## 4 Influence of the System Parameters
### Glrt  
Figure 3 shows the ROC curves relating the probability of false alarm (\(P_{\text{fa}}\)) and the probability of detection (\(P_{\text{d}}\)) for GLRT. It can be seen that the performance of the sensing scheme under the _R-model_ is poor for \(N_{\text{q}}=4\), changing slightly from \(N_{\text{q}}=8\) to \(N_{\text{q}}=256\). Following [8], this suggests that 3 bits per sample are enough for the transmission of the sample values collected by each CR to the FC, a result that  
\begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{c}{_C-Model_ and _R-Model_} \\ \hline Signal-to-noise ratio & \(\text{SNR}=-10\) dB \\ Number of primary transmitters & \(p=1\) \\ Number of CRs & \(m=6\) \\ Number of samples collected by each CR & \(n=50,100\) \\ Impulsive to thermal noise power ratio & \(K=0\) \\ Signal-to-noise ratio & \(\text{SNR}=-10\) \\ MA-filter length & \(L=1\)-\(20\) \\ ADC dynamic range & \(D=2\) \\ ADC overdrive factor & \(f_{\text{od}}=1\)-\(2\) \\ Number of quantization levels & \(N_{\text{q}}=4,8,256\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Reference System Parameters.  
can be useful to the analysis of the necessary bandwidth and traffic over the reporting control channel. Still referring to Figure 3, one can observe that, for a given false alarm probability, the GLRT detection probability under the _C-model_ is slightly overestimated if compared with the results produced under the _R-model_. In other words, ignoring the signal processing tasks typically performed in digital receivers may lead us to optimistic results. Figure 3 also shows that the ROC curves under the _R-model_ suffer little or no influence of variations of \(L\) and \(f_{\text{od}}\). These results support the choice of the following system parameters to mimic a real CR using the GLRT strategy: (i) number of quantization levels \(N_{\text{q}}=8\), which corresponds to a 3-bit quantization of the sample values; (ii) low-pass MA receive filter length was chosen by assumption as \(L=0.2n\); and (iii) AGC overdrive factor \(f_{\text{od}}=1.2\), which corresponds to a value that will produce a signal clipping in approximately \(1.2\%\) of the time for un-quantized Gaussian-distributed sample values, and around \(3\%\) for \(N_{\text{q}}=8\). This value of \(3\%\) was obtained experimentally.  
Since the influence of increasing the number \(n\) of collected samples per CR is a performance improvement, considering as fixed the remaining system parameters, from this point on we present simulation results only for \(n=50\). This is to avoid polluting unnecessarily the graphs (clusters of curves for \(n=50\) and \(n=100\) are not necessarily separated from each other in the cases of other detection techniques, as they are in the case of GLRT in Figure 3).  
In what concerns the effect of increasing the SNR, we also know that it has no influence on \(P_{\text{fa}}\), although it produces an increase in \(P_{\text{d}}\), pushing up the positions of the ROC curves. This motivates the presentation of results with a fixed value of SNR, which was chosen to be small (\(-10\) dB) to represent a more degrading, but yet realistic, situation from the perspective of spectrum sensing performance. For instance, IEEE 802.22 requires that the presence of digital TV transmissions should be sensed with \(0.9\) detection probability with a sensitivity of \(-114\)dBm, which may be translated into very low SNR levels [25].  
Figure 3: ROC curves for GLRT under parameter variations.