# A Random Subcarrier-Selection Method Based on Index Modulation for Secure Transmission
## 2 System Model  
### Conventional Random Subcarrier-Selection Method  
The frequency increments of the conventional uniform FDA are linearly assigned to all antenna elements, resulting in an insecure coupling effect [7]. For mitigating this problem, a random subcarrier-selection method was proposed in [18], where a \(N_{T}\)-element antenna array was employed at the transmit side and each element transmitted the same subcarrier symbol. The used subcarriers were randomly selected from the all-subcarrier set of OFDM. Assuming that the number of total subcarriers is \(N_{s}\), then we have the following:  
\[S_{sub}=\{f_{m}|f_{m}=f_{c}+m\Delta f,(m=0,1,\ldots,N_{s}-1)\}, \tag{1}\]  
where \(f_{c}\) is the reference frequency, \(m\) denotes the subcarrier index, and \(\Delta f\) stands for the sub-channel bandwidth. The random subcarrier-selection method PSS Plus RP is described below. Firstly, the authors constructed a random subcarrier index set defined as  
\[S_{p}=\{k_{p}|k_{p}\text{ {is a prime number}},k_{p}\in(0,1,2,\ldots,N_{s}-1)\}. \tag{2}\]  
In order to make a more random subcarrier index distribution over the transmit antennas, the authors proposed a random procedure (RP) by prime modulo operation and block interleaving. A constant \(p\) was chosen, which was the largest prime less than \(\sqrt{N_{T}}\). Then, the set \(S_{p}\) was partitioned into \(p\) subsets by taking all set elements modulo \(p\), which is represented as  
\[K=K_{0}\cup K_{1}\cup\cdots\cup K_{p-1}. \tag{3}\]  
Finally, the authors permuted the element order in the set \(K\) by block interleaving [18], which is depicted in Figure 1. In addition, the values of \(I\) and \(J\) should satisfy the inequality  
\[(J-1)I<N_{T}<JI. \tag{4}\]  
It is worth pointing out that the block interleaving would be repeated until the randomness of the subcarrier index was greater than a certain threshold. To measure the random degree of the selected subcarriers, a random metric (RM) \(\delta_{I}\) was defined in [18]. Suppose that the indices of the selected subcarriers are represented as  
\[\eta=\{\eta(1),\eta(2),\cdots,\eta(N_{T})\}. \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(X\sim\mathcal{CN}(0,\sigma^{2})\) & Sum. \\ \(\left(\begin{array}{c}n\\ k\end{array}\right)\) & The distribution of a circularly symmetric complex Gaussian. \\ The binomial coefficient. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations used in this manuscript.  
The subcarrier index spacing vector can be derived as  
\[\Delta\eta=[|\eta(1)-\eta(2)|,|\eta(2)-\eta(3)|,\cdots,|\eta(N_{T}-1)-\eta(N_{T})|]. \tag{6}\]  
Then, \(\delta_{I}\) can be expressed as  
\[\delta_{I}=\frac{1}{N_{T}-1}\sum_{i=1}^{N_{T}-1}(\Delta\eta(i)-\bar{\Delta} \eta)^{2}, \tag{7}\]  
where \(\Delta\eta(i)\) is the \(i\)-th element of \(\Delta\eta\). In addition, \(\bar{\Delta}\eta\) denots the average value of \(\Delta\eta\), which is given by  
\[\bar{\Delta}\eta=\frac{1}{N_{T}-1}\sum_{i=1}^{N_{T}-1}\Delta\eta(i), \tag{8}\]  
Figure 2 draws the flow chart of PSS Plus RP. As shown in Figure 2, firstly, \(N_{T}\) subcarriers were selected whose index was a prime number from the given subcarrier set in (1). Next, a prime number \(p\) was chosen, which was the largest prime less than \(\sqrt{N_{T}}\). Then, these \(N_{T}\) indices were partitioned into \(p\) groups using the modulo-\(p\) operation. Finally, the block interleaving operation was performed, as shown in Figure 1, on this \(p\)-group index and the RM was computed. If RM was larger than the predefined threshold, the index output after block interleaving was the desired index. Otherwise, the block interleaving operation was repeated until RM was larger than the predefined threshold. We could see that there were a number of unused subcarriers causing the waste of spectrum. Additionally, a proper threshold was hard to obtain and the repeated block interleaving operation increased the computational complexity.