# Absolute Distance Measurement Based on Self-Mixing Interferometry Using Compressed Sensing
## 1 Introduction  
The self-mixing phenomenon refers to the effect in which an emitted laser beam is partially reflected by a target and mixed with the light inside the laser cavity, resulting in modulation of both the output laser power and the optical frequency [1]. The modulated power or modulated frequency can be used to estimate the target's properties. A measurement sensor with SMI does not require a beam splitter, reference mirror, and external photodetector, so it has the characteristics of low cost, simplicity and compact structure. SMI sensors are widely used in the field of distance [2, 3], displacement [4, 5], vibration [6, 7], and velocity [8, 9] measurements. Moreover, the use of laser diodes as a light source can further reduce size and weight [2, 10]. These benefits make SMI-based sensors suitable for deployment as nodes in a measurement network. For example, distance measurement sensors based on SMI can be used to monitor the structural deformation of space optical communication satellites or quantum satellites, which have strict requirements for precise structures.  
Most already reported methods for SMI distance sensing depend on the beat frequency of fringes induced by linear current modulation, due to the approximate linear relation between beat frequency and distance [2]. The fringe-counting method [11] in the time domain and the spectrum analysis method [12] in the frequency domain are commonly used to obtain the beat frequency. When compared, the spectrum analysis method achieves higher measurement accuracy and can adapt to measurements with a lower signal-to-noise ratio [13, 14]; therefore, most signal-processing work for SMI distance measurement focused on spectrum estimation algorithms to obtain the modulated frequency.  
A high sampling rate for distance measurement is set to ensure accuracy and a wide range of measurements. The high sampling rate places a burden on sampling device capability, energy consumption, data transmission capability, and data storage capacity,which are sensitive to sensor nodes or satellites with limited resources [15]. To address this issue, compression algorithms are typically applied to satellites to reduce data size [16; 17]. However, they do not fundamentally solve the problem. Compression algorithms place an additional computational burden on satellites, and the demand for sampling devices with a high sampling rate is not alleviated.  
In the traditional sampling approach based on the Nyquist sampling theorem, a large effort is made to obtain complete information at the sampling stage, and useless information is discarded at the compression stage. Unlike the traditional approach, the novel theory of compressed sensing (CS) provides a new approach for data acquisition and data compression at the same time [18]. In the CS method, fewer data are sampled at the satellite, reducing the burden on sampling devices and subsequent transmission. No other complex computations are required at the satellite, and the recovery algorithm for sampled data is implemented on the ground with ample computing resources. The CS theory works on the premise that if a signal is sparse in some transform domain, far fewer measurements are needed to reconstruct the signal compared to the Nyquist sampling theorem [19]. The SMI signal is cosine-like, with only a few indexes of the spectrum having non-zero values. When spectrum estimation algorithms are utilized for SMI distance measurement, only the index corresponding to the maximum amplitude and a few other indexes are concerned [20]. Therefore, the SMI signal can be regarded as a sparse signal in the Fourier domain. In light of this, we used CS for SMI distance measurement in our work.