# A Random Subcarrier-Selection Method Based on Index Modulation for Secure Transmission
## 2 System Model
### The Proposed Random Subcarrier-Selection Method Based on Index Modulation  
It could be seen from the description of the conventional scheme in Section 2.1 that the complexity of PSS Plus RP mainly resulted from block interleaving, which was repeated until the random degree was greater than the predefined value. Meanwhile, the spectrum was wasted since a large number of subcarriers were unused. For alleviating this problem, this paper proposed a novel random subcarrier-selection method based on index modulation (RSCS-IM) under a new FDA OFDM communication architecture. The array structure of the proposed FDA OFDM transmitter based on IM is shown in Figure 3, where the legitimate transmitter [PERSON] equipped a \(N_{T}\)-element linear antenna array with  
Figure 1: The principle of block interleaving. element space \(d\). The legitimate receiver [PERSON] and \(N\) eavesdroppers [PERSON] all employed a single antenna. In the conventional scheme [18], all antennas of [PERSON] transmitted the same subcarrier symbol per OFDM symbol. While in our scheme, all antennas of [PERSON] transmitted different subcarrier symbols per OFDM symbol to [PERSON] by selecting multiple subcarriers from the subcarrier set using IM. If the first antenna was taken as the reference, [PERSON] was located at \((\theta_{0},R_{0})\), which was represented as the red star. The positions of the \(N\) eavesdroppers were \((\theta_{1},R_{1}),(\theta_{2},R_{2}),\cdots,(\theta_{n},R_{n})\) respectively, which were represented as the black stars. Suppose there were \(N_{s}\) orthogonal subcarriers in our FDA OFDM-IM system and the set of subcarriers was  
\[S_{sub}=\{f_{m}|f_{m}=f_{c}+m\Delta f,m=0,1,\cdots,N_{s}-1\}, \tag{9}\]  
where \(f_{c}\) is the reference frequency, \(m\) denotes the subcarrier index, and \(\Delta f\) stands for the subchannel bandwidth, \(T_{s}=1/\Delta f\) denotes the period of an OFDM symbol. It was assumed that \(N_{s}\Delta f\ll f_{c}\) and \(d=\lambda/2\), where \(\lambda=c/f_{c}\) denotes the wave length and \(c\) is the light speed. The subcarrier frequency assigned to the \(n\)-th antenna was \(f_{\eta(n)}\), where \(f_{\eta(n)}\in S_{sub}\) and \(\eta(n)\in\{0,1,\cdots,N_{s}-1\}\). \(\eta(n)\) was determined by the bits used for index selection. It is worth pointing out that for different \(n_{1},n_{2}\), \(\eta(n_{1})\
eq\eta(n_{2})\). In a far-field scenario, the received signal at \((\theta_{0},R_{0})\) could be represented by  
\[s(t)=\sum_{k=1}^{N_{T}}x_{k}e^{i\varphi_{k}}e^{2\pi f_{\eta(k)}(t-\frac{R_{k}}{ c})}+\sum_{k=1}^{N_{T}}n_{k}, \tag{10}\]  
where \(f_{\eta(k)}=f_{c}+\eta(k)\Delta f\), \(R_{k}=R_{0}-(k-1)d\sin\theta_{0}\). \(\varphi_{k}\) denotes the initial phase of the \(k\)-th antenna and \(n_{k}\) is the received additive white Gaussian noise (AWGN) of the \(k\)-th antenna with the distribution of \(n_{k}\sim\mathcal{CN}(0,\sigma^{2})\). So, we can rewrite (10) as  
\[\begin{split} s(t)&=\sum_{k=1}^{N_{T}}x_{k}e^{i \varphi_{k}}e^{i2\pi(f_{c}+\eta(k)\Delta f)(t-\frac{R_{k}}{c})}+\sum_{k=1}^{N_ {T}}n_{k}\\ &=\sum_{k=1}^{N_{T}}x_{k}e^{i\varphi_{k}}e^{i2\pi f_{c}t}e^{i2\pi \eta(k)\Delta f_{t}}e^{-i2\pi f_{\eta(k)}\frac{R_{k}}{The concrete random subcarrier-selection method based on index modulation (RSCS-IM) will be described in this part. It was assumed that the number of total orthogonal subcarriers and antenna array elements were \(N_{s}\) and \(N_{T}\), respectively. For convenience, suppose \(N_{s}=2^{m_{1}}\) and \(N_{T}=2^{m_{2}}\), where both \(m_{1}\) and \(m_{2}\) are positive integers and \(m_{1}>m_{2}\)  
Figure 3: Array Structure of the FDA OFDM-IM Transmitter. Figure 2: Flow chart of PSS Plus RP. Firstly, the index of all the orthogonal subcarriers was divided into a matrix whose dimension was \(N_{T}\times\frac{N_{s}}{N_{T}}\), i.e.,  
\[I_{sub}=\left( \begin{array}{cccc}1&N_{T}+1&\cdots&N_{s}-N_{T}+1\\ 2&N_{T}+2&\cdots&N_{s}-N_{T}+2\\ \vdots&\vdots&\ddots&\vdots\\ N_{T}&2N_{T}&\cdots&N_{s}\end{array} \right).