## 2 Background and Proposed System Model
### Glrt
#### 2.3.3 GLRT with SF and WSF  
For the GLRT with SF, denoted as GLRT-SF, the SUs collect the samples and send them to the fusion center, via CC, reaching a cooperative decision on the occupation state of the sensed PU band. From this point, the FC computes the SCM from the matrix of all received samples \(\mathbf{Y}_{\text{FC}}\in\mathbb{C}^{LU\times N}\), its eigenvalues, as well as the test statistic, and makes the decision on the occupation state of the sensed PU band. One can write the matrix \(\mathbf{Y}_{\text{FC}}\) according to:  
\[\mathbf{Y}_{\text{FC}}^{\text{T}}=[\mathbf{Y}_{1}^{\text{T}},\mathbf{Y}_{2}^{ \text{T}},\mathbf{Y}_{3}^{\text{T}},\ldots,\mathbf{Y}_{u}^{\text{T}},\ldots, \mathbf{Y}_{U}^{\text{T}}], \tag{16}\]  
with \(\mathbf{Y}_{u}\) being a submatrix of the received signal samples at the FC coming from the \(u\)-th SU, which is identical to the \(u\)-th matrix in (10) considering an error-free control channel. Then, at the FC, the \(LU\times LU\)-dimensional complex SCM \(\mathbf{R}_{\text{FC}}\) is:  
\[\mathbf{R}_{\text{FC}}=\mathbf{Y}_{\text{FC}}\mathbf{Y}_{\text{FC}}^{\text{ T}}/N, \tag{17}\]  
and the FC computes the GLRT-SF statistic as [6]:  
\[T_{\text{GLRT}_{\text{SF}}}=\frac{LU\lambda_{1}}{\sum_{i=1}^{LU}\lambda_{i}}, \tag{18}\]  
with \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{LU}\) denoting the eigenvalues of \(\mathbf{R}_{\text{FC}}\) in (17). The FC makes an occupation decision favoring \(\mathcal{H}_{1}\) when \(T_{\text{GLRT}_{\text{SF}}}>\gamma_{\text{FC}}\) or favoring \(\mathcal{H}_{0}\) otherwise, where \(\gamma_{\text{FC}}\) is a decision threshold previously established at the FC.  
The weighting scheme for received samples, denoted as GLRT-WSF, in [20] confers robustness to GLRT-SF against the adopted DN, then it is also used here under the DN model proposed in (2). Therefore, the matrix of weighted received samples at the FC for the GLRT-WSF can be written as:  
\[\mathbf{Y}_{\text{FC}}^{\prime}=[\mathbf{Y}_{1}^{\prime},\mathbf{Y}_{2}^{ \prime\text{T}},\ldots,\mathbf{Y}_{U}^{\prime\text{T}}]^{\text{T}}, \tag{19}\]  
where \(\mathbf{Y}_{u}^{\prime}=f_{u}\mathbf{Y}_{u}\) represents the weighted received samples at the FC related to the \(u\)-th SU and \(f_{u}\) is the \(u\)-th weight computed according to:  
\[f_{u}=\frac{1}{\vartheta_{u}}=\left(\frac{1}{L-1}\sum_{l=2}^{L}\lambda_{l,u} \right)^{-\frac{1}{2}}, \tag{20}\]where \(\hat{\sigma}_{u}\) is the square root of the maximum likelihood estimate of the noise power of the \(u\)-th SU in (14). The FC computes the modified SCM \(\mathbf{R^{\prime}_{FC}}\in\mathbb{C}^{LU\times LU}\) from the weighted samples in (20) as:  
\[\mathbf{R^{\prime}_{FC}}=\frac{1}{N}\mathbf{Y^{\prime}_{FC}}\mathbf{Y^{\prime} _{FC}}. \tag{21}\]  
The GLRT-WSF statistic is calculated as:  
\[T_{\text{GLRT}_{\text{WSF}}}=\frac{LU\lambda_{1}^{{}^{\prime}}}{\sum_{i=1}^{LU }\lambda_{i}^{{}^{\prime}}}, \tag{22}\]  
with \(\{\lambda_{1}^{{}^{\prime}},\lambda_{2}^{{}^{\prime}},\lambda_{2}^{{}^{\prime }},\ldots,\lambda_{LU}^{{}^{\prime}}\}\) standing for the eigenvalues calculated from (21).  
We consider the GLRT-SF as the blind benchmark technique or the state-of-the-art method since it has been largely studied in the last decade.