## 4 Analysis and Comparison
### Measure Metrics
#### 4.1.2 Decision Rule  
Another way to improve \(P_{d}\) is to set up a proper decision rule. The simple method is to compare \(m_{n}\) and \(T_{h}\). A complex one can be designed based on the characteristics of the response of each algorithm. Here are two examples of decision rule design.  
Auto-correlation. According to Figure 10, \(m_{n}\) is large throughout the short preamble period. Instead of using only a single point, multiple points can be used together for packet  
Figure 14: Misdetection rate for different thresholds.  
Figure 13: False alarm rate for different thresholds.  
detection. The engineers modified the simple method by setting two judgment conditions in MATLAB sample code. The schematic diagram is shown in Figure 15. First, it quantifies the \(m_{n}\) by the threshold. \(m_{n}\) is set to 1 when it exceeds \(Th\); otherwise, it is 0. Then, it is determined whether the number of 1 in the window is greater than 1.5 times the length of a symbol in STF to judge if there are enough points greater than \(Th\). Second, it is determined whether the distance between the first point and the other points is less than 3 times the length of a symbol in STF. The packet is considered detected only when both conditions are met. The comparison between the simple decision rule and the complex one is shown in Figure 16. The complex decision rule achieves a higher detection rate because it reduces \(P_{f}\) well, and the set conditions can be mostly satisfied when the packet arrives, so it will not cause \(P_{m}\) to increase too much.  
Cross-correlation. According to Figure 12, there are 10 peak points in the response of the cross-correlation algorithm. Moreover, the distances between these points are equal, which is the length of one short symbol. According to [15], the authors choose to check all the 10 peaks for detection, as shown in Figure 17. When the first point exceeds the threshold, they record the index of this point. Then, when the next point is detected, they judge whether the distance between this point and the previous point is exactly the length of a symbol. If checked, they continue to do so until all 10 points are detected. Otherwise, it is considered as no packet. Figure 18 shows the comparison result, which shows that the simple rule can achieve a better detection rate. It seems to conflict with the previous result. The reason is that the complex approach is too restrictive and results in a high \(P_{m}\). Thus, in this case, the simple rule is more suitable than a complex one. Appropriate relaxation of restrictions can have a better performance, which will be studied further in the future work.  
Figure 16: Detection rate comparison of different decision rules for auto-correlation.  
Figure 15: Complex decision rule for auto-correlation.  
Merely comparing the \(m_{n}\) and the \(T_{h}\) is the simplest rule, which may increase \(P_{f}\) because of the large singularity. However, that does not mean that a more complex decision rule is necessarily good, since it will lead to an increase in \(P_{m}\). Thus, a suitable decision rule is necessary to achieve high \(P_{d}\).