# On Performance Analysis of Protective Jamming Schemes in Wireless Sensor Networks
## 4 Analysis on Eavesdropping Probability
### Analysis of Fri-Jam Schemes
#### 4.2.3 Case III: FJ-PC Scheme  
We next analyze the case of FJ-PC scheme, in which jammers are placed in grid as the same as FJ-Reg scheme. We then assign the emitting power of jammers according to the different layers (as shown in Figure 1). We denote the layer number by \(k\), which is ranging from 1 to \(n\). The emitting power of jammers at the same layer is assigned with the same value. Specifically, we assign the emitting power at jammers in FJ-Reg scheme according to the following rule. **Property 1**. : _We assign the emitting power of jammers at the kth layer according to the following equation:_  
\[P_{j}(k)=P_{j}\cdot\zeta^{1-k}, \tag{12}\]  
_where \(P_{j}\) is the transmitting power of the jammers at the first layer and \(\zeta\) is the power control factor._  
In FJ-PC scheme, the transmission probability of a legitimate user cannot be derived directly by using the existing approaches in [31; 32; 33; 34] since the cumulative interference from jammers in FJ-PC scheme is quite different from that in FJ-Reg scheme. In particular, we have the following lemma to calculate the average cumulative interference. **Lemma 2**. : _In FJ-PC scheme, the average cumulative interference from power controlled jammers to a legitimate transmitter is_  
\[E[I_{c}]=\frac{\sum\limits_{k=1}^{n}\left(I_{k_{J_{0}}}+I_{k_{J_{k}}}+2\sum \limits_{t=t_{1}}^{t_{k-1}}I_{k_{J_{x}}}\right)}{\sum\limits_{k=1}^{n}2k}, \tag{13}\]  
_where \(I_{k,t_{x}}\) is the interference at \(t_{x}\), which can be calculated by_  
\[I_{k,t_{x}} =\sum\limits_{m=1}^{k}\sum\limits_{v=1}^{m}\left(2P_{j}(m)\left( \sqrt{\left[\left(v-\frac{1}{2}+t_{x}\right)\cdot 2d\right]^{2}+\left[\left(k-m+ \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\right.\right. \tag{14}\] \[\quad+\left.\left.\sqrt{\left[\left(v-\frac{1}{2}+t_{x}\right) \cdot 2d\right]^{2}+\left[\left(k+m-\frac{1}{2}\right)\cdot 2d\right]^{2}- \alpha}\right)\right)\] \[\quad+\sum\limits_{m=1}^{k}\sum\limits_{v=k-m+1}^{k+m}2P_{j}(m) \sqrt{\left[\left(m-\frac{1}{2}+t_{x}\right)\cdot 2d\right]^{2}+\left[\left(w- \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\] \[\quad+\left.\left.\sqrt{\left(\left(s-\frac{1}{2}+t_{x}\right) \cdot 2d\right)^{2}+\left[\left(q+k+\frac{1}{2}\right)\cdot 2d\right]^{2}- \alpha}\right)\right)\]  
We present the proof in Appendix B. We then derive the transmission probability \(P(C)\) of a legitimate user, which is given by Theorem 4. **Theorem 4**. : _In FJ-PC scheme, the transmission probability \(P(C)\) is_  
\[P(C)=\int\limits_{r>0}e^{-\mu\beta_{p}r^{\alpha}(\sigma^{2}+E[I_{c}])-\pi r^{2 }\lambda(\rho(\beta_{p},\alpha)+1)}2\pi\lambda rdr, \tag{15}\]  
_where \(\rho(\beta_{p},\alpha)={\beta_{p}}^{-2/\alpha}\int_{\beta_{p}}^{\infty}\frac{1 }{1+\mu^{\alpha/2}}d\mu\), \(\beta_{p}=\frac{\beta}{Pt}\) and \(E[I_{c}]\) is given by Equation (13)._  
Proof. : The SINR of the receiver at a random distance \(r\) from its nearest transmitter can be expressed as \(\mathrm{SINR}=\frac{Phr^{-\alpha}}{\sigma^{2}+I_{t}+I_{c}}\), where \(I_{c}\) is the cumulative interference caused by power controlled jammers on the receiver.