# Optimized Gateway Placement for Interference Cancellation in Transmit-Only LPWA Networks
## 5 Numerical and Simulation Results
### Simple Scenario  
To test the abilities of the algorithms introduced above, we start from the simple scenario of three SNs on the corner of isosceles triangle and one GW. In this and the following sections, we assume the capture threshold, \(\tau=3\) dB, which is a system parameter, depended upon the sensitivity of the radio receiver. The path loss exponent \(n=3.2\). In Section 2.2, we introduce the interference cancellation process by assuming that the cancellation of a signal with received power \(P\), leaves a residual interference power of \(z*P\), where \(0<z<1\), is called the residual power factor, and \(z=0.1\). In WBG algorithm, if an OP is inside of the capture circle and inside of ICD circle, set an edge with weight value of \(\alpha\); if an OP is inside of the capture region and outside of ICD circle, i.e., it is in the crescent, the edge is set by weight value of \(\beta\). Otherwise, in PGL algorithm, if the pixel locates in the ICD circle, that is, outside of the decoding crescent but inside of the capture circle, add \(\alpha\) to the gray level of this pixel; if the pixel locates in the decoding crescent, add \(\beta\) to the gray level of this pixel. In the two algorithms, we take \(\alpha=1\) and \(\beta=3\). It is obvious that these values satisfy \(z<\frac{1}{\tau^{2}}\), which means there exists intersection regions between the capture and ICD regions in WBG algorithm.  
(1) Optimized GWs Placement  
By algorithm of WBG, we can get an optimized placement of GW at the red-filled circle, labeled 1\({}^{\sharp}\) GW in Figure 10b, which is a magnified view of the center of Figure 10a. Figure 10c shows the coordinates of all SNs and GW positions. The 1\({}^{\sharp}\) GW is exactly in the region overlapped by the maximum number of crescents, where the average contentions for all of SNs is the least, which means it can decode the most packets in two-way collision. Figure 10d-f presents the contention of each SN by three cases of the GW placement, respectively. We can see in Figure 10d that contention of each SN is zero after capture and IC by the GW, which means that any packet transmitted by any of the three SNs could be decoded successfully in two-way collision. At the same time, it is obvious that the number of contention with capture only shown by the red line with circles in Figure 10d is more than that achieved by WBG algorithm.  
(2) Placement GWs Manually  
To certify the results of the WBG algorithm, we place the GW location manually in the capture region but outside of the decoding crescent, as shown by 2\({}^{\sharp}\) GW in Figure 10a,b. In Figure 10e, 1\({}^{\sharp}\) SN's contention is 2, which means the packet transmitted by it will be lost in any two-way collision. However, a packet transmitted by No. 3 could be decoded in any two-way collision with contention 0. Furthermore, it has to be noticed that the lines with capture only and capture with IC coincide because the GW is outside of all IC regions, but in one capture circle.  
The 3\({}^{\sharp}\) GW in Figure 10a,b, the blue-filled circle in the very center, is outside of all capture circles. Thus, the contention of each SN is 2, as shown in Figure 10f, i.e., none of packets transmitted by any SN can be decoded successfully in a two-way collision, whether by only capture or capture with IC.  
Figure 10: Contentions of each SN when a GW is placed at different locations. 1\({}^{\sharp}\) GW is at the location by WBG placement, 2\({}^{\sharp}\) GW is at the location of capture circle only and 3\({}^{\sharp}\) GW is at location of out of any crescentm as shown in (**a**,**b**) by zooming in. (**c**) The coordinate location of SNs and GWs in different strategy. (**d**–**f**) The contention of each SN with three cases of the GW placement, respectively.