

$1 \le v$	$L_D = \text{0 dB}$
$0 \le v < 1$	$L_D = 20log(0.5 + 0.62v)$
$-1 \le v < 0$	$L_D = 20log(0.5 e^{0.95v})$
$-2.4 \le v < -1$	$L_D = 20log\left(0.4 \sqrt{0.1184 - \left(0.1v + 0.38\right)^2}\right)$
$v < -2.4$	$L_D = 20log\left(\frac{-0.255}{v}\right)$

 

In a Real Situation - When Diffraction Is From a Real Hill

In past years, engineers had found that the Fresnel diffraction-loss formula always gave calculated values more pessimistic than the measured data. This is because the diffraction loss formula is based on the knife-edge scenario, however the curvature of the earth is not a knife-edge. Also the bending of the curvature is different from every hilltop. The knife-edge diffraction formula does not include the curvature factor. Therefore, in this section, a method [10] to predict the diffraction loss L more realistic is presented.

The slope of the hillside can gauge the bending of the curvature of the hilltop. In Lee Prediction Model, the slope of the hilltop is used to calculate the effective antenna gain, G_effh, when the mobile is in a non-shadow area. In the area of transition to or from shadow, the diffraction loss is affected by effective antenna height gain, G_effh, also. Therefore, use the G_effh to adjust the diffraction loss from the formula is the right approach.

In logically thinking, the G_effh of a mobile at top of different hills are different due to the hillside slope. The more steep is the hillside, the more gain is the G_effh. From the knife-edge diffraction loss formula, the loss is obtained based on the diffraction parameter v which is a function of r₁, r₂, h_p and $\lambda$ only, as shown the formula in Eq.(8). This equation does not involve the parameter of curvature. Therefore, we have added a correction factor to the knife-edge diffraction loss.

Now the prediction tool has taking the Max G_effh shown in Eq.(4) into the calculation of diffraction loss (Eq.(5)) as, \[ \begin{align} L &= L_D(v) - Max G_{effh} \tag{5'} \end{align} \] Where Max G_effh is the effective antenna height gain, G_effh, calculated from a maximum effective antenna height h'_e which is measured the height at the base station from the intersected point of a line that is drawing from the tip of the hill along the slope of the hillside to the base station [10] . In Figure 7, h'_e is depicted in the figure. Now comparing the calculation from Eq.(5') with the measurement data, we have found a great match.

Multiple Knife–Edge Case with Single Knife–Edge Check

When a signal path is blocked by multiple knife–edges, Lee model compares diffraction losses calculating from adding all knife–edges together and from each one separately. The greatest loss value is then used for the diffraction loss component of the signal strength prediction.

The multiple knife–edge evaluation involves these steps:

1. For each knife–edge, Lee model calculates the obstruction height (h_p) using the Epstein–Petersen method [11]. Lee models consider two different scenarios—when there are two knife–edges and when there are three or more knife–edges.
2. The parameter v is evaluated for each single knife–edge assuming no other knife-edge, and the individual single knife-edge diffraction loss is computed from the appropriate formula in Table 2.
3. The individual values L_i of two or three knife-edge are calculated as shown in Table 2 and summed the individual values for all knife–edges as a total diffraction loss (L_t) using the formulas in Table 2.
4. The “single knife–edge check” is then used to compare the greatest diffraction loss caused by any individual knife–edge with the total diffraction loss. Lee model uses the greatest of these values for the diffraction loss component of the signal strength prediction. This ensures that the total loss caused by the multiple knife–edge calculation is at least as great as the loss caused by any one knife–edge considered individually.

Fourth Component of The Model - Frequency Offset Adjustment

 

Table 3 Frequency offset adjustment algorithms

f, frequency	Urban	Non-Urban
$150 MHz \le f \le 450 MHz$	$A_f = -30log\left(\frac{450 MHz}{850 MHz}\right) + \left[-20log\left(\frac{(f) MHz}{450 MHz}\right)\right]$	$A_f = -20log\left(\frac{f}{850 MHz}\right)$
$451 MHz \le f \le 850 MHz$	$A_f = -30log\left(\frac{f}{850 MHz}\right)$	$A_f = -20log\left(\frac{f}{850 MHz}\right)$
$851 MHz \le f \le 2400 MHz$	$A_f = -30log\left(\frac{f}{850 MHz}\right)$	$A_f = -30log\left(\frac{f}{850 MHz}\right)$

 

The reference frequency for the Lee point-to-point model is 850 MHz, and the model is valid within the frequency range from 150 MHz to 2400 MHz. For purposes of the frequency offset adjustment, Lee model categorizes the environment types into two classes, as follows:

1. Urban, which includes dense urban and urban.
2. Non–urban, which includes commercial suburban, residential suburban, cluttered rural, open rural, deciduous forest, and evergreen forest.

