# Evaluation of BDS/GPS Multi-Frequency RTK Positioning Performance under Different Baseline Lengths
## 2 Multi-Frequency RTK Positioning Mathematical Model
### Kalman Filtering Theory for Short Baseline RTK  
Kalman Filtering (KF) is a linear filtering method based on the principle of linear unbiased minimum variance estimation for filtering. By introducing the idea of state equation, the new state parameters are recursively derived from the state valuation of the previous epoch combined with the observed values of the current epoch in a state transfer matrix. The Kalman Filtering is mainly divided into two parts: state prediction and parameter updating. For epoch \(n-1\) and \(n\), the prediction models for the state parameters and covariance matrix are as follows [12]:  
\[\mathbf{\hat{\kappa}}^{-}(n)=\mathbf{F}(n-1)\mathbf{\hat{\kappa}}^{-}(n-1) \tag{4}\]  
\[\mathbf{P}^{-}_{\mathbf{P}_{\mathbf{F}(n)}}=\mathbf{F}(n-1)\mathbf{P}^{-}_{ \mathbf{P}_{\mathbf{F}(n-1)}}\mathbf{F}^{T}(n-1)+\mathbf{Q}(n-1) \tag{5}\]  
where \(\mathbf{x}\) and \(\mathbf{P}_{\mathbf{P}}\) are the estimated state vector and its covariance matrix and \(\mathbf{F}\) and \(\mathbf{Q}\) are the transition matrix and the covariance matrix of the system noise. In short baseline RTK positioning mode, \(\mathbf{x}\) is defined as:  
\[\mathbf{x}=(\mathbf{r}^{T}_{r},\mathbf{v}^{T}_{r},\mathbf{B}^{T}_{1},\mathbf{B }^{T}_{2},\mathbf{B}^{T}_{3})^{T} \tag{6}\]where \(\mathbf{r}\) and \(\mathbf{v}\) are the three-dimensional position vector and velocity of the rover station and \(\mathbf{B}_{i}=\left(\mathbf{B}_{i}^{1},\mathbf{B}_{i}^{2},\cdots,\mathbf{B}_{ i}^{m-1}\right)^{T}\) is the DD carrier-phase biases in cycle. The parameter update portion of the Kalman Filter is as follows:  
\[\mathbf{K}(n)=\mathbf{P}_{\mathbf{r}_{\mathbf{j}(n)}}^{-}\mathbf{G}^{T}(n) \left[\mathbf{G}(n)\mathbf{P}_{\mathbf{r}_{\mathbf{j}(n)}}^{-}\mathbf{G}(n)^ {T}+\mathbf{R}(n)\right]^{-1} \tag{7}\]  
\[\mathbf{\hat{x}}=\mathbf{\hat{x}}^{-}(n)+\mathbf{K}(n)\left[\mathbf{y}(n)- \mathbf{G}(n)\mathbf{\hat{x}}^{-}(n)\right] \tag{8}\]  
\[\mathbf{P}_{\mathbf{P}_{\mathbf{r}(n)}}=\left[\mathbf{I}-\mathbf{K}(n) \mathbf{G}(n)\right]\mathbf{P}_{\mathbf{r}_{\mathbf{j}(n)}}^{-} \tag{9}\]  
where \(\mathbf{K}\) is the Kalman Gain matrix, \(\mathbf{G}\) is the design matrix, and \(\mathbf{R}\) is the covariance matrix of the observations. The variance of single-difference observations is usually calculated first from the elevation of the satellite or signal-to-noise ratio (SNR), and the covariance matrix of DD measurements can be obtained under the law of error propagation. The measurement vector \(\mathbf{y}\) is also defined by DD carrier-phase and pseudorange measurement as [8]:  
\[\mathbf{y}=\left(\mathbf{\Phi}_{1}^{T},\mathbf{\Phi}_{2}^{T},\mathbf{\Phi}_{3 }^{T},\mathbf{P}_{1}^{T},\mathbf{P}_{2}^{T},\mathbf{P}_{3}^{T}\right)^{T} \tag{10}\]  
where:  
\[\Phi_{i}=\left(\mathbf{\[PERSON]}_{rb,i}^{12},\mathbf{\[PERSON]}_{rb,i}^{13},\mathbf{ \[PERSON]}_{rb,i}^{14},\cdots,\mathbf{\[PERSON]}_{rb,i}^{1m-1}\right)^{T} \tag{11}\]  
\[P_{i}=\left(\mathbf{P}_{rb,i}^{12},\mathbf{P}_{rb,i}^{13},\mathbf{\Phi}_{rb,i }^{14},\cdots,\mathbf{P}_{rb,i}^{1m-1}\right)^{T} \tag{12}\]  
In Equations (11) and (12), \(\mathbf{\[PERSON]}_{rb,i}^{1m-1}\) and \(\mathbf{P}_{rb,i}^{1m-1}\) are the DD carrier-phase and pseudorange.