# Hard Fusion Based Spectrum Sensing over Mobile Fading Channels in Cognitive Vehicular Networks
## 3 Local Sensing with Energy Detection  
The spectrum sensing techniques often employed for local sensing are energy detection, cyclostationary detection and matched filter. The matched filter (also referred to as coherent detector) is widely regarded as the optimum approach but it relies heavily on the accurate _priori_ knowledge about the PU signal, which is hard to be obtained. Cyclostationary detection can distinguish the PU signal from noise at very low signal-to-noise ratio (SNR) but still needs some _priori_ knowledge about the PU signal. Energy detection [24; 25] is the most practical method because it merely estimates the signal energy on the considered band and produces good performance without any _priori_ information about the PU signal, as illustrated in Figure 2. Furthermore, energy detection is a viable choice for vehicular networks on account of its high mobile environment and low latency tolerance. When digital energy detector is adopted, the corresponding energy statistic at the \(i\)-th SVU, denoted by \(e_{i}\), is expressed as  
\[e_{i}=\sum_{k=0}^{L-1}|x_{i}(k)|^{2}. \tag{4}\]  
Figure 2: Digital energy detector model. where \(L\) corresponds to the number of samples within a sensing interval. Let \(\lambda\) denote the decision threshold for energy detector and \(u_{i}\) denote the local decision, the decision rule at each SVU is represented as  
\[u_{i}=\left\{ \begin{array}{l l l}+1&\quad\quad if\quad e_{i}\geq\lambda\\ -1&\quad\quad if\quad e_{i}<\lambda\end{array} \right.. \tag{5}\]  
This means that when \(e_{i}\geq\lambda\), SVU \(i\) makes its individual decision \(u_{i}=+1\) which tells the PU signal is detected (\(\mathcal{H}_{1}\)); otherwise, it makes a decision \(u_{i}=-1\) which declares that the PU signal is not detected (\(\mathcal{H}_{0}\)). When \(L\) is sufficiently large, based on the Central Limit Theorem (CLT), the energy statistic \(e_{i}\) in (4) can be described by a Gaussian distribution under both hypotheses \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\)[24]. Let \(E_{s}\ =\sum_{k=0}^{L-1}\left|s(k)\right|^{2}\) denote the transmitted signal energy by the PU. The corresponding mean and variance are given by  
\[\left\{ \begin{array}{l l l}\mathcal{H}_{0}&\quad\quad\mu_{0}=L\sigma_{n}^{2 }&\quad\quad\sigma_{0}^{2}=L\sigma_{n}^{4}\\ \mathcal{H}_{1}&\quad\mu_{1}=\left(L+\eta_{i}\right)\sigma_{n}^{2}&\quad \quad\sigma_{1}^{2}=\left(L+2\eta_{i}\right)\sigma_{n}^{4}\end{array} \right., \tag{6}\]  
where  
\[\eta_{i}=\frac{1}{L}\sum_{k=0}^{L-1}\left|h_{i}(k)\right|^{2}\frac{E_{s}}{ \sigma_{n}^{2}}=\frac{1}{L}\sum_{k=0}^{L-1}\left|h_{i}(k)\right|^{2}\eta_{s}, \tag{7}\]  
In the above equation, the term \(\eta_{i}\) denotes the instantaneous SNR experienced by the \(i\)-th SVU. It is straightforward to see that \(\eta_{i}\) is very different from (sensing) period to period, since it relies heavily on the fading characteristics of the wireless channel. In addition, the term \(\eta_{s}=\frac{E_{s}}{\sigma_{n}^{2}}\) denotes the local average SNR. To be emphasized, the SNR (\(\eta_{s}\)) is \(L\) times more than the average SNR measured at the energy detector output, which can be expressed as \(\eta_{0}=\frac{E_{s}}{L\sigma_{n}^{2}}\)[26]. Subsequently, the probabilities of false alarm, detection and miss detection for a given threshold (\(\lambda\)) at the \(i\)-th SVU can be deduced as follows, respectively  
\[P_{fi}(\lambda)=P\left(e_{i}\geq\lambda\left|\mathcal{H}_{0}\right.\right)= \mathcal{Q}\left(\frac{\lambda-L\sigma_{n}^{2}}{\sqrt{L}\sigma_{n}^{2}}\right), \tag{8}\]  
\[P_{di}(\lambda)=P\left(e_{i}\geq\lambda\left|\mathcal{H}_{1}\right.\right)= \mathcal{Q}\left(\frac{\lambda-(L+\eta_{i})\sigma_{n}^{2}}{\sqrt{L+2\eta_{i} \sigma_{n}^{2}}}\right), \tag{9}\]  
\[P_{mi}(\lambda)=1-P_{di}(\lambda)=\mathcal{Q}\left(\frac{(L+\eta_{i})\sigma_{ n}^{2}-\lambda}{\sqrt{L+2\eta_{i}\sigma_{n}^{2}}}\right), \tag{10}\]  
where \(\mathcal{Q}(x)\) stands for the standard \(\mathcal{Q}\)-function, i.e., \(\mathcal{Q}(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp(-\frac{t^{2}}{2})dt\).