## 4 Performance and Simulation of 2D Maximum Energy Search Approach
### Algorithm Baseline and Theoretical Model  
Following the introduction in [PERSON] and [PERSON] [8; 14], the incoherent integration is used after the coherent summation in periodogram analysis to improve the carrier estimation performance. The carrier estimation algorithm (used for the direct carrier parameter estimation without using any previous information) is illustrated in Figure 6.  
The theoretical model of the carrier acquisition for the 2D maximum energy search approach is derived by [PERSON], et al. [14], and the basic functions are listed here for the following interpretation and comparison. The periodogram statistics is computed in Equation (15):  
\[E_{l}(f_{k})=\left|\sum_{i=0}^{N-1}\overline{z}_{i}\times\exp\left(-j\times 2 \pi\times f\times i\right)\right|^{2} \tag{15}\]  
The statistic \(E_{l}(f_{k})\) is non-central chi square with 2M (\(M=T\Delta F\)) degrees of freedom and the non-centrality parameter is \(\lambda=2\times T\times\text{CNR}\), provided that \(l\) is the true carrier Doppler rate index  
Figure 6: The block diagram of the 2D maximum energy search approach in Mars entry, descent, and landing (EDL) flights.  
and \(f_{k}\) is the true Doppler frequency. The standard probability density function for non-central chi square is shown in Equation (16):  
\[f(x|H_{1})=\frac{1}{2}exp\bigg{\{}-\frac{1}{2}(x+\lambda)\bigg{\}}\bigg{(}\frac {x}{\lambda}\bigg{)}^{\frac{M-1}{2}}I_{M-1}(\sqrt{\lambda x}), \tag{16}\]  
where \(I_{M-1}(\cdot)\) is the modified Bessel function of the first kind, and of order \(M-1\).  
If \(l\) does not agree with the true carrier Doppler rate index and/or \(f_{k}\) does not agree with the true Doppler frequency, the statistic \(E_{l}(f_{k})\) is central chi square with 2M degrees of freedom, and the corresponding probability density function is shown in Equation (17):  
\[f(x|H_{0})=\frac{x^{M-1}}{2^{M}\Gamma(M)}\exp{(-\frac{x}{2})}, \tag{17}\]  
where \(\Gamma(\cdot)\) is the Gamma function, and the cumulative distribution function is denoted as \(F(x|H_{0})\).  
The carrier is correctly acquired provided that the associated statistics, \(E_{l}(f_{k})\), is the largest one among the full search space (the detection threshold is set to 0 to maximize the acquisition possibility). So, the acquisition possibility is:  
\[P_{acq}=\int_{0}^{\infty}f(x|H_{1})\times[F(x|H_{0})]^{N_{f}-1}dx, \tag{18}\]  
and the possibility of missed carrier acquisition is:  
\[P_{Miss}=1-P_{acq}, \tag{19}\]  
where \(N_{f}\) is the total cell number in the full 2D search space, \(N_{f}=N_{FFT}\times N_{r}\), \(N_{FFT}\) is the FFT cell number, and \(N_{r}\) is the Doppler rate cell number.  
It could be seen that the carrier detection performance could be predicted using Equations (16)-(18), given the processing parameter value of FFT resolution \(\Delta F\) (inverse to the coherent integration time \(\Delta T\)), the full incoherent time \(T\) (thus, the number of data segments for incoherent summation, \(M\), could be computed by \(M=T\Delta F=T/\Delta T\)), and the full Doppler (corresponding to \(N_{FFT}\)) and Doppler rate (corresponding to \(N_{r}\)) search range.