## 2 System Model
### Cooperative Channels Sensing  
Spectrum sensing is a key function of cognitive radio for determining licensed channel occupancy; this is done by detecting the existence of PUs in the channels. Energy detection technique has been commonly employed to detect the existence of PUs' signals in the spectrum bands by measuring energy of the received signal waveform over a specified observation time. The received signal is first filtered by a Band Pass Filter (BPF) to limit the noise bandwidth. The filtered output signal of bandwidth \(B_{w}\) is converted to discrete samples \(N_{o}\) by an Analogue-to-Digital Converter (ADC) and then passed through an integrator for an observation interval \(\tau\). The final average energy of the observed samples \(N_{o}\) from the output of the integrator \(Y=\sum_{\tau=1}^{N_{o}}\lvert y(\tau)\rvert^{2}\) is compared with a threshold \(\lambda\) to determine the existence or otherwise of a PU signal [4]. If \(Y<\lambda\), then a PU's signal is absent and the channel is considered available, otherwise, a PU's signal is considered to be present and the channel is being occupied. Thus, the received signal at the SU \(Y\) can be expressed as [4]:  
\[Y=\left\{ \begin{matrix}x(\tau),H_{o}\\ z(\tau)+x(\tau),H_{1}\end{matrix} \right. \tag{3}\]  
where \(x(\tau)\) denotes zero-mean Additive White Gaussian Noise (AWGN), \(z(\tau)\) denotes the received signal waveform. \(H_{o}\) denotes the null hypothesis which indicates the absence of a PU's signal, while \(H_{1}\) denotes a hypothesis which indicates the presence of a PU's signal. Thus, the test statistics \(Y\) from the output of the integrator follow a chi-square distribution and can be approximated to a Gaussian distribution using central limit theorem, which when the number of samples is large given as [29]:  
\[Y\approx\left\{ \begin{matrix}\mathcal{N}(N_{o}\sigma_{n}^{2},2n\sigma_{n}^{4} ),H_{0}\\ \mathcal{N}(N_{o}(\sigma_{n}^{2}+\sigma_{z}^{2}),2N_{o}(\sigma_{n}^{2}+\sigma_ {z}^{2})^{2}),\;H_{1}\end{matrix} \right. \tag{4}\]  
where \(\sigma_{n}^{2}\) denotes received the noise signal's variance and \(\sigma_{z}^{2}\) denotes the received signal's variance \(z(\tau)\). Optimal PU detection can be achieved through a Maximum _A Posteriori_ (MAP) scheme which models the PU behaviors into On and Off states using a two-state Markov chain that has been widely adopted [35; 36; 37]. Therefore, the probability of detection \(P_{d}\) which suggests the presence of a PU in the considered channel can be expressed as:  
Figure 3: Illustration of pairwise and groupwise constraints. \[P_{d}(\lambda)=P_{r}[Y>\lambda\backslash H_{1}]P_{on}\] \[=\mathbb{Q}\left(\frac{\lambda-2B_{w}T(\sigma_{n}^{2}+\sigma_{x}^{ 2})}{\sqrt{4B_{w}T(\sigma_{n}^{2}+\sigma_{x}^{2})^{2}}}\right)\cdot P_{on} \tag{5}\]  
Similarly, the probability of false alarm \(P_{f}\) which falsely indicates the presence of a PU's signal in the considered channel can be expressed as:  
\[P_{f}(\lambda)=P_{r}[Y>\lambda\backslash H_{0}]P_{off}\] \[=\mathbb{Q}\left(\frac{\lambda-2B_{w}T\sigma_{n}^{2}}{\sqrt{4B_{ w}T\sigma_{n}^{4}}}\right)\cdot P_{off} \tag{6}\]  
where \(\mathbb{Q}(. )\) is the generalized Marcum Q-function, \(P_{on}\) and \(P_{off}\) are probabilities that the channel is in busy or idle states respectively. Cooperative spectrum sensing enhances the PU detection through exploitation of SUs' observed signals spatial diversity. Each of the \(MN_{l}\) senses set of channels \(S_{ch}=\{Ch_{\mathrm{z}}|\mathrm{z}=1,2,3,...,n_{\mathrm{z}}\}\), makes a local decision \(LD_{i}=\{D_{\mathrm{z}}|\mathrm{z}=1,2,3,...,n_{\mathrm{z}}\}\) on the existence of PUs in the channels or otherwise and then reports its sensing result to the cluster head \(CH_{j}\) for decision fusion and final cooperative decision \(CD_{\mathrm{j}}=\{D_{\mathrm{z}}|\mathrm{z}=1,2,3,...,n_{\mathrm{z}}\}\).