## 2 Signal Model and Problem Formulation  
Following the general description in [PERSON], et al. [12], the complex envelope of the received signal including the downlink carrier from the spacecraft could be modeled as:  
\[\widetilde{r}(t)=\widetilde{s}(t)+\widetilde{n}(t), \tag{1}\]  
where \(\widetilde{s}(t)\) is the complex envelope of the downlink carrier, and \(\widetilde{n}(t)\) is the complex envelope of the narrow band thermal noise process. For the downlink carrier signal \(\widetilde{s}(t)\) modulated by the spacecraft movement, it could be treated as:  
\[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]  
where \(A\) is the amplitude of the carrier, \(\omega_{c}\) is the angular frequency of the transmitting carrier onboard the spacecraft, and \(\theta(t)\) is the Doppler phase induced by the spacecraft movement. We also choose a cubic phase model to describe the Doppler phase as follows:  
\[\theta(t)=\theta_{0}+2\pi\bigg{(}f_{0}t+\frac{1}{2}f_{1}t^{2}+\frac{1}{6}f_{2 }t^{3}\bigg{)}, \tag{3}\]  
where \(\theta_{0}\) is the initial phase, \(f_{0}\) is the constant Doppler frequency in Hz, \(f_{1}\) is its first derivative (Doppler rate) with units of Hz/s, and \(f_{2}\) is the second derivative of the Doppler frequency (second order Doppler rate) in Hz/s\({}^{2}\). Usually, the transmitting angular frequency \(\omega_{c}\) is the designed parameter in a space mission. Its value could vary due to the oscillator drift when the space-to-ground communication is in one-way mode. Without loss of generality, we regard \(\omega_{c}\) as the deterministic parameter, and the unknown parameters are \(A\), \(\theta_{0}\), \(f_{0}\), \(f_{1}\), and \(f_{2}\), which are of interests to be estimated. After complex mixing and sampling with sampling frequency \(F_{s}\), \(\widetilde{r}(t)\) could be digitized into the in-phase and quadrature signal components as:  
\[I_{i}=Acos(\theta_{i})+n_{I_{i}},\ i=1,\cdots,N \tag{4}\]  
\[Q_{i}=Asin(\theta_{i})+n_{Q_{i}},\ i=1,\cdots,N \tag{5}\]where \(\theta_{i}\) is the sampled Doppler phase at time instant \(iT_{s}\), \(T_{s}\) = 1/\(F_{s}\) is the sampling interval, and \(N\) is the sample number. The thermal noises are digitized and modeled as additive white Gaussian noise (AWGN) samples and their variance are equal. \[var(n_{I_{i}})=var(n_{Q_{i}})=\sigma^{2}, \tag{6}\]  
Thus, the carrier-to-noise density ratio (CNR) could be computed as:  
\[\text{CNR}=\frac{P}{N_{0}}=\frac{A^{2}}{2\sigma^{2}T_{s}}, \tag{7}\]  
where \(P\) is the received carrier power, and \(N_{0}\) is the thermal noise spectral density in the receiver. We still consider this parameter estimation problem in the maximum likelihood (ML) estimation framework as analyzed by [PERSON] and [PERSON] (2008); [PERSON] and [PERSON] (2008). The ML estimates of the signal parameters are those values that simultaneously maximize the conditional joint probability density of the observation vector, conditioned on the signal parameters. This could be formulated as:  
\[\left(\hat{A},\hat{\theta}_{0},\hat{f}_{0},\hat{f}_{1},\hat{f}_{2}\right)= \text{max}\Bigg{\{}\left(\frac{\pi N_{0}}{T_{s}}\right)^{-N}\exp\left[-\sum_{ i=-N/2}^{(N/2)-1}\lvert\widetilde{r}_{i}-\widetilde{s}_{i}\rvert^{2}\right] \Bigg{\}}, \tag{8}\]  
where \(\widetilde{r}_{i}=I_{i}+j\times Q_{i}\) is the digitized raw complex signal sample, and \(\widetilde{s}_{i}=\widetilde{s}(t_{i})\). [PERSON], et al.