## Theory
### The CRLB When There Are Interfering Signals and Noise in the Measurement Interferometer
Step (1) is used to depress the noises and the interfering signals in the auxiliary interferometer and measurement interferometer. The Hanning window filtering method had been proved to be an effective method to depress the low and high frequency noise in an FSI-based system [10]. In our experiments, the measurement signal was very weak, and its frequency was varying with the measured distance, so we must use an adaptive filtering method to depress the GWN. The wavelet threshold filtering method is adopted in our paper. The mechanism of wavelet threshold filtering is based on the different properties of the wavelet coefficients of signals and noises on scales. To eliminate the noise, corresponding rules are adopted to deal with the nonlinear processing of the wavelet coefficients of the noise [24]. Step (2) and Step (3) are used to delete the erroneous sampling points caused by the mode-hopping of the ECL. Firstly, we get all of the sampling points in the auxiliary interferometer. The set of all of the sampling points can be described as:  
\[Ep=Ep_{max}+Ep_{min} \tag{17}\]  
where \(Ep\) denotes a set of all of the sampling points, \(Ep_{max}\) denotes a set of all of the maximum extreme points, and \(Ep_{min}\) denotes a set of all of the minimum extreme points. Then, we find that the amplitude and the time interval of the erroneous sampling points is obviously smaller than the correct sampling points. Firstly, we set the amplitude thresholds to delete part of the erroneous sampling points, and the rest of the sampling points can be written as:  
\[Ep_{1}=\{x|A(x_{k})>\ 0.6\ A_{ma}+\ 0.4\ A_{mi}\lor A(x_{k})<\ 0.6A_{mi}\ +\ 0.4A_{mn},\ x\in\ Ep,\ k=\ 1,\ 2,\ \ldots,\ S\} \tag{18}\]  
where \(Ep_{1}\) denotes a set of the sampling points that contains part of the erroneous sampling points, \(A(x_{k})\) is the amplitude of the extreme point, \(A_{ma}\) is the maximum amplitude of the auxiliary interference signal, \(A_{mi}\) is the minimum amplitude of the auxiliary interference signal, \(k\) is the index of the extreme points, and \(S\) is the length of \(Ep\). Then, we set the time interval thresholds to delete part of the erroneous sampling points, and the rest of the sampling points can be written as:  
\[Ep_{2}=\ \{x|(T(x_{k})-\ T(x_{k-1}))\ >\ 0.6\Delta T_{me}\,,\ x\in\ Ep_{1},\ k =2,\ 3,\ \ldots,\ K\} \tag{19}\]  
where \(x\) denotes the sampling point in \(Ep_{1}\), \(T(x_{k})\) is the time interval of the extreme point, \(\Delta T_{me}\) denotes the mean time interval of all of the points in \(Ep_{1}\), and \(K\) is the length of \(Ep_{1}\). Lastly, we find that the rest of the erroneous sampling points do not follow the rule that the maximum and minimum values occur alternately. So, the set of the correct sampling points can be described as:  
\[Ep_{3}=\ \{x|(A(x_{k})\ -\ A(x_{k-1}))\ (A(x_{k-1})\ -\ A(x_{k-2}))\ <\ 0,\ x\in Ep_{2},\ k =3,\ 4,\ \ldots,\ M\} \tag{20}\]  
where \(Ep_{3}\) denotes a set of the correct extreme points, \(k\) is the index of the correct extreme points, and \(M\) is the length of the \(Ep_{2}\). Thus, the measurement errors that are caused by the erroneous sampling points and the interfering signals are eliminated. In addition, the phase-matching mosaic algorithm and the segmentation mosaic algorithm are used to eliminate the phase jump and improve the data-processing speed. After sampling the measurement signal with \(Ep_{2}\), phase jump occurs at the starting and ending point of each mode-hopping signal in the measurement interferometer [25]. To solve the problem, firstly we find the starting point and ending point of each mode-hopping signal by the time interval. Then, we use the Hilbert translation to exact the phase of the 10 points before the starting point and the 10 points after the ending point and compare their phases one by one. Lastly, we stitch the closest two points. When the tuning bandwidth, tuning speed, and sampling frequency of the data acquisition card are set, the required number of the sampling points is fixed.