# GNSS Signal Compression Acquisition Algorithm Based on Sensing Matrix Optimization
## 4 Simulation Analysis  
Taking the GPS L1 signal as an example, the data rate \(R_{\rm b}\) = 50 bps, code rate \(R_{\rm c}\) = 1.023 MHz, sampling rate \(f_{\rm s}\) = 2\(R_{\rm c}\), code phase \(\tau\) = 400, code length \(N\) = 1024, measurement length \(M\) = 512, maximum iteration times \(t_{\rm max}\) = 100, the deviation range of Doppler frequency \(f_{\rm d}=\pm\)12 KHz. Table 2 gives the detail configuration of the test bed. Figure 3 shows the histogram of the absolute off-diagonal entries in the Gram matrices of different compression acquisition algorithms with the case of \(M\) = 128, \(N\) = 256, \(t_{\rm max}\) = 50. It can be seen that the off-diagonal entries in the Gram matrix constructed by the proposed CG algorithm are closer to 0 than the existing compression acquisition algorithms. Figure 4 shows the coherence of the sensing matrices. We can see that the coherence of the GP and the Elad is greater than the CG, which reduces the recovery probability of \begin{table}
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**Hardware** & **Parameters** \\ \hline CPU & Intel I7 10750H CPU @ 2.60 GHz \\ RAM & Kingston DDRA 2666 MHz 16 GB \\ Hard Disk & Seagate SATA 7200 RPM 1 TB \\ Graphics Card & NVIDIA GTX 1660 Ti \\ \hline \hline \end{tabular}
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Table 2: Configuration of the test bed. Figure 3: Histogram of the off-diagonal entries in the Gram matrix. Figure 2: The complexity of different acquisition algorithms. received GNSS signal. The proposed CG algorithm optimizes the objective function by using the modified conjugate gradient method, which can reduce the mutual coherence between the measurement matrix and the sparse basis. The mutual coherence of the three equivalent dictionaries is given in Table 3. As described in Section 3.1, the quality of sensing matrix optimization depends on the size of the off-diagonal element. As shown in Table 3, the mutual coherence of the optimized sensing matrix is 40.1% lower than the GP, and 23.5% lower than the ELAD. Thus, the column incoherence of the optimized sensing matrix is better than the Gaussian matrix, which means it is more beneficial to signal recovery and reconstruction. Figure 5 shows the acquisition results of several compression algorithms when SNR is \(-11\) dB. In practice, the greater the ratio of the highest correlation peak to the second highest correlation peak, the higher the detection probability. As we can see from Figure 5, the ratio of the highest correlation peak to the second correlation peak of the GP, ELAD, and CG is 2.055, 3.766, and 6.797, respectively, which reiterates what is previously presented in Section 3.2: the detection probability of the compression acquisition algorithm can be improved by sensing matrix optimization. Figure 6 shows the detection probability of several algorithms with different SNRs. The acquisition performance of the compression algorithms is determined by the detection probability when the SNR is the same. As illustrated in Figure 5(a), if the detection probability reaches 90%, the SNR of the GP, ELAD, and CG must be more than \(-13\) dB, \(-13\) dB, and \(-16\) dB, respectively. Hence, the CG method performs about 3 dB better than the others. When the \(M/N\) and code length changed, we can obtain similar conclusions, which means with the change of the \(M/N\) and code length, the detection probability of the proposed algorithm is higher than the other algorithms when the SNR is the same. The low column incoherence of the sensing matrix limits the reconstruction performance of the GP and ELAD acquisition algorithms, which affects the detection probability of acquisition. As analyzed in Section 3.2, conjugate gradient optimization makes the Gram matrix closer to the target matrix \(\mathbf{G_{t}}\), which makes the mutual coherence \(\mu(\mathbf{\Theta})\) smaller and improves the SNR of the acquisition algorithm. Figure 4: The comparison of coherence. (**a**) GP vs CG. (**b**) ELAD vs CG.