# Design of a SIMO Deep Learning-Based Chaos Shift Keying (Dlcsk) Communication System
## 4 Simulation Results
### BER Performance under Multi-Path Rayleigh Fading Channels  
Figure 10 illustrates a comparison between the BER performance of the SISO DLCSK and conventional DCSK over multi-path Rayleigh fading channels, for \(\beta=50\) and \(\sigma_{tr}^{(n)}\in[19,23]\) dB. We consider two cases corresponding to different path gain ratios and different path delays. In the first case, a two-path channel is considered (\(L=2\)) in which the two paths have similar average power gain. In this case, the average power gain in each path is 0.5, (i.e., \(E(\alpha_{1}^{2})=E(\alpha_{2}^{2})=0.5\)), with \(\tau_{1}=0\), and \(\tau_{2}=2\). In the second case, three paths (\(L=3\)) are considered with different average power gains. The average power gains are \(E(\alpha_{1}^{2})=1/7\), \(E(\alpha_{2}^{2})=2/7\), and \(E(\alpha_{3}^{2})=4/7\) with \(\tau_{1}=0,\tau_{2}=3\), and \(\tau_{3}=6\). The average power gain of the third path is 3 dB below the second path, and for the second path it is 3 dB below the first path. When the receiver is already accustomed to chaotic signals, we do not need to transmit a reference signal, it means less energy will be used to transmit one bit. As shown in Figure 10, the SISO DLCSK overcomes the BER inefficiency of the existing DCSK under Rayleigh fading channels. For further benchmark comparisons, we consider the antipodal CSK system. The BER performance of DLCSK is close to antipodal  
Figure 8: Sensitivity and specificity of LSTM-based classifier under AWGN channel for different values of bifurcation parameters, \(\beta=50\), and \(\sigma_{tr}^{(n)}\in[19,23]\) dB.  
CSK when \(E_{b}/N_{0}\leq 10\) dB, and both systems have similar performance when \(E_{b}/N_{0}\) is more than 10 dB.  
Figure 11 compares the BER performance of the SIMO DLCSK with the SISO-DLCSK under Rayleigh fading channels. The number of receive antennas is \(M=\{1,3,5\}\), and the spreading factor is \(\beta=50\). The results show that the SIMO DLCSK attains a better BER performance due to the diversity gain. For example, when \(M=1\), to achieve the target  
Figure 10: BER of SISO DLCSK for two-path and three-path Rayleigh fading channels, \(\sigma_{tr}^{(n)}\in[19,23]\) dB and \(\beta=50\), in comparison with the benchmark modulations.  
Figure 9: BER of SISO DLCSK under AWGN channel, for \(\beta=50\) and different numbers of training epochs (E).  
\(BER=10^{-2}\), the required SNR will be about 20 dB. If we increase the number of antennas, i.e., \(M=5\), the required SNR to achieve the \(BER=10^{-2}\) will only be about 15 dB. Figure 11 also compares the BER performance of the SIMO DLCSK with a recently published LSTM-aided OFDM DCSK in [40]. The LSTM-aided NN calculates the correlations between chaotic modulated OFDM-DCSK signals in order to retrieve the data. For a fair comparison, we consider a SIMO DLCSK with \(M=5\) and an LSTM aided OFDM DCSK with \(K=5\), where \(K\) indicates the number of independent sub-channels in such a system. The SIMO DLCSK with \(M=5\) independent sub-channels can be fairly compared to LSTM-aided OFDM DCSK with \(K=5\) sub-channels. The LSTM-aided receiver sends reference signals that do not carry useful information. The proposed DLCSK scheme shows an expected gain compared to the LSTM-aided OFDM DCSK because of using indirect channel estimation and reducing the burden of the additional reference sample transmission.