## 3 The System Model of CP-SC-FDE in UAV LOS Communication  
The transceiver structure of CP-SC-FDE in UAV LOS communication is shown in Figure 1. At the transmitting end, the CP is appended before the preambles and data symbols, respectively. Then, in the framing module, they are grouped into a block. The receiver consists of modules for synchronization, channel estimation, and equalization. The synchronization module only considers time synchronization relevant to this study. Let us consider a CP-SC-FDE system where preambles and data symbols, represented by vectors \(\mathbf{p}=[p_{0},...,p_{M-1}]^{T}\) and \(\mathbf{x}=[x_{0},...,x_{N-1}]^{T}\) respectively, are grouped into a block. \(M\) and \(N\) represent the length of the preambles and data symbols, respectively. Moreover, a CP of length \(C\) is appended to \(\mathbf{x}\) and \(\mathbf{p}\). Thus, a block can be represented as \(\mathbf{s}=[p_{M-C},...,p_{M-1},p_{0},...,p_{M-1},\)\(x_{N-C},...,x_{N-1},x_{0},...,x_{N-1}]^{T}\), as shown in Figure 2. The preambles are used to perform time synchronization and channel estimation. Figure 1: The transceiver structure of the CP-SC-FDE. Figure 2: The block of the CP-SC-FDE and CP/CS-SC-FDE. Assume that the multipath channel is \(\mathbf{h}=[h_{0},..,h_{L-1}]^{T}\), through which the resulting block is transmitted. \(L\) represents the length of the multipath channel and \(h_{i}\) is the \(i\)-th path of the multipath channel. Moreover, the received signal is expressed as  
\[r_{n}=\sum_{i=0}^{L-1}h_{i}s_{n-i}+v_{n},\qquad n=0,..,M+N+2C-1, \tag{1}\]  
where \(v_{n}\) is discrete thermal noise. At the receiver of UAV communication system, the local preambles are used to perform a sliding correlation with the received signal to complete the time synchronization. The sliding correlation result, with \(v_{n}\) not considered, is represented as  
\[u_{n} =\sum_{j=0}^{M-1}p_{j}r_{n+j}\] \[=\sum_{j=0}^{M-1}p_{j}\sum_{i=0}^{L-1}h_{i}s_{n+j-i}\] \[=\sum_{i=0}^{L-1}h_{i}\sum_{j=0}^{M-1}s_{n+j-i}p_{j},\qquad n=0,..,N+2C-1. \tag{2}\]  
When \(C\leq n\leq C+L-1\), \(u_{n}\) can be represented as  
\[u_{n} =u_{C+l}\] \[=\sum_{i=0}^{L-1}h_{i}\sum_{j=0}^{M-1}s_{C+l+j-i}p_{j}\] Figure 2.0 Figure 2.1  
If it is LOS communication, the PwMP of the multipath channel is the first path, i.e.,  
Figure 2.2  
According to the correlation property of the preambles, the correlation peak is  
Figure 2.3  
which corresponds to the start of the preambles. According to it, the CP is removed. Due to the CP, the discrete time-domain linear convolution is converted to discrete-time cyclic convolution, i.e.,  
Figure 2.4 Figure 2.5  
It can be expressed as a matrix  
Figure 2.6  
where \(\mathbf{G}\) denotes the cyclic convolution matrix \[\mathbf{G}=\left[ \begin{array}{cccccc}h_{0}&0&...&0&h_{L-1}&...\\...&h_{0}&0&...&0&h_{L-1}\\ h_{L-1}&...&h_{0}&0&...&0\\ 0&h_{L-1}&...&h_{0}&0&...\\...&0&h_{L-1}&...&h_{0}&0\\ 0&...&0&h_{L-1}&...&h_{0}\end{array} \right]_{M\times M}. \tag{8}\]  
\(\mathbf{G}\) is a right-circulant matrix [40], and each of its row vectors is a right-circulant shift of the previous row vector. The first column corresponds to the channel vector \(\mathbf{h}\). In addition, according to the properties of the right circulant matrix, \(\mathbf{G}\), \(\mathbf{G}\) can be decomposed into  
\[\mathbf{G}= \mathbf{F}^{H}\mathbf{H}\mathbf{F}, \tag{9}\]  
where \({}^{H}\) denotes the Hermitian transpose, \(\mathbf{F}\) is the Discrete Fourier Transform (DFT) matrix, and \(\mathbf{H}\) is a diagonal matrix with its eigenvalues given by  
\[\mathbf{H}=diag(\mathbf{F}[\mathbf{h}^{T},\mathbf{0}_{1\times(M-L)}]^{T}). \tag{10}\]  
If the influence of noise is not considered, the time domain result of channel estimation using the preambles is \(\left[\mathbf{h}^{T},\mathbf{0}_{1\times(M-L)}\right]^{T}\).