## 2 Materials and Methods  
### Optimization Problem of DPE  
GNSS consists of a network of satellites orbiting the Earth, each transmitting predefined messages to the receivers on the ground. The signals captured by receivers are composed of a combination of structured plane waves, representing the line-of-sight (LOS) signals from visible satellites, distorted by thermal noise and possibly affected by external interference. Considering there are \(M\) visible satellites, the received complex baseband signal sampled at time \(t_{k}\) can be expressed as  
\[y(t_{k})=\sum_{i=1}^{M}a_{i}s_{i}(t_{k}-\tau_{i})e^{j2\pi f_{d_{i}}t_{k}}+n(t_ {k}), \tag{1}\]  
where \(a_{i}\) and \(s_{i}\) denote the signal amplitude and the known PRN codes of the \(i\)-th satellite, respectively. \(\tau_{i}\) and \(f_{d_{i}}\) denote the time delay and Doppler shift of the \(i\)-th satellite. \(n(t_{k})\) denotes the noise, generally assumed to be zero-mean additive white Gaussian noise (AWGN). If a receiver observes \(K\) snapshots, the signal model can be rearranged into matrix form as  
\[\mathbf{y}=\mathbf{S}(\mathbf{\tau},\mathbf{f}_{d})\mathbf{a}+\mathbf{n}, \tag{2}\]  
where \(\mathbf{y}=[y(t_{1}),\cdots,y(t_{K})]^{\top}\) is the \(K\times 1\) received signal vector, \(\mathbf{a}\) is the \(M\times 1\) vector of complex signal amplitudes, \(\mathbf{n}\) is the \(K\times 1\) AWGN vector, and \(\mathbf{S}(\mathbf{\tau},\mathbf{f}_{d})=[\mathbf{s}(\tau_{1},f_{d_{1}})^{\top},\cdots,\mathbf{s}(\tau_{M_{t}},f_{d_{M}})^{\top}]\) is the \(K\times M\) local signal replica matrix, which is a function of parameter vectors \(\mathbf{\tau}=[\tau_{1},\cdots,\tau_{M}]^{\top}\) and \(\mathbf{f}_{d}=[f_{d_{1}},\cdots,f_{d_{M}}]^{\top}\). Each component \(\mathbf{s}(\tau_{i},f_{d_{i}})\) is defined by the delayed and Doppler-shifted signal envelope:  
\[\mathbf{s}(\tau_{i},f_{d_{i}})=\Big{[}s_{i}(t_{1}-\tau_{i})e^{2\pi f_{d_{i}}t _{1}},\cdots,s_{i}(t_{K}-\tau_{i})e^{2\pi f_{d_{i}}t_{K}}\Big{]}.\]  
Conventional 2 SP-based receivers handle each satellite signal independently, conducting a two-dimensional search to estimate parameters such as \(\tau_{i}\) and \(f_{d_{i}}\). Based on a collection of \(K\) snapshots, the optimization problem can be expressed as  
\[\tau_{i},\hat{f}_{d_{i}}=\operatorname*{arg\,max}_{\tau_{i},f_{d_{i}}}\big{\|} \mathbf{y}^{\mathrm{H}}\mathbf{s}(\tau_{i},f_{d_{i}})\big{\|}^{2}, \tag{3}\]  
where \({}^{\mathrm{,H}}\) stands for the Hermitian transpose operation and the operator \(\|\cdot\|^{2}\) denotes the L\({}_{2}\)-norm of a vector. After determining these parameters, the receiver calculates its position through trilateration. This widely used 2 SP is favored for its computational simplicity and ease of implementation. However, when the signal channel deteriorates, significant errors may occur in the estimated carrier and code phase, leading to tracking loop loss of lock and rendering traditional GNSS receivers inoperative. DPE leverages the relationship between the PVT state of a receiver and the time delay and Doppler shift of the received GNSS signal. Essentially, \(\tau_{i}\) and \(f_{d_{i}}\) can be expressed as functions of the PVT state \(\gamma\):  
\[\tau_{i}(\mathbf{\gamma}) =\frac{||\mathbf{p}_{i}-\mathbf{p}||}{c}+\delta t-\delta t_{i},\] \[f_{d_{i}}(\mathbf{\gamma}) =-\frac{f_{c}}{c}\bigg{(}(\mathbf{v}_{i}-\mathbf{v})^{\top}\frac{\mathbf{p}_{ i}-\mathbf{p}}{||\mathbf{p}_{i}-\mathbf{p}||}+c(\delta t-\delta t_{i})\bigg{)}, \tag{4}\]  
where \(\mathbf{\gamma}=\big{[}\mathbf{p}^{\top},\delta t,\mathbf{v}^{\top},\delta t\big{]}^{\top}\) is the PVT state of the receiver, \(\mathbf{p}=[x,y,z]^{\top}\) denotes the position of the receiver, and \(\mathbf{v}=[\hat{x},\hat{y},\hat{z}]^{\top}\) denotes the velocity of the receiver, \(\delta t\) and \(\delta t\) are the receiver clock bias and drift. Note that this is a simplified model. First, atmospheric-related errors, including ionospheric and tropospheric delays, are not explicitly modeled in the signal delay term \(\tau_{i}\).