## 2 Signal Model and Problem Formulation
(2008) give that the corresponding ML estimation for each parameter of interest, which are given without derivation, are as follows:  
\[\hat{\theta}_{0}=\text{arg}[\sum_{i}\overline{z}_{i}e^{-j2\pi\theta_{i}t_{i}}], \tag{9}\]  
\[\hat{A}=\frac{1}{N}\Big{\lvert}\overline{z}_{i}e^{-j2\pi\theta_{i}t_{i}}\Big{\rvert}, \tag{10}\]  
\[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]0  
where:  
\[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]1  
Although ML estimation does not designate which parameter should be estimated first, the given maximum energy search approach follows the interpretation that the raw signal samples are first compensated by a hypothesized Doppler dynamic phase model; then, the modified samples are used to make a periodogram estimation, as shown in Equation (13) (2008):  
\[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]2  
If we use a second-order hypothesized Doppler dynamic phase model to compensate for the raw signal samples, then this approach is a two-dimensional (2D) maximum energy search. This is shown in Figures 1 and 2. For each possible frequency rate value, the periodogram estimation is made, and the highest energy as well as the corresponding estimated frequency is selected for this frequency rate trial. After all the possible frequency rate trials have been made, a two-dimensional signal energy matrix has been acquired, within which the global maximum energy could be found, and the corresponding frequency index and the frequency rate index are the corresponding ML estimation values. The 2D search approach is an open loop batch processing method. It computes the entire image of the signal for a given block of raw data, which allows immediate tracking recovery after a temporary signal loss ([PERSON] and [PERSON], 2008). It also does not rely on any a priori information, and thus provides more robust tracking capabilities, although a priori Doppler dynamic phase information may help improve the estimation performance in the carrier estimation case [14]. Besides, the 2D search approach tries to find the maximum energy among all the potential frequency cells, while Equation (13) could also be interpreted as:  
\[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]3  
which means that the maximum energy could also be searched among frequency rate cells, the second-order frequency rate cells, or there could be a joint search. When all of the Doppler parameters \((\hat{f}_{0},\hat{f}_{1},\hat{f}_{2})\) have been correctly estimated and demodulated, the value of statistics \(L(f_{0},f_{1},f_{2})\) would equal to \(N\times A^{2}\). The estimation performance largely relies on the length of the coherent integration time, which corresponds to the raw signal sample numbers, \(N\). If a longer coherent integration could be made over something other than frequency cells, estimation performance may be improved. Another potential factor that limits the 2D search approach is that its searching space is two-dimensional, which means that the chance that the energy of a noise-only frequency cell exceeds the energy of a signal-plus-noise frequency cell seems to be large. If the size of the searching space could be reduced, then the estimation performance may also be improved. \[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]4 Block diagram of two-dimensional (2D) maximum energy search. \[\widetilde{s}(t)=A\exp\{j[\omega_{c}t+\theta(t)]\}, \tag{2}\]5 An example of maximum energy search in the two-dimensional (2D) space of frequency cell \(\times\) frequency-rate cell.