# Hard Fusion Based Spectrum Sensing over Mobile Fading Channels in Cognitive Vehicular Networks
## 4 Cooperative Sensing with Counting Rule
### Global Probability of False Alarm and Misdetection  
Assume that the reporting channels do not interfere with each other, and the delay is negligible. Once the fusion center decodes \(\{z_{i}=g_{i}u_{i}+w_{i};i=1,2,\cdots,N\}\) and gets \(\{v_{i};i=1,2,\cdots,N\}\), a global test statistic based on the counting rule is calculated linearly as follows:  
\[\Lambda=\sum_{i=1}^{N}v_{i}. \tag{36}\]And, the decision rule at the FC is represented as  
\[v_{0}=\left\{ \begin{array}{lcl}1&&if&\Lambda\geq T\\ 0&&if&\Lambda<T\end{array} \right.. \tag{37}\]  
where \(T\) is the global decision threshold of counting rule, which can be in the form of \(T=\lceil\alpha N\rceil\) (\(0<\alpha\leq 1\)). This means that \(T\) or more SVUs decide the hypothesis \(\mathcal{H}_{1}\), then the global decision is \(\mathcal{H}_{1}\). From the preceding analysis, we can see that \(v_{i}\) is a stochastic variable from a Bernoulli distribution with its winning probability \(P_{Fi}\left(P_{Di}\right)\) under the hypothesis \(\mathcal{H}_{0}\left(\mathcal{H}_{1}\right)\). It is noteworthy that the pairs \(\left(P_{Fi},P_{Di}\right)\) of different SVUs are not really the same because the pairs \(\left(P_{fi},P_{di}\right)\) or \(\gamma_{i}\) are different. So \(\left\{v_{i},i=1,2,\cdots,N\right\}\) is a set of independent and non-identically distributed random variables. In consequence, their sum \(\Lambda=\sum\limits_{i=1}^{N}v_{i}\) may be not conformed to a Binomial distribution. This makes it difficult to derive the exact distribution of \(\Lambda\). For this reason, instead of relying on the exact distribution of \(\Lambda\), we exploit a computationally inexpensive gaussian approximation for the sum of independent but not identically distributed random variables, known as the Lindberg-Feller CLT [28]. Theorem 1: _Lindberg-Feller Central Limit Theorem (LF-CLT)_  
_Assume that \(\left\{X_{i},i=1,2,\cdots,N\right\}\) is a set of independently and non-identically distributed random variables with mean \(\mathrm{E}[X_{i}]=\mu_{i}\) and variance \(\mathrm{D}[X_{i}]=\delta_{i}^{2}\). Further, assume that the two following regularity conditions are satisfied_  
\[\mathrm{D}[X_{i}]>\beta_{1}, \tag{38}\]  
_and_  
\[\mathrm{E}\left[\left|X_{i}-\mathrm{E}[X_{i}]\right|^{3}\right]<\beta_{2}, \tag{39}\]  
_where \(\beta_{1}\) and \(\beta_{2}\) are two positive values. Then, for sufficiently big \(N\), \(\sum\limits_{i=1}^{N}X_{i}\) converges asymptotically to a Gaussian distribution characterized by_  
\[\sum\limits_{i=1}^{N}X_{i}\rightarrow\mathcal{N}\left( \begin{array}{cc}\sum \limits_{i=1}^{N}\mu_{i},&\sum\limits_{i=1}^{N}\delta_{i}^{2}\end{array}  \right). \tag{40}\]  
For the applicability of LF-CLT, we will show how the above-mentioned Lindberg-Feller conditions are satisfied in the Appendix A. Consequently, in a large cognitive vehicular network, the LF-CLT can be used to approximately describe the distribution of the global statistic (\(\Lambda\)) under both hypotheses \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\). Because \(\left\{v_{i},i=1,2,\cdots,N\right\}\) are all independent of each other, for a large number of SVUs, \(N\), on the basis of the LF-CLT [28], \(\Lambda\) is asymptotically Gaussian distributed with mean  
\[\mu=\mathrm{E}\left[\Lambda\right]=\sum\limits_{i=1}^{N}\mathrm{E}\left[v_{i} \right]=\left\{ \begin{array}{cc}\sum\limits_{i=1}^{N}P_{Fi}&\mathcal{H}_{0} \\ \sum\limits_{i=1}^{N}P_{Di}&\mathcal{H}_{1}\end{array} \right., \tag{41}\]  
and variance  
\[\sigma^{2}=\mathrm{D}\left[\Lambda\right]=\sum\limits_{i=1}^{N}\mathrm{D} \left[v_{i}\right]=\left\{ \begin{array}{cc}\sum\limits_{i=1}^{N}P_{Fi}(1-P_ {Fi})&\mathcal{H}_{0}\\ \sum\limits_{i=1}^{N}P_{Di}(1-P_{Di})&\mathcal{H}_{1}\end{array}