# In-Band Full-Duplex Relaying for SWIPT-Enabled Cognitive Radio Networks
## 2 System Model and Problem Formulation
### Information and Power Transfer Model  
To be energy-efficient, the data transmission of CRN is considered in a time block when the link between PT and PR sharing the same spectrum is sensed to be active. Accordingly, ST and FDR are allowed to harvest energy from RF signals transmitted by PT, and then, they use the harvested energy for their own data transmission. We consider a TSR protocol for joint energy harvesting and information processing [22], as illustrated in Figure 2. According to the TSR protocol, ST and FDR harvest energy from the PT's signal for a fraction \(\alpha\) (\(0\leq\alpha\leq 1\)) of a certain time slot \(k\) of duration \(T\). The remaining \((1-\alpha)\) fraction of the time slot is used for information transmission. As a result, the amount of energy harvested at ST and FDR can be respectively expressed as:  
\[E_{\mathbf{s}}=\eta P_{\mathrm{P}}\left|f_{p_{\mathrm{S}}}\right|^{2}\alpha T \text{ and }E_{\mathbf{r}}=\eta P_{\mathrm{P}}\left\|\mathbf{h}_{p}\right\|^{2}\alpha T, \tag{1}\]  
where \(0<\eta<1\) denotes the energy conversion efficiency and \(P_{\mathrm{P}}\) is the transmit power of PT. Correspondingly, the maximum available power at ST and that at FDR for the time duration \((1-\alpha)T\) are respectively calculated as:  
\[P_{\mathbf{s}}=\frac{\alpha}{1-\alpha}\eta P_{\mathrm{P}}\left|f_{p_{\mathrm{S }}}\right|^{2}\text{ and }P_{\mathbf{r}}=\frac{\alpha}{1-\alpha}\eta P_{ \mathrm{P}}\left\|\mathbf{h}_{p}\right\|^{2}. \tag{2}\]  
Herein, we suppose that the energies harvested by ST and FDR in the first phase time of \(\alpha T\) are completely used for information transmission in the second phase time of \((1-\alpha)T\). The received signals at FDR, SR, and PR during the time duration \((1-\alpha)T\) at the \(k^{\text{th}}\) time slot are respectively given as:  
\[\mathbf{y}_{\mathbf{r}}[k] =\mathbf{H}\mathbf{x}[k]+\sqrt{P_{\mathbf{r}}\rho}\mathbf{H}_{0} \mathbf{W}\mathbf{s}[k]+\mathbf{n}_{\mathbf{r}}[k], \tag{3}\] \[y_{\mathbf{s}}[k] =\sqrt{P_{\mathbf{r}}}\mathbf{g}_{\mathbf{s}}\mathbf{W}\mathbf{s} [k]+\mathbf{n}_{\mathbf{s}}[k],\] (4) \[y_{\mathrm{p}}[k] =\sqrt{P_{\mathrm{r}}}\mathbf{g}_{p}\mathbf{W}\mathbf{s}[k]+ \sqrt{P_{\mathrm{p}}}f_{pp}x_{\mathrm{p}}[k]+\mathbf{n}_{\mathrm{p}}[k], \tag{5}\]  
where \(\mathbf{H}\triangleq\left[\mathbf{h}_{\mathbf{s}}\,\mathbf{h}_{p}\right]\) and \(\mathbf{x}[k]\triangleq\left[\sqrt{P_{\mathrm{r}}}x_{\mathrm{s}}[k]\,\,\sqrt {P_{\mathrm{p}}}x_{\mathrm{p}}[k]\right]^{H}\), with \(x_{\mathrm{g}}[k]\) and \(x_{\mathrm{p}}[k]\sim\mathcal{CN}(0,1)\) being the transmit symbols of ST and PT in the \(k^{\text{th}}\) time slot, respectively; the elements of \(\mathbf{n}_{\mathbf{r}}[k]\), \(\mathbf{n}_{\mathbf{s}}[k]\), and \(\mathbf{n}_{\mathrm{p}}[k]\sim\mathcal{CN}(0,\sigma^{2})\) denote the additive white Gaussian noise (AWGN) at FDR, SR and PR, respectively. At FDR, a beamforming matrix \(\mathbf{W}\) is applied to combat co-channel interference, yielding  
Figure 1: A cognitive radio network with a full-duplex relay (FDR). the transmit signal \(\mathbf{s}[k]=\left[x_{\mathbf{s}}[k-\tau]\ x_{\mathbf{p}}[k-\tau]\right]^{H}\), where \(\tau\) accounts for the time delay caused by the relay processing. The system is assumed to adopt a zero-forcing (ZF) detector \(\mathbf{A}^{H}\) and a ZF beamforming \(\mathbf{W}\) at FDR:  
\[\mathbf{A}^{H}\triangleq\left(\mathbf{H}^{H}\mathbf{H}\right)^{-1}\mathbf{H}^ {H},\text{ and }\mathbf{W}\triangleq c\mathbf{G}\left(\mathbf{G}^{H}\mathbf{G} \right)^{-1},\]  
where:  
\[\mathbf{G}^{H}=\left[\mathbf{g}_{s}^{H}\quad\mathbf{g}_{p}^{H}\right]^{H}, \text{ and }c=\frac{1}{\sqrt{\text{tr}\left(\left(\mathbf{G}^{H}\mathbf{G} \right)^{-1}\right)}}.\]  
Herein, the power control factor \(c\) is to ensure that the maximum transmit power at FDR does not exceed the harvested power in the corresponding transmission block.