3.2.4. Quantization

Another limitation of the sensors we used was the level of quantization of the recorded values. Especially, the coarse resolution of time was a limiting factor when using our radar detectors, as they were only able to record timestamps up to a resolution of one second. This led to the necessity of examining the influence of the quantization on the results using our synthetic dataset. We conducted this experiment by again using 200 vehicles and three sensor distance values of 50 m, 100 m, and 150 m, as we expected that the distance could change the outcome. We neglected all previously-discussed errors by setting them to zero mean and variation. We applied a quantization to the time values of both sensors. We implemented the quantization by means of a factor value iterating from 1–10, which defined the partition of a second to which the recorded values were rounded. With a factor of 10, the values were rounded to a tenth of a second, while with a factor of one, the values were discretized to seconds just as the case in our real-world measurement.

**Figure 11.** Influence of quantization on the offset estimation (**a**) and on the trajectory RMS error (**b**). Continuous lines show the mean values, while dashed lines show the standard deviation of the error.

As we can see from Figure 11, the quantization of the values did have a significant effect on both the sensor offset estimation and the trajectory reconstruction. Note that the lower values of the fraction of a second corresponded to a coarser quantization. As one would expect, the lower values not only led to a higher mean offset estimation error, but also to a higher standard deviation of this value. There was also some change due to the used distance, as a very coarse quantization led to higher errors when low distances were used. This can be explained by the fraction of the quantization time out of the complete passing time of the vehicles. If the quantization of the time values was in the magnitude of a second, then the slopes computed in the EM-algorithm were altered much more when short time ranges were used, thus leading to higher resulting errors. When we examine the outcome of the RMS trajectory error, we can clearly see that the errors in the trajectory reconstruction were a lot higher than in the offset estimation. This again can be explained by the compensation effect of using many different vehicles to estimate the sensor offset, while the errors of individual vehicles remained quite high. We can also see that the sensor distance did not affect the trajectory error. The other metrics, like the number of iterations and the matching sensitivity, were largely unaffected by quantization.
