## 2 Traditional Eigenvalue-based Cooperative Spectrum Sensing  
Consider a discrete-time memoryless multiple input multiple output (MIMO) model, where each of the \(m\) sensors (antennas) in a CR or each single-sensor CR collects \(n\) samples of the received signal from \(p\) primary transmitters during the sensing period. Those samples are arranged in a matrix \(\mathbf{Y}\in\mathbb{C}^{m\times n}\). Analogously, consider that the signals transmitted by the \(p\) transmitters are organized in a matrix \(\mathbf{X}\in\mathbb{C}^{p\times n}\). Let \(\mathbf{H}\in\mathbb{C}^{m\times p}\) be the channel matrix with elements \(\{h_{ij}\}\), \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,p\), representing the channel gain between the \(j\)-th primary transmitter and the \(i\)-th sensor (antenna or CR). It is assumed that these gains do not vary during the sensing period. Finally, let \(\mathbf{V}\in\mathbb{C}^{m\times n}\) be the matrixwith thermal noise samples corrupting the signal received by the \(m\) sensors. Therefore, the matrix with received samples under this MIMO channel model is given by  
\[\mathbf{Y}=\mathbf{H}\mathbf{X}+\mathbf{V} \tag{1}\]  
In eigenvalue-based cooperative spectrum sensing, a spectral hole is detected by applying a binary hypothesis test where the test statistic is built from the eigenvalues of the received signal ensemble covariance matrix \(\mathbf{R}_{\mathbf{e}}=\mathbb{E}[\mathbf{Y}\mathbf{Y}^{\dagger}]\), where \(\mathbb{E}[\cdot]\) is the expected value operator. Since the ensemble covariance matrix is normally unknown, it is replaced by its maximum-likelihood estimate, the sample covariance matrix  
\[\mathbf{R}=\frac{1}{n}\mathbf{Y}\mathbf{Y}^{\dagger} \tag{2}\]  
where \((\cdot)^{\dagger}\) denotes the conjugate and transpose operations. The eigenvalues \(\{\lambda_{1}\geq\lambda_{2}\geq\cdots\lambda_{m}\}\) of \(\mathbf{R}\) are computed and then, the test statistics for the GLRT, MMED, MED and ED are formed. In [21, 22, 23], binary hypothesis tests for multiple sources are deduced. Here, for simplicity reasons, we consider just a single primary transmitter, and in this case the following test statistics apply [14]:  
\[T_{\text{GLRT}}=\frac{\lambda_{1}}{\frac{1}{m}\text{tr}(\mathbf{R})}=\frac{ \lambda_{1}}{\frac{1}{m}\sum\limits_{i=1}^{m}\lambda_{i}} \tag{3}\]  
\[T_{\text{MMED}}=\frac{\lambda_{1}}{\lambda_{m}} \tag{4}\]  
\[T_{\text{MED}}=\frac{\lambda_{1}}{\sigma^{2}} \tag{5}\]  
\[T_{\text{ED}}=\frac{\|\mathbf{Y}\|_{F}^{2}}{mn\sigma^{2}}=\frac{1}{m\sigma^{2} }\sum\limits_{i=1}^{m}\lambda_{i} \tag{6}\]  
where \(\sigma^{2}\) is thermal noise variance at the input of each sensor, and \(\text{tr}(\cdot)\) and \(\|\cdot\|_{F}\) correspond to the trace and the Frobenius norm of the matrix, respectively. The sensing process is then concluded by comparing the test statistic with a threshold pre-defined according to the desired performance of the sensing process. If the test statistic is greater than the threshold, the channel is deemed occupied; otherwise the channel is declared vacant. The performance of a sensing technique is usually measured in terms of the probability of false alarm (\(P_{\text{fa}}\)) and the probability of detection (\(P_{\text{d}}\)). \(P_{\text{fa}}\) is the probability of inferring that a sensed band is occupied when it is in fact vacant, _i.e._, \(P_{\text{fa}}=\text{Pr}[T>\gamma|\mathcal{H}_{0}]\). \(P_{\text{d}}\) is the probability of inferring that a sensed band is occupied when it is indeed occupied, _i.e._, \(P_{\text{d}}=\text{Pr}[T>\gamma|\mathcal{H}_{1}]\). Here \(\gamma\) is the decision threshold, and \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\) are the hypotheses that the primary transmitted signal is absent and present, respectively. It is thus desirable that \(P_{\text{fa}}\) is minimized and \(P_{\text{d}}\) is maximized. However, these objectives are conflicting ones: increasing the threshold decreases \(P_{\text{fa}}\), but also decreases \(P_{\text{d}}\). As a consequence, a tradeoff must be adopted so that the correct threshold is established.