# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## 3 Proposed-Method Analysis  
For NJPE, the limited number of symbols in a slot is utilized for calculation, which causes an estimation error. In this section, we derived the NCRB of the proposed NJPE, analyzed the detection performance of the proposed jammer detector, and calculated the detection probability.  
The characteristic of the NCRB of the proposed NJPE is described in Lemma 1.  
**Lemma 1**.: _The estimation error of NJPE is inversely proportional to the number of symbols in a slot. When the SNR is large, the estimation error only depends on the number of symbols._  
Proof.: Let the estimated noise power of the proposed NJPE be \(\hat{\theta}\) for actual power \(\theta\). The variance of \(\hat{\theta}\) is bounded by Fisher information (\(I(\theta)\)), which is given as [24]  
\[\text{Var}\big{[}\hat{\theta}\big{]}\geq\frac{1}{I(\theta)}, \tag{13}\]  
where \(I(\theta)\) is calculated as (see Appendix B for detail derivation)  
\[I(\theta)=-\text{E}\bigg{[}\frac{\partial^{2}\ln f_{Z}(z|\theta)}{\partial \theta^{2}}\bigg{]}=\frac{N{P_{r}}^{2}}{\partial^{2}(\theta^{2}+4P_{r}\theta+2 {P_{r}}^{2})}. \tag{14}\]  
where \(N\) denotes the number of symbols, and \(P_{r}\) denotes the received symbol power. Thus, the Cramer-Rao bound (CRB) is obtained as  
\[\text{Var}\big{[}\hat{\theta}\big{]}\geq CRB(\theta)=\frac{\theta^{2}\Big{(} \theta^{2}+4P_{r}\theta+2{P_{r}}^{2}\Big{)}}{N{P_{r}}^{2}}. \tag{15}\]  
NCRB is defined as \(NCRB(\theta)\stackrel{{\text{def}}}{{=}}CRB(\theta)/\theta^{2}\)[25]. NCRB is written as  
\[NCRB(\theta)=\frac{\theta^{2}+4P_{r}\theta+2{P_{r}}^{2}}{N{P_{r}}^{2}}=\frac{1 +4\Omega+2\Omega^{2}}{N\Omega^{2}}, \tag{16}\]  
where \(\Omega=P_{r}/\theta\), which is SNR. In Equation (16), NCRB is inversely proportional to \(N\) and is functional of \(\Omega\). When \(\Omega\) is large, NCRB becomes \(2/N\).  
The detection probability (\(P_{D}\)) of the proposed jammer detector was calculated from threshold \(V_{T}\), which was obtained from \(P_{fa}\). Because the proposed NJPE followed gamma distribution, \(P_{D}\) needed to be calculated on the basis of the conditional probability of the gamma function in Equation (7). If the jammer existed in the observation slot, \(\theta\) in Equation (7) became the sum of the noise power and jammer power (\(\sigma_{N}{}^{2}+\sigma_{J}{}^{2}\)). Then, \(P_{D}\) with threshold \(V_{T}\) was obtained by an integration over the region greater than \(V_{T}\), which was given as  
\[P_{D}=\int_{V_{T}}^{\infty}f_{Z}\Big{(}z\big{|}{\sigma_{N}{}^{2}}+\sigma_{J}{ }^{2}\Big{)}dz=\frac{1}{\Gamma\Big{(}\alpha\Big{(}{\sigma_{N}{}^{2}}+\sigma_{ J}{}^{2}\Big{)}\Big{)}}\Gamma\Bigg{(}\alpha\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2} \Big{)},\frac{V_{T}}{\beta\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2}\Big{)}} \Bigg{)}, \tag{17}\]  
where \(\alpha\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2}\Big{)}\) and \(\beta\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2}\Big{)}\) were defined as in Equations (8) and (9), respectively. Thus, in Equation (17), as \(\alpha\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2}\Big{)}\) increased and \(\beta\Big{(}{\sigma_{N}{}^{2}}+\sigma_{J}{}^{2}\Big{)}\) decreased, \(P_{D}\) increased. In other words, a larger SNR provided better detection probability. In the next section, we outline the computer simulations that were executed to show the detection performance of the proposed and conventional schemes.