# Secure Mimo Communication System with Frequency Hopping Aided OFDM-DCSK Modulation
## 2 MIMO-FH-OFDM-DCSK System
### Transmitter  
At the transmitter, firstly, we use the binary phase shift keying (BPSK) to modulate the user data. Secondly, the BPSK symbols \([a_{1},a_{2},\ldots]\) are encoded by STBC, which can be expressed as [19]:  
\[\mathbf{A}=\left[ \begin{array}{cc}a_{1}&-a_{2}^{*}\\ a_{2}&a_{1}^{*}\end{array} \right], \tag{1}\]  
where \((\cdot)^{*}\) represents the conjugate operation. For the matrix \(\mathbf{A}\), in the row direction, the symbols are delivered from the same antenna, while in the column direction, the symbols are delivered in the same time slot. After composing the serial data stream that each antenna needs to send, the transmitter will conduct the serial to parallel (S/P) conversion and perform the FH-OFDM-DCSK modulation. As shown in Figure 2, each data stream delivered via the antenna 1 of the MIMO communication system is first modulated by the chaotic sequences. Then, we use the FH module to make chaotic modulation symbols hop in the frequency domain. Subsequently, the symbols generated by the FH module are modulated using the inverse fast Fourier transform (IFFT) through the OFDM module. After the parallel to serial (P/S) conversion and adding a cyclic prefix (CP), the resultant signals are transmitted to the wireless channels [18]. Figure 1: Transceiver structure of the 2 \(\times\) 1 MIMO-FH-OFDM-DCSK system. To be more explicit, let \(\mathbf{b}_{1}\) represent the first row of symbols in matrix \(\mathbf{A}\), which can be expressed as follows:  
\[\mathbf{b}_{1}=[a_{1},-a_{2}^{*}]. \tag{2}\]  
Similarly, we can obtain an expression for \(\mathbf{b}_{2}\):  
\[\mathbf{b}_{2}=[a_{2},a_{1}^{*}]. \tag{3}\]  
The symbols modulated by DCSK can be expressed as:  
\[\mathbf{g}_{\mathbf{k}}=\mathbf{b}_{1}\times c_{k}, \tag{4}\]  
where \(c_{k}\) represents the chaotic sequence, and \(\mathbf{g}_{\mathbf{k}}=[g_{0,k},g_{1,k},\cdots,g_{n,k},\cdots,g_{N-1,k}]\) represents the symbols modulated by DCSK at antenna 1 and \(g_{N-1,k}=a_{N-1}\times c_{k}\), where \(N\) is the number of subcarriers, \(a_{N-1}\) represents the \((N-1)\)th symbol in \(\mathbf{b}_{1}\), \(g_{0,k}=c_{k}\) denotes the \(k\)th chaotic reference symbol, and \(g_{N-1,k}\) denotes the \(k\)th chaotic symbol carrying symbol \(a_{N-1}\). Note that the chaotic sequences are generated by the second order Chebyshev polynomial function (CPF) [16], which can be expressed as follows:  
\[c_{k+1}=1-2c_{k}^{2},(0\leq k\leq\beta-1), \tag{5}\]  
where \(\beta\) is the length of chaotic sequences. The modulation process of the second row of symbols in matrix \(\mathbf{A}\) is similar to the first row of symbols in matrix \(\mathbf{A}\). With reference to [18], the FH module is defined as a matrix represented by \(\mathbf{U}\). Without loss of generality, here we set the length of the chaotic sequences to be the same as that of the number of the subcarriers, i.e., \(\beta=N\). Then, after the FH, we can obtain the symbols as:  
\[\mathbf{F}_{1}=\mathbf{G}_{1}\cdot\mathbf{U}, \tag{6}\]  
where the symbols modulated by DCSK at antenna 1 are represented by \(\mathbf{G}_{1}=[\mathbf{g}_{0}^{T},\mathbf{g}_{1}^{T},\cdots,\mathbf{g}_{ \mathbf{k}}^{T},\cdots,\mathbf{g}_{\mathbf{\beta-1}}^{T}]\)\((\cdot)^{T}\) represents the transposition operation. \(\mathbf{F}_{1}\) represents the symbols obtained after \(\mathbf{G}_{1}\) hopping in antenna 1, \(\mathbf{U}\) represents the hopping matrix used at antenna 1, the generation process of \(\mathbf{U}\) can be expressed as follows [18]:  
\[\mathbf{U}=\left[ \begin{array}{cc}\mathbf{0}&\mathbf{E}_{m}\\ \mathbf{E}_{N-m}&\mathbf{0}\end{array} \right], \tag{7}\]  
Figure 2: Structure of the FH-OFDM-DCSK transmitter with the antenna 1.  
where \(\mathbf{E}_{m},\mathbf{E}_{N-m}\) denote the identity matrices of order \(m\) and order \(N-m\), respectively; thus, \(\mathbf{U}\) can be equivalent to an identity matrix \(\mathbf{E}_{N}\) that has undergone several elementary transformations.