## 4 Numerical Analysis and Results
### BER Analysis  
Figure 2 presents BER results for different numbers of users over the AWGN channel. The parameters used for the simulations are \(\beta=150\), \(\rho_{(p)}=\{0.9,0.8,0.7,0.6\}\) where the main user always takes the value of \(\rho_{(d)}=0.9\). We can notice the tiny gap for a single user transmitting at a sub-Nyquist rate (\(\rho_{(d)}=0.9\)) in comparison with normal Nyquist transmissions (\(\rho_{(d)}=1\)), labeled as the normal DCSK, due to the ISI introduced by the sinc filter functioning in sub-Nyquist regime. For \(U=1\) the ISI introduced by the matched filters results in an approximate 1 dB loss of BER, but with a gain of 10% of the bandwidth. This means that slightly raising the energy for transmission, in order to compensate for the  
1 dB, would allow a gain of 10% in bandwidth usage in both the reference and the data sequences. The analytical results match the simulation results proving the validity of the new covariance derived in (34). The same observation is perceived when the number of users is increased. We notice that up to three users with different sampling parameters can be accepted in the system with an acceptable BER for noisy environments. In the setting used for the simulations that lead to these results with sampling parameters \(\rho_{(p)}=\{0.9,0.8,0.7\}\), we obtain a slot of length (0.6 \(\times\)\(\beta\)) of extra bandwidth. As a comparison, for \(U=4\) users, the BER performance offered by the MU-FTN-DCSK over an AWGN channel matches that of the regular DCSK over the Nakagami-m fading for a channel parameter \(m=0.5\)[3] due to the severe interference caused by the simultaneous transmissions of all users in the sub-Nyquist regime. Nonetheless, the bandwidth gain here is \(2\beta\) if we sum the individual gains from different users transmitting at a sub-Nyquist rate in both the reference and data signals. An optimization in the choice of the spreading factor would improve these results since it can significantly contribute to the reduction of ISI. In this sense, Figure 3 provides insights into the impact of the spreading factor relative to the BER performances of the MU-FTN-DCSK system for an SNR = 25 dB. The sampling parameter takes the values \(\rho_{(p)}=\{0.9,0.8,0.7,0.6,0.5\}\) with \(\rho_{(d)}=0.9\) for the desired user. As observed in this figure, increasing \(\beta\) positively impacts the system's capability to reduce the ISI for a certain range before reaching a plateau of BER performances centered around certain values depending on the number of users under consideration. Notably, after these specific values, higher BER values are observed. However, the value of \(\beta\) is also limited by the desired data rate, which creates a trade-off. We also notice that a system with a superior number of users and higher spreading factors achieves the same BER as a system with fewer users with reduced spreading factors for certain values of BER. Hence,  
Figure 2: BER performance of MU-FTN-DCSK over AWGN channel with different number of users \(U=\{1,2,3,4\}\), \(\beta=150\), \(\rho_{(d)}=0.9\), \(\rho_{(p)}=\{0.8,0.7,0.6\}\). trading the two configurable parameters for optimal access is possible with the new design of MU-FTN-DCSK. The BER performance of the MU-FTN-DCSK over multi-path Rayleigh fading channels with dissimilar channel gains is presented in Figure 4. We can notice from the BER performance of a single user that the degradation caused by the ISI is comparable in magnitude to the degradation for the AWGN situation. However, now the impact of the floor introduced by the extra Chaos-Chaos interference on the BER, derived in (34), is more apparent, regardless of the number of users in the system. Nonetheless, it can be seen that such high ISI has a nearly identical impact on the BER for the numbers of users superior to two. Moreover, the BER performances in this figure confirm the system's capability to combat multi-path loss.