# Optimization Scheme for Modulation of Data Transmission Module in Endoscopic Capsule
## 2 Principle of the MIP-MDCSK Scheme
### IP-MDCSK and General MIP-MDCSK Sharing One Reference Signal over In-Body Channel 2.2.1. IP-MDCSK  
The system based on IP-MDCSK with \(n_{c}\) bits is achieved by index modulation, and the \(m_{c}\) bits are achieved by MDCSK. Thus, the total number of transmitted bits is \(m_{c}+n_{c}\), and the symbol duration is \((1+2^{n_{c}})R\), with \(\log_{2}M=m_{c}\). The chaotic generator generates the reference signal \(c_{x,k}\), with \(c_{y,k}\) being its Hilbert transform. Thus, the transmitted signal is \(t_{s}=a_{s}c_{x,k}+b_{s}c_{y,k}\), where \(s=1,\ldots M\) is the symbol in the signal space [27], and \(a_{s}\) and \(b_{s}\) are the orthogonal coordinates of the chaotic symbol \(t_{s}\). Assume that \(a_{l}\) time slots are chosen from a total of \(P\) time slots to transmit the information-bearing signals, while the remaining \(P-a_{l}\) time slots remain unused. \(a_{l}\) takes values within the range from \(1\) to \(a_{l}\), where \(a_{l}\) is less than P. As a result, the total is given by \(u=\sum_{p=1}^{P}\frac{p!}{d!(P-a_{l})!}\) for indexes, and t is the number of indexes selected. Hence, the output of the transmitter is expressed as  
\[\mathbf{s_{x0}}=[\underbrace{c_{x}}_{reference\,\ \  
where \(\delta(.)\) is the impulse function, \(\alpha_{l}\) and \(\tau_{l}\) are the gain and delay of the lth path, respectively, and \(L\) is the number of paths. \(W_{k}\) is the noise for in-body whose PDF follows Equation (2). When \(\alpha_{l}\) and \(L\) paths are one, the channel has only one path.  
At the receiver, both MDCSK and index positions are detected. For the index part, position information-bearing signals \(r_{inf}=r_{al}\). After obtaining the position of information-bearing signals, the reference signal \(\tilde{c}_{x,k}\) on the receiver correlates with the received information-bearing signal \(\tilde{t}_{s}\), and then the \(z_{a,\rho}\) is obtained. Moreover, \(z_{b,\rho}\) can be obtained by correlating \(\tilde{t}_{s}\) and \(\tilde{c}_{y}\), where \(\tilde{c}_{y,k}\) is the Hilbert transform of \(\tilde{c}_{x,k}\). The received symbol is decided by \(\arctan(z_{a,\rho}/z_{b,\rho})\). We assume that the delay \(\tau\) is much shorter than the chaotic sequence duration, i.e., \(0<\tau<<R\). Thus, the inter-symbol interference (ISI) present can be neglected [27]. After finding the indexed position, the decision variable of \(z_{a,\rho}\) and \(z_{b,\rho}\) can be written as  
\[z_{a,\rho} =\sum_{k=1}^{R}(\sum_{l=1}^{L}\alpha_{l}c_{x,k}+W_{k})(\sum_{l=1 }^{L}\alpha_{l}(a_{s}c_{x,k}+b_{s}c_{y,k}) \tag{8}\] \[+W_{k+R}),\] \[z_{b,\rho} =\sum_{k=1}^{R}(\sum_{l=1}^{L}\alpha_{l}c_{y,k}+\widetilde{W}_{k} )(\sum_{l=1}^{L}\alpha_{l}(a_{s}c_{x,k}+b_{s}c_{y,k})\] (9) \[+W_{k+R}).\]  
Similarly, when \(a,b\
eq a_{l}\), the corresponding decision variable can be expressed as  
\[z_{ao,n} =\sum_{k=1}^{R}(\sum_{l=1}^{L}\alpha_{l}c_{x,k}+W_{k})W_{k+R}, \tag{10}\] \[z_{bo,n} =\sum_{k=1}^{R}(\sum_{l=1}^{L}\alpha_{l}c_{y,k}+\widetilde{W}_{k} )W_{k+R}, \tag{11}\]  
\(a_{l}\) can be detected using the maximum comparator as  
\[\hat{a}_{l}=\arg\max_{m=1,\ldots,p}(|z_{a,\rho}+z_{a,n}|), \tag{12}\]  
where \(W_{k}\) is caused by the reference signal and \(\widetilde{W}_{k}\) is caused by the Hilbert transform of \(W_{k}\).