# On Performance Analysis of Protective Jamming Schemes in Wireless Sensor Networks
## Appendix B  
Proof of Lemma 2.: We assume that the transmitter is located between \(k\)th layer and \(k+1\)th layer of the jammers. Since the jammers are distributed in a symmetric manner (around the eavesdropper), we only need to calculate the cumulative interference from one corner of the plane (e.g., the north-east corner). Summing the cumulative interference from all the four corners, we can obtain the cumulative inference from all the jammers. Firstly, we consider the interference caused from friendly jammers at the center location between 1 st layer and 2 nd layer as shown in Figure 1 (see the red dashed line). Interference caused by jammers at the \(m\)th layer, which is surrounded by the jammers at the \(k\)th layer is calculated as the following equation,  
\[ \begin{split} I_{j}(m)&=P_{j}(m)\Bigg{\{}2\sqrt{ \left(\frac{1}{2}\cdot 2d\right)^{2}+\left[\left(k-m+\frac{1}{2}\right)\cdot 2d \right]^{2}-\alpha}\\ &+2\sqrt{\left(\frac{3}{2}\cdot 2d\right)^{2}+\left[\left(k-m+ \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left[\left(m-\frac{1}{2}\right)\cdot 2d\right]^{2}+\left[ \left(k-m+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\\ &+2\sqrt{\left[\left(m-\frac{1}{2}\right)\cdot 2d\right]^{2}+\left[ \left(k-m+\frac{3}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left[\left(m-\frac{1}{2}\right)\cdot 2d\right]^{2}+\left[ \left(k+m-\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\\ &+2\sqrt{\left[\left(m-\frac{3}{2}\right)\cdot 2d\right]^{2}+ \left[\left(k+m-\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left(\frac{1}{2}\cdot 2d\right)^{2}+\left[\left(k+m- \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\Bigg{\}}.\end{split}  \tag{18}\]  
Summing all the terms in Equation (18), we then obtain the simplified expression of \(I_{j}(m)\) as follows,  
\[ \begin{split} I_{j}(m)&=2P_{j}(m)\Bigg{\{}\sum\limits_{ n=1}^{m}\left(\sqrt{\left[\left(v-\frac{1}{2}\right)\cdot 2d\right]^{2}+\left[ \left(k-m+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\right.\\ &+\sqrt{\left[\left(v-\frac{1}{2}\right)\cdot 2d\right]^{2}+ \left[\left(k+m-\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\Bigg{\}}\\ &+\left.\sum\limits_{w=k-m+1}^{k+m}\sqrt{\left[\left(m-\frac{1}{ 2}\right)\cdot 2d\right]^{2}+\left[\left(w-\frac{1}{2}\right)\cdot 2d\right]^{2}- \alpha}\Bigg{\}}.\end{split}  \tag{19}\]  
Interference caused by jammers at the \(q\)th layer, which is placed outside of \(k\)th layer is calculated by this equation,  
\[ \begin{split} I_{j}(q)&=P_{j}(q)\Bigg{\{}2\sqrt{ \left(\frac{1}{2}\cdot 2d\right)^{2}+\left[\left(q-k-\frac{1}{2}\right)\cdot 2d \right]^{2}-\alpha}\\ &+2\sqrt{\left(\frac{3}{2}\cdot 2d\right)^{2}+\left[\left(q-k- \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left(\left(q-\frac{1}{2}\right)\cdot 2d\right)^{2}+ \left[\left(q-k-\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\\ &+2\sqrt{\left(\left(q-\frac{1}{2}\right)\cdot 2d\right)^{2}+ \left[\left(q-k+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left(\left(q-\frac{1}{2}\right)\cdot 2d\right)^{2}+ \left[\left(q+k+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}+\cdots\\ &+2\sqrt{\left(\frac{1}{2}\cdot 2d\right)^{2}+\left[\left(q+k+ \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\Bigg{\}}.\end{split}  \tag{20}\]From Equation (B3) we can get the simplified expression of \(I_{j}(q)\) as follows,  
\[ \begin{split} I_{j}(q)&=2P_{j}(q)\Bigg{\{}\sum_{s=1}^{q }\Bigg{(}\sqrt{\left(\left(s-\frac{1}{2}\right)\cdot 2d\right)^{2}+\left[\left(q-k- \frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\\ &+\sqrt{\left(\left(s-\frac{1}{2}\right)\cdot 2d\right)^{2}+ \left[\left(q+k+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha}\Bigg{)}\\ &+\sum_{z=q-k-1}^{q+k}\sqrt{\left(\left(q-\frac{1}{2}\right) \cdot 2d\right)^{2}+\left[\left(z+\frac{1}{2}\right)\cdot 2d\right]^{2}-\alpha} \Bigg{\}}.\end{split} \] (B4)  
Therefore, summing Equations (B2) and (B4), we obtain the cumulative interference of jammers at the location \(t_{0}\) as the following equation,  
\[