# Channel Model for Estimating Received Power Variations at a Mobile Terminal in a Cellular Network
## 2 The Channel Model
### Directional Derivative of Received Power with Respect to Angle \(\theta\)  
The antenna gain varies with the angle, particularly in cellular systems where antennas exhibit directional radiation patterns. By differentiating with respect to \(\theta\), the objective is to analyze how the gain increases or decreases as the terminal's angular position changes relative to the serving antenna, as expressed in Equation (8):  
\[\frac{\partial P_{\text{Rx}}}{\partial\theta}=\frac{d}{d\theta}\left[-\min\left( 12\cdot\left(\frac{\theta+\gamma_{h}}{\text{HPBW}_{h}}\right)^{2},\text{FBR} \right)\right] \tag{8}\]  
To evaluate this partial derivative, it is important to recognize that the minimum function (min) selects the smaller value between two terms. In this context, the derivative depends on which term is smaller. If the minimum corresponds to the first term containing \(\theta\), the expression can be further differentiated. Conversely, if the minimum is \(FBR\), its derivative is zero, since \(FBR\) is constant within each base station sector when differentiating with respect to the angle \(\theta\).  
Now, assuming the case where  
\[12\cdot\left(\frac{\theta_{1}+\gamma_{h}}{\text{HPBW}_{h}}\right)^{2}<\text{ FBR} \tag{9}\]  
the left term can be differentiated with respect to \(\theta\) as follows:  
\[\frac{\partial P_{Rx}}{\partial\theta}=\frac{d}{d\theta}\left(12\cdot\left( \frac{\theta+\gamma_{h}}{\text{HPBW}_{h}}\right)^{2}\right) \tag{10}\]  
By applying the chain rule, we obtain  
\[\frac{d}{d\theta}\left(12\cdot u^{2}\right)=12\cdot 2u\cdot\frac{du}{d\theta} \tag{11}\]  
where \(u=\frac{\theta+\gamma_{h}}{HPBW_{h}}\). Now, we differentiate \(u\) with respect to \(\theta\):  
\[\frac{du}{d\theta}=\frac{1}{HPBW_{h}} \tag{12}\]  
Substituting \(u\) and its derivative into Equation (11), we get  
\[\frac{d}{d\theta}\left(12\cdot\left(\frac{\theta+\gamma_{h}}{\text{HPBW}_{h}} \right)^{2}\right)=\frac{24\cdot\left(\theta+\gamma_{h}\right)}{\text{HPBW}_{ h}^{2}} \tag{13}\]  
Therefore, the derivative of the given expression, assuming that the condition (9) holds, is  
\[\frac{\partial P_{Rx}}{\partial\theta}=-\frac{24\cdot\left(\theta_{1}+\gamma_ {h}\right)}{HPBW_{h}^{2}}[\text{W}/\text{rad}] \tag{14}\]  
Figure 1 illustrates the relationship between the partial derivative of the received power with respect to the angle \(\theta\) and the half-power beamwidth (HPBW), derived using the method outlined in Section 2.1. In the figure, the green dots represent the actual power measurements across various HPBW values, with multiple samples for each HPBW. The yellow line shows the trend of these measurements, calculated from the theoretical derivative given in Equation (14), which is explained below.  
It can be observed that as the HPBW increases, the variation in power level decreases. This behavior aligns with Equation (14), which indicates that the power variation is inversely proportional to the HPBW and diminishes with the square of this angle. In other words, a greater number of sectors with directional antennas (i.e., smaller HPBW) leads to larger fluctuations in received power compared to an omnidirectional antenna, which shows less variation. In Figure 1, the green points represent the experimental data obtained from measurements across different antenna configurations, while the yellow line corresponds to a simple linear regression fitted to these data. This regression helps illustrate the general trend of decreasing power variation as HPBW increases. The methodology used to acquire and process the experimental data is detailed in a subsequent section.