# A Capacity Enhancement Method for Frequency-Hopping Anti-Jamming Communication Systems
## 2 Materials and Methods
### FH Capacity Boost Signal Design
#### 2.2.3 FH Capacity Boost Signal Spectral Analysis  
To derive the FH capacity boost signal spectrum, s(t) is divided into steady state and alternating waves [25]. * Steady State Wave Analysis  
Here, the steady state wave is the statistically averaged component of the random sequence s(t) and a weighted average of the probability of occurrence of each code element. When both Information A and Information B are equal-probability sources, the number of frequency-hopping points is M, and Information A adopts BPSK modulation; there are a total of 2M possible scenarios for the waveform of s(t) in each frequency-hopping cycle, and the probability of occurrence of each scenario is 1/2M. The statistical average of each frequency-hopping symbol is \(\rm{v(t)=0}\). Therefore, the s(t) signal contains no steady-state waves, and the s(t) signal spectrum contains no discrete spectral components. * Alternating Wave Analysis  
Alternating waves are defined according to the power spectral density Equation:  
\[\rm{P_{S}}(\omega)=\lim_{T\to\infty}\frac{E[|S_{T}(\omega)|^{2}]}{T} \tag{5}\]  
where E denotes the statistical average and T is the interception time, which is set equal to the length of \((2N+1)\) code elements, \(\rm{T=(2N+1)T_{s}}\). First, we find the spectral function, \(\rm{S_{T}(t)}\), of the function (1) of the random signal s(t). The truncated function, \(\rm{s_{T}(t)}\), of the random signal, \(\rm{s(t)}\), can be expressed as  
\[\rm{s_{T}(t)=} \sum\limits_{n=-N}^{+N}a(n)g_{a}(t-nT)b\omega_{0}(n)g_{b}(t-nT) \cos\omega_{0}t\] \[+\sum\limits_{n=-N}^{+N}a(n)g_{a}(t-nT)b\omega_{1}(n)g_{b}(t-nT) \cos\omega_{1}t \tag{6}\] \[+\ldots+\sum\limits_{n=-N}^{+N}a(n)g_{a}(t-nT)b\omega_{M}(n)g_{b }(t-nT)\cos\omega_{M}t\]  
For simplicity, the carrier term is initially ignored and the first term of the truncation function is taken to compute the spectrum, \(\rm{S_{T_{0}}(\omega)}\). \[\rm{s_{T_{\omega_{0}}}(t)=s_{T_{0}}(t)\cdot\cos\omega_{0}t} \tag{7}\]  
\[\rm{s_{T_{0}}(t)=\sum\limits_{n=-N}^{+N}a(n)g_{a}(t-nT)b\omega_{0}(n)g_{b}(t- nT)} \tag{8}\]  
\[\rm{S_{T_{0}}(\omega)=\int_{-\infty}^{\infty}s_{T_{0}}(t)e^{-j\omega t}dt= \sum\limits_{n=-N}^{+N}a(n)b\omega_{0}(n)\cdot e^{-j\omega nT}G_{ag}(\omega)} \tag{9}\]  
where \(\rm{G_{ab}(\omega)=\int_{-\infty}^{\infty}g_{a}(t)g_{b}(t)e^{-j\omega t}dt}\). \[\left|S_{T_{0}}(\omega)\right|^{2}=\sum\limits_{k=-N}^{+N}\sum\limits_{n=-N}^{+ N}a(n)b(n)\cdot e^{-j\omega(n-k)T}G_{ab}(\omega)G_{ab}^{\ast}(\omega) \tag{10}\]\[\mathrm{E}[|\mathrm{S}_{\mathrm{T}}(\omega)|^{2}]=\sum_{k=-N}^{+N}\sum_{n=-N}^{+N} \mathrm{E}[\mathrm{a}(n)b(n)]\cdot\mathrm{e}^{-\mathrm{j}\omega(n-k)T}\mathrm{G} _{\mathrm{ab}}(\omega)\mathrm{G}_{\mathrm{ab}}{}^{*}(\omega) \tag{11}\]  
1. When \(k\
eq n\), \(b(n)\cdot b(k)=0,a(n)\cdot a(k)=\left\{ \begin{matrix}1&0.5\\ -1&0.5^{\prime}\end{matrix} \right.\mathrm{E}[\mathrm{a}(n)^{2}]=0,E[b(n)^{2}]=0\). 2. When \(k=n\), \(a(n)\cdot a(k)=a(n)^{2}=1\), \(b(n)\cdot b(k)=b(n)^{2}=\left\{ \begin{array}{cc}1&\mathrm{P}=\frac{1}{M}\\ 0&1-\mathrm{P}\end{array} \right.\), and \(\mathrm{E}[\mathrm{a}(n)^{2}]=1,E[b(n)^{2}]=\frac{1}{M}\). \(\mathrm{E}[\mathrm{a}(n)^{2}]\), \(E[b(n)^{2}]\) has a non-zero value only at \(k=n\). \[\mathrm{E}[|\mathrm{S}_{\mathrm{T}}(\omega)|^{2}]=\sum_{n=-N}^{+N}\frac{1}{M} |\mathrm{G}_{\mathrm{ab}}(\omega)|^{2}=\frac{(2N+1)}{M}|\mathrm{G}_{\mathrm{ ab}}(\omega)|^{2} \tag{12}\]  
T is the interception time; let it be equal to \((2N+1)\) code element lengths, \(T=(2N+1)T_{s}\). \[\mathrm{P}_{S}(\omega)=\lim_{N\to\infty}\frac{(2N+1)}{(2N+1)T_{s}M}|\mathrm{G }_{\mathrm{ab}}(\omega)|^{2}=\frac{\omega_{S}}{2\pi M}|\mathrm{G}_{\mathrm{ ab}}(\omega)|^{2} \tag{13}\]  
When the carrier term is added, the power spectral density function of \(\cos\omega_{0}t\) is \(\frac{\pi}{2}[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]\).