# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 4 Simulation Results
### BER Performance under AWGN Channel  
Figure 4 shows the simulated results of the BER performance of the DLCSK, non-coherent DCSK, chaotic switching CSK, and antipodal CSK over the AWGN channel. It is noteworthy that the simulated coherent CSK systems are plotted assuming a perfect chaotic synchronization and only provide benchmark data for evaluation. Antipodal CSK is a chaotic modulation scheme with one basis function that can theoretically achieve the BER performance of binary phase shift keying (BPSK) under AWGN channels. In our simulations, the Logistic map is used for antipodal CSK, such that if \"0\" is to be sent, a chaotic signal \(\tilde{\mathbf{x}}\) is transmitted, and if \"1\" is to be sent, -\(\tilde{\mathbf{x}}\) is transmitted. The results are also compared with those of chaotic switching, a special case of CSK with two basis functions, in which the transmitted samples are obtained from the Chebyshev and the Logistic maps. The BER curve of the chaotic switching scheme is also simulated assuming a correlation receiver with perfect chaotic synchronization.  
The proposed SISO DLCSK scheme shows a gain compared to conventional DCSK in the lower SNRs, when the DLCSK system is trained at relatively lower SNR values (Training SNR \(\sigma_{tr}^{(n)}\in[11,15]\) dB). Since the auto and cross correlation properties of the chaotic signals are similar in this experiment, according to Equation (3), this gain comes from reducing the  
Figure 4: BER curve of the SISO DLCSK under AWGN channel for different \([\sigma_{tr,min},\sigma_{tr,max}]\), \(\beta=50\), in comparison with the benchmark modulations.  
third term, i.e., \(\int_{0}^{T}n\bar{x}\,dt\). When the training process is performed under different channel conditions, NN can indirectly estimate noise distribution, for use in iterative minimization of the cross-entropy cost function. As an important result, DLCSK shows a more robust behaviour in the test phase.  
When \(\sigma_{tr}\) is relatively high, the LSTM can only grasp the clean signal. For proper training, the SNR value must help the LSTM learn both clear and noisy samples. The results show that under the AWGN channels, when the \(\sigma_{tr}^{(n)}\in[19,23]\) dB, the performance of the SISO DLCSK is close to the conventional DCSK system. Therefore, for better BER performance in high-SNR conditions, the receiver should be trained at higher SNRs. These results seem trustworthy because the above-mentioned training options allow us to design a receiver with a flexible data rate depending upon specific channel conditions.  
Figure 5 depicts Monte Carlo simulations of the BER performances obtained from the SISO DLCSK system for \(\beta=50\) under an AWGN channel. The DLCSK system is trained over a limited SNR range, i.e., \([19,23]\) dB, and tested over the whole SNR range. The other parameters for both case 1 and 2 are selected according to Table 1. The results show that the BER performance of DLCSK has a low sensitivity to changes in the hyper-parameters, such as the number of training samples \(S\) and the number of hidden units \(H\). There is a trade-off between security and classification accuracy. When a system is trained with \(\rho_{train}=3.6\), using \(\rho_{test}=3.6\) results in more non-periodic behavior. However, reducing the \(\rho_{test}\) may result in a better BER performance.