## 2 Model  
### Centralized Eigenvalue-based Spectrum Sensing  
In what concerns the baseband memoryless linear discrete-time MIMO fading channel model, assume that there are \(m\) sensors (e.g., antennas) in a CR, or \(m\) single-sensor CRs, each one collecting \(n\) samples of the received signal from \(p\) primary transmitters during the sensing period. Consider that these samples are arranged in a matrix \(\mathbf{Y}\in\mathbb{C}^{m\times n}\). Similarly, consider that the transmitted signal samples from the \(p\) primary transmitters are arranged 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 receiver). Finally, let \(\mathbf{V}\) and \(\mathbf{V}_{\text{IN}}\in\mathbb{C}^{m\times n}\) be the matrices containing thermal noise and IN samples that corrupt the received signal, respectively. The matrix of collected samples is then  
\[\mathbf{Y}=\mathbf{H}\mathbf{X}+\mathbf{V}+\mathbf{V}_{\text{IN}} \tag{1}\]  
In eigenvalue-based sensing, spectral holes are detected using test statistics based on the eigenvalues of the sample covariance matrix of the received signal matrix \(\mathbf{Y}\). If a multi-sensor device is used to decide upon the occupation of a given channel in a non-cooperative fashion, or even in a centralized cooperative scheme with data-fusion (see Section 1), matrix **Y** is formed, and the sample covariance matrix  
\[\mathbf{R}=\frac{1}{n}\mathbf{Y}\mathbf{Y}^{\dagger} \tag{2}\]  
is estimated, where \((\cdot)^{\dagger}\) means complex conjugate and transpose. The eigenvalues \(\{\lambda_{1}\geq\lambda_{2}\geq\cdots\lambda_{m}\}\) of **R** are then computed, and assuming a single primary transmitter (\(p=1\)), the test statistics for GLRT, MMED, MED, and ED are respectively calculated according to [6]:  
\[T_{\text{GLRT}}=\frac{\lambda_{1}}{\frac{1}{m}\text{tr}(\mathbf{R})}=\frac{ \lambda_{1}}{\frac{1}{m}\sum_{i=1}^{m}\lambda_{i}} \tag{3}\]  
\[T_{\text{MMED}}=\frac{\lambda_{1}}{\lambda_{m}} \tag{4}\]  
\[T_{\text{MED}}=\frac{\lambda_{1}}{\sigma^{2}} \tag{5}\]  
\[T_{\text{ED}}=\frac{\|\mathbf{Y}\|_{F}^{2}}{mn\sigma^{2}}=\frac{\sum_{i=1}^{m }\lambda_{i}}{m\sigma^{2}}, \tag{6}\]  
where \(\sigma^{2}\) is the thermal noise power that is assumed to be known and the same in each sensor input, and \(\text{tr}(\cdot)\) and \(\|\cdot\|_{F}\) are the trace and the Frobenius norm of the underlying matrix, respectively. All the eigenvalue based methods rely on the fact that the covariance matrix in the presence of noise only is a diagonal matrix with all its elements equal to \(\sigma^{2}\), and, hence, has a single eigenvalue also equal to \(\sigma^{2}\). In the presence of a primary user, this is no longer true, and these methods try to identify this situation. As we can see in Equation (3), with GLRT we calculate the ratio between the largest eigenvalue and the average of all the remaining eigenvalues. In MMED we consider the ratio between the largest and the smallest eigenvalues. In MED we assume that the noise variance \(\sigma^{2}\) is known, and compare the largest eigenvalue with \(\sigma^{2}\). In all these methods, the test statistic should be equal to one, in case only Gaussian noise is present. In the conventional model, when a centralized cooperative sensing with single-sensor (e.g., single antenna) CRs is considered, matrix **Y** is presumed to be available at the fusion center as if no signal processing is needed before each row of **Y** is forwarded to the FC by each CR. A more realistic model was originally proposed in [8] and called the _implementation-oriented model._ It considers the main signal processing tasks performed by each CR before the collected sample values are sent to the FC. The diagram shown in Figure 1, which adopts a direct-conversion receiver (DCR) architecture [15], was the main reference for constructing such a model.