# Channel Model for Estimating Received Power Variations at a Mobile Terminal in a Cellular Network
## 2 The Channel Model  
This section presents the theoretical modeling of the wireless communication channel using vector calculus, applied to the analysis of the received power in mobile networks. Beginning with Friis' equation to describe propagation under ideal conditions, modifications are introduced to account for the directional characteristics of transmitting antennas. Directional derivatives of the received power with respect to angle and distance are developed, analyzing how these variables influence coverage and system performance. Finally, the concept of the directional derivative relative to the movement of the mobile terminal, along with the gradient of the power scalar field, is introduced--both essential tools for the optimization and planning of cellular networks in urban environments. The following describes the application of vector calculus to derive expressions for the rate of change of the received power at the mobile terminal within the radio network. To do this, Friis' equation is used as the foundation. The Friis equation [10] provides a method to calculate the received power in wireless communication systems, specifically predicting signal behavior under ideal line-of-sight (LoS) conditions. In practical scenarios, however, the equation can be adjusted to incorporate additional factors such as system-specific gains or losses, rates of change, and other real-world effects. The following equation expresses this relationship:  
\[P_{\mathrm{Rx}}=\frac{P_{\mathrm{Tx}}\cdot G_{\mathrm{Tx}}\cdot G_{\mathrm{ Rx}}\cdot\lambda^{2}}{(4\pi R)^{2}} \tag{1}\]  
where \(P_{Rx}\) is the received power at the mobile terminal, measured in watts [W]; \(P_{Tx}\) is the transmitted power emitted by the base station, measured in watts [W]; \(G_{Tx}\) is the dimensionless gain of the transmitting antenna; \(G_{Rx}\) is the dimensionless gain of the antenna located at the mobile terminal; \(\lambda\) is the wavelength of the transmitted signal, measured in meters [m], which is related to the signal frequency through the equation \(\lambda=\frac{c}{f}\), where \(c\) is the speed of light in vacuum (\(3\times 10^{8}\) [m/s]) and \(f\) is the transmission frequency; and \(R\) is the distance between the transmitting antenna and the receiving antenna, measured in meters [m]. If [PERSON] equation is expressed in decibels, it is as follows:  
\[P_{\mathrm{Rx}}\left[\mathrm{dBm}\right]=P_{\mathrm{Tx}}\left[\mathrm{dBm} \right]+G_{\mathrm{Tx}}\left[\mathrm{dBi}\right]+G_{\mathrm{Rx}}\left[ \mathrm{dBi}\right]+20\log_{10}(\lambda)-20\log_{10}(4\pi R) \tag{2}\]  
The antenna gain can be decomposed into two main components: the vertical gain, which affects the coverage range of the cellular cell, and the horizontal gain, which determines the coverage breadth. In urban environments, where high antenna density reduces the importance of extended range, the contribution of the vertical gain to power estimation becomes negligible. As a result, antenna specifications primarily emphasize horizontal plane angles, as these have the most significant impact on the overall gain. The transmitter gain reflects the combined contribution of both the horizontal and vertical plane gains, as expressed in Equation (3):  
\[G_{\mathrm{Tx}}\left[\mathrm{dB}\right]=G_{\mathrm{Tx},h}\left[\mathrm{dB} \right]+G_{\mathrm{Tx},\varphi}\left[\mathrm{dB}\right] \tag{3}\]  
For simplification purposes, this study is conducted in two dimensions, considering only the gain in the horizontal plane. This approach is further justified by the fact that, in urban areas with high antenna density, network design prioritizes coverage extent and capacity over the range influenced by the vertical plane gain.