# Compensation of Frequency Drift in Frequency-Sweep Polarization-Modulation Ranging System
## 2 Principle and Method
### Multi-Frequency Interval-Based Measurement of Frequency Interval  
Although frequency error can be partially eliminated through the reciprocating sweeping method, measurement of N requires a higher accuracy of frequency. The measurement error of N can be expressed as:  
\[\Delta\text{N}=\left[\frac{\Delta f_{x}}{f_{x}}-\frac{\Delta f_{\text{b}}\cdot f _{x}}{f_{\text{b}}^{2}}\right]=\left[-\frac{\Delta f_{\text{b}}\cdot f_{x}}{ f_{\text{b}}^{2}}\right]. \tag{9}\]  
The above equation indicates that the absolute error of N is determined by the relative error of \(f_{\text{b}}\). However, in the FSPM method, the impact of relative error of \(f_{\text{b}}\) is often 2N times larger than the relative error of \(f_{x}\), which means the accuracy of \(f_{x}\) may not meet the accuracy requirement of \(f_{\text{b}}\). If \(f_{x}\) is around 2.5 GHz, \(l\) is about 100 m, a frequency measurement error of 2500 Hz only contributes a ranging error of 100 \(\upmu\)m, but the resulting relative error of \(f_{\text{b}}\) contributes a ranging error of 0.36 m. To ensure the accuracy of N, \(\Delta f_{\text{b}}\) should be less than 90 Hz, which cannot be satisfied by the reciprocating sweeping method.  
In order to reduce the measurement error of \(f_{\text{b}}\), we used the characteristic of equal spacing of the \"in phase\" frequencies in our multi-frequency interval-based approach. For example, \(f_{\text{b}}\) can be expressed as follows:  
\[f_{\text{b}}=\frac{\left(f_{x+u}-f_{x}\right)}{u}, \tag{10}\]  
where \(f_{x}\) is an arbitrary \"in phase\" frequency in sweeping band, \(f_{x+u}\) is the \(u\)th \"in phase\" frequency on its upper side. Therefore, N can be rewritten as:  
\[\text{N}=\left[\frac{f_{x}}{f_{\text{b}}}\right]=\left[\frac{u\cdot f_{x}}{f_ {x+u}-f_{x}}\right]. \tag{11}\]  
Figure 3: Influence of frequency drift during reciprocating sweeping.  
Hence, taking the frequency drift and random measurement error into account, the multi-frequency interval can be expressed as:  
\[f_{x+u}-f_{x}=\ u\cdot f_{b}\pm\Delta f_{u}\pm(|v_{x+u}|+|v_{x}|), \tag{12}\]  
where \(\Delta f_{u}\) represents the frequency drift. The measurement error of the wavelength multiple can be expressed as:  
\[\Delta\text{N}=\left[\frac{\Delta f_{x}}{f_{\text{b}}}-\frac{f_{x}\cdot \Delta(f_{x+u}-f_{x})}{u\cdot f_{b}^{2}}\ \right]\approx\left[-\frac{f_{x}\cdot\Delta(f_{x+u}-f_{x})}{u\cdot f_{b}^{2}} \ \right]=\left[-\frac{f_{x}\Delta(\pm\Delta f_{u}\pm(|v_{x+u}|+|v_{x}|))}{u\cdot f _{b}^{2}}\ \right]. \tag{13}\]  
Therefore, errors caused by frequency drift and random measurement error are \(u\) times narrower than the general method.  
In the above method, \(u\) can be determined by the following equation:  
\[u\ \left[\ \frac{f_{x+u}-f_{x}}{f_{x+1}-f_{x}}\ \right]. \tag{14}\]  
According to the expression of \(\Delta u\):  
\[\Delta u=\bigg{[}\frac{\Delta(f_{x+u}-f_{x})}{f_{b}}-\frac{\Delta(f_{x+1}-f_{ x})}{u\cdot f_{b}}\ \bigg{]}=\bigg{[}-\frac{\pm\Delta f_{u}\pm(|v_{x+u}|+|v_{x}|)}{f_{b}}-\frac{ \pm\Delta f_{1}\pm(|v_{x+1}|+|v_{x}|)}{u\cdot f_{b}}\bigg{]}. \tag{15}\]  
The items are small quantities, that is, \(\pm\Delta f_{u}\pm(|v_{x+u}|+|v_{x}|)/f_{b}\ \ll\ 1\), \(\ \pm\Delta f_{1}\pm(|v_{x+1}|+|v_{x}|)/(uf_{b})\ll 1\), so \(u\) can be accurately acquired.