## 4 Performance Metric for Spatial-Temporal Spectral Hole-Sensing Model
Given \(S(i)\) in Equation (1), and Equation (2) as i.i.d., the mean and variance of \(T(S)\) can be expressed as,  
\[E\left[T(S)\right]=E\left[|S(i)|^{2}\right],\qquad\quad E\left[T(S)\right]= \frac{1}{N}var\left[|S(i)|^{2}\right]. \tag{17}\]  
Using the central limit theorem (CLT), we obtain the distribution of \(T(S)\) given each of the defined hypotheses for HD and FD sensing slots. For a large \(N\), the probability density functions (pdf) of \(T(S)\) given \(H_{00}\), \(H_{10}\), \(H_{01}\), and \(H_{11}\) are complex-valued Gaussian with the following mean and variance,  
\[E\left[T(S)|H_{00}\right]=\sigma_{n}^{2},\qquad\quad var\left[T(S)|H_{00} \right]=\frac{1}{N}\left[\sigma_{n}^{\mathcal{A}}\right] \tag{18}\]  
\[E\left[T(S)|H_{10}\right]=\sigma_{n}^{2}(\gamma_  
where \(\gamma_{p}\) is the instantaneous sensing SNR at the FD-SU TX, and is expressed as \(\frac{\sigma_{h_{1}}^{2}\sigma_{\gamma_{1}}^{2}}{\sigma_{\gamma_{1}}^{2}}\). The parameter \(\gamma_{1}\) is the INR at the FD-SU TX due to the SU transmission, and is expressed as \(\frac{X^{2}\sigma_{\gamma_{2}}^{2}}{\sigma_{\gamma_{1}}^{2}}\). \(X\) is an SIS factor that quantifies the degree of imperfect SIS, \(X\in[0,1]\). \(X\) represents the ratio between the residual self-interference and the transmitting power of FD-SU TX [28]. If \(X=0\) (perfect SIS), FD-SU TX can completely suppress its residual self-interference. Substituting the statistical properties under each hypothesis (18)-(21), we obtain,  
\[Pr\left\{T(S)>\phi_{1}\left|\right.H_{00}\right\}=Q\left[\left(\frac{\phi_{1} }{\sigma_{n}^{2}}-1\right)\sqrt{N}\right] \tag{22}\]  
\[Pr\left\{T(S)>\phi_{1}\left|\right.H_{01}\right\}=Q\left[\left(\frac{\phi_{1} }{\left(1+\gamma_{P}\right)\sigma_{n}^{2}}-1\right)\sqrt{N}\right] \tag{23}\]  
\[Pr\left\{T(S)>\phi_{2}\left|\right.H_{10}\right\}=Q\left[\left(\frac{\phi_{2} }{\left(1+\gamma_{1}\right)\sigma_{n}^{2}}-1\right)\sqrt{N}\right] \tag{24}\]  
\[Pr\left\{T(S)>\phi_{2}\left|\right.H_{11}\right\}=Q\left[\left(\frac{\phi_{2} }{\left(1+\gamma_{1}+\gamma_{P}\right)\sigma_{n}^{2}}-1\right)\sqrt{N}\right]. \tag{25}\]  
The sensing performance during HD and FD sensing slots can be formulated either under the QoS constraint for the PU, i.e., guaranteed protection level to the PU, or the QoS constraint for the FD-SU TX, i.e., guaranteed usability rate of spatial-temporal spectral holes [5]. In the proposed work, we consider the QoS constraint for the PU, i.e., the probability of finding the non-availability of spatial-temporal spectral holes is fixed to the desired value (\(P_{F.nST}^{0}\), and \(P_{F.nST}^{1}\)), and the corresponding probability of missing the availability of spatial-temporal spectral holes (\(P_{Mi.ST}^{0}\), and \(P_{Mi.ST}^{1}\)), given the sensing parameters, is obtained. Under the considered constraints, (\(P_{Mi.ST}^{0}\), and \(P_{Mi.ST}^{1}\)) for the FD-SU TXs in \(R1\) and \(R2\) can be expressed as  
\[P_{Mi.ST}^{0}= \begin{cases}Q\left[\frac{\sqrt{N\left(\gamma_{p}+1\right)^{2}} Q^{-1}\left(P_{F.nST}^{0}\right)+\left(N\gamma_{p}\right)}{\sqrt{N}}\right]P_{ OFF},&S_{B}\\ Q\left[\frac{\sqrt{N\left(\gamma_{p}+1\right)^{2}}Q^{-1}\left(P_{F.nST}^{0} \right)+\left(N\gamma_{p}\right)}{\sqrt{N}}\right]P_{OFF}+P_{F.nST}^{0}P_{ON},& S_{A}\end{cases}  \tag{26}\]  
\[P_{Mi.ST}^{1}= \begin{cases}Q\left[\frac{\sqrt{N\left(\gamma_{1}+\gamma_{P}+1 \right)^{2}}Q^{-1}\left(P_{F.nST}^{1}\right)+\left(N\gamma_{p}\right)}{\sqrt{N }\left(\gamma_{1}+1\right)}\right]P_{OFF},&S_{B}\\ Q\left[\frac{\sqrt{N\left(\gamma_{1}+\gamma_{P}+1\right)^{2}}Q^{-1}\left(P_{F.nST }^{1}\right)+\left(N\gamma_{p}\right)}{\sqrt{N}\left(\gamma_{1}+1\right)} \right]P_{OFF}+P_{F.nST}^{1}P_{ON},&S_{A},\end{cases}  \tag{27}\]  
where \(Q(. )\) and \(Q^{-1}(. )\) denote the complementary distribution function of the standard Gaussian and its inverse, respectively.