## 4 Energy Detection Model
\begin{table}
\begin{tabular}{c l} \hline \hline
**Index** & **Description** \\ \hline \(H_{0}\) & Hypothesis which determines the absence of the PU signal \\ \(H_{1}\) & Hypothesis which determines the presence of the PU signal \\ \(y_{i}(n)\) & Averaged received signal for _i-th_ SU and for _n-th_ sample \\ \(w_{i}(n)\) & AWGN signal for _i-th_ SU \\ \(\sigma_{n_{i}}^{2}\) & Variance of AWGN signal for _i-th_ SU without NU variations and DT adaptation \\ \(\sigma_{NULT}^{2}\) & AWGN variance (interval) for ED with NU variations and DT adaptation \\ \(\sigma_{NULT}^{2}\) & AWGN variance (interval) for ED with NU variations \\ \(x(n)\) & Transmitted _i-th_ signal from the PU \\ \(\tau_{i}\) & Energy test statistic signal level of the detected signal \\ \(\lambda_{i}\) & Detection threshold of _i-th_ SU _for ED_ without NU variations and DT adaptation \\ \(\lambda_{dji}\) & Detection threshold based on CDR and without NU variations and DT adaptation \\ \(\lambda_{dji}\) & False alarm threshold based on CFAR and without NU variations and DT adaptation \\ \(\lambda_{i}^{IDT}\) & DT (interval) for ED without NU variation \\ \(\lambda_{i}^{NDUT}\) & DT (interval) for ED with DT adaptation and NU variation \\ \(P_{d_{i}}\) & Detection probability for ED without NU and DT \\ \(P_{f_{d_{i}}}\) & False alarm probability for ED without NU and DT \\ \(P_{f_{d_{i}}}^{IDT}\) & Detection probability for ED with DT adaptation \\ \(P_{f_{d_{i}}}^{IDT}\) & False alarm probability for ED with DT adaptation \\ \(P_{f_{d_{i}}}^{NI}\) & Detection probability for ED with NU variation \\ \(P_{f_{d_{i}}}^{NI}\) & False alarm probability for ED with NU variation \\ \(P_{f_{d_{i}}}^{NUDUT}\) & Detection probability for ED with DT adaptation and NU variation \\ \(P_{f_{d_{i}}}^{NDUT}\) & False alarm probability for ED with DT adaptation and NU variation \\ \(\rho\) & NU factor \\ \(\rho^{\prime}\) & DT factor \\ \(Q\) & Standard Gaussian complementary CDF \\ \(Q^{-1}\) & Inverse standard Gaussian Complementary CDF \\ \(p\) & Average received PU signal power at position of SU \\ \(N\) & Overall sample quantity for ED without DT adaptation and NU variation \\ \(N^{DT}\) & Overall sample quantity for ED with DT adaptation \\ \(N^{NUI}\) & Overall sample quantity for ED with NU variation \\ \(N^{NDUT}\) & Overall sample quantity for ED with DT adaptation and NU variation \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters with their corresponding descriptions. Figure 3: Block diagram of the energy detection process. multipath channels have been neglected. This is due to OFDM spreading of the wideband signal into many modulated narrowband subcarriers, which results in exposing those subcarriers to flat fading. Hence, the impact of noise received at the location of SU has been modelled in this analysis as additive white Gaussian noise (AWGN). AWGN is assumed to be an independent and identically distributed random process of zero mean and variance, \(\sigma_{n_{i}}^{2}\), with a power spectral density, \(w_{i}(n)\ \sim\mathcal{N}(\sigma_{n_{i}}^{2})\). The total variance of the received signal (\(\sigma_{y_{i}}^{2}\)) can be expressed as:  
\[\sigma_{y_{i}}^{2}=\sigma_{x}^{2}+\sigma_{n_{i}}^{2}=\sigma_{n_{i}}^{2}(1+SNR) \tag{3}\]  
where \(\sigma_{x}^{2}\) represents the variance of transmitted signal from the PU which can be expressed as \(\sigma_{x}^{2}=SNR\ \sigma_{n_{i}}^{2}\). Assuming that there is no deterministic knowledge about the signal \(x(n)\) besides the average received power of the PU signal (which is characteristic of ED as a SS method), the average received power is expressed as:  
\[P=\frac{1}{N}\sum_{n=1}^{N}\left|x(n)\right|^{2}\approx\sigma_{x_{i}}^{2} \tag{4}\]  
and the relation (3) can be transformed to \(\sigma_{y_{i}}^{2}=P+\sigma_{n_{i}}^{2}\).