# AI-Enabled Compressive Spectrum Classification for Wideband Radios
## 2 Materials and Methods
### Compressed Sensing  
The WB signal can be represented in the frequency domain by taking the discrete Fourier transform of the \(z(t)\) in (1):  
\[z_{f}=\sum_{l=1}^{M}D_{h}^{(m)}S_{f}^{(m)}+w_{f} \tag{3}\]  
here \(z_{f}\) is equivalent frequency domain WB signal, \(D_{h}^{(m)}\) is a channel matrix, and \(S_{f}^{(m)}\), \(h_{f}^{(m)}\), and \(w_{f}\) are acquired frequency-domain samples of \(s_{k}(t)\), \(h_{k}(t)\), and \(v(t)\), consecutively.  
WB signal \(z_{f}\) in (2) could be expressed in a more generalized form as given below:  
\[z_{f}=H_{f}S_{f}+w_{f} \tag{4}\]  
here \(S_{f}=[(S_{f}^{(1)})^{T}....(S_{f}^{(N)})^{T}]^{T}\) represents the NB received signals in frequency domain, and \(H_{f}=[(D_{f}^{(1)})....(D_{f}^{(M)})\) is the corresponding channel matrix. From Equation (3), it can be noticed that the WB sensing needs to measure the \(S_{f}\) when \(H_{f}\) and \(z(t)\) are provided. Since a sparse WB signal is at its disposal, the receiver can exploit the compressed sensing, a sub-Nyquist signal recovery technique, to alleviate the Nyquist rate A/D sampling requirements. A number of computationally efficient techniques, for example, Orthogonal Matching Pursuit (OMP) [21] or Basis Pursuit [35], were used to estimate the received signal accurately from the samples acquired at the sub-Nyquist sampling rate. The receiver needs to collect the compressed time domain samples of the WB signal and, therefore, \(S_{c}\) measurement matrix is used to obtain a sample vector \(x_{t}\) of size \(K\times 1\) from the WB signal \(z(t)\) as given:  
\[x_{t}=S_{c}z_{t} \tag{5}\]  
where \(z_{t}\) is the discrete-time representations of \(z(t)\) at the Nyquist sampling rate with \(K<=M\) and \(s_{c}\) is the \(k\times m\) projection matrix. Various schemes are reported in the literature for compressive sampler design such as random sampler [36] and non-uniform  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Targeted Modulated Signal** & **Jamming Signal** \\ \hline
16-QAM & QPSK \\
4-PAM & BPSK \\ QPSK & QPSK \\ BPSK & BPSK \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimal jamming strategy in a coherent scenario [34].  
sampler [37]. It should be noted that \(z_{t}=F_{M}^{-1}z_{f}\), and given \(K\) compressive samples, the spectrum \(S_{f}^{-}\) could now be obtained in (4), as below:  
\[x_{t}=S_{c}^{T}F_{M}^{-1}H_{f}S_{f}+\bar{w}_{f} \tag{6}\]  
here \(\bar{w}_{f}=S_{c}^{T}F_{M}^{-1}w_{f}\) shows white Gaussian noise vector. Due to the under-utilization of the spectrum by primary users in cognitive radio networks framework, the signal vector \(Sf\) is sparsely populated. The sparsity of the signal \(Sf\) is calculated using the \(p\)-norm such as \(||Sf||p\), \(pin[0,2)\), with \(p=0\) indicating definite sparsity. Therefore, (6) presents a linear regression problem with a sparsely populated WB (\(\hat{S_{f}}\)) which could be approximated by solving the linear convex optimization problem:  
\[\hat{S_{f}}=\underset{S_{f}}{argmin}||\hat{S_{f}}||_{1},\quad s.t.\quad x_{t} =S_{c}^{T}F_{M}^{-1}H_{f}\hat{S_{f}} \tag{7}\]  
The problem in (7) is solved using a variety of methods, including Convex Programming, as in BP [21] which is opted in this work to estimate the sparse WB signal. After obtaining the reconstructed WB using compressive sensing, the SCF is computed for each NB signal in WB, which is given in Section 2.2.