# A New Propagation Channel Synthesizer for UAVs in the Presence of Tree Canopies
## 3 The Proposed Channel Synthesizer
### Discussion and Workflow  
Following Sections 3.2 and 3.3, summing the coherent and incoherent scattered fields enables to obtain the resulting time series of received signal levels at the position of interest, i.e., at the ground terminal.  
Assuming only the plane wave illuminating the tree canopies enables to save some computation time as the incident direction \(\hat{\mathbf{k}}_{\text{inc}}\) is the same throughout the scenario for one particular UAV position.  
As the UAV moves, \(\hat{\mathbf{k}}_{\text{inc}}\) changes and, as a consequence, the canopy's equivalent scattering amplitude per unit volume in the forward scatter direction \(F^{eq}(\hat{\mathbf{k}}_{\text{inc}},\hat{\mathbf{k}}_{\text{inc}})\) and the complex propagation constant \(K\) inside the canopies should be updated, see (6). However, obtaining \(F^{eq}(\hat{\mathbf{k}}_{\text{inc}},\hat{\mathbf{k}}_{\text{inc}})\) is, as explained above in Section 2.2, computationally very demanding. Following the analysis in [4; 5; 6], we may, however, update these values only when the UAV's position is changed significantly, e.g., when the elevation or azimuth w.r.t. the ground terminal changes by more than \(5^{\circ}\). This enables us to provide a time series with a very small sampling step while keeping the calculations within reasonable computation times.  
Similar situation is with the equivalent scattering cross section per unit volume \(\sigma^{eq}(\hat{\mathbf{k}}_{\text{out}},\hat{\mathbf{k}}_{\text{inc}})\), which, moreover, depends also on the terminal position due to the scattering direction \(\hat{\mathbf{k}}_{\text{out}}\). Fortunately, \(\sigma^{eq}(\hat{\mathbf{k}}_{\text{out}},\hat{\mathbf{k}}_{\text{inc}})\) applies only in the incoherent field calculations and, following the analysis in [4; 5; 6], it dominates only in other than the forward scatter direction and its dependence on \(\hat{\mathbf{k}}_{\text{out}}\) is quite weak. A significant decrease in the resulting computation time can, thus, be achieved when \(\sigma^{eq}(\hat{\mathbf{k}}_{\text{out}},\hat{\mathbf{k}}_{\text{inc}})\) is for a particular tree calculated only for one selected backscatter direction, \(\sigma^{eq}(-\hat{\mathbf{k}}_{\text{inc}},\hat{\mathbf{k}}_{\text{inc}})\), and this value is used for all the \(\hat{\mathbf{k}}_{\text{out}}\) and \(\hat{\mathbf{k}}_{\text{inc}}\) directions. In this way, the reduction of the synthesizer's computation time is very significant while the overall resulting accuracy is well preserved.  
Another reduction of the computation time is also apparent from [4; 5; 6]. The coherent scattered field obtained by the new modified PO/MST model dominates over the incoherent field only in a narrow area around the forward scatter direction. This means only selected trees located around the forward scatter direction need to be taken into account. However, such selection must be performed very carefully and based on the particular scenario of interest, otherwise discontinuities in the resulting time series may appear.  
Now, in Figure 4, we can summarize the whole workflow of the new proposed propagation channel synthesizer where each block clarifies its functionality. Following this diagram, the synthesizer has been implemented in the MATLAB environment for the validation purposes described below in Sections 4 and 5.  
Figure 4: Workflow of the new proposed propagation channel synthesizer. The change of the UAV and/or terminal positions is denoted by the time input sampling. The blue entities symbolize the input data while the yellow processes denote the corresponding calculations performed by the synthesizer. The resulting output field is symbolized by \(\mathbf{E}_{\text{tot}}\).