# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## Appendix B  
In this appendix, the CRB of the proposed NJPE using Fisher information is derived. CRB is defined as \(\text{Var}\Big{[}\delta\Big{]}\geq 1/I(\theta)\)[24], and \(I(\theta)\) is defined as  
\[I(\theta)=\text{E}\Bigg{[}\Bigg{(}\frac{\partial\ln f_{Z}(z|\theta)}{ \partial\theta}\Bigg{)}^{2}\Bigg{]}=-\text{E}\Bigg{[}\frac{\partial^{2}\ln f _{Z}(z|\theta)}{\partial\theta^{2}}\Bigg{]},\] (A28)  
where \(\ln f_{Z}(z|\theta)\) can be written as  
\[\begin{array}{ll}\ln f_{Z}(z|\theta)&=\ln\Bigg{(}\frac{1}{\Gamma(\alpha( \theta))\beta(\theta)^{\alpha(\theta)}}z^{\alpha(\theta)-1}\exp(-z/\beta( \theta))\Bigg{)}\\ &=-\ln\Gamma(\alpha(\theta))-\alpha(\theta)\ln\beta(\theta)+(\alpha(\theta)- 1)\ln z-\frac{z}{\beta(\theta)}.\end{array}\] (A29)  
From [31], Equation (A29) can be rewritten as  
\[\begin{array}{ll}\ln f_{Z}(z|\theta)=&-\alpha(\theta)\ln\alpha(\theta)+ \frac{1}{2}\ln\alpha(\theta)+\alpha(\theta)-\frac{1}{2}\ln 2\pi\\ &-\alpha(\theta)\ln\beta(\theta)+(\alpha(\theta)-1)\ln z-\frac{z}{\beta( \theta)}.\end{array}\] (A30)  
The second derivative of \(\theta\) in Equation (A30) is given as  
\[\begin{array}{ll}\frac{\partial^{2}\ln f_{Z}(z|\theta)}{\partial\theta^{2}} &=\frac{\partial^{2}\alpha(\theta)}{\partial\theta^{2}}\bigg{(}\ln z-\ln \theta+\frac{1}{2\alpha(\theta)}\bigg{)}-\frac{\partial\alpha(\theta)}{ \partial\theta}\bigg{(}\frac{1}{\partial}+\frac{\partial\alpha(\theta)}{ \partial\theta}\frac{1}{2\alpha(\theta)^{2}}\bigg{)}\\ &\quad\quad+\Bigg{(}\frac{\partial^{2}\beta(\theta)}{\partial\theta^{2}} \frac{1}{\beta(\theta)^{2}}-\bigg{(}\frac{\partial\beta(\theta)}{\partial \theta}\bigg{)}^{2}\frac{2}{\beta(\theta)^{3}}\bigg{)}(z-\theta)-\frac{ \partial\beta(\theta)}{\partial\theta}\frac{1}{\beta(\theta)^{2}}.\end{array}\] (A31)  
From Equations (A28) and (A31), \(I(\theta)\) can be calculated as  
\[\begin{array}{llll}I(\theta)&=&-\text{E}\Bigg{[}\frac{\partial^{2}\ln f_{Z }(z|\theta)}{\partial\theta^{2}}\Bigg{]}\\ &=&-\frac{\partial^{2}\alpha(\theta)}{\partial\theta^{2}}\bigg{(}\text{E}[\ln Z ]-\ln\theta+\frac{1}{2\alpha(\theta)}\bigg{)}+\frac{\partial\alpha(\theta)}{ \partial\theta}\bigg{(}\frac{1}{\partial}+\frac{\partial\alpha(\theta)}{ \partial\theta}\frac{1}{2\alpha(\theta)^{2}}\bigg{)}\\ &=&-\Bigg{(}\frac{\partial^{2}\beta(\theta)}{\partial\theta^{2}}\frac{1}{\beta( \theta)^{2}}-\bigg{(}\frac{\partial\beta(\theta)}{\partial\theta}\bigg{)}^{2} \frac{2}{\beta(\theta)^{3}}\bigg{)}(\text{E}[Z]-\theta)+\frac{\partial\beta( \theta)}{\partial\theta}\frac{1}{\beta(\theta)^{2}},\end{array}\] (A32)where \(\mathrm{E}[Z]\) is equal to \(\theta\) by Equation (A22), and \(\mathrm{E}[\ln Z]\) is calculated using second-order [PERSON]'s approximation, and calculated as  
\[\mathrm{E}[\ln Z]=\ln\mathrm{E}[Z]-\frac{\mathrm{Var}[Z]}{2\mathrm{E}[Z]^{2}}= \ln\theta-\frac{1}{2\alpha(\theta)}.\] (A33)  
Using Equation (A33), Equation (A32) can be rewritten as  
\[I(\theta)=\frac{\partial\alpha(\theta)}{\partial\theta}\bigg{(}\frac{1}{ \theta}+\frac{\partial\alpha(\theta)}{\partial\theta}\frac{1}{2\alpha(\theta )^{2}}\bigg{)}+\frac{\partial\beta(\theta)}{\partial\theta}\frac{1}{\beta( \theta)^{2}}.\] (A34)  
Fisher information \(I(\theta)\) is calculated as  
\[I(\theta)=\frac{{NP_{r}}^{2}\Big{(}\theta^{2}+4{P_{r}}\theta+2{P_{r}}^{2} \Big{)}+2\theta^{2}(\theta+{P_{r}})^{2}}{\theta^{2}(\theta^{2}+4{P_{r}}\theta +2{P_{r}}^{2})^{2}}.\] (A35)  
When the number of samples is sufficiently large, Equation (A35) is simplified as  
\[I(\theta)=\frac{{NP_{r}}^{2}}{\theta^{2}(\theta^{2}+4{P_{r}}\theta+2{P_{r}}^{ 2})}.\] (A36)  
Therefore, CRB is obtained as  
\[\mathrm{Var}\Big{[}\hat{\theta}\Big{]}\geq CRB(\theta)=\frac{\theta^{2}\Big{(} \theta^{2}+4{P_{r}}\theta+2{P_{r}}^{2}\Big{)}}{{NP_{r}}^{2}}.\] (A37)