## 5 Influence of IN
### Detecting and Combating IN  
Although some eigenvalue-based sensing schemes are robust against IN, particularly with the _R-model_, as shown in this paper, one might consider additional countermeasures to further reduce IN influence. Detecting and removing IN influence is an active research topic in audio, image processing, and radio communications [12; 13; 14], and is beyond the scope of this contribution. In what follows we investigate the efficacy of two simple strategies for combating IN in eigenvalue-based spectrum sensing techniques, assuming that IN presence is _perfectly known_. This assumption aims at decoupling IN-detection performance from spectrum sensing performance, directing the attention towards the latter. Furthermore, it is particularly useful for determining the spectrum sensing performance gain under the _R-model_ and measure its intrinsic ability for combating IN. In other words, small performance improvements brought by the IN countermeasures are an indication of the inherent IN immunity of the _R-model_. Particularly, we investigate the following heuristic strategies for combating IN when it is present: _muting the samples under IN_ and _eliminating from cooperation those CRs under IN_. In the second countermeasure, the effective number of cooperating CRs is found as  
\[m_{E}=E[X|x\geq 2](1-\Pr[X<2])+2\Pr[X<2] \tag{8}\]  
where \(X\) is the random variable that models the number of CRs _not_ hit by IN (see details in the Appendix). Figure 11: ROC curves for ED with and without (w/o) moderate or strong, concentrated IN. Figures 12-15 show the simulation results considering the above-mentioned IN countermeasure strategies for GLRT, MMED (or ERD), MED (or RLRT), and ED. As previously emphasized, these figures were inserted close to the corresponding results from subsection 5.2 to facilitate comparisons. The system parameters for moderate IN are those in Table 2. In CR elimination, the number of CRs, \(m\), was modified so that the effective number of cooperating CRs, \(m_{E}\), was made as close as possible to 6. We have not chosen to keep \(m=6\), since \(m_{E}\) would be smaller than six, pushing the ROC curves towards \(P_{\rm fa}=P_{\rm d}\) and approximating them from each other. This would prevent a clear view of the effect of the CR elimination countermeasure. In what concerns GLRT, we compare all ROC curves in Figure 12 with the corresponding curves in Figure 8. Under the _C-model_, we can see that both CR elimination and sample muting significantly improve performance, with an advantage of muting, which pushes the ROC curve with IN towards the neighborhood of the curve without IN. In other words, the IN countermeasures are effective under the _C-model_. Under the _R-model_, however, the IN countermeasures produce only marginal improvements for both moderate and strong concentrated IN. This indeed is evidence that the implementation-oriented model has inherent IN robustness. Figure 14: Effect of muting samples and CR elimination under moderate or strong concentrated IN on MED (or RLRT). Figure 15: Effect of muting samples and CR elimination under moderate or strong concentrated IN on ED. The above comparisons and conclusions closely hold for MMED (or ERD), as can be seen by comparing all ROC curves in Figure 13 with the corresponding curves in Figure 9. A small improvement can be observed for both IN countermeasures under the _R-model_, in the case of moderate IN. A small performance reduction is observed under the _R-model_ in the case of strong concentrated IN and CR elimination; a marginal gain is observed for the sample muting. The improvement under the _C-model_, however, is noticeable larger in all situations. Once more, this is evidence that the implementation-oriented model has inherent IN robustness. For MED (or RLRT), we compare all ROC curves in Figure 14 with the corresponding curves in Figure 10. Now we can see that the performance improvements caused by the IN countermeasures under the _C-model_, though evident, are not as large as in the cases of GLRT and MMED (or ERD).