## 3 Spatial-Temporal Spectral Hole-Sensing Model  
In this section, we formulate a spatial-temporal spectral hole-sensing model for FD-SU TXs deployed in CR-IoT spectrum-heterogeneous environment. As shown in Figure 4, we consider a primary network, i.e., a single primary transmitter (PU-TX), multiple primary receivers (PU-RXs), and a secondary network, i.e., \(z\) number of IoT devices or FD-SU pairs (FD-SU TXs and FD-SU RXs) [5; 10; 16]. The proposed system can be considered for both downlink and uplink scenarios, i.e.,  
Figure 3: Time-slotted frame structure of FD-SU TXs. Figure 2: Network model with PU and full duplex-enabled secondary user transmitter (FD-SU TX). PU-TX as base station (BS) and mobile station (MS). Because there is no difference in the sensing of PU signal at a given time and frequency, the results of the paper are valid for both scenarios. The proposed system depends only on the transmission range of PU-TX, and do not depend on the number of PU-RXs. The performance of channel sensing in temporal domain with multiple PUs has been proposed [33]. However, the authors did not consider the variation of signal power of each PU-TX due to the presence of active surrounding PU-TXs. Given the stochastic nature of PU \(ON/OFF\) pattern, the analytical modelling of (two dimensional) spatial-temporal spectral holes sensing framework in the presence of multiple PUs is challenging. To the best of our knowledge, the analysis of spatial-temporal spectral holes with multiple PU-TXs has not been addressed [29; 30; 31; 32]. The spectrum sensing system model [33] with multiple PUs can be extended for the temporal-spatial analysis. However, we consider a single user primary network as the main contribution of the proposed work is to consider the traffic variations of PU in both time and space domains, and characterize and evaluate the sensing and UoS performance of FD-SU pairs present at different spatial positions. The consideration of multiple PUs (cellular users) is added in the future work. In the proposed work, a single PU-TX case is based on the assumption that a single PU-TX has a much higher signal power than any other PU-TXs (in a given frequency and time) in the considered geographical area. Moreover, the transmission range of the primary network is much greater than the secondary IoT network employing low power transmission. In addition, secondary receivers only sense the signal energy, i.e., do not distinguish (decode) the signals from different PU-TXs. Therefore, we can realistically assume that any FD-SU pair is effected by a single PU-TX. We also consider that a single FD-SU pair is assigned to a time-slotted primary channel at a time. Hence, the proposed model do not include the scheduling mechanism between the FD-SU TXs, and only considers the transmission-scheduling of the individual FD-SU pairs in each time slot (illustrated in Figure 3). At a given time and frequency band, we consider only one SU pair by assuming the orthogonal multiple access, e.g., TDMA, FDMA, OFDMA. We considers that all the FD-SU TXs are equipped with the sensing capability. The transmission radius of an FD-SU TX is \(r_{0}\). The transmission region of the PU (with radius \(D1\) from PU-TX) is \(R1\), and is determined by the receiving sensitivity of PU [29; 30]. If a primary receiver (PU-RX) is outside \(R1\), it will not work properly. The FD-SU pairs are randomly distributed in an area (with radius \(D2\) from PU-TX), which is closely related to the sensing sensitivity of FD-SU TXs. Any FD-SU TX, outside this area, will not be able to detect the primary signal. The \(R2\) is the region outside of \(R1\). We considers a typical scenario (i.e., \(D2>D1\)), in which FD-SU TXs have a higher sensing sensitivity than the receiver sensitivity of PU-RXs (to avoid interference with the primary transmission) [29; 30; 31; 32; 33]. Interference to the PU-RX is not permitted. Hence, FD-SU TXs in \(R1\) have the temporal spectral opportunities only, and the traffic variations of PU are considered in the time domain.