## 3 Novel Framework for Wideband Cooperative Spectrum Sensing  
### Analysis of SU Spectral Efficiency for Spectrum Sensing  
Due to the need for periodic spectrum sensing (PSS), a typical frame structure of a CR network will at least consist of a sensing slot and a data transmission slot, as depicted in Figure 3.  
It is assumed that \(\tau\) is used for implementing compressed spectrum sensing, and \(T-\tau\) is reserved for transmission. In PSS, the \(i\)-th SU data transmission occurs in following two cases: correct detection absence of PU and missing detection presence of PU. When the former occurs, the amount of transmission data is \(\mathcal{D}_{i,1}=P_{\mathcal{H}_{0}}(1-p_{f,i})(T-\tau)r_{0}\), where \(P_{\mathcal{H}_{0}}\) denotes the prior probability of the absence of PU and \(r_{0}\) means transmission rate in the absence of PU, \(r_{0}=Wlog_{2}\big{(}1+g^{2}\sigma_{s}^{2}/\sigma_{n}^{2}\big{)}\), \(W\) is the available channel bandwidth, \(log_{2}\) means binary logarithm, and \(g\) is channel gain between SU. In the case of missing detection of PU, the amount of transmission data is \(\mathcal{D}_{i,2}=P_{\mathcal{H}_{1}}(1-p_{d,i})(T-\tau)r_{1}\), where \(P_{\mathcal{H}_{1}}\) denotes the prior probability of the presence of PU and \(r_{1}\) means transmission rate in the presence of PU, \(r_{1}=Wlog_{2}\big{(}1+g^{2}\sigma_{s}^{2}/\sigma_{n}^{2}++h^{2}\sigma_{s}^{2} \big{)}\). Then, the transmission spectral efficiency at \(i\)-th SU is derived as:  
\[\mathcal{C}=\frac{1}{WT}\Big{[}P_{\mathcal{H}_{0}}(1-p_{f,i})(T-\tau)r_{0}+P_ {\mathcal{H}_{1}}(1-p_{d,i})(T-\tau)r_{1}\Big{]} \tag{20}\]  
As to be highlighted, \(\mathcal{C}\) is relied on the \(p_{f,i}\) and \(p_{d,i}\). It is equivalent to say that \(\mathcal{C}\) is dependent on \"\(I\)\", which is applicable for all any SU. Obviously, for a given frame length \(T\), the longer the sensing time \(\tau\), the shorter the data transmission time \(T-\tau\). Since \(Q(x)\) is monotonically decreasing in \(x\), for a target \(p_{d,i}^{th}\) at SU, increasing the sensing time results in a lower false alarm probability. Hence, sensing time \(\tau\) plays a crucial role in SU transmission spectral efficiency. Our fundamental purpose is to design  
Figure 3: Frame structure of periodic spectrum sensing. the optimal sensing time \(\tau\) to maximize the SU transmission spectral efficiency under the condition of sufficient protection to PU. Mathematically, the optimization problem can be described as:  
\[ \begin{array}{c}\max\limits_{\tau}:\mathcal{C}\\ s.t.,\,p_{d,i}\geq p_{d,i}^{th}\end{array}  \tag{21}\]  
where \(p_{d,i}^{th}\) is target detection probability with which PU is defined as being adequately protected. **Theorem 2. **\(\mathcal{C}\) _is a decreasing function of \(p_{d}^{i}\)._  
**Proof**. For a given \(\tau\), we have:  
\[\frac{d\mathcal{C}}{dp_{d,i}}=-\frac{1}{WT}\left(P_{\mathcal{H}_{0}}(T-\tau)r _{0}\frac{dp_{f,i}}{dp_{d,i}}+P_{\mathcal{H}_{1}}(T-\tau)r_{1}\right) \tag{22}\]  
\[\frac{dp_{f,i}}{dp_{d,i}}=\frac{dp_{f,i}}{d\lambda_{i}}\left/\frac{dp_{d,i}}{d \lambda_{i}}=\exp\left(-\frac{2\lambda_{i}-\left\|\Phi hs\right\|_{2}}{2\sigma _{n}^{2}}\right)>0\right. \tag{23}\]  
Obviously, \(\frac{d\mathcal{C}}{dp_{d,i}}<0\), \(\mathcal{C}\) is a decreasing function of \(p_{d,i}\). Thus, Theorem 2 is proven. When choosing the equality constraint \(p_{d,i}=p_{d,i}^{th}\), the optimal spectral efficiency can be achieved with compression ratio \(\mu_{i}\) and sensing SNR \(\gamma_{i}\) at the \(i\)-th SU. Then, we have:  
\[\mathcal{C}=\frac{(T-\tau)r_{0}}{WT}P_{\mathcal{H}_{0}}\Big{\{}1-Q\Big{(}Q^{ -1}(p_{d,i}^{th})+\sqrt{\mu_{i}\gamma_{i}}\Big{)}\Big{\}}+\frac{(T-\tau)r_{1} }{WT}P_{\mathcal{H}_{1}}(1-p_{d,i}^{th})\square \tag{24}\]  
**Theorem 3. **_For any target detection probability \(p_{d,i}^{th}\), there exists one optimal sensing time \(\tau_{opt}\) that yields the maximal SU spectral efficiency_. **Proof**.