# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 3 SIMO DLCSK System Model
### Transmitter  
The transmitter structure is similar to conventional coherent CSK transmitters. Thus, the proposed DLCSK demodulator can easily work with existing CSK transmitters. In the training phase, the transmitter generates two sets of the chaotic signals, \(\{\tilde{\mathbf{x}}^{(n)}\}_{n=1}^{N}\) and \(\{\tilde{\mathbf{x}}^{(n)}\}_{n=1}^{N}\) using the Chebyshev and the Logistic maps, respectively. Each one of these sets includes the number of \(N\) chaotic signals with length \(\beta\) samples, i.e., \(\tilde{\mathbf{x}}^{(n)}=\{\tilde{\mathbf{x}}^{(n)}_{1},\ldots,\tilde{\mathbf{x}}^{(n)}_{ \beta}\}\) and \(\tilde{\mathbf{x}}^{(n)}=\{\tilde{\mathbf{x}}^{(n)}_{1},\ldots,\tilde{\mathbf{x}}^{(n)}_{ \beta}\}\). In other words, we use two sets of the chaotic signals, each with length \(S=N\beta\) samples, where \(N\) and \(\beta\) are the number and the length of the chaotic signals, respectively. All of the chaotic signals \(\tilde{\mathbf{x}}^{(n)}\) and \(\tilde{\mathbf{x}}^{(n)}\) with known binary labels \(j^{(n)}\in\{0,1\}\) are transmitted repeatedly to train the receivers. All the \(M\) antennas obtain the corresponding altered signals, i.e., \(\tilde{\mathbf{r}}^{(n)}_{m}=\{\tilde{\mathbf{r}}^{(n)}_{m,1},\ldots,\tilde{\mathbf{r}}^{( n)}_{m,\beta}\}\) and \(\tilde{\mathbf{r}}^{(n)}_{m}=\{\tilde{\mathbf{r}}^{(n)}_{m,1},\ldots,\tilde{\mathbf{r}}^{( n)}_{m,\beta}\}\), for input to the NN. After the number of \(2N\) transmissions, we have a set of received vectors \(\{\mathbf{r}^{(n)}_{m}\}_{n=1}^{2N}=\{\tilde{\mathbf{r}}^{(n)}_{m},\tilde{\mathbf{r}}^{( n)}_{m}\}_{n=1}^{N}\) with known class labels \(\{j^{(n)}\}_{n=1}^{2N}=\{0^{(n)},1^{(n)}\}_{n=1}^{N}\) to train the NN through the supervised learning framework. Therefore, training set of the \(m\)th DLCSK receiver can be expressed as  
\[\mathcal{D}_{m}=\{\mathbf{r}^{(n)}_{m},j^{(n)}\}_{n=1}^{2N}, \tag{5}\]  
where \(2N\) is the number of chaotic signals in the training set. Assuming a Rayleigh fading channel, the \(r_{m,t}\) (\(t\)th sample of the vector \(\mathbf{r}_{m}\)) can be modeled as a complex-valued random variable, i.e., \(r_{m,t}=\Re(r_{m,t})+\Im(r_{m,t})\), where the operators \(\Re(. )\) and \(\Im(. )\) represent the real and imaginary parts of a complex number, respectively. Therefore, the vector \(\mathbf{r}_{m}\) can be separated into two vectors, i.e., \(\mathbf{r}_{m}=[\Re(\mathbf{r}_{m}),\Im(\mathbf{r}_{m})]\), before being fed into the classifier, and the training set can be rewritten as  
\[\mathcal{D}_{m}=\{[\Re(\mathbf{r}_{m}),\Im(\mathbf{r}_{m})]^{(n)},j^{(n)}\}_{n=1}^{2N}. \tag{6}\]  
Once the DL-based receiver is trained, it can be used for online demodulation and data detection. In the test/deployment phase, the DLCSK modulator maps a transmission bit \(b^{(z)}\in\{0,1\}\), \((1\leq z\leq Z)\), to a chaos waveform, where \(b^{(z)}\) denotes \(z^{th}\) transmission bit and \(Z\) is the number of data bits. Thus, in the test phase, the transmitter generates two sets of the chaotic signals, i.e., \(\{\tilde{\mathbf{x}}^{(z)}\}_{z=1}^{Z}\) and \(\{\tilde{\mathbf{x}}^{(z)}\}_{z=1}^{Z}\) using the Chebyshev and the Logistic maps, respectively. Each of these sets includes the number of \(Z\) chaotic signals with length \(\beta\) samples, i.e., \(\tilde{\mathbf{x}}^{(z)}=\{\tilde{x}^{(z)}_{1},\ldots,\tilde{\mathbf{x}}^{(z)}_{\beta}\}\) and \(\tilde{\mathbf{x}}^{(z)}=\{\tilde{x}^{(z)}_{1},\ldots,\tilde{x}^{(z)}_{\beta}\}\). According to the Equation (1), and depending on the current bit \(b^{(z)}\), the signal \(\mathbf{s}^{(z)}_{b}\) is transmitted, i.e.,  
\[\mathbf{s}^{(z)}_{b}= \begin{cases}\sqrt{E_{b}}\;\tilde{\mathbf{x}}^{(z)}&\text{ if }b^{(z)}=0\\ \sqrt{E_{b}}\;\tilde{\mathbf{x}}^{(z)}&\text{ if }b^{(z)}=1.\end{cases}  \tag{7}\]  
Since in every practical communication system, the chaos generator circuits may operate under different environmental conditions, it is essential to consider a parameter mismatch between the training and testing phases.