## 2 Background and Proposed System Model
### Correlation Model  
This work assumes a scenario where the received samples at the \(L\) antennas of \(u\)-th SU are possibly correlated and partially follows the correlation model adopted in [23]. Therefore, in order to produce correlation on the received signal samples, consider, at first, the matrix \(\mathbf{B}_{u}\in\mathbb{C}^{L\times Z}\) of uncorrelated slow-FS sensing channel gains that affect the received signals at the \(L\) antennas of \(u\)-th SU given by:  
\[\mathbf{B}_{u}^{\mathrm{T}}=[\mathbf{b}_{1,u}^{\mathrm{T}},\mathbf{b}_{2,u}^ {\mathrm{T}},\ldots,\mathbf{b}_{L,u}^{\mathrm{T}}], \tag{3}\]  
with \(\mathbf{b}_{l,u}\in\mathbb{C}^{1\times Z}\) being the gains of the time variant impulse response of sensing channels between the \(l\)-th antenna of the \(u\)-th SU and the PU transmitter with \(Z\)-tap and T being the matrix transposition. In (3), the gain vectors \(\mathbf{b}_{J,u}^{\mathrm{T}}\) and \(\mathbf{b}_{J,u}^{\mathrm{T}}\) are uncorrelated for \(t\
eq J\). It is possible to obtain the correlated channel gains matrix \(\mathbf{H}_{u}\in\mathbb{C}^{1\times Z}\) from (3) by making:  
\[\mathbf{H}_{u}=[\mathbf{h}_{1,u}^{\mathrm{T}},\mathbf{h}_{2,u}^{\mathrm{T}}, \ldots,\mathbf{h}_{l,u}^{\mathrm{T}},\cdots\mathbf{h}_{L,u}^{\mathrm{T}}]^{ \mathrm{T}}=(\mathbf{B}_{u}^{\mathrm{T}}\mathbf{Q})^{\mathrm{T}}\in\mathbb{C} ^{L\times Z}, \tag{4}\]where \(\mathbf{h}_{l,u}=[h_{l,u}(1),h_{l,u}(2),h_{l,u}(3),\ldots,h_{l,u}(Z)]\). The matrix \(\mathbf{Q}\in\mathbb{R}^{L\times L}\) can be written as [25]:  
\[\mathbf{Q}= \begin{bmatrix}q_{11}&q_{12}&\ldots&q_{1L-1}&q_{1L}\\ 0&q_{22}&\ldots&q_{2L-1}&q_{2L}\\ \vdots&&\ddots&&\vdots\\ 0&0&\ldots&q_{L-1L-1}&q_{L-1L}\\ 0&0&\ldots&0&q_{LL}\end{bmatrix} . \tag{5}\]  
The matrix in (5) can be obtained from a matrix \(\mathbf{A}\)[25]:  
\[\mathbf{A}= \begin{bmatrix}1&\rho&\ldots&\rho^{[2-L]}&\rho^{[1-L]}\\ \rho&1&\ldots&\rho^{[3-L]}&\rho^{[2-L]}\\ \vdots&&\ddots&&\vdots\\ \rho^{[L-2]}&\rho^{[L-3]}&\ldots&1&\rho\\ \rho^{[L-1]}&\rho^{[L-2]}&\ldots&\rho&1\end{bmatrix} , \tag{6}\]  
in which \(\mathbf{A}=\mathbf{Q}^{\mathrm{T}}\mathbf{Q}\) and \(\rho\), with \(0\leq\rho\leq 1\), is the correlation coefficient that leads to a given correlation between received signals at different antenna pairs of each SU. For \(\imath\
eq\jmath\), the vectors \(\mathbf{h}_{\jmath,u}^{\mathrm{T}}\) and \(\mathbf{h}_{\imath,u}^{\mathrm{T}}\) in \(\mathbf{H}_{u}\) are now correlated. The elements of the \(i\)-th line and \(j\)-th column of \(\mathbf{A}\), with \(i,j=1,2,\ldots,L\), are formulated as:  
\[a_{ij}=\rho^{[i-j]}. \tag{7}\]  
The \(a_{ij}\) parameter denotes the correlation between signals at different antenna pairs of each SU. Clearly, the correlation between received signals at a given antenna pair decreases as \(|i-j|\) increases if \(0<\rho<1\), i.e., received signal samples at farther apart antenna pairs are less correlated than those at closer apart antenna pairs, as expected. Likewise, the element of the \(i\)-th line and \(j\)-th column of matrix \(\mathbf{Q}\) in (5), is:  
\[q_{ij}= \begin{cases}\sqrt{a_{ij}},&i=1,j=i,\\ a_{ij}/q_{ii},&i=1,j\
eq i,\\ \sqrt{a_{ij}-\sum_{k=1}^{j-1}q_{kj}^{2}},&i>1,j=i,\\ (a_{ij}-\sum_{k=1}^{j-1}q_{ki}q_{kj})/q_{ii},&i>1,j\
eq i.\end{cases}  \tag{8}\]  
The \(z\)-th element of channel gain, \(h_{l,u}(z)\), that affects the received signal at the \(l\)-th antenna of \(u\)-th SU in (1) is forming matrix \(\mathbf{H}_{u}\) in Equation (4). Therefore, signal samples received at antenna pairs \(k\) and \(j\) of \(u\)-th SU in (1), given by \(y_{k,u}(n)\) and \(y_{j,u}(n)\) for \(n=1,2,\ldots,N\), have a non-zero correlation under hypothesis \(\mathcal{H}_{1}\) if \(\rho>0\). Notice also that the correlation model adopted here does not take into account information as, for instance, operation frequencies, separation distances between antenna pairs, and other information related to the propagation environment, as the angle of arrival of received signals and temporal and frequency dispersion [26]. Moreover, ref.