## 2 System Design
### Ftn-Dcsk  
The FTN-DCSK system is constructed using a Sinc filter with a sub-Nyquist sampling rate to encode the chaotic sequence at the output of the DCSK system, as shown in the block diagram in Figure 1a. We use the normalized Chebychev map for chaos generation  
\[e_{i,k}^{p}=\left\{ \begin{array}{rl}&x_{ijk}^{p}&\text{for }2i\beta+1<k\leq(2i \beta+1)\beta\\ &b_{i}^{p}x_{i,k-\beta}^{p}&\text{for }(2i\beta+1)\beta<k\leq(i+1)2\beta, \end{array} \right. \tag{1}\]  
where \(x_{i,k}^{p}\) is the \(i\)th state for \(i\)th symbol and \(k\) is the index of chaotic sample, for the \(p\)th user, with a spreading factor \(\beta\) for the reference and data bearing signals, and the \(b_{i}^{p}\in[-1,1]\) are the data bits. We use a normalized Sinc filter \(g_{\rho(p)}^{k}\) to encode the chaotic sequences with a different sampling parameter \(\rho_{(p)}\) for every user, expressed as  
\[g_{\rho(p)}^{k}(t)=\sqrt{\rho_{(p)}}sinc(t-k\rho_{(p)}.T), \tag{2}\]  
where \(k\) is the chaotic sample being encoded, \(\rho_{p}\) is the sampling parameter for \(p\)th user, and \(T\) is the typical Nyquist period. The multiplication by the sampling parameter's square root is for normalizing the filters at both ends of the transmission. As a result, the continuous-time transmitted signal \(s_{i,k}^{p}(t)\), for the \(p\)th user, has the following form  
\[s_{i,k}^{p}(t)=e_{i,k}^{p}g_{\rho_{(p)}}^{k}(t). \tag{3}\]  
Figure 1: Block diagram of the MU-FTN-DCSK communication system, (**a**) Random \(p\)th user’s transmitter, (**b**) Frame shapes for different users with different sampling parameters, (**c**) Receiver structure for the desired user. For detection, the Sinc filter is first applied at the receiver, followed by the DCSK detection scheme. The multi-user system design is shown in Figure 1, where a different FTN sampling parameter is used for each user. The detection is based on selecting the desired user's signal by passing the combined signal, summed in the channel, through the filter with the designated sampling parameter \(\rho_{d}\), followed by the delay loop and the product and summation blocks. The index \(d\) is used to separate the symbol of the desired user from the rest of the users transmitting in the same channel. Hence, the received signal \(r^{d}_{i,k}(t)\) during \(2\beta\) samples, for the desired user with a sampling parameter \(\rho_{(d)}\), corresponding to the \(k\)th sinc pulse of the \(i\)th transmitted symbol, is expressed as  
\[\mathrm{r^{d}_{i,k}(t)=}\sum_{l=1}^{L}a^{d}_{i,l}s^{d}_{i,k-\tau_{l,l}}(t)+\sum_{k^{\prime}=-\infty}^{\infty}\sum_{p=1}^{U-1}\sum_{l^{\prime}=1}^ {L}\alpha^{p}_{i^{\prime},l^{\prime}}s^{p}_{i^{\prime},k^{\prime}-\tau_{p,l^{ \prime}}}(t)+W(t),\ \ \mathrm{i^{\prime}=\lfloor\frac{k^{\prime}}{2\beta}\rfloor}, \tag{4}\]  
where \(U\) is the total number of users, \(\alpha^{p}_{i^{\prime},l^{\prime}}\) is the \(p\)th user's channel coefficient over the \(l^{\prime}\)th channel tap, and \(l=\{1,\ldots,L\}\), \(\tau_{i^{\prime},l^{\prime}}\) is the \((l^{\prime})\)th channel tap delay for the \((i^{\prime})\)th symbol. Similarly, \(\alpha^{d}_{i,l}\) is the desired user's channel coefficient over the \(l\)th channel tap, \(l=\{1,\ldots,L\}\), \(\tau_{i,l}\) is the \(l\)th channel tap delay for the \(i\)th symbol, \(\lfloor.\rfloor\) is the floor operator, and \(W\) is the noise signal. The detection is based on applying the matched filter to the received signal \(r^{d}_{i,k}(t)\) in order to retrieve the different \(\epsilon^{d}_{i,k}\) samples and is expressed as  
\[r^{d}_{i,n}=\langle r^{d}_{i,k}(t),g^{n}_{\mathcal{P}_{(d)}}(t)\rangle. \tag{5}\]  
\(\langle.\rangle\) is the dot product. For filters with different sampling factors \(\rho_{(d)}\) at reception for the desired user and \(\rho_{(p)}\) for any random user with a different sampling parameter, the detection matrix is derived as follows,  
\[