## 5 PAPR-Aware Transmission Strategies
### Pre-Equalization for Multisine PSK Modulation  
To compensate the signal distortion caused by the frequency-selective channel, a pre-equalizer is introduced. The envelope of the pre-equalized signal when applying multisine PSK is  
\[A_{w}\left(t\right)\leq a\left\|\mathbf{h}\circ\mathbf{w}\right\|_{1}\text{ if }\varphi_{i}\left(t\right)-\varphi_{j}\left(t\right)=0. \tag{19}\]  
Thus the received signal PAPR is  
\[PAPR_{w}=\leq\frac{2\left\|\mathbf{h}\circ\mathbf{w}\right\|_{1}^{2}}{\left\| \mathbf{h}\circ\mathbf{w}\right\|^{2}}. \tag{20}\]  
The PAPR of the received pre-equalized signal reaches its maximum when \(\left|h_{i}\right|\left|w_{i}\right|=\left|h_{j}\right|\left|w_{j}\right|\) and \(\angle w_{n}=-\angle h_{n}\). Without changing the transmit power, \(\left\|\mathbf{w}\right\|^{2}=N_{t}\), we can derive \(\mathbf{w}_{PAPR}\) as the element-wise inverse of the channel  
\[\mathbf{w}_{PAPR}\propto\mathbf{\hat{h}}, \tag{21}\]  
with \(\mathbf{\hat{h}}=\mathbf{h}\circ^{-1}\) and normalizing constant \(\sqrt{\frac{N_{t}}{\left\|\mathbf{\hat{h}}\right\|^{2}}}\) multiplied to \(\mathbf{\hat{h}}\). Based on the Cauchy-Schwarz inequality, the SNR of each tone (8) reaches a maximum when \(w_{n}\) is proportional to the conjugate of the channel coefficient \(h_{n}\), namely matched pre-equalization. Ensuring that transmit power does not change, the matched pre-equalizer is  
\[\mathbf{w}_{m}\propto\mathbf{h}^{*}, \tag{22}\]  
with normalisation constant \(\sqrt{\frac{N_{t}}{\left\|\mathbf{h}\right\|^{2}}}\). Since the choice of the pre-equalizer influences both the PAPR and power level of the received signal, the crux of designing the transmission strategy for more converted DC power is whether the PAPR or the power level of the received signal impacts the DC output power the most. A high PAPR signal indeed improves the PCE of the rectifier, yet a signal of lower power not only decreases the RF power that can be received by the rectifier, but also the PCE of the rectifier as addressed in Section 4. To achieve as high as possible DC output, we propose a balanced pre-equalizer, ensuring the same transmit power, that is a linear combination of \(\mathbf{w}_{PAPR}\) and \(\mathbf{w}_{m}\). This pre-equalizer is determined by two balancing coefficients \((p,q)\) as below  
\[\mathbf{w}_{bl}=p\mathbf{w}_{PAPR}+q\mathbf{w}_{m},\ p\in[0,1]. \tag{23}\]  
The detailed computation of coefficients \((p,q)\) is addressed in Appendix A ensuring no change in the transmit power. We denote the balanced pre-equalizer with coefficient \(p\) as \(\mathbf{w}_{\mathbf{p}}\). The optimal balancing coefficients are determined by a thorough search by computing the maximal DC power using the nonlinear power transfer model of this work. The computation time is around one hour. Potential relaxed solution to reduce the computation time can be studied in future work. For comparison, the pre-equalizer achieving maximal channel capacity is also obtained. This pre-equalizer can be derived by solving the optimization problem as below  
\[\underset{\mathbf{w}_{C}}{\text{maximize }}C_{PSK}\ (10)\] \[\text{subject to }\|\mathbf{w}_{C}\|^{2}=N_{t}\]  
The solution of this optimization problem is  
\[\mathbf{w}_{C}=\mathbf{1}, \tag{24}\]  
which can be understood intuitively as the information is in the phase. The water-filling pre-equalizer to maximize the channel capacity is addressed in Appendix B. Finally, we also compare our proposed pre-equalizers with the low-complexity transmission strategy for WPT proposed in [23] by which the pre-equalizer is proportional to \(|\mathbf{h}|^{3}\triangle\mathbf{h}^{*}\). Figure 6 plots the channel response of a two-tone multisine signal of 10 MHz, and the results of the considered pre-equalizers. It can be observed that the matched pre-equalizer assigns more power to the channel with higher gain, while the pre-equalizer maximizing the PAPR assigns more power to the channel with lower gain.