## Theory
### The CRLB When There Are Interfering Signals and Noise in the Measurement Interferometer  
Ignoring the influence of the erroneous sampling points in the auxiliary interferometer, when the system measures the non-cooperative target and there is noise in the measurement interferometer, Equation (4) turns into:  
\[ \begin{array}{c}I_{m}(k)=\ A_{m}\cos\Bigl{(}2\pi f_{0}\tau_{m}\ +\ \frac{\pi\tau_{m}}{\tau_{a}}\Bigr{)}+Noise_{m}(k)+I_{i}(k)=\ A_{m}\cos( \zeta k+\varphi)+Noise_{m}(k)+I_{i}(k),\\ k=\ 1,\ 2,\ \ldots,\ N\end{array}  \tag{12}\]  
where \(Noise_{m}(k)\) denotes the Gaussian white noise (GWN) with an expectation of 0 and variance \(\sigma^{2}\), \(I_{i}(k)\) denotes the interfering signals, and \(\zeta\) is \(\frac{\pi\tau_{m}}{\tau_{a}}\), \(\varphi\) is \(2\pi f_{0}\tau_{m}\). The interfering signals \(I_{i}(k)\) are caused by the reflected lights from the circulator and the fiber end in Figure 1.  
As shown in Figure 4, when our system measures a non-cooperative target, the measurement signal (the red line) becomes very weak, and the interfering signals (the blue line) are much stronger than the measurement signal. In addition, the frequencies of the interfering signals are not varying with the measured distances and are lower than the frequency of the measured signal, so we can use an appropriate low pass filter to delete them. Thus, the SNR of the measurement interferometer output can be shown as \(A_{m}^{2}/\sigma^{2}\). The Cramer-Rao lower bound (CRLB) is a method in the statistics and the variance of any unbiased estimator cannot be lower than the CRLB [23]. Therefore, we use the CRLB to evaluate our system. The likelihood function of \(I_{m}(k)\) can be expressed as:  
\[ \begin{array}{c}\mathrm{L}(I_{m}(k)|\zeta)\ =\ \frac{1}{\left(2\pi\sigma^{2} \right)^{N/2}}exp\{-\frac{1}{2\sigma^{2}}\sum_{k=1}^{N}(I_{m}(k)-A_{m}\cos( \zeta k\ +\ \varphi))^{2}\}\end{array}  \tag{13}\]  
The Fisher information is obtained as:  
\[ \begin{array}{c}\mathrm{E}(\frac{\partial^{2}\mathrm{lnL}(I_{m}(k)|\zeta)}{ \partial\zeta^{2}})\ =\ SNR\sum_{k=1}^{N}(k\sin(\zeta k\ +\ \varphi))^{2}\end{array}  \tag{14}\]  
The CRLB on the variance of the measured distance is:  
\[ \begin{array}{c}\mathrm{Var}\{L_{m}\}\ \geq\ \frac{6\pi f^{2}L_{a}{}^{2}}{\pi^{2} n_{air}^{2}N^{3}SNR}\end{array}  \tag{15}\]  
Figure 4: Measured result without cooperative target. Accordingly, when there is GWN in the measurement interferometer, the CRLB on the standard deviation of measured distance can be shown as:  
\[\Delta L_{m2}\ \geq\ \frac{n_{f}L_{a}}{\pi n_{air}N^{2}}\sqrt{\frac{6(N\ -\ 1)}{SNR}} \tag{16}\]  
When the measured target is not a reflector or a corner prism and the incident angle is not zero, the SNR should be low, and the error caused by the GWN in the measurement interferometer can't be ignored. The Adaptive Filtering Method, the Phase-Matching Mosaic Algorithm, and the Segmentation Mosaic Algorithm  
According to the analysis above, completely eliminating the erroneous sampling points of the auxiliary interferometer and the interfering signals in the measurement interferometer are essential in order to ensure the accuracy of our system. Thus, an adaptive filtering method and a phase-matching mosaic algorithm are proposed to ensure the accuracy, and a segmentation mosaic algorithm is used to improve the data-processing speed. The adaptive filtering method can be described as following. 1. Firstly, we make use of the Hanning window and the wavelet threshold filtering to depress the noises (such as the GWN) and the interfering signals in the auxiliary interferometer and measurement interferometer. 2. Then, we get the sampling points (all of the maximum and minimum extreme points of the auxiliary interference signal) and set appropriate thresholds to delete most of the erroneous sampling points caused by mode-hopping signals. 3. Lastly, according to the rule that the maximum and minimum values occur alternately, we remove the rest of the erroneous sampling points.