## 4 Mathematical Description of the Principles of ED Using SLC in MIMO-OFDM Systems
### Mathematical Formulation of the Role of Detection Threshold on ED Process  
As shown in Equations (7) and (8), the detection and false alarm probability depend on the proper DT settings. The choice of the threshold is an essential step in the sensing process that determines whether the PU signal is present or not. For the FDT, the set DT value is constant, even when noise power fluctuation exists in the wireless channel. In the FDT approach, the DT based on guaranteeing a constant false alarm rate (CFAR) is often chosen for the specified false alarm probability (\(P_{fa}\)), noise variance \(\big{(}\sigma_{w}^{2}\big{)}\), and the number of Rx branches. Following the CFAR principles, the threshold of false alarm (\(\lambda_{fa}\)) for the concept of ED employing SLC can be expressed as:  
\[\lambda_{fa}=\left[Q^{-1}\Big{(}P_{fa}\Big{)}+\sqrt{RN}\right]\sqrt{RN}2 \sigma_{w}^{2} \tag{10}\]  
In the case of the CFAR approach, the threshold values are selected to ensure the highest detection probability. The CFAR approach is used in the ED process when, for the maximum utilization of the channel, the false alarm probability must be set to some small and predefined fixed values (\(P_{fa}\leq 0.1\)), whereas the detection probability needs to be maximized (\(P_{d}\geq 0.9\)). Similarly, to guarantee the priority of PU transmission, the expression for the detection threshold level selection based on the constant detection rate (CDR) principles is derived from Equation (7):  
\[\lambda_{d}=\Bigg{[}Q^{-1}(P_{d})\sqrt{\left(1+\frac{P}{R\sigma_{w}^{2}} \right)}+\sqrt{RN}\left(1+\frac{P}{2R\sigma_{w}^{2}}\right)\Bigg{]}\sqrt{RN}2 \sigma_{w}^{2} \tag{11}\]  
The CDR method is used to reduce the leverage of interference during the observed SS period. Based on Equations (10) and (11), it can be concluded that DT based on CFAR principles (Equation (10)) does not require information about PU Tx power. The result is a greater use of CFAR principles in real implementations. Additionally, CFAR principles enhance the throughput of the SU signal, but it does not guarantee sufficient security for the licensed PU in terms of possible transmissions overlapping with the SU. Consequently, using the DT set on CFAR principles is not the most appropriate approach to the selection of DT. As fluctuations in noise power in real wireless channels represent regular phenomena, more precise signal detection can be accomplished through DDT selection during the ED process. To obtain maximum detection accuracy, the DT level should be adjusted dynamically in time according to the NU fluctuations of the detected signal. This yields an approach to realizing ED that is based on the principles of DDT selection. Such an approach to ED can reduce the number of errors in the SS process and thus provide sufficient protection of the PU signal against overlapping in transmissions with SU. This results in better spectrum usability. The DDT principles achieved through the dynamic selection of DT represent the most appropriate solution for precise signal detection in the targeted frequency spectrum. However, the implementation of DDT principles in realistic wireless systems is very demanding. For the mathematical formulation of the DDT concept, the DT adaptation level is expressed with the DDT factor \(\rho^{\prime}\) (\(\rho^{\prime}\geq 1\)) in the further analyses. The values of the DDTfactor determine the level of DDT adaptation and unlike the fixed values based on FDT, the DDT values are assumed to be in the interval \(\left[\lambda_{d}^{DDT}/\rho^{\prime},\ \rho^{\prime}\lambda_{d}^{DDT}\right]\).