## 3 Performance Analysis of System
### BER Performance Analysis of GCI-DCSK SWIPT  
#### 3.2.1 BER of Index Bits  
It is obvious that the index bits are incorrectly recovered when one or more carrier statuses are incorrectly judged. The symbol error rate (SER) of the index symbol can be deduced by adopting the simple idea of excluding the case of judging the carriers correctly, which is given by  
\[P_{i}^{SER} =\Pr\Bigl{[}\min\Bigl{(}|D_{i}^{1}|\Bigr{)}>\max\Bigl{(}|D_{j}^{1} |\Bigr{)}\Bigr{]}\] \[=1-\int_{0}^{+\infty}F_{max|D_{j}^{1}|}(y)f_{min|D_{i}^{1}|}(y) \mathrm{d}y\] \[=1-\int_{0}^{+\infty}\Bigl{[}f_{|D_{j}^{1}|}(y)\Bigr{]}^{n-p_{2}} \cdot p_{2}\cdot f_{|D_{i}^{1}|}(y)\cdot\Bigl{[}1-F_{|D_{j}^{1}|}(y)\Bigr{]}^{ p_{2}-1}\mathrm{d}y, \tag{18}\]  
where \(\min\bigl{(}|D_{i}^{1}|\bigr{)}\) and \(\max\bigl{(}|D_{j}^{1}|\bigr{)}\) represent the minimum and maximum absolute values of the active carrier correlator and the inactive carrier correlator, respectively. \(F_{|D_{j}^{1}|}(y)\) is the cumulative distribution function (CDF) of \(|D_{j}^{1}|\), with \(1\leq j\leq n-1\), and \(f_{|D_{i}^{1}|}(y)\) is the probability density function (PDF) of \(|D_{i}^{1}|\) with \(1\leq i\leq n-1\). Both \(D_{i}^{1}\) and \(D_{j}^{1}\) follow the normal distribution; hence both \(|D_{i}^{1}|\) and \(|D_{j}^{1}|\) follow the folded normal distribution. It is also derived that \(F_{max[D_{j}^{1}]}\) is the maximum order statistic. Based on the characteristics of the folded normal distribution, the CDF of the active carrier and the inactive carrier, and the PDF of the active carrier are derived as  
\[F_{[D_{i}^{1}]}(y)=\frac{1}{2}\text{erf}\left(\frac{y-E[D_{i}^{1}]}{\sqrt{2 Var[D_{i}^{1}]}}\right)+\frac{1}{2}\text{erf}\left(\frac{y+E[D_{i}^{1}]}{ \sqrt{2 Var[D_{i}^{1}]}}\right), \tag{19}\]  
\[f_{[D_{i}^{1}]}(y)=\frac{1}{\sqrt{2\pi\big{(}Var[D_{i}^{1}]\big{)}}}\cdot\left[ \exp\left(-\frac{\big{(}y-E[D_{i}^{1}]\big{)}^{2}}{2\big{(}Var[D_{i}^{1}] \big{)}}\right)+\exp\left(-\frac{\big{(}y+E[D_{i}^{1}]\big{)}^{2}}{2\big{(} Var[D_{i}^{1}]\big{)}}\right)\right], \tag{20}\]  
\[F_{[D_{i}^{1}]}(y)=\text{erf}\Big{(}y\div\sqrt{2 Var[D_{j}^{1}]}\Big{)}. \tag{21}\]  
Substituting (19)-(21) into formula (18), and setting \(t=\frac{y}{N_{0}\cdot\theta}\), the SER of the index bits is given by  
\[P_{i}^{SER}=1-\frac{p_{2}}{\sqrt{2\pi\Big{(}\frac{\beta_{1}}{ \beta}\delta+\frac{\beta_{1}}{4}\Big{)}}}.\] \[\int_{0}^{+\infty}\left[\text{erf}\left(\frac{t}{\sqrt{2\Big{[} \frac{\mu^{2}\beta_{1}}{\beta^{2}}\delta+\frac{\mu^{2}\beta_{1}}{2\beta} \delta+\frac{\beta_{1}}{2\beta}\delta+\frac{\beta_{1}}{4}\Big{]}}}\right) \right]^{n-p_{2}}.\] \[\left[\frac{1}{2}\left(\text{erfc}\left(\frac{t-\frac{\beta_{1}} {\beta}\delta}{\sqrt{\frac{2\beta_{1}}{\beta}\delta+\frac{\beta_{1}}{2}}} \right)+\text{erfc}\left(\frac{t+\frac{\beta_{1}}{\beta}\delta}{\sqrt{\frac{ 2\beta_{1}}{\beta}\delta+\frac{\beta_{1}}{2}}}\right)\right)\right]^{p_{2}-1}.\] \[\left[\exp\left(-\frac{\left(t-\frac{\beta_{1}}{\beta}\delta \right)^{2}}{\frac{2\beta_{1}}{\beta}\delta+\frac{\beta_{1}}{2}}\right)+\exp \left(-\frac{\left(t+\frac{\beta_{1}}{\beta}\delta\right)^{2}}{\frac{2\beta_{ 1}}{\beta}\delta+\frac{\beta_{1}}{2}}\right)\right]dt, \tag{22}\]  
where \(\delta=\frac{p\gamma_{b}}{[p_{2}+1+\mu^{2}(n-p_{2})]}\), \(\gamma_{b}=\frac{P_{i}[p_{2}+1+\mu^{2}(n-p_{2})]\sum_{i=1}^{n}\alpha_{v_{2}}^{2 }E_{i}^{v}}{d_{tr}^{2}pN_{0}}\) and \(p=p_{1}+p_{2}\). The BER of the index bits is derived as  
\[P_{i}^{BER}=\frac{P_{i}^{SER}}{p_{1}}. \tag{23}\]