# Double-Threshold Cooperative Spectrum Sensing Algorithm Based on Sevcik Fractal Dimension
## 3 Simulation  
Simulation experiments are under AWGN environment without channel fading effects. The signal of the PU is the modulated QPSK signal. Signal parameters are set the same as that of Section 2.1. Bandwidth \(B=20\) kHz and carrier frequency \(f_{c}=10\) kHz. The sampling frequency at the cognitive receiver is set as \(f_{s}=40\) kHz. Baseband symbol rate \(R_{b}=1000\) bit/s. Assume that the number of collaborative SUs is 10. Simulation results are averaged over 1000 channels.  
In Figure 5, we compare the detection probability of the proposed method with the Katz fractal dimension scheme (Katz), the box dimension method (Box dimension) and the energy detection algorithm. According to Figure 2, the noise fractal dimension is basically maintained, fluctuate around a mean value under different noise powers. Hence, the decision threshold can be determined by the value of the noise fractal dimension, and correspondingly the false alarm probability is a constant which is set as \(P_{f}=0.03\). The values of the threshold and the correction in different schemes are shown in Table 1.  
Observing Figure 5, the detection probabilities of the three fractal dimension algorithms all outperform energy detection. The proposed scheme achieves 100% detection probability \(P_{d}\) at \(-20\) dB, whereas Katz fractal dimension and box dimension attains 100% detection probability at \(-17\) dB  
Figure 5: Detection probability comparison of the proposed method with the other three methods. SNR: signal-to-noise ratio.  
and \(-13\) dB, respectively. It can be inferred that the double-threshold CSS algorithm based on SFD improves the detection performance in low SNR environment.  
Figure 6 investigates the influence of the number of SUs on the detection performance when applying the proposed algorithm. We consider \(M=1,5,10\) and let the average SNR at each secondary receiver be the same. As seen from the figure, the detection performance of the proposed scheme gets better when the number of cognitive users involved in collaborative sensing increases. We compare the detection performance of the proposed algorithm with that of the single threshold CSS algorithm based on SFD in Figure 7, where the threshold in the single threshold detection equals to the average noise SFD \(\lambda_{S}=1.765\). The double-threshold method achieves \(100\%\) detection probability at \(-20\) dB, which is \(1\) dB higher than that of the single threshold method. It can be demonstrated that the proposed approach benefits from higher detection performance compared with the single threshold CSS scheme based on SFD.  
In Figure 8, the detection performance of the proposed algorithm is plotted when the noise uncertainty is \(1\) dB, \(2\) dB, and \(5\) dB, respectively. As expected, noise uncertainty had little impact on the detection performance of the proposed method, which means the proposed scheme is insensitive to noise uncertainty. Furthermore, even in the case of \(5\) dB noise uncertainty, the proposed scheme still obtains \(100\%\) detection probability when the SNR is greater than \(-20\) dB, and it can achieve more than \(90\%\) detection probability when the SNR is greater than \(-21\) dB. Therefore, the algorithm presented by this paper can not only be insensitive to noise uncertainty, but also achieve high detection performance at low SNR.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fractal Dimension Schemes** & **Threshold 1** & **Threshold 2** & **Correction** \\ \hline Sevcik & \(\lambda_{S1}=1.760\) & \(\lambda_{S2}=1.830\) & \(\Delta_{S}\)= \(-0.9\) \\ [PERSON] & \(\lambda_{K1}=1.004\) & \(\lambda_{K2}=1.007\) & \(\Delta_{K}\)= \(-0.9\) \\ Box dimension & \(\lambda_{B1}=1.385\) & \(\lambda_{B2}=1.425\) & \(\Delta_{B}\)= \(-0.98\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The values of the threshold and the correction in different schemes.  
Figure 6: Detection probability comparison of different numbers of secondary users (SUs).