## 2 System Model
### Frame Structure of FD-SU TXs  
The FD-SU TXs consist of imperfect SIS enabled narrowband antennas, i.e., \(Ant-1\) and \(Ant-2\). As shown in Figure 2, \(Ant-1\) (a receive antenna) performs the sensing procedure to locate the spectral holes and \(Ant-2\) (a transmit antenna) transmits when spectral holes are available [22; 23; 24; 25; 26]. In traditional time-slotted FD frame structure, sensing procedure is continuous, i.e., extends to the entire duration. The drawbacks of such frame structure are: (i) change of PU state needs to be consider in the sensing procedure when PU and SU are not synchronized (PU state, in each frame, is not consistent), which normally degrades the detection performance [25], and (ii) time required to make a decision regarding the presence of PU extends to the entire time-slotted frame [26]. The considered time-slotted frame structure for FD-SU TX is shown in Figure 3, in which sensing duration is divided into \(n\) consecutive short sensing slots (labelled as \(T_{S,0}\), \(T_{S,1}\)... and \(T_{S,n-1}\)), and a sensing procedure is performed at each sensing slot. It is assumed that the sensing-slots (in each time-slotted frame)  
Figure 1: Activity model for primary user (PU).  
are of same duration. The number of samples in each sensing-slot is \(N\), and is expressed as \(M/n\), where \(M\) is the number of total samples in each time-slotted frame. The constraint for an asynchronized case is much relax as the duration of each sensing-slot is shorter. Therefore, a consistent PU state, in each sensing-slot, is a more realistic assumption than in the traditional time-slotted FD frame structure, and is considered in the proposed system. The initial (first) sensing-slot in each time-slotted frame is HD (FD-SU TX performs sensing only) to avoid collision at the start. If a PU is not detected at the end of HD sensing slot (\(T_{S,0}\) in Figure 3), FD-SU TX initiates its transmission and sensing simultaneously. If a PU is detected at the end of HD sensing slot, FD-SU TX do not transmit and continue to perform the sensing procedure until channel is available in the next time-slotted frame (which also starts with a HD sensing-slot). The reason of continuous sensing is that we do not introduce any switching mechanism for sensing procedure at the receiving antennas of resource constraint IoT devices (FD-SU TXs). Similarly, if a PU is detected at the end of any FD sensing slot, FD-SU TX aborts its transmission until the next time-slotted frame. The motivation behind the considered time-slotted frame structure is to account for the tradeoff between the sensing efficiency and timelines in detecting the PU. An increment in the duration of sensing slots reduces sensing errors; however, it increases the delay of detecting the PU. Conversely, a reduction in the duration of sensing slots reduces the delay of detecting the PU, with the cost of increased sensing errors. The sensed signal during the FD sensing slots (\(T_{S,1}\)-\(T_{S,n-1}\) in Figure 3) is corrupted by a self-interference signal. The formulation of the sensing procedure at FD-SU TX is considered for both types of sensing slots: (i) HD sensing slots (when FD-SU TX is not transmitting) and (ii) FD sensing slots (when FD-SU TX is transmitting). The energy detection (ED) scheme is most commonly used for channel sensing because of its low computational and implementation complexities. Moreover, it does not require prior knowledge of the PU signal parameters [24; 25; 26; 27; 28; 29; 30; 31; 32]. The secondary network does not guarantee on-time services, i.e., quality of service (QoS) is not guaranteed. Hence, it is assumed that the considered FD-SU TX always has packets to transmit [27].