## 5 Experiments and Discussions
### Software Simulations  
Before simulations, we have to define the successful detection as \(\hat{S}_{x}\supseteq S_{x}\) rather than \(\hat{S}_{x}=S_{x}\), where \(\hat{S}_{x}\) is the the detected support. Because that if the band-limited signal occupies two bands like \(X_{1}(f)\)as shown in Figure 12, this signal will have two neighboring supports after calculating. In general, before the CS algorithms solving the Equation (18), we need to know the largest possible sparsity. In order to avoid missing alarm, we set the sparsity two times of \(N\). So estimating the success detection equals to deciding whether \(S_{x}\) is included in \(\hat{S}_{x}\).  
However, acquiring the solution \(S_{x}\)is NP-hard. Using some known CS algorithms can not promise 100% correct solution. However, the detection performance can be improved by increasing the sensing time. A straightforward way is to increase the parameter \(m\). From [11], for practical applications, the value of \(m\) is roughly  
\[m\approx 4N\log(M/2N) \tag{21}\]  
To evaluate the performance of the specifications we choose in Table 1 for the proposed prototype, we did some simulations on Matlab. The bandwidth we plan to sense is 1 GHz, and it is so wide that we can not generate a such wideband noise for hardware experiment. The signal to noise ratio is defined as \(10\log(\left\|x(t)\right\|^{2}\Big{/}\left\|w(t)\right\|^{2})\), where \(x(t)\) is the wideband signal and \(w(t)\) is the wideband additive white Gaussian noise. We conducted two simulations, and each experiment is repeated over 500 realizations. The first one is to find the relationship between detection performance and SNR. As shown in Figure 13a, we choose three different empirical values of \(m\) according to Equation (21). We can see that the detection performance becomes better with SNR increasing. When the SNR is up to \(-5\) dB, the successful detection rate is more than 80%. Besides, we can also find the detection probability can improve a lot with \(m\) increasing when the SNR is between \(-10\) dB and \(-5\) dB. In other SNR intervals, this effect is not significant.  
As for another simulation, we introduced to use Sparse Bayesian Learning (SBL) [27] owning to its adaptation to number of signals. SBL-based method takes advantage of the probability density distribution of the data to achieve a best matching between the calculational results and the samples, which means that SBL techniques do not depend on the information maximum number of signals. This property of SBL-based techniques is suitable in our situation, we intend to apply SBL algorithm  
Figure 11: Block diagram of spectrum sensing algorithm.  
Figure 12: Schematic of a signal occupying two bands.  
to solve the problem in SCMWC. Figure 13 depicts the probability of detection between OMP and SBL algorithm in different SNR. We set \(m=50\) in this situation. We can see that SBL-based method has better performance, it can increase around \(20\%\) between\(-10\) dB and \(-5\) dB. However, SBL algorithm takes up more time than OMP, which is not appropriate in real time detection. Above all, these two simulations proves the feasibility of our prototype, and indicate the performance of the proposed specifications.