# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## Appendix A  
In this appendix, we calculate the parameters of the probability distribution for the proposed NJPE. Let the NJPE output be a random variable \(Z\), and its sample mean and variance be \(M\) and \(S^{2}\), respectively. NJPE (\(Z\)) is expressed as Equation (6). We rewrite the equation as  
\[Z=M-\sqrt{M^{2}-S^{2}}. \tag{1}\]  
The multiplication of the received symbol and its complex conjugate is set as a random variable \(X\). When the number of observation samples is \(N\), \(X_{1},~{}X_{2},~{}\dots,~{}X_{N}\) are assumed as independent and identically distributed (i.i.d.) random variables. Then, sample mean \(M\) and sample variance \(S^{2}\) are defined as  
\[M=\frac{1}{N}\sum_{i=1}^{N}X_{i}, \tag{2}\]  
\[S^{2}=\frac{1}{N-1}\sum_{i=1}^{N}(X_{i}-\text{E}[X])^{2}. \tag{3}\]  
The mean and variance of \(M\) and \(S^{2}\) are obtained as follows [27]. \[\text{E}[M]=\mu, \tag{4}\]  
\[\text{Var}[M]=\frac{\mu_{2}}{N}, \tag{5}\]  
\[\text{E}\Big{[}S^{2}\Big{]}=\mu_{2}, \tag{6}\]  
\[\text{Var}\Big{[}S^{2}\Big{]}=\frac{\mu_{4}}{N}-\frac{{\mu_{2}}^{2}}{N}+\frac {2{\mu_{2}}^{2}}{N(N-1)}, \tag{7}\]  
where \(\mu\) and \(\mu_{2}\) denote the mean and variance of \(X\), respectively; \(\mu_{3}\) and \(\mu_{4}\) are the third and fourth central moment of \(X\), respectively. In FHSS, if the length of each slot was designed to be shorter than coherence time, and the channel gain was assumed to be constant in a slot. Then, random variable \(X\) follows gamma distribution [28]. When the number of samples is sufficiently large, \(M\) follows Gaussian distribution by the central limit theorem (CLT). Without loss of generality, the distribution of the sample variance was assumed to be Gaussian distribution [29]. As in Equation (1), i.e., \(Z=M-\sqrt{M^{2}-S^{2}}\), since \(M\) and \(S^{2}\) follow Gaussian distributions, the distribution of \(Z\) cannot be obtained in a closed form because of the square and square-root terms. This paper approximated the distribution of \(Z\) as gamma distribution on the basis of simulations. The parameters of gamma distribution were determined by the mean and variance of random variable \(Z\). The mean of \(Z\) could be calculated by second-order Taylor's approximation. Let \(\text{g}(\cdot)\) be \(\sqrt{\cdot}\). Then, the mean of \(Z\) is  
\[ \begin{split}&\text{E}[Z]=\text{E}[M]-\text{E}\Big{[}\text{g} \Big{(}M^{2}-S^{2}\Big{)}\Big{]}\\ &\approx\text{E}[M]-\text{g}\Big{(}\text{E}\Big{[}M^{2}-S^{2} \Big{]}\Big{)}-\frac{\text{Var}\big{[}M^{2}-S^{2}\big{]}}{2}\text{g}^{\prime }\Big{(}\text{E}\Big{[}M^{2}-S^{2}\Big{]}\Big{)}.\end{split}  \tag{8}\]In Equation (A4), \(\mathrm{E}\left[M^{2}-S^{2}\right]\) and \(\mathrm{Var}\!\left[M^{2}-S^{2}\right]\) are calculated as  
\[\mathrm{E}\left[M^{2}-S^{2}\right] =\mathrm{E}\!\left[M^{2}\right]-\mathrm{E}\!\left[S^{2}\right]\] (A9) \[=\mathrm{Var}\!\left[M\right]+\mathrm{E}\!\left[M\right]^{2}- \mathrm{E}\!\left[S^{2}\right]\] (A10)  
where  
\[\mathrm{Var}\!\left[M^{2}\right] =\mathrm{E}\!\left[\left(M^{2}\right)^{2}\right]-\mathrm{E}\! \left[M^{2}\right]^{2}\] (A11) \[=\frac{\mu_{4}}{N^{3}}+\frac{(2N-3)\mu_{2}}{N^{3}}+\frac{4\mu_{3 }\mu}{N^{2}}+\frac{4\mu_{2}\mu^{2}}{N},\] \[\mathrm{Cov}\!\left(M^{2},S^{2}\right) =\mathrm{E}\!\left[M^{2}S^{2}\right]-\mathrm{E}\!\left[M^{2} \right]\mathrm{E}\!\left[S^{2}\right]\] (A12) \[=\frac{\mu_{4}}{N(N-1)}+\frac{2\mu_{3}\mu}{N-1}+\mu_{2}\mu^{2}+ \mu^{2}.\]  
From Equations (A9) and (A10), the mean of \(Z\) can be rewritten as  
\[\mathrm{E}\!\left[Z\right]=\mu-\sqrt{\mu^{2}-\mu_{2}}.\] (A13)  
The variance of \(Z\) can be calculated by linearizing the square root of Equation (A1).