# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 7 Comparison with the Case of Myopic SUs  
In this section, we consider the detection problem in a network of myopic SUs, which are users that are capable of covering only one source at a time. As a consequence, the complexity is reduced; however, at the price of a considerable performance loss, as we will demonstrate. For this case, we formulate the problem and show through simulation results that it performs poorly in the case of intensive connectivity.  
In a relatively dense environment, each SU receives a signal from several sources. A naive approach to detecting whether one source emits is to find the nearest SU to the specific source and rely only on this user for the detection process.  
The SU uses a simple binary Neyman-Pearson hypothesis testing based on its (aggregated) received signal strength. If we denote SU \(y_{i}\) to be the nearest user to source \(s_{\mu}\), then the observations under the two different hypotheses are given by  
\[y_{i}=\begin{cases}\text{G}_{i\mu}+\sum\limits_{v\in\mathcal{N}(i)\backslash \mu}\text{G}_{i\mu}s_{v}+\eta_{i},&H_{1}\\ -\text{G}_{i\mu}+\sum\limits_{v\in\mathcal{N}(i)\backslash\mu}\text{G}_{i\mu} s_{v}+\eta_{i},&H_{0}\end{cases} \tag{39}\]  
Figure 7: Error detection rate as a function of sources’ \(T\) and SUs’ \(R\) factors for fixed \(\texttt{SNR}=0\) dB. The network consists of 10 sources and 10 SUs, and the SUs’ and sources’ densities range from 0.1 to 1. (**a**) The error detection rate \(\mathbb{P}_{\text{err}}\) derived from the population dynamics algorithm according to (37), and (**b**) the total error detection rate \(\mathbb{P}_{\text{tot}}\), which includes the probability of a node to be isolated. We have assumed that \(r_{c}=0.5\) and the environmental factor \(\gamma=4\). The regions (between contours) with the same colors have the same error detection rate (in dB). As expected, since \(R\) and \(T\) are proportional to densities \(n_{s}\) and \(n_{t}\), the total error detection rate becomes higher when the number of SUs placed within the area of an emitting source is small.  
where \(H_{1}\) denotes the emitting hypothesis (\(s_{\mu}=+1\)) and \(H_{0}\) is the null hypothesis (\(s_{\mu}=-1\)). Therefore, the likelihood under hypothesis \(H_{1}\) is  
\[\mathbb{P}(y_{i}|H_{1})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\Biggl{[}-\frac{ 1}{2\sigma_{i}^{2}}\bigl{(}y_{i}-\mathrm{G}_{i\mu}\bigr{)}^{2}\Biggr{]} \tag{40}\]  
and, respectively, under hypothesis \(H_{0}\) is  
\[\mathbb{P}(y_{i}|H_{0})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\Biggl{[}-\frac {1}{2\sigma_{i}^{2}}\bigl{(}y_{i}+\mathrm{G}_{i\mu}\bigr{)}^{2}\Biggr{]} \tag{41}\]  
The logarithmic likelihood ratio test is then given by  
\[\mathcal{L}(y_{i})=\log\biggl{(}\frac{\mathbb{P}(y_{i}|H_{1})}{\mathbb{P}(y_{ i}|H_{0})}\biggr{)}\overset{H_{1}}{\underset{H_{0}}{\overset{H_{1}}{\rightarrow}}}\tau \tag{42}\]  
where the SU decides, according to a threshold \(\tau\). Different criteria can be used to find the optimal threshold, but we do not deal with this topic in this paper.  
The above-mentioned analysis shows that the multi-source detection task based on the measurements of the energy intensity field is a very hard problem since each measurement is composed of information from all individual sources.  
Due to the nonlinearity of the energy intensity with respect to the distance, it is very difficult to decode each individual component [56]. In Figure 5b, a comparison of the aforementioned analysis for the myopic case detection and the population dynamics algorithm is depicted. It can be seen that the population dynamics algorithm and the message-passing algorithm (since their behavior is very similar) are superior to the simple scenario of the myopic SU detection of a single source.