# A High-Resolution Ultrasonic Ranging System Using Laser Sensing and a Cross-Correlation Method
## 5 Results and Discussion  
### TOF by the Cross-Correlation Method  
The reference signal is the signal that is transmitted by the transducer and first passes through the laser beam. The echo wave is the wave that is reflected from the target. The cross-correlation coefficient is calculated at each point of the ultrasound signal. The laser sensing system makes it possible to record both the reference signal and echoes without disturbing the ultrasound. Please note that they are all recordings of the same ultrasound pulse at different times. Although the ultrasound propagates in the opposite direction after it is reflected, the signal series recorded by the laser do not need to be reversed, as shown in Figure 6.  
Figure 6: Ultrasound excited by sinusoidal burst signal: (**a**) 5-cycle 1 MHz burst signal; (**b**) ultrasound signal detected by laser sensing system; (**c.d**) zooms of the signal.  
The reference signal is Ref(\(t\)), and the echoes are \(S(t)\) as follows:  
\[S(t)=\sum a_{i}\text{Ref}(t-\tau_{i}), \tag{10}\]  
where \(i\) is the \(i\)th echo, \(a_{i}\) is the reflection coefficient and \(\tau_{i}\) is the time delay of the echo to the reference signal. Usually, we consider the first echo as the signal reflected from the target. The cross-correlation coefficient \(C(\tau)\) is calculated by Equation 8, and the result is shown in Figure 7b. The envelope of the cross-correlation coefficient is calculated by the Hilbert transform. The first peak of the envelope refers to the moment that the ultrasound propagates through the laser light, which is the starting point of TOF. The TOF is defined as the duration \(t\) between peaks. The distance is the distance between the reference plane and the reflected plane, as shown in Figure 3. Although the TOF can be calculated by the envelope of the ultrasound signal, which is shown in Figure 7c, the signal distortion would cause error and distortion. Compared to Figure 7c, the envelope of the cross-correlation coefficient shows better smoothness and symmetry. With a stable and clean reference signal, an accurate TOF can be calculated according to the cross-correlation method. A plane wave is emitted by the transducer. At each test point, the ultrasound can be seen as propagating along a straight line.  
Figure 7: (**a**) The reference signal and echo wave in the ultrasound signal; (**b**) the cross-correlation coefficient and envelope of the ultrasound; (**c**) zooms of (**a**); (**d**) zooms of (**b**).  
There are some factors that impact on the performance of the proposed method. First, the sampling rate determines the time resolution of the cross-correlation method. However, a high sampling rate leads to expensive equipment and large data sizes. In this experiment, the sampling rate is 1 GHz, and we interpolate it to 2.5 GHz using linear interpolation. Interpolation is commonly used in cross-correlation calculations to improve the measurement resolution, without affecting the accuracy [43; 44]. The time resolution is 0.4 ns, in theory. Second, stochastic noise is the main noise in the proposed method. Two ways to reduce stochastic noise in the experiment are bandwidth limitation and average sampling. The central frequency of the transducer is 1 MHz. As such, the limiting frequency of the filter is set at 20 MHz. Eight average times are suitable in this experiment, considering both the SNR and time consumption. In addition, pulse repetition frequency (PRF) here is 200 Hz. On one hand, the PRF determines the maximum ranging distance, which is 3.75 m when the sound velocity is 1500 m/s. On the other hand, at each measurement point, the detection signal is the mean of eight sampling results. Thus, the PRF should not be too small.  
Although the positions of the transducer and laser were fixed, temperature variation, vibration, and electrical noise would cause a shift in the signal. The repeatability of the TOF measurements in 50 continuous tests at distances of 30 cm and 80 cm is shown in Figure 8. Both were less than 4 ns at a distance up to 80 cm.