## 2 MWC System Model
### Reconstruction  
Convert (5) to matrix form:  
\[Y(f)=A(f)Z(f) \tag{7}\]  
\[z_{i}(f)=X\big{(}f+(i-L_{0}-1)f_{p}\big{)},1<i<2L_{0}+1,f\in\mathcal{F}_{s} \tag{8}\]  
Figure 2: MWC Sampling Component Framework.  
Therefore, signal reconstruction operation can be regarded as the process of acquiring unknown \(Z(f)\) from known \(Y(f)\) and \(A(f)\). The solution of (7) is an NP-hard problem. Since most of the elements \(z_{i}(f)\) in \(Z(f)\) are 0, meaning that \(Z(f)\) has sparsity, this process can be regarded as the recovery of support set \(\Psi\) of \(Z(f)\), where \(\Psi=|\text{supp}(Z(f))|\)  
Since the sampling front-end continuously generates observation vector \(Y_{i}(f)\), the solution of (7) is an Infinite Measurement Vectors (IMV) problem. Using CTF module can be transformed into the Multiple Measurement Vectors (MMV) problem [23]. Before using the CTF module, the covariance matrix \(Q\) needs to be constructed first.  
\[Q=\int_{f\in\mathcal{F}_{x}}y(f)y^{H}(f)df=\sum y[n]y^{T}[n] \tag{9}\]  
The architecture of the CTF module is shown in Figure 3. The CTF module is made up of two parts, eigenvalue decomposition and support set recovery. The main purpose of eigenvalue decomposition is to reduce the dimensionality of the sampling matrix. The matrix \(V\) is constructed based on the decomposition result of \(Q\), which satisfies \(Q=VV^{H}\). We set the support set of \(U\) as the same as that of \(Z(f)\). So, the solution of \(Z(f)\) can be converted into the following:  
\[\min\|U\|_{1}s.t.V=AU \tag{10}\]  
After solving the support set of \(U\), the device has an initial overview of the current spectrum availability. The device can further communicate with the outside world by reconstructing the signal based on the support set. The reconstructed signal can be based on the following equation.  
\[z_{i}[n]=\left\{\begin{array}{cc}A_{\Psi}^{\dagger}y[n]&i\in\Psi\\ 0&i\
otin\Psi\end{array}\right. \tag{11}\]  
where \(\Psi\) is the support set of \(U\) and \(Z(f)\), and \(A_{s}^{\dagger}\) is the pseudo-inverse of \(A_{s}\).  
There are many algorithms used for signal reconstruction, but two commonly used algorithms are the greedy and convex optimization algorithms. Convex optimization algorithms are highly accurate but also imply high complexity and slow computational speed. Compared to convex optimization algorithms, greedy algorithms are easier to implement with higher execution speeds and lower computational complexity. Therefore, in this paper, we use the OMP algorithm as the basis and the OMP algorithm to form the final solution through multiple iterations, continuously screening the optimal solution in each iteration. This means that the final solution of the greedy algorithm does not skip out of the range of local optimal solutions. To achieve the balance between the global optimal solution and the local optimal solution, the OMP algorithm needs to be optimized.  
In the above analysis, only \(x(t)\) is mentioned. In fact, noise also appears in the system at the same time, that is, the sampled signal \(y_{i}[n]\) is noisy. The MWC system does not handle the noise between sampling and support set recovery. Noise also enters the process of signal reconstruction. In the OMP algorithm, there is no separation operation between the useful signal and the noise. The presence of a large amount of noise will interfere with the support set recovery, thus affecting the device's judgment of spectrum resources and the accuracy of acquiring information about the outside world. The MWC system is based on compressed sensing theory. After sampling and reconstruction, the energy of noise is  
Figure 3: CTF Module Framework.  
weakened to some degree. Although the recovery signal's Signal-to-Noise Ratio (SNR) was improved, its accuracy could not be guaranteed. As a result, it is necessary to integrate the denoising of the signal between sampling and support set recovery to reduce the impact of noise on the sensing and communication of devices.