## 3 Spatial-Temporal Spectral Hole-Sensing Model
The FD-SU TXs in \(R2\) have the spatial spectral opportunities (can access the channel anytime). However, FD-SU TXs lose their spatial spectral opportunities when detect a PU-TX. The FD-SU TXs in \(R2\), near to the PU-RXs, can cause harmful interference and are referred to be present in the \(NC\,Region\). The \(NC\,Region\) is determined by the interference constraint from the PU-RXs, and the peak transmission power of the FD-SU TXs [29]. An appropriate power control, which is beyond the scope of this paper, is an essential requirement to allow these FD-SU TXs to exploit their spectral opportunities. Hence, FD-SU TXs in the \(NC\,Region\) are not considered in the proposed work. In the considered network, there are two potential signal sources, i.e., PU-TX and FD-SU TX. Thus, we define four tests of the hypothesis as follows, \begin{tabular}{l l l} \(\bullet\) & \(H_{00}\) & PU-TX is \(OFF\) and FD-SU TX is not transmitting. \\ \(\bullet\) & \(H_{01}\) & PU-TX is \(ON\) and FD-SU TX is not transmitting. \\ \(\bullet\) & \(H_{10}\) & PU-TX is \(OFF\) and FD-SU TX is transmitting. \\ \(\bullet\) & \(H_{11}\) & PU-TX is \(ON\) and FD-SU TX is transmitting. \\ \end{tabular}   
The traditional temporal spectral-holes during HD and FD sensing slots are respectively expressed as following hypothesis testing problems,  
\[S(i)= \begin{cases}n(i),&H_{00}\\ h_{1}s_{1}(i)+n(i),&H_{01}\end{cases}  \tag{1}\]  
\[S(i)= \begin{cases}g_{1}s_{2}(i)+n(i),&H_{10}\\ h_{1}s_{1}(i)+g_{1}s_{2}(i)+n(i),&H_{11}\end{cases}  \tag{2}\]  
Here, \(n(i)\) denotes the \(i\)-th noise signal sample at FD-SU TX, and is considered to be complex Gaussian with zero mean and variance \(\sigma_{g_{1}}^{2}\). \(s_{1}(i)\) is the \(i\)th sample of the transmitted signal of PU-TX, and is assumed to be phase-shift keying (PSK)-modulated with variance \(\sigma_{s_{1}}^{2}\)[34; 35]. \(s_{2}(i)\) is the \(i\)-th sample of the transmitted signal of FD-SU TX (before SIS) and is also assumed to be PSK-modulated with variance \(\sigma_{s_{2}}^{2}\). \(h_{1}\) is the complex Gaussian channel coefficient between PU-TX and FD-SU TX with zero mean and variance \(\sigma_{h_{1}}^{2}\). \(g_{1}\) represents the self-interference channel between the \(Ant-1\) and \(Ant-2\) of FD-SU TX, which follows the complex Gaussian random variable with zero mean and variance \(\sigma_{g_{1}}^{2}\). In (1) and (2), \(i=0,\,1\,\ldots\,\,N-1\) is the sample index. In general, the traditional spatial spectral holes for SU node can be expressed as the following hypothesis-testing problem [29; 30; 31; 32],  
\[ \begin{cases}S_{A}:&\quad\text{$SU\,node\,is\,present\,in\,R2$}\\ S_{B}:&\quad\text{$SU\,node\,is\,present\,in\,R1$},\end{cases}  \tag{3}\]  
where hypothesis \(S_{A}\) denotes the availability of spatial spectral holes, and hypothesis \(S_{B}\) denotes the non-availability of spatial spectral holes. From a joint (two-dimensional) sensing perspective, spatial-temporal spectral holes for SU node can be described as the following hypothesis testing problem [29; 30; 31; 32],  
\[ \begin{cases}O_{A}:&H_{0}\cup S_{A}\\ O_{B}:&H_{1}\cap S_{B},\end{cases}  \tag{4}\]  
where \(O_{A}\) denotes the case where spatial-temporal spectral holes are available, either because of the absence of PU (\(H_{0}\)), or the PU is present and the SU node locates in \(R2\) (\(S_{A}\)). Similarly, \(O_{B}\) denotes the case that spatial-temporal spectral holes are not available because of the presence of PU (\(H_{1}\)) and the SU node locates in \(R1\) (\(S_{B}\)). Figure 4: Cognitive radio (CR)-Internet of Things (IoT) spectrum-heterogeneous environment. Considering Equations (1), (2), and (4), we model the availability of spatial-temporal spectral holes for FD-SU TX as the following composite hypothesis testing problems  
\[O_{A}^{0}:S(i)= \begin{cases}n(i)&\left\{H_{00}\right\}\\ h_{1}s_{1}(i)+n(i)&\left\{H_{01}\right\},\qquad S_{A},\end{cases}  \tag{5}\]  
\[O_{A}^{1}:S(i)= \begin{cases}g_{1}s_{2}(i)+n(i)&\left\{H_{10}\right\}\\ h_{1}s_{1}(i)+g_{1}s_{2}(i)+n(i)&\left\{H_{11}\right\},\qquad S_{A},\end{cases}