## 2 The Principle of DS Positioning with LEO Communication Satellites
### The Principle of DS Positioning  
Given the unique nature of DS positioning, user positioning can be accomplished by monitoring the DS variations of a single satellite at different times. Alternatively, observing the DS of multiple satellites simultaneously can also achieve user positioning. Assuming that only frequency changes due to relative motion are considered, continuous observation of the DS of satellite A is conducted, as depicted in Figure 2. The DS of the signals received by the user at times t1, t2, and t3 are fd1, fd2, and fd3, respectively, as shown in Equation (5).  
\[f_{di}=\frac{f}{c}\frac{\mathbf{V}_{si}\cdot\left(\mathbf{P}-\mathbf{S}_{i} \right)}{\left\|\mathbf{P}-\mathbf{S}_{i}\right\|}=\frac{f}{c}\frac{V_{siz} \left(P_{x}-S_{ix}\right)+V_{siy}\left(P_{y}-S_{iy}\right)+V_{siz}\left(P_{z}- S_{iz}\right)}{\sqrt{\left(P_{x}-S_{ix}\right)^{2}+\left(P_{y}-S_{iy} \right)^{2}+\left(P_{z}-S_{iz}\right)^{2}}}, \tag{5}\]  
where \(i=1,2,3\) denotes the three positions of satellite A; \(f\) is the transmission frequency of the satellite signals; \(c\) is the speed of light; \(\mathbf{V}_{si}\) the vector velocity of satellite A; \(\mathbf{S}_{i}\) is the spatial position of the satellite; \(\mathbf{P}=\left(P_{x},P_{y},P_{z}\right)\) is the position of the test point. The meanings of the other letters in Equation (5) are shown in Figure 2.  
Figure 1: Schematic diagram of DS.  
Linearize Equation (5) around the predicted position \(\mathbf{P}_{0}=(P_{x0},P_{y0},P_{z0})\) to derive Equation (6), as follows:  
\[\begin{array}{rl}f_{di}(S_{ix},S_{iy},S_{iz})&=f_{di0}(P_{x0},P_{y0},P_{z0})+ \frac{\partial f_{di}}{\partial S_{x}}(P_{x}-P_{x0})+\frac{\partial f_{di}}{ \partial P_{y}}(P_{y}-P_{y0})+\frac{\partial f_{di}}{\partial P_{z}}(P_{z}-P_ {z0})+h.o.t\\ &\left\{\begin{array}{l}\left[\frac{V_{ix}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{ 0}-\mathbf{S}_{i})^{T}(P_{x0}-\mathbf{S}_{iz})}{d_{i}^{3}}\right](P_{x}-P_{x0} )\\ \\ +\left[\frac{V_{xiy}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{0}-\mathbf{S}_{i})^{T}( P_{x0}-\mathbf{S}_{iy})}{d_{i}^{3}}\right](P_{y}-P_{y0})\\ \\ +\left[\frac{V_{xiz}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{0}-\mathbf{S}_{i})^{T}( P_{z0}-\mathbf{S}_{iz})}{d_{i}^{3}}\right](P_{z}-P_{z0})\end{array}\right\}+h.o.t \end{array}\right.,\end{array} \tag{6}\]  
where \(d_{i}=\sqrt{\left(P_{x0}-S_{ix}\right)^{2}+\left(P_{y0}-S_{iy}\right)^{2}+\left( P_{z0}-S_{iz}\right)^{2}}\); Equation (6) is the Taylor expansion of the DS positioning equation at the predicted position \(\mathbf{P}_{0}\); \(h.o.t\) is the higher-order term of the Taylor expansion. For Equation (6), assuming that the higher-order terms are not considered, there are three unknowns: \(P_{x0},P_{y0},P_{z0}\). Thus, neglecting temporal and other inaccuracies, observing the satellite at three distinct times suffices to determine the user's position.  
To facilitate comprehension and computation, we recast Equation (6) into state space form, as follows:  
\[\mathbf{f}_{d}=\mathbf{f}_{d0}+\mathbf{A}(\mathbf{P}-\mathbf{P}_{0}), \tag{7}\]  
where \(\mathbf{f}_{d}=(f_{d1},f_{d2},f_{d3})^{T}\), \(\mathbf{f}_{d0}=(f_{d10},f_{d20},f_{d30})^{T}\). The state transition matrix \(\mathbf{A}\) is as follows:  
\[\mathbf{A}=\frac{f}{c}\begin{bmatrix}a_{x1}&a_{y1}&a_{z1}\\ a_{x2}&a_{y2}&a_{z2}\\ a_{x3}&a_{y3}&a_{z3}\end{bmatrix}, \tag{8}\]  
where \(a_{xi}=\frac{V_{xiy}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{0}-\mathbf{S}_{i})^{T}( P_{x0}-S_{iz})}{d_{i}^{3}}\), \(a_{yi}=\frac{V_{xiy}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{0}-\mathbf{S}_{i})^{T}( P_{y0}-S_{iz})}{d_{i}^{3}}\), \(a_{zi}=\frac{V_{xiz}}{d_{i}}-\frac{\mathbf{V}_{zi}(P_{0}-\mathbf{S}_{i})^{T}( P_{y0}-S_{iz})}{d_{i}^{3}}\).  
Figure 2: The principle of DS positioning.