## 4 Performance Analysis
### Performance Comparison  
From the optimal solution in (15)-(18), the following theorem can be obtained. **Theorem 1**. : _If \(g(P_{d}^{\mathrm{max}})\leq\rho_{c}^{\mathrm{ND}}\leq 1\), we have \(P_{d,\mathrm{I}}^{opt}=P_{d}^{\mathrm{max}}\) and \(P_{d,\mathrm{E}}^{opt}=0\), where_  
\[g(x)=\frac{2P_{b}|h_{bc}|^{2}+x|h_{dc}|^{2}+\sigma_{c,\mathrm{cov}}^{2}-\sqrt{ (x|h_{dc}|^{2}+\sigma_{c,\mathrm{cov}}^{2})^{2}+4P_{b}|h_{bc}|^{2}\sigma_{c, \mathrm{cov}}^{2}}}{2P_{b}|h_{bc}|^{2}}. \tag{23}\]  
**Proof. ** In (15), \(\frac{\rho_{c}^{\text{ND}}c_{c_{c}}^{2}\omega_{d}^{\text{max}}}{(1-\rho_{c}^{ \text{ND}})^{2}P_{\text{j}}|_{\text{hc}}|^{2}+(1-\rho_{c}^{\text{ND}})\rho_{d}^{ \text{max}}|_{\text{hc}}|^{2}}\) is an increasing function of \(\rho_{c}^{\text{ND}}\) and is equal to \(P_{d}^{\text{max}}\) when \(\rho_{c}^{\text{ND}}=g(P_{d}^{\text{max}})\). Therefore, when \(g(P_{d}^{\text{max}})\leq p_{c}^{\text{ND}}\leq 1\), we have \(P_{d\text{J}}^{\text{opt}}=P_{d}^{\text{max}}\) and \(P_{d\text{J}}^{\text{opt}}=0\). By the results in Theorem 1, the performance comparison between the proposed scheme in (15)-(18) and the BL scheme in (19)-(22) can be derived as shown in the following theorem. Theorem 2: _The optimal and BL schemes perform as shown in Table 1._  
Proof. : This proof is composed of three parts. (1) When \(0\leq\rho_{c}^{\text{ND}}\leq g_{2}(0)\), \(P_{d\text{J}}^{\text{opt}}\) in (15) is larger than \(0\), and \(P_{d\text{J}}^{\text{BL}}\) in (19) is equal to \(0\). Therefore, when \(0\leq\rho_{c}^{\text{ND}}\leq g_{2}(0)\), we have \(R_{d}^{\text{O}}\geq R_{d}^{\text{BL}}=0\). (2) When \(g_{2}(0)<\rho_{c}^{\text{ND}}<g_{2}(P_{d}^{\text{max}})\), \(P_{d\text{J}}^{\text{BL}}\) in (19) is smaller than \(P_{d\text{J}}^{\text{opt}}\) in (15) and is larger than \(0\). Therefore, when \(g_{2}(0)<\rho_{c}^{\text{ND}}<g_{2}(P_{d}^{\text{max}})\), we have \(R_{d}^{\text{O}}>R_{d}^{\text{BL}}>0\). (3) When \(g_{2}(P_{d}^{\text{max}})\leq\rho_{c}^{\text{ND}}\leq 1\), \(P_{d\text{J}}^{\text{opt}}\) in (15) is larger than \(0\) and is equal to \(P_{d\text{J}}^{\text{BL}}\) in (19). Therefore, when \(g_{2}(P_{d}^{\text{max}})\leq\rho_{c}^{\text{ND}}\leq 1\), we have \(R_{d}^{\text{O}}=R_{d}^{\text{BL}}>0\). The detailed explanation of the conditions in Theorem 2 is as follows. When the CU demands a large amount of energy, specifically when \(g(P_{d}^{\text{max}})\leq\rho_{c}^{\text{ND}}\leq 1\), the transmission of energy signals becomes unnecessary to achieve the optimal D2D data rate, i.e., \(R_{d}^{\text{O}}=R_{d}^{\text{BL}}>0\). This occurs because, under this condition, the energy harvested from the DT's interference power, i.e., \(P_{d\text{J}}|h_{dc}|^{2}\) in (3), is sufficient to offset the data rate reduction caused by the same interference term, i.e., \(P_{d\text{J}}|h_{dc}|^{2}\) in (4). Thus, additional energy signal transmission does not further enhance the D2D data rate in this scenario. On the other hand, when the CU's energy demand is moderate, specifically when \(g(0)<\rho_{c}^{\text{ND}}<g(P_{d}^{\text{max}})\), energy signal transmission becomes critical for achieving the optimal D2D data rate, as \(R_{d}^{\text{O}}>R_{d}^{\text{BL}}>0\). This indicates that energy signal transmission is necessary to maximize the D2D performance in situations where the CU's energy requirement falls within this range. Finally, when the CU requires only a small amount of energy, i.e., \(0\leq\rho_{c}^{\text{ND}}\leq g(0)\), the BL scheme fails to enable any data transmission by the DT without the assistance of energy signals. In contrast, the proposed scheme in (15)-(18) enables data transmission even under this condition. This demonstrates the effectiveness of the proposed energy signal transmission approach in such low-energy-demand scenarios. In summary, as the CU's energy requirement decreases, the role of energy signal transmission becomes increasingly significant in improving the D2D data rate.