## 4 Mathematical Description of the Principles of ED Using SLC in MIMO-OFDM Systems  
During the ED of signals admitted at all \(R\) Rx chains of SU using SLC techniques, the process of squaring and combining the signal is performed to form the overall detected energy of the signal, referred as test statistics (\(\Lambda_{SLC}\)). The aggregate test statistics are expressed as follows:  
\[\Lambda_{SLC}=\sum_{r=1}^{R}\Lambda_{r}=\sum_{r=1}^{R}\sum_{n=1}^{N}|\mathbf{y_{r} }(n)|^{2}, \tag{3}\]  
where \(\Lambda_{r}\) presents the test statistics of the signal received at \(r\)-\(th\) Rx antenna of the SU. In order to decide whether a signal is present or not in an analysed frequency band, the total test statistics are compared to the detection threshold \(\lambda(n)\). The process is executed for each sensing sample \(n\) used during ED and can be expressed as:  
\[\begin{array}{ll}&H_{0}\\ &<\\ \Lambda_{SLC}(n)&>\\ &H_{1}\end{array}\lambda(n),\ \forall n\in\{1,\ \dots,\ N\} \tag{4}\]  
where \(\lambda(n)\) can be dynamically adjusted according to the level of noise in the wireless channel or it can be statically assigned and fixed during the whole ED SS period.  
According to [16; 23], the test statistic from Equation (4) can be represented using normal distribution as:  
\[\Lambda_{SLC}\sim\mathcal{N}\left(\begin{array}{c}\sum_{r=1}^{R}\sum_{n=1}^ {N}\mathbb{E}\Big{[}|\mathbf{y_{r}}(n)|^{2}\Big{]},\\ \sum_{r=1}^{R}\sum_{n=1}^{N}\mathbb{V}ar\Big{[}|\mathbf{y_{r}}(n)|^{2}\Big{]}\end{array} \right), \tag{5}\]  
where \(\mathbb{V}ar\) [ ] represents the expectation operator, while \(\mathbb{E}\) [ ] represents the variance operator.  
Figure 2: The ROC curve indicating the interdependence among the detection and false alarm probability.  
For the _n-th_ SS sample, \(h_{r}(n)\) is used to express the channel gain of the signal received at the _r-th_ Rx antenna. Assuming for simplicity that there is an equal channel gain during _n-th_ sample period on every of \(R\) Rx antenna, the complex channel gain is assumed to be \(h_{r}(n)=h\). Similarly, to reduce the complexity, the noise variance at the _r-th_ Rx antennas can be assumed to be equal in every of \(R\) Rx antenna and thus is expressed as \(2\sigma_{wr}^{2}(n)=2\sigma_{w}^{2}\). Consequently, in the _n-th_ sensing interval, the total PU Tx power is equal to \(P=\sum_{r=1}^{R}\lvert\boldsymbol{h}\rvert^{2}2\sigma_{r}^{2}(n)\). Thus, the relationship between the average SNR values and PU Tx power at the SU site is given as \(\overline{\gamma_{SLC}}\approx\frac{P}{R2\sigma_{w}^{2}}\).  
According to this knowledge, the overall test statistics from Equation (3) can be approximated as:  
\[\Lambda_{SLC}\,\sim\,\left\{\begin{array}{rl}\mathcal{N}\left(RN\big{(}2 \sigma_{w}^{2}\big{)},\ RN\big{(}2\sigma_{w}^{2}\big{)}^{2}\right):&H_{0}\\ \mathcal{N}\Big{(}N\big{(}2\sigma_{w}^{2}\big{)}(R+\gamma_{SLC}),\ N\big{(}2 \sigma_{w}^{2}\big{)}^{2}(R+2\gamma_{SLC}\big{)}\Big{)}:&H_{1}\end{array} \right., \tag{6}\]  
According to Equation (6), the entire test statistic at the SU site depends on several parameters that include (_R_) representing the number of Rx chains, the \(2\sigma_{w}^{2}\) representing noise variance, the \(N\) indicating the overall number of detection samples, and the \(\gamma_{SLC}\) representing signal-to-noise ratio.  
The following sections present analytical models expressing interdependence of the detection and false alarm probabilities and the role of the level of DDT and NU factors to the ED process. The initial parts of the analytical models presented in the following sections have been partially explained in [19] for SISO and in [22] for MIMO-OFDM systems.