# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 2 Traditional Correlation Receivers  
This section presents a statistical study on a typical CSK system equipped with the correlation receivers for a deeper identification of its unsolved problems. In contrast with regular communications systems in which the basis functions are periodic and orthogonal (e.g., sinusoidal functions), in a chaotic communications system, the basis functions are not orthogonal necessarily and differ from symbol to another. Consider now a coherent CSK system using two basis functions [10]. The elements of the signal set are given by  
\[\mathbf{s}_{b}=s_{b1}\mathbf{\tilde{x}}+s_{b2}\mathbf{\tilde{x}}, \tag{1}\]  
where \(b\) is the index of the current transmission bit, the weights \(s_{b1}\) and \(s_{b2}\) are the elements of the signal vector, and the basis functions \(\mathbf{\tilde{x}}\) and \(\mathbf{\tilde{x}}\) are discrete-time chaotic signals with \(\beta\) samples, i.e., \(\mathbf{\tilde{x}}=\{\tilde{x}_{1},\ldots,\tilde{x}_{\beta}\}\) and \(\mathbf{\tilde{x}}=\{\tilde{x}_{1},\ldots,\tilde{x}_{\beta}\}\). A binary bit data symbol is spread by a chaotic signal with the bit duration \(T_{b}\). Thus, we have \(T_{b}=\beta T_{c}\), where \(T_{c}\) is the time between each chaotic sample (chip). The transmitted sample functions are \(\mathbf{s}_{1}=\sqrt{E_{b}}\ \mathbf{\tilde{x}}\) and \(\mathbf{s}_{2}=\sqrt{E_{b}}\ \mathbf{\tilde{x}}\), displaying symbols \"0\" and \"1\", respectively. The corresponding signal vectors are (\(s_{11}\ s_{12}\))=(\(\sqrt{E_{b}}\ 0\)) and (\(s_{21}\ s_{22}\))=(\(0\ \sqrt{E_{b}}\)), where \(E_{b}\) stands for the average energy per bit. With two basis functions, the receiver should be configured with at least two correlators. The message is detected by correlating the received signal with two reference signals \(\mathbf{\tilde{x}}^{\prime}\) and \(\mathbf{\tilde{x}}^{\prime}\), and forming the corresponding observation signals (so-called decision variables) \(D_{b1}\) and \(D_{b2}\). If \(D_{b1}>D_{b2}\), then the decision is \"1\", and if \(D_{b1}<D_{b2}\), then the decision is \"0\". Consider now the output of a correlator under a noisy channel. In a coherent CSK system, the reference signals \(\mathbf{\tilde{x}}^{\prime}\) and \(\mathbf{\tilde{x}}^{\prime}\) are derived from the noisy received signal (\(\mathbf{s}_{b}+\mathbf{n}\)). The decision variable \(D_{b1}\), at the output of the correlator, can be written as  
\[\begin{split} D_{b1}&=\int_{T_{s}}^{T}[\mathbf{s}_{b}+ \mathbf{n}]\mathbf{\tilde{x}}^{\prime}\ dt\\ &=\int_{T_{s}}^{T}[s_{b1}\mathbf{\tilde{x}}+s_{b2}\mathbf{\tilde{x}}+\bm {n}]\mathbf{\tilde{x}}^{\prime}\ dt\\ &=s_{b1}\int_{T_{s}}^{T}\mathbf{\tilde{x}}\mathbf{\tilde{x}}^{\prime}\ dt+s_{b2}\int_{T_{s}}^{T}\mathbf{\tilde{x}}\mathbf{ \tilde{x}}^{\prime}\ dt\\ &+\int_{T_{s}}^{T}\mathbf{n}\mathbf{\tilde{x}}^{\prime}\ dt,\end{split} \tag{2}\]  
where \(T_{s}\) is the synchronization transient time for each bit duration. Note that \(D_{b1}\) is a random variable, whose mean value depends on In contrast with the traditional demodulation methods that often use synchronization (coherent schemes) or delayed reference transmission (non-coherent schemes) for reference recovery, the proposed DLCK utilizes DL for training and recovery of the reference signals \(\vec{x}^{\prime}\) and \(\vec{x}^{\prime}\). By using DL, it not only mitigates the need for chaotic synchronization during transmissions, but also can reduce the effect of the last term in Equation (4) using an indirect channel estimation method. It should be noted that our main focus is not on the auto and cross-correlation problems, or selecting proper chaotic maps. Therefore, we have chosen the Chebyshev map and logistic map to make chaotic sequences simpler in terms of generating.