## 2 System Model
### Sensor Network Model
The instantaneous transmit power of the \(m\)-th SSN from IT or MT at time \(t\in\left[t_{0}^{\prime},t_{0}^{\prime}+\tau_{rp}\right]\), denoted by \(p_{m}(t)\), can be adjusted in a continuous way but is also limited by a maximum value \(P_{m}^{\rm max}\), i.e., \(0\leq p_{m}(t)\leq p_{m}^{\rm max}\). In the case of the simultaneous communications by IT and MT SSNs over an AWGN multiple-access channel, \(p_{m}(t)\) should satisfy an average power constraint given as follows to mitigate the interference among IT and MT SSNs [26]:  
\[\left\{ \begin{array}{ll}\frac{1}{n_{IT}}\frac{n_{IT}^{IT}}{m=1}p_{m}(t)\leq P _{av}^{IT},&m\in\mathcal{N}_{IT}\\ \frac{1}{n_{MT}}\sum\limits_{m=1}^{n_{MT}}p_{m}(t)\leq P_{av}^{MT},&m\in \mathcal{N}_{MT}\end{array} \right. \tag{3}\]  
where \(P_{av}^{IT}\) and \(P_{av}^{MT}\) are the total average power assigned to IT and MT SSNs, respectively. We assume that each SSN knows its distance \(d_{m\to b}\) from bottleneck SU \(b\) via the CCC and the channel path gain \(h_{m\to b}\) from the \(m\)-th SSN to the bottlenecked SU \(b\) can be expressed as \(h_{m\to b}=(d_{m\to b})^{-\kappa}\), where \(\kappa\geq 2\) is the path-loss exponent and \(b\in\mathcal{N}_{S}\). Thus, the channel capacity between the \(m\)-th SSN and bottlenecked SU \(b\) can be characterized by a concave function of the transmit power and channel conditions as follows [27]:  
Figure 2: The time-slotted frame structure adopted by dedicated spectrum sensor nodes (SSNs) in time-slot duration \(T_{F}\). SSR: spectrum sensing results. \[C_{m\to b}=W\log_{2}\left(1+\frac{p_{m}(t)K|h_{m\to b}|^{2}}{N_{0}W}\right) \tag{4}\]  
where \(K\) is a constant that depends on the transmission frequency and \(N_{0}\) is the noise power spectral density. Under this channel capacity formulation, the signal-to-noise ratio (SNR) between the \(m\)-th SSN and bottlenecked SU \(b\) is given by:  
\[\gamma_{m}=\frac{p_{m}(t)K|h_{m\to b}|^{2}}{N_{0}W} \tag{5}\]  
**Remark 3**. : _It is assumed that the data sequence of the SSR will be forwarded by ET SSNs to neighbor MT SSNs via a single-hop fashion during the slot \(\tau_{f}^{ET}\). Let \(t_{0}^{\prime\prime}\) and \(t_{0}^{\prime\prime}+\tau_{f}^{ET}\) denote the starting time and the terminal time of the slot \(\tau_{f}^{ET}\), respectively. The transmit power \(p_{n}(t)\) of the \(n\)-th SSN from ET at time \(t\in\left[t_{0}^{\prime\prime},t_{0}^{\prime\prime}+\tau_{f}^{ET}\right]\) is limited to a maximum transmit power \(P_{n}^{\max}\), for \(0\leq p_{n}(t)\leq P_{n}^{\max}\) and \(n\in\mathcal{N}_{ET}\). It is clear that the neighbor MT SSN with the shortest distance will be selected by an ET SSN as the next-hop SSN to save energy during the SSR forwarding period. In other words, the selection metric for the next-hop SSN by an ET SSN is only dependent on the distance between neighbor MT SSN and itself. Owing to the scenario of the rectangular grid based sensor network, it should be admitted that the shortest distance between ET SSN and neighbor MT SSN is equal to \(d\) for any ET SSN. Therefore, the \(n\)-th SSN from ET is inclined to hold the same transmit power \(p_{n}(t)\) at time \(t\). Under an AWGN multiple-access channel, the average power constraint should also be satisfied for all ET SSNs, i.e., \(\sum_{n\in\mathcal{N}_{ET}}p_{n}(t)\leq n_{ET}P_{av}^{ET}\), where \(P_{av}^{ET}\) is the total average power assigned to ET SSNs. Thus, our work in this paper primarily focuses on the distributed power control for IT and MT SSNs._  
**Remark 4**. : _The channel capacity \(C_{m\to b}\) in Equation (4) will be rigorously guaranteed if the channel state information (CSI) including the channel path gain \(h_{m\to b}\) and constant \(K\) is perfectly known at the \(m\)-th SSN which transmits the data sequence of the SSR. In practice, the perfect knowledge of the CSI measured at the \(m\)-th SSN side cannot be available because of time-varying wireless channel impairments along with hardware limitations [28; 29].