## 2 PWC-Based \(m\)CAP-DM with 2D Chaotic Encryption  
### Principle  
Figure 1a,b illustrate the block diagrams of the transmitter and receiver of the proposed PWC-based \(m\)CAP-DM with 2D chaotic encryption, respectively. In the transmitter, the input bits are firstly divided into \(G\) groups through a bit splitter and every group of \(b\) bits is used to generate an \(m\)CAP subblock of length \(N\). Subsequently, every \(b\) bits is further partitioned into the index bits \(b_{i}\) and the constellation bits \(b_{c}\), i.e., \(b=b_{i}+b_{c}\). More specifically, the \(b_{i}\) bits are used to select the indices of \(k\) subbands out of \(N\) subbands in total via a chaotic index selector, while the \(b_{c}\) bits are employed to generate the corresponding constellation symbols to perform DM through a chaotic constellation mapper. Assuming the selected \(k\) subbands adopt constellation mode 1 while the remaining \(N-k\) subbands adopt constellation mode 2, the two constellation sets corresponding to constellation modes 1 and 2 can be denoted by \(\mathcal{M}_{1}=[S_{1}^{1},S_{2}^{1},\cdots,S_{M_{1}}^{1}]^{T}\) with size \(M_{1}\) and \(\mathcal{M}_{2}=[S_{1}^{2},S_{2}^{2},\cdots,S_{M_{2}}^{2}]^{T}\) with size \(M_{2}\), respectively, where \((\cdot)^{T}\) denotes the transpose operation. Since \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\) are two distinguishable constellation sets, we have \(\mathcal{M}_{1}\cap\mathcal{M}_{2}=\varnothing\). The detailed principle of DM with 2D chaotic encryption is introduced in the following subsection. Subsequently, the \(m\)CAP block can be created by concatenating \(G\) subblocks through a block creator. Before executing upsampling, PWC encoding is performed to mitigate the SNR imbalance. After passing through the in-phase (I) and quadrature (Q) filters, the transmitted PWC-based \(m\)CAP-DM signal with 2D chaotic encryption is obtained. Specifically, the impulse responses of a pair of orthogonal I and Q filters for the \(n\)-th (\(n=1,\cdots,m\)) subband can be, respectively, expressed as  
\[f_{n}^{I}(t)=g(t)cos(2\pi f_{c,n}t), \tag{1}\]  
\[f_{n}^{Q}(t)=g(t)sin(2\pi f_{c,n}t), \tag{2}\]  
where \(g(t)\) is the impulse response of the baseband shaping filter and \(f_{c,n}\) denotes the center frequency of \(n\)th subband. The root raised cosine filter (RRCF) was adopted as the baseband shaping filter in this work [10]. In the receiver, as shown in Figure 1b, the received signal first passes through \(m\) pairs of matched filters and the impulse responses of the matched filters for the \(n\)th subband are given by  
\[r_{n}^{I}(t) =f_{n}^{I}(-t), \tag{3}\] \[r_{n}^{Q}(t) =f_{n}^{Q}(-t). \tag{4}\]  
After downsampling, frequency-domain equalization (FDE) and PWC decoding are performed to recover the data symbols on each subband. Subsequently, the \(m\)CAP block is divided into \(G\) subblocks by a block splitter. The signal of each subblock can be detected by a low-complexity LLR detector. Letting \(y_{g}^{\eta}\) (\(g=1,\cdots,G;\eta=1,\cdots,N\)) be the input signal, the corresponding LLR value for the \(g\)th subblock is given by  
\[\lambda_{g}^{\eta}=\mathcal{C}+\ln\left(\sum_{i=1}^{M_{1}}\exp\left(-\frac{ \left|y_{g}^{\eta}-S_{1}^{1}\right|^{2}}{N_{0}}\right)\right)-\ln\left(\sum_{ j=1}^{M_{2}}\exp\left(-\frac{\left|y_{g}^{\eta}-S_{j}^{2}\right|^{2}}{N_{0}} \right)\right), \tag{5}\]  
where \(\mathcal{C}=\ln(k)-\ln(N-k)\) and \(N_{0}\) denotes the noise power. After subblock detection in each subblock, the index bits and the constellation bits can be recovered via a chaotic index decoder and a chaotic constellation demapper, respectively. Finally, the obtained index bits and constellation bits are combined by a bit combiner to yield the output bits.