# LEO-Based Satellite Constellation for Moving Target Detection
## 2 Materials and Methods
### Pre-Compensation of Spaceborne Echo for LEO Satellite  
In literature [10], the moving target detection method is demonstrated with BeiDou satellites, in which the platform speeds are much lower than that of low orbit satellite, and the Doppler ambiguity problem can be ignored when integrating multiple echoes in DFR domain. However, this method is not applicable for the LEO constellation scenario. On one hand, the pulse repetition frequency (PRF) of the satellite illuminator is generally preset. Additionally, its adjustable dynamic range is limited. Thus, the high-speed movement of LEO satellites could apparently broaden the Doppler bandwidth (Bd) related to the observation scene and thus to reduce the target detectable interval, as indicated in Figure 2a. On the other hand, the extended Doppler bandwidth may exceed the given PRF, which leads to the Doppler ambiguity issue, as indicated in Figure 2b. In this case, the decoupling method, such as the KT transform [13], is no longer applicable owing to the Doppler ambiguity of the observation scene. On the other hand, the LEO satellites adopt the self-transmitting and self-receiving operation mode instead of the multiple-transmitting and single-receiving operation mode,  
Figure 2: The relation between the extended Doppler bandwidth and the given PRF. (**a**) The limited target detectable interval as a result of the Doppler bandwidth expansion; (**b**) The Doppler ambiguity issue caused by the extended Doppler bandwidth. where the configuration does not possess the extreme asymmetry in respect of the GEO (Geostationary Orbit) or MEO (Medium Earth Orbit) satellites. Thus, the LEO-based multi-source echoes cannot be concentrated on one cell in DFR domain. For the multiple-transmitting and single-receiving operation mode, DFR of a moving target can be defined as [13]:  
\[ \begin{array}{ll}f_{dt,k}=&-\frac{1}{\lambda}\Big{\{}-|\mathbf{P}_{T,k}- \mathbf{P}_{Tg}|^{-3}\cdot[(\mathbf{P}_{T,k}-\mathbf{P}_{Tg})(\mathbf{v}_{T,k }-\mathbf{v}_{Tg})^{\mathsf{T}}]^{2}+|\mathbf{P}_{T,k}-\mathbf{P}_{Tg}|^{-1} \cdot(\mathbf{v}_{T,k}-\mathbf{v}_{Tg})^{2}+|\mathbf{P}_{T,k}-\mathbf{P}_{R}|^ {-3}\cdot[(\mathbf{P}_{T,k}-\mathbf{P}_{R})(\mathbf{v}_{T,k}-\mathbf{v}_{R})^{ \mathsf{T}}]^{2}\\ &-|\mathbf{P}_{T,k}-\mathbf{P}_{R}|^{-1}\cdot(\mathbf{v}_{T,k}-\mathbf{v}_{R}) ^{2}-|\mathbf{P}_{Tg}-\mathbf{P}_{R}|^{-3}\cdot[(\mathbf{P}_{Tg}-\mathbf{P} _{R})(\mathbf{v}_{Tg}-\mathbf{v}_{R})^{\mathsf{T}}]^{2}+|\mathbf{P}_{Tg}- \mathbf{P}_{R}|^{-1}\cdot(\mathbf{v}_{T,k}-\mathbf{v}_{R})^{2}\Big{\}}\end{array}  \tag{3}\]  
While that of the self-transmitting and self-receiving operation mode is given by:  
\[f_{dt,k}=-\frac{2}{\lambda}\{-|\mathbf{P}_{T,k}-\mathbf{P}_{Tg}|^{-3}\cdot[( \mathbf{P}_{T,k}-\mathbf{P}_{Tg})(\mathbf{v}_{T,k}-\mathbf{v}_{Tg})^{\mathsf{ T}}]^{2}+|\mathbf{P}_{T,k}-\mathbf{P}_{Tg}|^{-1}\cdot(\mathbf{v}_{T,k}- \mathbf{v}_{Tg})^{2}\} \tag{4}\]  
In Equations (3) and (4), \(\lambda\) is the wavelength, \(\mathbf{P}_{T,k}\) and \(\mathbf{P}_{R}\) stand for the position vector of the \(k\)th transmitting platform and the receiving platform, respectively. \(\mathbf{v}_{T,k}\) and \(\mathbf{v}_{R}\) denote the corresponding velocity vector in turn. \(\mathbf{P}_{Tg}\) and \(\mathbf{v}_{Tg}\) are the position vector and the velocity vector of the target. Under the condition of bistatic configuration with GEO or MEO satellite as its transmitter, Equation (3) can be approximately expressed as:  
\[f_{dt,k}\approx-\frac{1}{\lambda}\Big{\{}-|\mathbf{P}_{Tg}-\mathbf{P}_{R}|^{ -3}\cdot[(\mathbf{P}_{Tg}-\mathbf{P}_{R})(\mathbf{v}_{Tg}-\mathbf{v}_{R})^{ \mathsf{T}}]^{2}+|\mathbf{P}_{Tg}-\mathbf{P}_{R}|^{-1}\cdot(\mathbf{v}_{Tg}- \mathbf{v}_{R})^{2}\Big{\}} \tag{5}\]  
From Equation (5), the target DFR mainly depends on the position vector of the receiving platform under asymmetric configuration. Therefore, the multiple-transmitting and single-receiving operation mode is conducive to the fusion processing [10].