## 3 Design of Physical Layer Security Communication System Using Proposed LPD Signal
### Precoding Scheme for Fading Channel  
According to the proposed secure transmission protocol, the channel fading coefficients from Bob to Alice can be estimated by Alice. Then, the channel coefficient from Alice to Bob can also be known based on the channel reciprocal principle. Therefore, a precoding scheme that exploits the channel characteristics could be employed to improve the receiving gain at Bob. Meanwhile, the precoding scheme can also optimize the power allocation of sub-band signals. As stated, the SNR of the sampled signal at Bob is given by  
\[SNR_{B,d}=\frac{\left|\sum\limits_{k=L}^{L+N-1}\alpha_{k}h_{k}\right|^{2} \cdot P_{s}}{NN_{0}\omega_{B}} \tag{7}\]  
when \(\Delta=0\). Weighted factor \(\alpha_{k}\) and channel fading coefficient \(h_{k}\) can be written as \(1\times N\) vectors by \(\mathbf{\alpha}=[\alpha_{L},\alpha_{L+1},\cdots,\alpha_{L+N-1}]\) and \(\mathbf{H}=[h_{L},h_{L+1},\cdots,h_{L+N-1}]\). The Cauchy-Schwarz inequality [27] states that for all vectors \(\mathbf{\alpha}\) and \(\mathbf{H}\) of an inner product space, the following equation holds true:  
\[|<\mathbf{\alpha},\mathbf{H}>|^{2}\leq||\mathbf{\alpha}||^{2}\cdot||\mathbf{H}||^{2} \tag{8}\]  
where \(<\mathbf{\alpha},\mathbf{H}>\) denotes the inner product of vectors \(\mathbf{\alpha}\) and \(\mathbf{H}\), and the notion \(||\cdot||\) denotes the Euclidean norm. Moreover, the equality holds only when \(\mathbf{\alpha}\) and \(\mathbf{H}\) are linearly dependent. It can be written by \(\mathbf{\alpha}=\lambda\mathbf{H}^{*}\), where \(\lambda\) is a nonzero constant. The superscript \(*\) denotes a conjugate operation. In addition, the weighted factor is constrained by \(\sum\
olimits_{k=L}^{L+N-1}|\alpha_{k}|^{2}=1\). In order to make the equality in Equation (8) hold true, the weighted factor \(\alpha_{k}\) should be given by \(\alpha_{k}=h_{k}^{*}/||\mathbf{H}||\). Then, the SNR of the sampled signal at Bob can be given by \(SNR_{B,d}=||\mathbf{H}||^{2}\cdot P_{s}/(NN_{0}\omega_{B})\). We can conclude that the optimal power allocation strategy for the frequency selective fading channel is to make the SNR over each sub-band identical.  
Receiving gain \(\eta\) versus sampling phase offset \(\Delta\) is shown in Figure 5. We assumed that channel coefficients for sub-band signals are independent, identically distributed, and subject to Rayleigh distribution. Let \(\tilde{h}_{k}\) denote the estimates of \(h_{k}\) that can be written by \(\tilde{h}_{k}=h_{k}+h_{k}^{e}\). Two different scenarios are explored: (1) perfect CSI, the channel coefficients are perfectly known as \(h_{k}^{e}=0\); (2) imperfect CSI, \(h_{k}^{e}\) is supposed to be a Gaussian random variable with zero mean, and the estimation error is defined by \(\rho=E\{|h_{k}^{e}|^{2}/|h_{k}|^{2}\}\).  
The simulation results show that the sampling phase offset plays an important role in the proposed scheme. It reveals that the accuracy requirement for sampling phase offset is higher with the increase in sub-carrier number \(N\). In addition, an estimation error of CSI may result in inaccurate precoding on [PERSON]'s side. It may lead to a performance loss of receiving gain on [PERSON]'s side, but will not influence the effectiveness of the under-sampling method.