## 4 Mathematical Description of the Principles of ED Using SLC in MIMO-OFDM Systems
### Mathematical Formulation of the Role of Noise Uncertainty in the ED Process  
Equations (12) and (13) reveal that the threshold values depend on the precise knowledge of noise variance (\(\sigma_{w}^{2}\)). As the noise level in a wireless channel is variable in time, predicting in advance exact information about the level of noise variance (\(\sigma_{w}^{2}\)) in every moment of the ED process is a highly demanding task. The NU, therefore, refers to the lack of knowledge on the exact information about the noise power during the ED process. Uncertainty in noise power reduces the precision and accuracy of the PU signal detection. However, to simulate the realistic operations of the MIMO-OFDM system, the impact of NU in the wireless channel must be considered in the ED performance evaluation. Therefore, these fluctuations in noise power are considered through the NU factor \(\rho\) (\(\rho\geq 1\)). To demonstrate the impact of NU variation, in the analysis, the scope of NU variation is assumed to be in the range \(\sigma_{wNul}^{2}\epsilon\left[\sigma_{ww}^{2}/\rho^{\prime},\ \rho\sigma_{w}^{2}\right]\). The detection and false alarm probabilities in a scenario that considers such a fluctuation in noise power can be expressed by:  
\[P_{d}^{NU}(\text{Pr}(\Lambda_{SLC}<\lambda)\ \approx H_{1})=\min_{\sigma_{wNul}^{2} \epsilon\big{[}\sigma_{ww}^{2}\big{]}}Q\Bigg{(}\frac{\lambda-RN^{NU}\big{(}2 \sigma_{wNul}^{2}\big{)}\left(1+\frac{P}{2R\sigma_{wNul}^{2}}\right)}{\sqrt{RN^{ NU}\left(1+\frac{P}{R\sigma_{wNul}^{2}}\right)\left(2\sigma_{wNul}^{2} \right)}}\Bigg{)}=Q\Bigg{(}\frac{\lambda_{fa}^{NU}-RN^{NU}\left(2\sigma_{w}^{2 }\right)\left(1+\frac{P}{2R\sigma_{w}^{2}}\right)}{\sqrt{RN^{NU}\left(1+\frac{P }{R\sigma_{w}^{2}}\right)\left(2\sigma_{w}^{2}\right)}}\Bigg{)} \tag{15}\]\[P_{fa}^{N\!U}(\Pr(\Lambda_{SLC}>\lambda)\ \approx H_{0})=\max_{\sigma_{a\!N\!U}^{2} \varepsilon\left[\sigma_{a\!N\!U}^{2}/\!\!\!\!/_{\rho}\ \sigma_{a\!N\!U}^{2}\right]}Q\Bigg{(}\frac{\lambda-RN^{N\!U}\!\left(2\sigma_{ w\!N\!U}^{2}\right)}{\sqrt{RN^{N\!U}\left(2\sigma_{w\!N\!U}^{2}\right)}} \Bigg{)}=Q\Bigg{(}\frac{\lambda_{fa}^{N\!U}-RN^{N\!U}\!\left(2\rho\sigma_{w}^{2 }\right)}{\sqrt{RN^{N\!U}\left(2\rho\sigma_{w}^{2}\right)}}\Bigg{)}, \tag{16}\]  
where \(\lambda_{d}^{N\!U}\) and \(\lambda_{fa}^{N\!U}\) represents the detection and false alarm thresholds, respectively. From Equations (15) and (16), it can be concluded that choosing adequate values of thresholds (\(\lambda_{fa}^{N\!U}\) and \(\lambda_{d}^{N\!U}\)) can significantly improve the ED process. The probability of detection \(\left(P_{d}^{N\!U}\right)\) can be defined with respect of the probability of a false alarm \(\left(P_{fa}^{N\!U}\right)\) in the channel impacted by NU as follows:  
\[P_{d}^{N\!U}=Q\Bigg{(}\frac{\rho Q^{-1}\left(P_{fa}^{N\!U}\right)-\sqrt{RN^{ N\!U}\left(\frac{P-1}{P}\right)}}{\sqrt{\frac{1}{P}+2\sigma_{S\!S\!S}^{2}}} \Bigg{)}=Q\Bigg{(}\frac{\rho Q^{-1}\left(P_{fa}^{N\!U}\right)-\sqrt{RN^{N\!U} \left(\frac{P}{2\rho\sigma_{w}^{2}}-\frac{P-1}{P}\right)}}{\sqrt{\frac{1}{P}+ \frac{P}{2\rho\sigma_{w}^{2}}}}\Bigg{)} \tag{17}\]  
If the NU factor is set to 1.00 (\(\rho=1.00\)), there is no fluctuation in noise power and Equation (17) is identical to Equation (9). If the NU factor is greater than 1 (\(\rho>1\)), the impact of NU is present in the wireless channel. Greater values for the NU correspond to the simulation of the wireless channel which is impacted by more fluctuations in noise power and vice versa. Mathematical Formulation of the Combined Role of Noise Uncertainty and Detection Threshold on the ED Process  
Involving DDT in the ED process considerably increases ED effectiveness and reduces the adverse effects caused by NU. The most realistic simulation approach to implementing the ED with the SLC method in the MIMO-OFDM transmission system is an analysis of the joint role of DDT adaptation and NU variation on ED performance.