## 2 System Model
### Cognitive Radio Wireless Sensor Network Energy Consumption Model
#### 3.3.3 Cognitive Radio Unit Energy Consumption  
The CR unit senses the licensed channels and detects vacant channels that can be used for data communications. Energy consumption for vacant channels detection comprises energy consumption for sensing sets of channels and reporting local decisions as well as receiving final cooperative decisions [47]. Energy consumption for sensing sets of channels \(\mathrm{E_{cs}}\) comprises of energy consumed for listening over the channels and receiving \(N_{o}\) observation samples, as well as energy required to process the signal samples (modulation, signal shaping _etc._) and make local decisions. If \(P_{ed}\) denotes the energy detector's circuit power consumption and \(E_{sp}\) denotes the energy consumption for processing the received \(N_{o}\) signal samples, then energy dissipation for sensing \(n_{z}\) sets of channels can be expressed as:  
\[\mathrm{E_{cs}}(n_{z},T_{cs})=\sum_{m=1}^{n_{z}}(T_{cs}P_{ed}+\mathrm{E_{sp}}) \tag{11}\]  
This suggests that the energy consumption for channel sensing is a function of channel sensing duration \(T_{cs}\) and it increases along with an increase in the number of channels \(n_{z}\). Minimum energy consumption can be achieved with minimum channel sensing time but accurate results may not be obtained. The Nyquist sampling theorem suggests that the sample frequency \(f_{s}\) of the received signal samples (\(N_{o}=2T_{cs}B_{w}\)) must be at least twice the bandwidth (\(f_{s}\geq 2B_{w}\)) [29]. Let \(P_{dt}\) denote the target probability of detection, \(P_{ft}\) denotes target probability of false alarms, and \(\delta_{s}\) denotes the average of the PU's SNR received on the channel. The channel sensing time \(T_{cs}\) can be expressed as [37]:  
\[T_{cs}=\frac{1}{\delta_{s}^{~{}2}f_{s}}\left(Q^{-1}\big{(}P_{ft}\big{)}-Q^{-1 }(P_{dt})\sqrt{2\delta_{s}+1}\right)^{2} \tag{12}\]  
In cooperative sensing, each member node \(MN_{i}\) senses sets of channels to detect vacant channels, makes local decisions on the existence of PU and then reports its result to the FC which is the clusterhead \(CH_{j}\) for the final cooperative decision. Let \(d_{i,j}\) denote the Euclidian distance between \(MN_{i}\) and \(CH_{j}\), \(E_{ec}\) denotes the energy consumption for running the radio electronics of \(MN_{i}\) and \(E_{am}\) denotes the energy consumption for amplifying the signal to be transmitted to \(CH_{j}\) so as to maintain an acceptable SNR level. Then the energy cost \(E_{rp}\) for reporting \(B_{ld}\)-bits packet of local decisions to the \(CH_{j}\) can be expressed as:\[\mathrm{E}_{rp}(B_{ld},d_{i,j})=\ B_{ld}(E_{ec}+E_{am}\ d_{i,j}) \tag{13}\]  
The energy cost for receiving the \(B_{ld}\) bits packet of final cooperative decision broadcast by the \(CH_{j}\) after performing a decision is mainly determined by the number of bits in the packet and energy consumed for running the radio electronics circuitry. Therefore, the energy consumption for receiving \(B_{ld}\)-bits of broadcasted packet can be expressed as:  
\[\mathrm{E}_{rx}(B_{ld})=\ B_{ld}E_{ec} \tag{14}\]  
Therefore, the energy consumed by member node \(MN_{i}\) is the energy consumption for cooperative channel sensing which comprises the energy consumption for sensing the set of channels, energy consumption for reporting local decisions and energy consumption for receiving the final cooperative decision which is given as:  
\[E_{MN}^{cs}=\frac{\mathrm{E}_{cs}(n_{z},T_{cs})}{\underset{Sensing}{ \mathrm{{channels}}}}+\frac{\mathrm{E}_{rp}\big{(}B_{ld},d_{i,j}\big{)}}{ \underset{Recision}{\mathrm{{reporting}}}}+\frac{\mathrm{E}_{rx}(B_{cd})}{ \underset{Receiving}{\mathrm{{copy.Decision}}}} \tag{15}\]  
Each \(CH_{j}\) performs data fusion upon receiving MNs' local decisions and then broadcasts the final cooperative decision.