## 4 The NLOS Communication Scheme Based on CP/CS-SC-FDE
### The Channel Estimation Algorithm of CP/CS-SC-FDE for UAV PC  
In a CP/CS-SC-FDE system, assume that the length of the CS is \(D\). Moreover, assume that in the multipath channel, the number of symbol intervals between the PwMP and the first and last path are \(E\) and \(F\), respectively. \(C\), \(D\), \(E\), and \(F\) satisfy  
\[\left\{ \begin{array}{l}C+D\leq M\\ C\geq F\\ D\geq E\end{array} \right. \tag{13}\]  
Then, \(\mathbf{s}=\left[p_{M-C},...,p_{M-1},p_{0},...,p_{M-1},p_{0},...,p_{D-1},x_{ N-C}\,...,x_{N-1},x_{0},...,x_{N-1},x_{0},...,\\ x_{D-1}\right]^{T}\). When \(C\leq n\leq L+C-1\),  
\[u_{n} =u_{C+l}\] \[=\sum_{i=0}^{L-1}h_{i}\sum_{j=0}^{M-1}s_{C+l+j-i}p_{j}\] \[=\sum_{i\
eq l}^{L-1}h_{i}\sum_{j=0}^{M-1}p_{l+j-i}p_{j}+h_{l} \sum_{j=0}^{M-1}p_{j}p_{j},\qquad 0\leq l\leq L-1. \tag{14}\]  
And,  
\[h_{E}=\max\{h_{l}|0\leq l\leq L-1\}. \tag{15}\]  
Then,  
\[u_{\max}=\max\{u_{n}|C\leq n\leq L-1+C\}=u_{C+E}, \tag{16}\]  
Figure 4: FFT region in LOS and NLOS channel based on CP/CS-SC-FDE. which corresponds to the start of the preambles. According to it, the CP and CS are removed. Due to the CP and CS, the discrete time-domain linear convolution is converted to discrete-time cyclic convolution, i.e.,  
\[{y^{\prime}}_{n}=\sum_{i=0}^{L-1}h_{i}p_{(n+E-i)\text{ mod }N}+v_{n},\qquad n=0,...,M-1. \tag{17}\]  
It can be expressed as a matrix  
\[\mathbf{y^{\prime}}=\mathbf{G}_{c}\mathbf{p}+\mathbf{v}, \tag{18}\]  
where the cyclic convolution matrix \(\mathbf{G}\) of (7) becomes \(\mathbf{G}_{c}\), i.e.,  
\[\mathbf{G}_{c}=\left[ \begin{array}{ccccccccc}h_{E}&h_{E-1}&...&h_{0}&0&...&0& h_{L-1}&...\\...&h_{E}&h_{E-1}&...&h_{0}&0&...&0&h_{L-1}\\ h_{L-1}&...&h_{E}&h_{E-1}&...&h_{0}&0&...&0\\ 0&h_{L-1}&...&h_{E}&h_{E-1}&...&h_{0}&0&...\\...&0&h_{L-1}&...&h_{E}&h_{E-1}&...&h_{0}&0\\ 0&...&0&h_{L-1}&...&h_{E}&h_{E-1}&...&h_{0}\\ h_{0}&0&...&0&h_{L-1}&...&h_{E}&h_{E-1}&...\\...&h_{0}&0&...&0&h_{L-1}&...&h_{E}&h_{E-1}\\ h_{E-1}&...&h_{0}&0&...&0&h_{L-1}&...&h_{E}\\ \end{array} \right]_{M\times M}. \tag{19}\]  
\(\mathbf{G}_{c}\) can be decomposed into  
\[\mathbf{G}_{c}{=}\mathbf{F}^{\text{H}}\mathbf{H}_{c}\mathbf{F}, \tag{20}\]  
where \(\mathbf{H}_{c}\) is a diagonal matrix with its eigenvalues given by  
\[\mathbf{H}_{c}=diag(\mathbf{F}[h_{E},...,h_{L-1},\mathbf{0}_{1\times(M-L)},h_ {0},...h_{E-1}]^{T}). \tag{21}\]  
If the influence of noise is not considered, the time-domain result of the channel estimation using the preambles is \([h_{E},...,h_{L-1},\mathbf{0}_{1\times(M-L)},h_{0},...h_{E-1}]^{T}\), which is the cyclic shift of \(\mathbf{h}\). By determining that \(E-i=-j\) in Formula (17), we obtain  
\[{y^{\prime}}_{n} =\sum_{j=0}^{L-1}h_{(E+j)\text{ mod }L}p_{(n-j)\text{ mod }M}+v_{n}\] \[=h_{n+E}\otimes p_{n}+v_{n},\qquad n=0,...,M-1. \tag{22}\]  
Moreover, it is converted to the frequency domain through a DFT, denoted by  
\[{Y^{\prime}}_{k}=H_{k}P_{k}+V_{k},\qquad k=0,...,M-1, \tag{23}\]  
where \({Y^{\prime}}_{k}\), \(P_{k}\), \(H_{k}\) and \(V_{k}\) stand for the DFT of \({y^{\prime}}_{n}\), \(p_{n}\), \(h_{n+E}\) and \(v_{n}\) respectively. In this CP/CS-SC-FDE system, the channel estimation algorithm is based on the frequency domain Least Square (LS) algorithm. Then,  
\[\hat{H}_{k}=\frac{{Y^{\prime}}_{k}}{P_{k}},\qquad k=0,...,M-1, \tag{24}\]  
which is converted to the time domain through an inverse DFT (IDFT), denoted by  
\[\hat{h}_{n}=\frac{1}{M}\sum_{k=0}^{M-1}\hat{H}_{k}e^{j2\pi\frac{\text{H}}{M}},\qquad n=0,...,M-1, \tag{25}\]Once the channel estimation for the preamble position is obtained, the channel estimation for the data position is calculated through an interpolation algorithm, i.e.,  
\[\hat{\mathbf{h}}_{d}=[\hat{h}_{d},\ldots,\hat{h}_{C-1},0_{1\times(N-C-D)},\hat{h }_{M-D},\ldots,\hat{h}_{M-1}]^{T}. \tag{26}\]  
Let \([\hat{\mathbf{h}}_{d}]_{n}\) denote the \(n\)-th element of \(\hat{\mathbf{h}}_{d}\). The pseudocode of the channel estimation algorithm is shown in Algorithm 1.