## 2 System Model of RSST-SR-DCSK
### Receiver Structure of RSST-SR-DCSK  
Figure 6 shows the receiver structure of RSST-SR-DCSK. During the \(i\)th symbol period when the system is recovering information, the demodulator first takes the received signal \(r_{i,k}\) and delays it by \(R\) to obtain \(r_{i-R,k}\). After the descrambling process, it becomes \(\hat{r}_{r-R,k}\), thus extracting the first \(R\) signals in a data frame, which is the reference signal. To correctly demodulate the data information \(b_{k+M-1}\), it is necessary to perform \(P\) correlation calculations between the delayed and descrambled reference signal \(\hat{r}_{r-R,k}\) and the actual received information \(r_{i,k}\). Finally, the \(P\) independently calculated correlation values are summed, and the accumulated signal value is compared with the decision threshold. To demodulate the \(N\)-dimensional information \(b_{0}\cdots b_{N-1}\), it is necessary to delay the received signal \(r_{i,k}\) and then pass it through \(2^{N}\) descramblers. Subsequently, we correlate the received signal \(r_{i,k}\) with the delayed and descrambled reference signal \(\hat{r}_{r-R,k}\). The descrambler with the maximum absolute value can be identified as the one chosen by the transmitter. Finally, we compare the result with the decision threshold. This process yields the N-dimensional information \(b_{0}\cdots b_{N-1}\), and finally, the information data \(b_{k}\cdots b_{k+M-1}\) are obtained. Assuming that \(N\) is set to 2, the working principle of the descrambler 1 is to swap the signal values corresponding to adjacent positions in the received signal after L after delay; the working principle of the descrambler 2 is to divide the received signal after L after delay into two segments equally, and swap the signal values corresponding to the even positions in the first half of the segment with the signal values corresponding to the odd positions in the second half; the working principle of the descrambler 3 is to divide the received signal after L after delay into four segments equally, and mirror invert the signal values in the first and third segments. The received signal is divided into four segments equally, and the signal values in the first and third segments are mirrored and inverted; the principle of operation of the disambiguator is that the received signal after the delay L is divided into two segments equally, and the signal values corresponding to the even positions in the front and rear segments are interchanged. To recover the information bits \(b_{k+M-1}\), the correlator at the receiving end correlates the signal \(\hat{r}_{r-R,k}\), formed after descrambling \(r_{i,k}\), with the delayed received signal \(r_{i-R,k}\). At the end of the \(i\)th bit time, the output of the correlator is  
\[y_{l}=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}r_{i,l}\hat{r}_{i-R,k} \tag{7}\]  
which are compared and sent to the threshold detector. The input to the detector at this time instant, denoted by \(y_{l}\), is given using  
\[r_{i,k}=S_{i,k}+\xi_{i,k} \tag{8}\]  
where \(\xi_{i,k}\) is the noise component, which follows a Gaussian distribution with a mean of 0 and a variance of \(N_{0}/2\). Here, the Gaussian approximation proposed in [27] is utilized to assume that each of the constituent terms of the judgment variable at the receiving end obeys a Gaussian distribution. Substituting Equation (8) into (7), we find  
\[ \begin{array}{l}y_{l}=\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}r_{i,l} \hat{r}_{i-R,k}\\ =\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}\left(b_{k+M-1}x_{i-R,k}+\xi_{i,k }\right)\left(x_{i-R,k}+\xi_{i-R,k}\right)\\ =\sum\limits_{p=1}^{P}\sum\limits_{i=0}^{R-1}\left(\underbrace{b_{k+M-1}x_{i- R,k}^{2}+\underbrace{b_{k+M-1}x_{i-R,k}\xi_{i-R,k}+x_{i-R,k}\xi_{i,k}}}_{\text{ integral interference}}+\underbrace{\xi_{i,k}\xi_{i-R,k}}_{\text{noise}}\right)\\ \end{array}  \tag{9}\]  
Figure 6: RSST-SR-DCSK receiver. Notice that the input to the detector consists of three components, namely, the required signal, inter-user interference, and noise.