## 2 Method Description  
As mentioned in Section 1, the advantage of methods based on folding and averaging is poor since SNR deterioration causes a decline of detection probability and an increased false alarm probability. In addition, the time cost of other unfolding methods such as ZP and dual-channel is unacceptable. For these reasons, the proposed DC-XFAST method folds local samples for fast acquisition, uses multi-peak verification to guarantee low false alarm probability and applies max-valuedetection to achieve high detection probability. As shown in Figure 1, the steps of the overall algorithm are as follows:  
Step 1. The incoming baseband signal is sampled by analog-to-digital converter at an appropriate rate and properly compensated for Doppler offset, then delivered to two channels. Step 2. Choose FN local code samples, fold them into N new samples, and zero-pad N new samples to 2N. Step 3. Feed the local blocks into FFT. Step 4. Group the baseband samples of both channels into blocks of 2N, feed it into FFT, conjugated the FFT results and multiplied with the FFT results of Step 3. Step 5. Perform IFFT on multiplication result of Step 4, the correlation result of each channels contains two peaks, find the two largest value in the absolute values of IFFT output for each channel. Their positions are preserved for verification. Step 6. Verify according to the position results in Step 5. If the validation condition is not met, shift \((\mathrm{F}-4)\mathrm{N}+1\) unfolded local samples and follow the same process again. When the condition is met, acquisition is over. In Step 2, we discussed about how local block is constructed. The above steps set forth a two-level acquisition. In the first level, the max-value tactic is adopted and the positions of the two highest peaks are preserved. The second level verifies the result in the first level. In this step, only the relative positions of the peaks are of interest. Step 5 and Step 6 in the above procedure deserve further discussion. In Step 5, we mentioned that the correlation result in each channel contains two peaks. To represent a clearer correlation process of the proposed method, the observation window is shown in Figure 2. The solid lines of channel 1 and channel 2 indicates two incoming signal blocks with length N.  
As plotted in Figure 2, the block length N is adjustable in our method. A bigger block would result in a longer coherent length, and higher detection probability. Block m and block m + 2 are fully contained in channel, block m \(-\) 1, m + 1, m + 3 are partially contained in channel. Theoretically, 'fully contained' blocks will produce full peaks, 'partially contained' blocks will produce partial peaks. In the case of channel 1, the block m has code phase lead k compared with the incoming signal block in channel 1, and block m + 1 has code phase lead \((\mathrm{N}+\mathrm{k})\). Therefore, correlation peaks produced by block m and block m + 1 would occur at k and \((\mathrm{N}+\mathrm{k})\) respectively. The code phase difference between block m \(-\) 1 and the incoming signal block is \(-\mathrm{k}\). After circular correlation, the correlation peak produced by block m \(-\) 1 would occur at \((\mathrm{N}+\mathrm{k})\). The locations of correlation peak produced by block m \(-\) 1 and m + 1 are the same. As we mentioned in Step 2, local blocks are added together term by term to produce a new block with length N, then zero pad the new block to 2N. As a result, the partial peaks of m \(-\) 1 and m + 1 are stacked together and produce a full peak, m + 1 and m + 3 each produce another full peak. Figure 1: System model of DC-XFAST. As can be deduced from Figure 2, the magnitudes of the partial peaks produced by block m \(-\) 1 and block m + 1 are k and N \(-\) k respectively. Therefore, the magnitude of the full peak that contains two partial peaks, is the same as the full peak produced by \"fully contained\" block. Correlation results between two channels and local blocks are illustrated in Figure 3, which confirms our theoretical analysis.