## 2 System Model and GD Test Statistics
### The Case \(s_{i}^{n}[k]\
eq s_{i}[k]\)  
In practice, the problem to satisfy an equality between the model signal \(s_{i}^{\text{mod}}[k]\) and incoming PU signal \(s_{i}[k]\) amplitudes is a very difficult problem. For simplicity, we consider a case when a relation between the amplitudes of the model signal \(s_{i}^{\text{mod}}[k]\) and incoming PU signal \(s_{i}[k]\) can be presented in the following form:  
\[s_{i}^{\text{mod}}[k]=\mu s_{i}[k] \tag{46}\]  
where \(\mu\) is the amplitude coefficient of proportionality. Under the condition given by Equation (46), the MGF of the GD partial decision statistics  
\[T_{GD}(\mathbf{X}_{k})=\sum_{i=1}^{M}s_{i}^{2}[k](2\mu-1)+\sum_{i=1}^{M}2s_{i} [k]\zeta_{i}[k](\mu-1)+\sum_{i=1}^{M}\eta_{i}^{2}[k]-\sum_{i=1}^{M}\xi_{i}^{2}[k] \tag{47}\]  
takes the following form:  
\[\mathcal{M}_{s_{GD}(\mathbf{X}_{k})}(I)=\frac{1}{\sqrt{\left(1-4\sigma^{4}\,I \right)^{M}}}\times\prod_{i=1}^{M}\frac{1}{1-E_{s}\ \sigma_{h}^{2}(2\mu-1)\,\alpha_{i}}\times\prod_{i=1}^{M}\frac{1}{1-2(\mu-1)\, \alpha_{i}\,I\sqrt{E_{s}\ \sigma_{h}^{2}\sigma^{2}}} \tag{48}\]  
Based on Equation (48), the mean \(m_{\eta_{i}}^{GD}\) and the variance \(Var_{\eta_{i}}^{GD}\) of the test statistics \(T_{GD}(\mathbf{X})\) under the hypothesis \(\mathcal{M}_{i}\) are defined using the following form:  
\[\left\{ \begin{array}{l}m_{\eta_{i}}^{GD}=\mathrm{E}\Big{[}T_{GD}(\mathbf{X} )\Big{|}\mathcal{M}_{1}\Big{]}=NM\,(2\mu-1)E_{s}\sigma_{h}^{2}\quad;\\ Var_{\eta_{i}}^{GD}=Var\Big{[}T_{GD}(\mathbf{X})\Big{|}\mathcal{M}_{1}\Big{]} =NM\,\{[(2\mu-1)E_{s}\sigma_{h}^{2}]^{2}+4(\mu-1)^{2}\,E_{s}\sigma_{h}^{2} \sigma^{2}+4\sigma^{4}\}\quad;\\ \end{array} \right.\sigma_{\xi}^{2}=\sigma_{\eta}^{2}=\sigma^{2} \tag{49}\]  
In the case of noise power uncertainty and under the condition given by Equation (46), Equation (49) allows us to define the probability of false alarm \(P_{FA}^{GD}\) and the probability of miss \(P_{miss}^{GD}\) for GD using the following form:\[\left\{ \begin{array}{ll}P_{{}_{FA}}^{{}_{GD}}=Q\Bigg{(}\frac{{ \mathit{THR}_{{}_{GD}}}}{{2\rho\sigma_{{}_{\pi}}^{2}\sqrt{NM}}}\Bigg{)}&;&\sigma_{ \xi}^{2}=\sigma_{\eta}^{2}=\sigma^{2}\\ \\ P_{{}_{FA}}^{{}_{GD}}=Q\Bigg{(}\frac{{\mathit{THR}_{{}_{GD}}}}{{ \rho\sigma_{{}_{\pi}}^{2}\sqrt{2 NM\left(1+\beta^{2}\right)}}}\Bigg{)}&;&\sigma_ {\xi}^{2}\
eq\sigma_{\eta}^{2}\\ \\ \left\{\begin{array}{ll}P_{{}_{miss}}^{{}_{GD}}=1-Q\Bigg{(}\frac{{ \mathit{THR}_{{}_{GD}}}-NM\left(2\mu-1\right)E_{{}_{s}}\sigma_{{}_{\eta}}^{2}}{ 2\rho^{-1}\sigma_{{}_{\pi}}^{2}\sqrt{NM\left[\left(\mu-1\right)^{2}SNR^{2}+2 \left(1+\beta^{2}\right)\right]}}\Bigg{)}&;&\sigma_{\xi}^{2}=\sigma_{\eta}^{2} =\sigma^{2}\\ \\ P_{{}_{miss}}^{{}_{GD}}=1-Q\Bigg{(}\frac{{\mathit{THR}_{{}_{GD}}}-NM\left(2 \mu-1\right)E_{{}_{s}}\sigma_{{}_{\eta}}^{2}}{\rho^{-1}\sigma_{{}_{\pi}}^{2} \sqrt{NM\left[\left(\mu-1\right)^{2}SNR^{2}+2\left(1+\beta^{2}\right)\right]} }\Bigg{)}&;&\sigma_{\xi}^{2}\
eq\sigma_{\eta}^{2}\\ \end{array}\right.\end{array} \right. \tag{50}\]  
Under delivering (51) we ignore the term \(\left(2\mu-1\right)^{2}SNR^{2}\) since CR networks operate at very low \(SNR\) values, _i.e._, \(SNR<<1\). Defining the threshold \({\mathit{THR}_{{}_{GD}}}\) in terms of the probability of false alarm \(P_{{}_{FA}}^{{}_{GD}}\) based on (50) and substituting it in (51), we obtain  
\[\left\{ \begin{array}{ll}P_{{}_{miss}}^{{}_{GD}}=1-Q\Bigg{(}\frac{{2\rho Q^{ -1}\left(P_{{}_{FA}}^{{}_{GD}}\right)-\sqrt{NM}\left(2\mu-1\right)SNR}}{{2\rho ^{-1}\sqrt{\left(\mu-1\right)^{2}SNR^{2}+1}}}\Bigg{)}&;&\sigma_{\xi}^{2}=\sigma _{\eta}^{2}=\sigma^{2}\\ \\ P_{{}_{miss}}^{{}_{GD}}=1-Q\Bigg{(}\frac{{\sqrt{2(1+\beta^{2})}}\ \rho Q^{-1} \left(P_{{}_{FA}}^{{}_{GD}}\right)-\sqrt{NM}\left(2\mu-1\right)SNR}{{\rho^{-1 }\sqrt{\left(\mu-1\right)^{2}SNR^{2}+2\left(1+\beta^{2}\right)}}}\Bigg{)}&;& \sigma_{\xi}^{2}\
eq\sigma_{\eta}^{2}\\ \end{array} \right.