# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## 4 Computer Simulations  
The validation of gamma approximation and the derived NCRB were verified through computer simulations. The detection probability and the BER of the proposed methods were evaluated alongside conventional schemes. For the validation of gamma approximation, the distribution of the NJPE output was fitted to gamma, normal, and Weibull, and the root mean squared errors (RMSEs) of moments for the three fitted distributions were compared to select best-fit distribution. The derived NCRB was compared with the NMSE of the proposed NJPE. The detection probability of the proposed jammer detector based on gamma distribution was compared with those of the conventional cell average (CA)-CFAR, smallest of (SO)-CFAR, and differential jammer rejection (DJR) [20; 21]. The BERs of the combining slots without jammed slots were tested alongside conventional methods. Experiment parameters are shown in Table 1, where \(\rho\) denotes the ratio of the bandwidths of FHSS system and jammer. A small \(\rho\) indicates a strong and narrowband jammer for the same jammer power. In order to verify gamma approximation, maximum-likelihood estimation was utilized to fit the distribution of the NJPE output to gamma, normal, and Weibull distributions [26], and the RMSEs of the \(n\)-th moment from 1 to 4 were compared to select the best-fit distribution. The PDFs of the three fitted distributions and the RMSEs are shown in Figure 2 and Table 2, respectively. In Figure 2, the black solid line denotes the PDF of the output of NJPE. Red-triangle, green-square, and blue-circle lines denote the PDFs of the fitted distributions with gamma, normal, and Weibull, respectively. Gamma distribution was well-matched with the distribution of the NJPE output. In Table 2, \(\mu_{n}\) denotes the \(n\)-th moment, and the gamma distribution displayed the smallest RMSEs from the three fitted distributions. In Figure 3, the NCRB of the proposed NJPE was compared with the NMSE of the simulated NJPE according to bit-energy-to-noise-power spectral-density ratio (\(E_{b}/N_{0}\)) with \(N=50\), and according to \(N\) with \(E_{b}/N_{0}=12\) dB. In Figure 3a, for a large SNR region, NMSE and NCRB converged to 0.04, which was also calculated by \(2/N\). In Figure 3b, NMSE and NCRB were inversely proportional to the number of symbols. These results were well-matched with Lemma 1. In Figure 4, the detection probabilities of the proposed jammer detector based on NJPE versus bit-energy-to-jammer-power spectral-density ratio (\(E_{b}/J_{0}\)) at \(P_{fa}=10^{-10}\) are shown for \(\rho=0.0005\), 0.005, and 0.05. The jammer with \(\rho=0.0005\) was a strong and narrowband signal. The red-triangle line denotes CA-CFAR, the green-asterisk line denotes SO-CFAR, the blue-hexagram line denotes DJR, and the magenta-diamond line denotes the proposed detector. The Gaussian-distribution-based detector is also depicted as a dotted-magenta-diamond line for comparison with the proposed gamma-distribution-based detector. When \(E_{b}/J_{0}\) was small, i.e., large jammer power, the jammer was easily detected, and the detection probability of jammer was close to 1; the inverse was also true. For the same \(E_{b}/J_{0}\), a small \(\rho\) in Figure 4a demonstrates larger detection probability than a large \(\rho\) in Figure 4c. The proposed method had about 15, 13, and 5 dB detection-probability gain, respectively, compared to the conventional CA-CFAR, SO-CFAR, and DJR. In addition, the proposed gamma-distribution-based detector had about 3 dB better gain than the Gaussian-distribution-based detector.