# Absolute Distance Measurement Based on Self-Mixing Interferometry Using Compressed Sensing
## 3 Experimental Results  
To demonstrate the feasibility of using CS in SMI distance measurement, experiments to evaluate the difference between the traditional method and the CS method were implemented. The experimental setting used in this study was the same as the experimental equipment used in our previous study [10], which showed that when utilizing the conventional approach with interpolated FFT in the 3 to 20 cm range, a resolution of approximately 60 \(\upmu\)m could be achieved at a signal-to-noise ratio of 30 dB. Figure 2 depicts the experimental setup. A VCSEL HVS6003-001 laser diode with an integrated phototransistor was utilized in the experiment. It was modulated by a sawtooth current with amplitude of 2 mA and frequency of 50 Hz. The target was mounted on a translation stage, and the external distance between the target and the laser could be changed by adjusting the stage. The SMI signal was collected with the phototransistor and sampled using a data acquisition (DAQ) device after amplification and filtering. The sampling rate was set to 100 kHz. In this setting, 2000 data points could be extracted from each sawtooth period and 50 sets of data could be obtained in 1 s. Distance measurement was carried out at 10 different positions in the range of 4 cm to 10 cm. The SMI signal of the laser modulated with the sawtooth wave is shown in Figure 3a, and the SMI signal extracted from the sawtooth edge is shown in Figure 3b. The cosine-like signal on the sawtooth wave in Figure 3a is the self-mixing signal, which could be extracted by subtracting the modulated sawtooth signal. The distance could be obtained via Equation (4) after calculating the frequency of the signal. In signal processing, 100 points in the dashed box at the beginning and end of each sawtooth period were removed to eliminate distortion in the SMI signal caused by the sudden change in the sawtooth signal. In this experiment, a Gaussian random matrix was multiplied by the SMI signal to simulate the CS sampling process. For the CS method, the sampling size \(M\) was set to 300, and the maximum number of iterations \(m\) in the recovery algorithm was set to 50. The spectrum of the raw data s and the spectrum of the CS recovered data \(s^{\prime}\) are shown in Figure 4a. The details of the non-zero values in the dashed box are shown in Figure 4b and the details of the zero values are shown in Figure 4c. The results in Figure 4 show that most of the amplitudes of the SMI signal in the frequency domain were so small that they could be considered zero. In other words, the SMI signal could be regarded as sparse in the Fourier domain. Indexes with larger amplitudes were reconstructed and others were treated as zero in the recovery signal. At each distance measurement, the data from 50 sawtooth cycles within one second was analyzed. The distance was calculated using both raw and recovered data. The difference between the calculated distances from two data sources was recorded. Figure 5 depicts the average difference and standard deviation of 50 measurements. Figure 3: SMI signal: (**a**) Signal with sawtooth current; (**b**) Extracted SMI signals. Figure 2: Schematic diagram of the experimental setup. The results in Figure 5 illustrate that the differences in distance between the CS method and the traditional method in the experiment were within 0.15 \(\upmu\)m. Compared with the measurement errors of approximately 60 \(\upmu\)m in the distance measurement system using the traditional method, the differences caused by CS sampling were much smaller. Moreover, there were 300 points sampled to recover the SMI signal in the CS method, which was one-sixth of the original signal, with 1800 points. This means that the CS method is a promising technology in SMI distance measurement, achieving fewer samples at the expense of minor errors. Some additional experiments were carried out to evaluate acceleration in the improved algorithm.