# A Fine Fast Acquisition Scheme for a Communication and Navigation Fusion System
## 4 Derivation of Acquisition Time  
The previous section discussed search space and probability characteristics of detection, which provided the fundamental parameters for the acquisition time analysis. MAT mentioned in Section 1 only consider the bi-dimensional search space, but the number of search space is not taken into account. Therefore, the MAT analyzed in this section consider three factors which are search space size, search space number and the detection performance. The derivation of MAT is based on the knowledge of Markov chain and flowchart (Mason Gain Formula). The search process is depicted using flowchart in Figure 8. The block is represented by pie and is also called status. Code ID, code phase estimation and frequency coarse estimation is obtained in the first dwell and fine frequency estimation is obtained in the second dwell. In the first dwell, the search starts with the first block and all the blocks are tested one by one. The H1-block is located at \(p\)-th block and \(\mathrm{p}\) is integer whose range is from 1 to \(\mathrm{N_{B}}\) and \(\mathrm{N_{B}}\) is the block number. If H1-cell in H1-block is successfully detected, the status switches into second dwell. We can assume that there is only one block in second dwell because the number of cells to be tested is dramatically reduced. Successful acquisition is declared if the H1-cell in the second dwell is successfully detected. The transfer between two status is represented by transfer function (TF) \(\mathrm{H(z)}\) or \(\mathrm{h(z)}\). The TF between two H0-blocks is represented by \(\mathrm{H_{0}(z)}\) which is shown in Figure 9a. Status will transfer to false alarm status when false alarm occurs. System can then detect this false alarm in tracking loop easily. The extra time spent in this process is called FAPT. The TF \(\mathrm{H_{1}(z)}\) from H1-block to H0-block has  
Figure 8: Flowchart of search procedure. similar structure as shown in Figure 9b. \(\rm{h_{1}(z)}\) is the TF from H1-block in second dwell to the H0-block next to H1-block in first dwell. All the TF is expressed as the equation below:  
\[\rm{H_{Bd}(z)}=P_{Bd1}z^{t_{1}},h_{Bd}(z)=P_{Bd2}z^{t_{2}}\]  
\[\rm{H_{Bm}(z)}=P_{Bm1}z^{t_{1}},h_{Bm}(z)=P_{Bm2}z^{t_{2}}\]  
\[\rm{H_{Be}(z)}=P_{Be1}z^{t_{1}},h_{Be}(z)=P_{Be2}z^{t_{2}}\]  
\[\rm{H_{Bcr}(z)}=P_{Bcr1}z^{t_{1}},h_{Bcr}(z)=P_{Bcr2}z^{t_{2}}\]  
\[\rm{H_{Bfa}(z)}=P_{Bfa1}z^{t_{1}},h_{Bfa}(z)=P_{Bfa2}z^{t_{2}}\]  
\[\rm{H_{P}(z)}=z^{t_{P}}\]  
\[\rm{H_{0}(z)}=H_{Bcr}(z)+H_{Bfa}(z)\Big{(}h_{Bcr}+h_{Bfa}(z)H_{P}(z)\Big{)}\]  
\[\rm{H_{1}(z)}=H_{Bm}(z)+H_{Be}(z)\Big{(}h_{Bcr}(z)+h_{Bfa}(z)H_{P}(z)\Big{)}\]  
Figure 8:0  
where \(\rm{P_{Bd1}}\), \(\rm{P_{Bm1}}\), and \(\rm{P_{Bel}}\) donate the probability of block detection, block miss detection and block error in the first dwell, \(\rm{P_{Bd2}}\), \(\rm{P_{Bm2}}\), and \(\rm{P_{Be2}}\) donate the corresponding probability in the second dwell, \(\rm{t_{1}}\) and \(\rm{t_{2}}\) is the dwell time in first and second dwell, \(\rm{t_{P}}\) is the FAPT. The search can start at any block, and the TF from the beginning to acquisition status is given as  
Figure 8:1  
Figure 8:2  
Figure 8:3  
where \(\rm{\pi_{i}}\) is the probability information about the beginning of the search. It is reasonable to assume that \(\rm{\pi_{i}}=1/N_{B}\). Then, Equation (27) can be rewritten as  
Figure 8:4  
The time taken (\(\rm{\overline{T}_{1}}\)) for the search space which has H1-block (H1-SS) can be obtained by calculating derivative of the TF in Equation (30)  
Figure 8:5  
Figure 8:6  
Figure 8:7  
If the H1-SS is the first H1-SS with successful acquisition, the block number \(\rm{N_{B}}\) is obtained according to the Equation (31)  
Figure 8:8  
Acquisition module can reduce the code phase search space size using CPPI after the first code successfully acquired.