## 4 Performance Metric for Spatial-Temporal Spectral Hole-Sensing Model  
In this section, we introduce the performance metric to guide the spatial-temporal spectral hole-sensing model. Closed-form expressions of sensing probabilities are proposed. The sensing probabilities are: (i) the probability of finding the non-availability of spatial-temporal spectral holes and (ii) the probability of missing the availability of spatial-temporal spectral holes. The probabilities of finding the non-availability of spatial-temporal spectral holes during HD and FD sensing slots are respectively expressed as  
\[P_{F.nST}^{0}\overset{\Delta}{=}Pr\left\{\Psi_{ST}^{0}=O_{B}^{0}\,|\,O_{B}^{0}\right\} \tag{9}\]  
\[P_{F.nST}^{1}\overset{\Delta}{=}Pr\left\{\Psi_{ST}^{1}=O_{B}^{1}\,|\,O_{B}^{1 }\right\}, \tag{10}\]  
where \(\Psi_{ST}^{0}\) and \(\Psi_{ST}^{1}\) denote the decision functions for spatial-temporal spectral holes during HD and FD sensing-slots, respectively. Based on the detail derivation in [29] and extending it separately for HD and FD sensing slots, the probabilities of finding the non-availability of spatial-temporal spectral holes are expressed by the temporal states of the PU and the spatial positions of FD-SU TX as  
\[P_{F.nST}^{0}=Pr\left\{T(S)>\phi_{1}\,|\,H_{01}\right\},\qquad S_{B} \tag{11}\]\[P_{F.nST}^{1}=Pr\left\{T(S)>\phi_{2}\,|\,H_{11}\right\},\qquad S_{B}, \tag{12}\]  
where \(P_{F.nST}^{0}\) and \(P_{F.nST}^{1}\) denote the probabilities of finding the non-availability of spatial-temporal spectral holes during HD and FD sensing-slots, respectively. \(\phi_{1}\) and \(\phi_{2}\) are the detection thresholds during HD and FD sensing slots, respectively. \(T(S)\) denotes the test statistics, and is expressed as \(\frac{1}{N}\sum\limits_{i=0}^{N-1}|S(i)|^{2}\). Similarly, the probabilities of missing the availability of spatial-temporal spectral holes during HD and FD sensing slots are respectively expressed as  
\[P_{Mi.ST}^{0}=Pr\left\{\Psi_{ST}^{0}=O_{B}^{0}\,|\,O_{A}^{0}\right\} \tag{13}\]  
\[P_{Mi.ST}^{1}=Pr\left\{\Psi_{ST}^{1}=O_{B}^{1}\,|\,O_{A}^{1}\right\}. \tag{14}\]  
In a similar fashion, extending the derivation in [29] separately for HD and FD sensing slots, the probabilities of missing the availability of spatial-temporal spectral holes are also expressed by the temporal states of the PU and spatial positions of the FD-SU TX as  
\[P_{Mi.ST}^{0}= \begin{cases}Pr\left\{T(S)>\phi_{1}\,|\,H_{00}\right\}P_{OFF},& S_{B}\\ Pr\left\{T(S)>\phi_{1}\,|\,H_{00}\right\}P_{OFF}+Pr\left\{T(S)>\phi_{1}\,|\,H_ {01}\right\}P_{ON},&S_{A}\end{cases}  \tag{15}\]  
\[P_{Mi.ST}^{1}= \begin{cases}Pr\left\{T(S)>\phi_{2}\,|\,H_{10}\right\}P_{OFF},& S_{B}\\ Pr\left\{T(S)>\phi_{2}\,|\,H_{10}\right\}P_{OFF}+Pr\left\{T(S)>\phi_{2}\,|\,H_ {11}\right\}P_{ON},&S_{A}.\end{cases}  \tag{16}\]  
It can be observed that the probabilities of finding the non-availability of spatial-temporal spectral holes (\(P_{F.nST}^{0}\), and \(P_{F.nST}^{1}\)) are related to the FD-SU TXs in \(R1\) only, because the FD-SU TXs in \(R2\) have spatial spectral holes, i.e., can access the channel anytime. The probabilities of missing the availability of spatial-temporal spectral holes (\(P_{Mi.ST}^{0}\), and \(P_{Mi.ST}^{1}\)) for the FD-SU TXs in \(R1\) are determined by the \(OFF\) state of the PU, and are represented with an event that the FD-SU TX detects the presence of the PU when the PU is in \(OFF\) state, i.e., temporal false alarms. Furthermore, \(P_{Mi.ST}^{0}\), and \(P_{Mi.ST}^{1}\) for the FD-SU TXs in \(R2\) are determined by each state of the PU, i.e., \(ON\), and \(OFF\) states, and are represented with the following two events; (i) FD-SU TX determines the presence of PU when PU is in the \(OFF\) state, i.e., temporal false alarms, and (ii) FD-SU TX determines the presence of the PU when the PU is in the \(ON\) state, i.e., spatial false alarms.