Table 3 gives the frequency adjustment algorithms for each of the three frequency ranges and within each of the two environment cases.

The Fifth Component of the Model - An Adjustment Factor

The initial signal strength prediction at each point along the radial assumes a set of standard conditions in making certain parameters at a base station transmit and at a mobile unit, as follows: \[ \begin{align} P_t &= \text{base station output power} \\ &= \text{10 Watts} \\ h_1 &= \text{base station antenna height} \\ &= \text{100 feet (∼ 30.5 meters)} \\ h_2 &= \text{mobile antenna height} \\ &= \text{10 feet (∼ 3 meters)} \\ g_b &= \text{base station antenna gain} \\ &= \text{6 dBd (dB over dipole)} \\ g_m &= \text{mobile antenna gain} \\ &= \text{0 dBd (dB over dipole)} \end{align} \]

The fifth component of Lee model in Eq.(1) also applies an adjustment factor $\alpha$ for the signal strength prediction to compensate for the difference between the standard conditions and the actual values of these parameters, which are defined as follows: \[ \begin{align} P_t' &= \text{actual base station output power in Watts } \\ h_1' &= \text{actual base station antenna height in feet} \\ h_2' &= \text{actual mobile antenna height} \\ &\phantom{{}={}} \text{(default at 5 feet or 1.5 meters)} \\ g_b' &= \text{actual base station antenna gain in dBd} \\ g_m' &= \text{mobile antenna gain in dBd} \\ &\phantom{{}={}} \text{(default at 0 dBd)} \end{align} \]

Note that the adjustment factor calculated here does not include an adjustment for the standard base station antenna height. Because the standard base station antenna height is used in determining the effective antenna height gain, whereas the actual base station antenna height is used in determining diffraction loss.

The adjustment factor $\alpha$ is given by the following equation; \[ \begin{align} \alpha &= 10log\left(\frac{P_t'}{P_t}\right) + 20log\left(\frac{h_1'}{h_1}\right) + 10log\left(\frac{h_2'}{h_2}\right) + (g_b' - g_b) + (g_m' - g_m) \tag{9} \end{align} \]

Water Enhancement

An optional water enhancement is available for use with any of the three modes of Lee Macrocell Model. Like effective antenna height gain, the water enhancement behaves as an additional attenuation factor that is applied to the calculation of Lee model for determining the signal strength.

When water enhancement is turned on, Lee model checks to see if the mobile is located on land but receives both reflected waves, one from water and one from land, as illustrated in Figure 8. In this case, Lee model compensates for the effect of the water reflected wave by adjusting the path loss at the mobile upward to approach the free space curve.

Figure 8: Effect of Water Reflected Waves

Over-the –water condition [12]

When the mobile is traveling on the other side of the water from the base station antenna as shown in Figure 8 there are three waves arrived at the mobile. Since there is no human made structures over the water, the reflected wave over the water is still considered as a speculated reflected wave, although the reflection point of the wave is at a distant location away from the mobile. The received signal strength from three waves can be expressed by extending from two waves shown as \[ \begin{align} P_r &= P_0\left(\frac{1}{4 \pi d/\lambda}\right)^2 \quad \Bigg\lvert 1 + a_1 e^{(j \Delta\phi_1)} + a_2 e^{(j \Delta \phi_2)}\Bigg\rvert^2 \tag{10} \end{align} \] where \[ \begin{align} a_1 \text{ and } a_2 &= \text{the reflection coefficients of water and land respectively} \\ \Delta \phi_1 \text{ and } \Delta \phi_2 &= \text{the phase differences between a direct path and} \\ &\phantom{{}={}} \text{a reflected path from water and from land respectively.} \end{align} \]

In a mobile environment, a₁ and a₂ are equal to -1 because of the energy of the signal is totally reflected with a phase reversed. Then Eq.(10) becomes \[ \begin{align} P_r &= P_0\left(\frac{1}{4 \pi d/\lambda}\right)^2 \quad \Bigg\lvert 1 - (\cos \Delta\phi_1 + \cos \Delta \phi_2) - j(\sin \Delta\phi_1 + \sin \Delta \phi_2)\Bigg\rvert^2 \\ &= P_0\left(\frac{1}{4 \pi d/\lambda}\right)^2 \cdot L_r \tag{11} \end{align} \] where L_r is the loss factor \[ \begin{align} L_r &= \Bigg\lvert 1 - (\cos \Delta\phi_1 + \cos \Delta \phi_2) - j(\sin \Delta\phi_1 + \sin \Delta \phi_2)\Bigg\rvert^2 \\ &= \Bigg\lbrack 1 - (\cos \Delta\phi_1 + \cos \Delta \phi_2)\Bigg\rbrack^2 + \Bigg\lbrack \sin \Delta\phi_1 + \sin \Delta \phi_2\Bigg\rbrack^2 \tag{12} \end{align} \]

