## 3 Experiments and Results
### Static Target Ranging  
The retroreflector, which served as the target for measurement, was securely mounted on the slider of the electric guide rail. Initially, the retroreflector was positioned at the furthest point from the collimator, and then it was sequentially shifted 25 mm closer, resulting in a total of 40 steps and 41 distinct positions. In total, 634 distance measurements were recorded at each position. Figure 6a illustrates the differences between the mean values of the distance measurements at adjacent positions. For the proposed method, the average value over 40 steps was \(-25.00\) mm, with a maximum ranging deviation of \(-0.56\) mm, corresponding to a delay deviation of \(-1.87\) ps. In contrast, the average value over 40 steps for the traditional method was \(-26.55\) mm, with a maximum deviation of \(-51.05\) mm. This discrepancy arose primarily from the mismatch between the theoretical phase discrimination model and the ranging system within the phase discrimination range of \((-T_{\text{C}}/2,\ T_{\text{C}}/2)\). Generally, the mismatch between the phase discrimination model and the actual system can be reduced by increasing the bandwidth of the receiver. However, this increase in bandwidth also raises the noise level, which, in turn, diminishes the signal-to-noise ratio. Figure 6b presents a histogram of the distance measurements. It indicates that the results obtained from the traditional method were dispersed, and those from the proposed method exhibited a higher degree of concentration. The ranging accuracy of the proposed method was higher than that of the traditional method by a factor of approximately 91. This improvement in ranging accuracy was primarily attributed to the fact that the characteristic parameters of the phase discriminative model in the proposed method matched the actual band-limited system. In contrast, the theoretical phase discriminative model of the traditional method was matched to the ideal ranging system, which introduced ranging deviations when used in the actual system. Figure 5: Z values and linear fitting. Figure 6: Measurement results of 40 steps of 25 mm. (**a**) Ranging accuracy. (**b**) Histogram of ranging results. Figure 7 shows the standard deviations of the distance measurements at the 41 positions, indicating that the ranging precisions of the two schemes were comparable. The maximum standard deviation (1\(\sigma\)) of the proposed method was 0.34 mm, which corresponded to a time-delay precision of 1.13 ps. The ranging precision (1\(\sigma\)) of the traditional method was 0.38 mm. Figure 8 shows the cumulative distance measurement results for the 40 steps. Because of the limited length of the guide rail, the range of motion for the retroreflector was restricted to 1000 mm. Figure 8a shows the theoretical distance set by the control terminal of the guide rail, along with the mean values of the distances measured using both the traditional and proposed methods. Figure 8b shows the deviations between the mean measurements of the distances and the theoretical distances. An analysis of Figure 8 revealed that the traditional ranging method exhibited a maximum measurement deviation of 62.95 mm and a minimum deviation of 0.41 mm. Furthermore, the measurement deviation exhibited a periodic pattern with a period of 600 mm, which corresponded to the width of the chip of 4 ns. In contrast, the proposed method exhibited an increasing trend in the measurement deviations within the 0-500 mm range but stabilized in the 500-1000 mm range, with a minimum deviation of 0.00 mm and a maximum deviation of 1.47 mm. Repeated ranging was conducted on static targets to verify the repetitive measurement accuracy of the proposed method. The measurements were repeated three times. Figure 9a shows the differences between the mean values of the measurements and the theoretical distances traveled for each position, with the three sets of results demonstrating a consistent trend.