# A Random Subcarrier-Selection Method Based on Index Modulation for Secure Transmission
## 3 Performance Analysis
### BER Performance Analysis of the Desired Position  
In what follows, the average BER of the desired position is analyzed. As shown in Figure 5, demodulation includes index demodulation and QAM demodulation, both of which need to know the index of the activated subcarriers. For all alternative orthogonal subcarriers, we could consider that the activated subcarriers conveyed the QAM symbol plus noise while the inactivated subcarriers just conveyed the noise. As a result, we adopted greedy detection to determine the index of activated subcarriers. Suppose that \(y_{i}(\alpha)\) denotes the symbol whose index is \(\alpha\) of the \(i\)-th row in (14), then,  
\[\hat{\alpha}=\operatorname*{arg\,max}_{\alpha}|y_{i}(\alpha)|^{2}. \tag{18}\]  
After determining the index of the activated subcarriers, ML detection was adopted to determine the constellation symbols, which was similar to the classical QAM demodulation, i.e.,  
\[\hat{s}(\hat{\alpha})=\operatorname*{arg\,min}_{x(\hat{\alpha})\in S}|y(\hat{ \alpha})-x(\hat{\alpha})|^{2}. \tag{19}\]  
where \(S\) is the set of standard constellation symbols and \(y(\hat{\alpha})\) denotes the received symbol. According to the analysis above, average BER could be represented as  
\[P_{s}=\frac{n_{i}+n_{Q}}{n_{t}}, \tag{20}\]  
where \(n_{i}\) and \(n_{Q}\) denote the number of wrong bits of index demodulation and QAM demodulation, respectively. \(n_{t}\) denotes the total bits of an OFDM symbol, including the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{3}{c}{**Complexity**} \\ \cline{2-4}  & **Addition** & **Multiplication** & **Modulo** \\ \hline RSCS-IM & \(N_{T}(log_{2}\frac{N_{b}}{N_{T}}-1)\) & \(\frac{N_{T}log_{2}\frac{N_{b}}{N_{T}}(log_{2}\frac{N_{b}}{N_{T}}-1)}{2}+N_{T}\) & — \\ PSS Plus RP & \(4N(N_{T}-1)\) & \(N(N_{T}-1)\) & \(N_{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complexity of PSS Plus RP and RSCS-IM. bits of index selection. As the index selection of each group, i.e., each row in (14) did not affect each other, taking an arbitrary group for example, we could get  
\[n_{t}=log_{2}(\frac{N_{s}}{N_{T}})+log_{2}M. \tag{21}\]  
It was assumed that \(P_{g}\) denoted the error probability of greedy detection. When greedy detection was wrong, the bit error rate of index was \(P_{i}\) and the counterpart of QAM modulation was \(P_{\mathbb{Q}1}\). Otherwise, the bit error rate of QAM modulation was \(P_{\mathbb{Q}2}\). So, \(n_{i}\) could be written as  
\[n_{i}=P_{g}P_{i}log_{2}(\frac{N_{s}}{N_{T}}), \tag{22}\]  
and \(n_{\mathbb{Q}}\) as  
\[n_{\mathbb{Q}}=P_{g}P_{\mathbb{Q}1}log_{2}M+(1-P_{g})P_{\mathbb{Q}2}log_{2}M. \tag{23}\]  
From the principal of greedy detection, when greedy detection was wrong, i.e., \(max\{|n(\hat{\alpha})|^{2}\}>|x(\alpha)+n(\alpha)|^{2}\), we have  
\[P_{g}=P(max\{|n(\hat{\alpha})|^{2}\}>|x(\alpha)+n(\alpha)|^{2}). \tag{24}\]  
In line with [20], a closed-form expression of (24) is  
\[P_{g}=1-\sum_{k=0}^{\frac{N_{\mathbb{k}}}{N_{\mathbb{T}}}-1}\left( \begin{array} []{c}N_{s}/N_{T}-1\\ k\end{array} \right)\frac{(-1)^{k}}{k+1}e^{\gamma_{\alpha}(\frac{1}{k+1}-1)}, \tag{25}\]  
where \(\gamma_{\alpha}\) denotes the SNR of activated subcarriers and \(\left( \begin{array}{c}N_{s}/N_{\mathbb{T}}-1\\ k\end{array} \right)\) is the binomial coefficient. When it came to \(P_{i}\), we knew that the index demodulation was turning the decimal index into a binary bit stream and the index met \(I\in[0,\frac{N_{s}}{N_{T}}-1]\). The essence of wrong greedy detection was detecting the correct index as one of the other \(\frac{N_{s}}{N_{T}}-1\) wrong indexes. Meanwhile, the probability of detecting the correct index as any other wrong index was the same.