# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## 2 Proposed Estimator and Detector
### Jammer Detector wirh Gamma Distribution  
Jammer detectors determine a jammed slot when an estimated noise power of a slot is greater than those of the other slots. A robust jammer-detection method is the constant false-alarm-rate (CFAR) detector, which calculates a threshold obtained from a given false-alarm-rate probability (\(P_{fa}\)) and compares the threshold with a test sample. The conventional CFAR detector of radar systems assumes that background noise follows Gaussian distribution. However, the output distribution of the proposed NJPE was not Gaussian because of the square and square-root terms in Equation (6).  
In this study, we approximated that the NJPE output followed gamma distribution on the basis of simulations, and derived the jammer detector based on gamma distribution. In the experiment section, we demonstrate that the detection performance of the derived jammer detector attained better detection probability than those of conventional Gaussian-distribution-based detectors.  
To calculate a relationship between threshold and false-alarm rate \(P_{fa}\), let an actual and an estimated noise-jammer power be \(\theta\) and \(z\), respectively. The conditional probability-density function (PDF) of \(z\) given \(\theta\) (\(f_{Z}(z|\theta)\)), follows gamma distribution and is given as (see Appendix A)  
\[f_{Z}(z|\theta)=\frac{1}{\Gamma(\alpha(\theta))\beta(\theta)^{\alpha(\theta)} }z^{\alpha(\theta)-1}\exp(-z/\beta(\theta)), \tag{7}\]  
where \(\alpha(\theta)\) and \(\beta(\theta)\) denote  
\[\alpha(\theta)=\frac{NP_{r,l}{}^{2}}{\theta^{2}+4P_{r,l}\theta+2P_{r,l}{}^{2}}, \tag{8}\]  
\[\beta(\theta)=\frac{\theta\Big{(}\theta^{2}+4P_{r,l}\theta+2P_{r,l}{}^{2} \Big{)}}{NP_{r,l}{}^{2}}, \tag{9}\]  
where \(N\) denotes the number of samples, and \(P_{r,l}\) denotes a received symbol power of the \(l\)-th slot (\(\text{E}_{\text{s}}\text{E}\big{[}|\textit{$\textit{$\textit{$\textit{$\textit{ $\textit{$\textit{$\textit{$\textit{$\textit{$\textit{$\textit{$ $\textit{$\textit{$\textit{$\textit{$\textit{$\textit{$\textit{$\textit{$ $}$}$\textit{$\textit{$ $\textit{$\textit{${$\textit{${$\textit{${${$}$}$}$\textit{$ $\textit{${$\textit{${${}}}${}${}${}${${}}}{${${${${${}}}}{${${${${$ $\textit{$\textit{$$$\textit{$$$$$$$}$}{$$$$$${$$$$$$$$$}{$$$$$$$$$$}{$$$$$$${$$$$$$${$$$$$$${$$$$$$$${$$$$$$$$$${$$$$$$$$$$${$$$$$$$$$$$$$${$$$$$$$$$$$$$${$$$$$$$$$$$$$${$$$$$$$$$$$$${$$$$$$$$$$$$${$$$$$$$$$$$$$${$$$$$$$$$$$$$$${$$$$$$$$$$$$$$$${$$$$$$$$$$$$$$$$$${$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$}$}$}$}$}$}$}$}$}$}$}$}$}$}$}$}$}$}\)))) from Equation (3). Then, a relationship between \(P_{fa}\) and threshold \(V_{T}\) of the proposed jammer detector is obtained as  
\[P_{fa}=\int_{V_{T}}^{\infty}f_{Z}(z|\theta)dz=\frac{1}{\Gamma(\alpha( \theta))}\Gamma\bigg{(}\alpha(\theta),\frac{V_{T}}{\beta(\theta)}\bigg{)}, \tag{10}\]  
where \(\Gamma(s,x)\) denotes an upper incomplete gamma function by  
\[\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}dt. \tag{11}\]  
Thus, for a given \(P_{fa}\), \(V_{T}\) is calculated by Equation (10), and a jammed slot can be determined.  
After detecting all jammed slots, the index set (\(S_{comb}\)) of the combining slots is obtained as  
\[S_{comb}=\{l\,|\,Z_{l}<V_{T},\,\,l\in\{1,2,\ldots,L\}\}, \tag{12}\]  
where \(Z_{l}\) denotes the estimated power of the noise and jammer in the \(l\)-th slot from Equation (6). If the only nonjammed slots based on \(S_{comb}\) were combined, a larger signal-to-noise ratio (SNR) and a better BER could be attained.