## 4 Numerical Analysis and Results
### BER Analysis
The impact of the sampling rate on the BER performances over AWGN and multi-path Rayleigh fading channels is presented for different numbers of users up to a maximum \(U=9\) in Figure 5 and Figure 6, respectively. An SNR of 30 and a sampling parameter \(\rho\) of [0.05,..., 0.95], are used for the primary user, while the other user's sampling parameters are switched in the same range of values, but different from the desired user's sampling parameters in order to enable detection. It is interesting to notice that for a single user, in the high SNR regime, the reduction in sampling rate to half the Nyquist limit (\(\rho=0.5\)) does not hinder the MU-FTN-DCSK from having a successful transmission with a good BER performance, especially for the AWGN channel. The same observation can be applied for up to three users in the system with relatively close sampling rates but for a reduced spectral efficiency. As a result, the ISI, despite impacting the system's performance considerably even in the absence of noise, does not prevent a successful transmission, even with a substantial decrease in sampling rates. Therefore, the combination of DCSK with FTN seems very promising in improving the spectral efficiency if the detection can be improved. Figure 3: BER performance of MU-FTN-DCSK over AWGN channel for different spreading factors and number of users \(U=\{1,2,3,4\}\), \(\beta=150\), \(\rho_{(d)}=0.9\), \(\rho_{(p)}=\{0.8,0.7,0.6,0.5\}\), and SNR = 25 dB. Figure 4: BER performance of MU-FTN-DCSK over Rayleigh channel for a dissimilar power distribution, \(U=\{1,2,3,4\}\), \(\beta=150\), \(\rho_{(d)}=0.9\), \(\rho_{(p)}=\{0.8,0.7,0.6\}\). Figure 5: BER performance for different sampling rates and number of users over the AWGN channel, \(U=\{1,\ldots 9\}\), \(\beta=100\), \(\rho_{(p)}=\{0.95,0.9,\ldots 0.5\}\), SNR = 30 dB. The BER performance of the MU-FTN-DCSK system is dictated by the extra interference between the chaotic signals, which does not exist in traditional chaos-based systems. The chaos-chaos interference is caused solely by the functioning of the sub-Nyquist filters in a similar manner to the ISI. The impact of this new form of interference can be seen in Figure 7, where a noise-free channel (\(N_{0}=0\)) is considered for the transmission. As seen here, the chaos-chaos interference results in a performance floor that establishes the best BER performance that can be obtained for a given number of users. The new form of interference depends on the filter and its sampling rate and is also cumulative to the group of users in the system. The introduction of the custom-selected sampling rates to enable FTN and multi-user transmission results in an additional feature presented by the MU-FTN-DCSK in terms of security. Indeed, the probability of detection goes to zero if the appropriate sampling rate is not used as presented in Figure 8. Using a sampling rate different from the desired users' rate enables perfect secrecy, as seen in the simulation results in Figure 8 for \(\rho_{(d)}\
eq 0.9\). This intuitively comes down to the fact that the use of an erroneous sampling rate subsequently alters the capability of detection of the basic DCSK design, given that the value of \(\beta\) changes by the change of the sampling rate. Hence, MU-FTN-DCSK can be seen as an improved version of DCSK in terms of security since it requires an extra parameter for successful detection. Figure 6: BER performance for different sampling rates and number of users over the dissimilar multi-path Rayleigh fading channel, \(U=\{1,\ldots 9\}\), \(\beta=100\), \(\rho_{(p)}=\{0.95,0.9,\ldots 0.5\}\), SNR = 30 dB. Figure 8: BER performances of MU-FTN-DCSK for a faulty sampling rate. Figure 7: BER performances of MU-FTN-DCSK for a noiseless channel.