Since \[ \begin{align} \cos \Delta\phi &= 1 - 2\sin^2 (\Delta\phi/2). \tag{13} \end{align} \] Substituting Eq.(13) into Eq.(12) and simplifying the equation yields \[ \begin{align} L_r &= 1 - \left[-2 + 2 - 2\sin^2\left(\frac{\Delta\phi_1 - \Delta\phi_2}{2}\right)\right] \\ &= 1 - 2\sin^2\left(\frac{\Delta\phi_1 - \Delta\phi_2}{2}\right) \\ &\approx 1 \tag{14} \end{align} \] It is because the value of $\sin \left(\frac{\Delta\phi_1 - \Delta\phi_2}{2}\right)$ is very small. From Eq.(14), the loss factor L_r becomes one. It means no loss. Therefore Eq.(10) becomes \[ \begin{align} P_r &= P_0\left(\frac{1}{4 \pi d/\lambda}\right)^2 \tag{15} \end{align} \] It is a free space path loss. We may conclude that the free space path loss will be observed when the propagation over the water occurs.

For treating the water enhancement, there are two options for determining where water is located, as follows:

1. Attribute—When the morphology is used as the basis for determining the location of water.
2. Terrain— If the elevation for any point along the terrain radial path is within 0.2 meters of the specified elevation for water base, then the point is considered to be located on water for implementing the water enhancement.

Determination of water location from the morphology of water is the more accurate method. Generally, terrain elevation should be used for this condition only when the morphology files of water files are not available. The enhancement of the Lee Macrocell Propagation model for the radio path over the water is insuring the inclusion of the result impacting from the water when designing a cellular system.

Conclusion

The Lee Macrocell Point-to-Point Prediction Model has given the physical explanation with experimental data as a model base to predict the receiving signal strength. In previous literature, the description of Lee model has been showed in pieces. This paper has given a whole picture of the point-to-point model. The other two models, the Lee microcell model and in-building model will be written in separate papers.

References

1. Lee, W C Y (1988). Lee's Model, Appendix VI in Coverage prediction for mobile radio systems operating in the 800/900 MHz frequency range. Vehicular Technology, IEEE Transactions on 37(1): 68-70. doi:10.1109/25.42678.
2. Lee, W C Y (1985). Estimate of Local Average Power of A mobile Radio Signal. Vehicular Technology, IEEE Transactions on 34(1): 22-77. doi:10.1109/T-VT.1985.24030.
3. U.S. Geological Survey Issued 7.5 minute Maps and Terrain Elevation Data tapes (a quarter-million scale; 250,000:1 map tapes). Available at: http://www.usgs.gov/.
4. Defense Map Agency (DMA) A 3-second arc (roughly 200 ft to 300 ft depending on the geographic locations) tape, and a planar 0.01-in tape (an elevation at intervals of 0.01 inch, i.e. 208 ft). Available at: http://www.archives.gov/research/guide-fed-records/groups/456.html.
5. Lee, W C Y (2005). Coverage and Antennas. In: 3rd Ed. Wireless & Cellular Telecommunication. New York: McGraw-Hill, . 349-424 978-0071436861
6. Lee, W C Y (1993). Prediction of Propagation Loss. In: 2nd Ed. Mobile Communications Design Fundamentals. New York: John Wiley & Sons, Inc., . 47-100 978-0471574460
7. Lee, W C Y (1997). Path Loss over Flat Terrain and Path Loss over Hilly Terrain and General Methods of Prediction. In: 2nd Ed. Mobile Communications Engineering: Theory and Applications. New York: McGraw-Hill, . 101-176 978-0070371033
8. Anderson(1958). Simplified Method For Computing Knife-edge Diffracion in the Shadow region. Antennas and Propagation, IRE Transactions on 6(3): 281-286. doi:10.1109/tap.1958.1144598.
9. Lee, W C Y (1997). Diffraction Loss in Path Loss over Hilly Terrain and General Methods of Prediction. In: 2nd Ed. Mobile Communications Engineering: Theory and Applications. New York: McGraw-Hill, . 142 978-0070371033
10. Lee, W C Y (2005). The Merit of the Point-to-Point Model in Coverage and Antennas. In: 3rd Ed. ccWireless & Cellular Telecommunication. New York: McGraw-Hill, . 382 978-0071436861
11. Epstein(1953). An Experimental Study of Wave Propagation at 850 Mc/s. Proceedings of the IRE 41(5): 595-611. doi:10.1109/JRPROC.1953.274401.
12. Lee, W C Y (2005). Land to Mobile Transmission Over Water in Coverage and Antennas. In: 3rd Ed. Wireless & Cellular Telecommunication. New York: McGraw-Hill, . 366 978-0071436861