# Absolute Distance Measurement Based on Self-Mixing Interferometry Using Compressed Sensing
## 2 Theoretical Analysis
### SMI Distance Measurement Using CS  
In comparison to the traditional sampling method, the CS method employs non-uniform sampling based on the measurement matrix and incorporates a signal recovery process. Figure 1 depicts block diagrams of the traditional sampling method and the CS method for SMI distance measurement. In the traditional spectrum estimation method, the SMI signal is acquired by analog-to-digital (AD) conversion. Following that, FFT and interpolated correction are applied to the sampled signal with the Hanning window added. The modulated frequency in Equation (3) can be obtained, and the external distance calculated, according to Equation (4). In the CS method, signal \(\mathbf{y}\) with fewer points than in the traditional method is sampled through control of the measurement matrix, which can be implemented by a demodulator [26]. To reduce spectral leakage as in traditional methods, a Hanning window is multiplied on each row of the measurement matrix. After that, the recovery coefficient vector \(\mathbf{s}^{\prime}\) can be solved using the recovery algorithm. The following steps in the CS method are consistent with the traditional method. In our work, the widely used Gaussian matrix was selected as measurement matrix to facilitate analysis, and the CS sampling process was simplified into the multiplication of the original signal and the measurement matrix. The OMP algorithm was used as the signal recovery algorithm for SMI distance measurement. When the Fourier basis is used as a sparse matrix and OMP is selected as the recovery algorithm, two improvements were made in our work regarding acceleration and robustness. * Acceleration  
Conjugate symmetry is present in the Fourier transform of real numbers. This feature can be used to accelerate the OMP recovery algorithm. When a column in the sensing matrix achieves its largest inner product with the residual vector during an iteration, not only the index of that column but also the symmetrical index is added to the selected atom index matrix. * Robustness  
The correction method in Equation (5) is concerned with the amplitudes of the main peaks and lateral peaks. However, the OMP algorithm selects the atoms on which the residual vector achieves maximum projection. This strategy does not guarantee that the lateral peaks are selected. To address this issue, an additional iteration is applied in the OMP algorithm. This finds the index of the maximum peak and adds the atoms at the lateral peaks to the selected matrix if they are not selected. The steps of the improved OMP for SMI signals are shown in Algorithm 1, and the differences from OMP are marked in red font. Figure 1: Block diagrams for SMI distance measurement: (**a**) Traditional method; (**b**) CS method. ```
1:Sensing matrix \(\mathbf{\Theta}\in\mathbb{R}^{M\times N}\); Sampling vector \(\mathbf{y}\); Maximum number of iterations \(m\). Initialization: Number of iterations \(t=0\); Residual vector \(\mathbf{r_{0}}=\mathbf{y}\); Selected atom matrix \(\mathbf{\Omega_{0}}=\mathcal{O}\); Selected index matrix \(\mathbf{\Lambda_{0}}=\mathcal{O}\).