# A Capacity Enhancement Method for Frequency-Hopping Anti-Jamming Communication Systems
## 2 Materials and Methods
### FH Capacity Boost Signal Design  
#### 2.2.1 FH Capacity Boost Signal Expression  
The FH capacity boost signal transmits Information A and Information B in two dimensions: phase and FM point, respectively.  
Information A uses BPSK modulation and Information B equivalently uses FSK modulation.  
Without channel coding, the FH capacity boost signal is denoted as  
\[\mathrm{s}(\mathrm{t})=\sum_{\mathrm{n}=-\infty}^{+\infty}\sum_{\mathrm{m}=0}^{ \mathrm{M}-1}\mathrm{a}(\mathrm{n})\mathrm{g}_{\mathrm{a}}(\mathrm{t}-\mathrm{ n}\mathrm{T})\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\mathrm{g}_{\mathrm{b}}( \mathrm{t}-\mathrm{n}\mathrm{T})\cos\omega_{\mathrm{m}}\mathrm{t} \tag{1}\]  
where the baseband Information A is expressed as \(\mathrm{a}(\mathrm{n})\). When \(\mathrm{A}=1\), \(\mathrm{a}(\mathrm{n})\) takes 1, indicating that the carrier phase is 0. When \(\mathrm{A}=0\), \(\mathrm{a}(\mathrm{n})\) takes \(-1\), which indicates that the carrier phase is \(\pi\). Information A is an equal probability source.  
\[\mathrm{a}(\mathrm{n})=\left\{\begin{array}{ll}1&\mathrm{A}=1\ \ \mathrm{p}_{ \mathrm{A}}(1)=1/2\\ -1&\mathrm{A}=0\ \ \mathrm{p}_{\mathrm{A}}(0)=1/2\end{array}\right. \tag{2}\]  
where the baseband Information B is expressed as \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\). The \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\) vector is obtained from the modulation mapping relation. \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\) is the coefficient before the carrier at each frequency point. When \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})=1\), it indicates that this carrier frequency is used for transmission. When \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})=0\), it indicates that this carrier frequency is not selected for transmission. \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\) to \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\) correspond to \(\omega_{0},\omega_{1}\ldots\omega_{\mathrm{M}-1}\). In each frequency-hopping cycle, only one \(\mathrm{b}\omega_{\mathrm{m}}(\mathrm{n})\) is 1 and the others are 0.  
When modulating a carrier with different frequency points using Information B, the M frequency points are divided into I groups of J frequency points each, which can modulate k bits of information.  
\[\mathrm{I}=\frac{\mathrm{M}}{\mathrm{J}}=2^{\mathrm{k}} \tag{3}\]  
The group label is i (the group label corresponds to the code element label), and \(0\leq\mathrm{i}\leq(\mathrm{I}-1)\). The frequency points within each group are labeled j, and \(0\leq\mathrm{j}\leq(\mathrm{J}-1)\). J is the number of frequency points in each group.  
Figure 2: Schematic diagram of information rate enhancement for frequency-hopping systems.  
Putting Information B in terms of one code element per k bits, the corresponding decimal value is \(\mathrm{i_{B}}\). Each Information B code element cycle corresponds to J frequency-hopping cycles. The jth frequency-hopping cycle of the current code element corresponding to \(\mathrm{b}_{\omega_{m}}(\mathrm{n})\) is  
\[\mathrm{b}_{\omega_{m}}(\mathrm{n})=\mathrm{b}_{\omega_{\mathrm{J+j}}}(\mathrm{ n})=\left\{\begin{matrix}1&(\mathrm{i}=\mathrm{i_{B}})\cap(\mathrm{j}=\mathrm{j_{B}}) &1/\mathrm{M}\\ 0&(\mathrm{i}\
eq\mathrm{i_{B}})\cap(\mathrm{j}\
eq\mathrm{j_{B}})&1-1/\mathrm{ M}\end{matrix}\right. \tag{4}\]  
where \(0\leq\mathrm{m}\leq(\mathrm{M}-1)\), \(0\leq\mathrm{i}\leq(\mathrm{I}-1)\), and \(0\leq\mathrm{j}\leq(\mathrm{J}-1)\).  
This demonstrates the correspondence between each k bits and code elements in Information B. The grouping of code elements can undergo real-time chaotic processing to improve the system's anti-interception performance.