# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 3 SIMO DLCSK System Model
### Channel Model and Estimation  
In an AWGN channel, a one-dimensional noisy version of the transmitted signal \(s_{b}\) can be observed at the receiver side. The \(m\)th receiver obtains the altered signals \(\mathbf{r}_{m}\) and makes a decision about the transmitted bit. In addition to the Gaussian noise effect, many other stochastic phenomena may occur in a practical communication channel. Conventional channel estimation approaches are sensitive in terms of the quality of the pilot signals. One way of augmenting DL models is through the use of learning channel variations [73]. If the LSTM-based classifier is trained using a dataset that contains signals transmitted under different channel conditions, the classifier will be resilient to channel changes, eliminating the need for instantaneous channel estimation [74; 75]. In this paper, we assume that the channel changes from one signal transmission to another, and the transmitted signal acts as a pilot that carries channel information. Thus, the NN can simultaneously obtain different chaotic maps and estimate the statistical distribution of the fading.  
Consider a multi-path fading channel, with \(L\) independent paths, where the channel coefficients follow a Rayleigh distribution. Therefore, the probability density function (pdf) of the channel coefficient \(\alpha\) can be given as [76],  
\[\mathrm{q}(\alpha|\delta)=\frac{\alpha}{\delta^{2}}e^{\frac{-\alpha^{2}}{2 \delta^{2}}}, \tag{8}\]  
where \(\delta>0\) is the scale parameter of the distribution representing the root mean square value of the received voltage signal. Considering a multi-antenna system with the number of \(M\) antennas, if the \(t\)th transmitted sample is shown by \(s_{b,t}\), the \(t\)th received sample at the \(m\)th antenna can be modeled as  
\[r_{m,t}=\sum_{l=1}^{L}\alpha_{m,l}s_{b,t-\tau_{m,l}}+n_{m,t},\ m=1,2,\ldots,M, \tag{9}\]  
where \(\alpha_{m,l}\) represents the channel gain of \(l\)th path between the transmit antenna and \(m\)th receive antenna, and \(\tau_{m,l}\) is the delay of the \(l\)th path. \(L\) is the number of paths, and \(n_{m,t}\) is independent noise at each antenna, which is assumed to be additive white Gaussian noise with zero mean and variance \(N_{0}/2\).  
Since we assume that the channel changes from one signal transmission to another, the channel gain \(\alpha_{m,l}^{(n)}\) and training SNR \(\sigma_{tr}^{(n)}\) changes after the \(n\)th channel realization. In particular, \(\sigma_{tr}^{(n)}\) is a Gaussian random variable, such that \(\sigma_{tr}^{(n)}\in[\sigma_{tr,min},\sigma_{tr,max}]\), where \(\sigma_{tr,min}\) and \(\sigma_{tr,max}\) are optional SNR values. The goal of training process is to train a NN with a complex-valued vector \(\mathbf{r}_{m}\). The input complex values are split into real and imaginary parts, i.e., \(\mathbf{r}_{m}=[\Re(\mathbf{r}_{m}),\Im(\mathbf{r}_{m})]\), before being fed into the classifier. Therefore, we have two feature vectors containing the channel coefficients and stochastic phases. Later, a Softmax layer estimates probability vectors \(p_{n,j}\) from input distribution, for the \(n\)th observation, where \(j\) shows the possible classes (i.e., \(j\in\{1,2\}\)), and optimizes the cross-entropy cost function in Equation (13). In the test phase, we use this trained network to detect unknown inputs.