# Energy Efficient GNSS Signal Acquisition Using Singular Value Decomposition (SVD)
## 5 Numerical Results
### Compression Performance  
The diagonal values are said to make up the singular value spectrum, and the importance of the singular values are given by its magnitude. To be more specific, the square root of each singular [45] value is proportional to the variance explained by each singular vector [46]. Assuming that small PODs are related to noise, we can use this assumption to reduce noise [46].  
Table 5 below compares the quality of the GNSS-SVD-C algorithm among several compression levels, as there is a direct relation with the PODs. Results show that increasing the compression increases the SNR, improving the quality of the signal, and the optimal value has to be calculated. The best approach is to determine the N number of columns from the scree plot (see Figure 6) and stop at that soft threshold. As can be seen, N = 40 columns for the Toeplitz matrix gives a SNR = 29.31 with an acquisition time of 2.70 s. The number of columns represent the rank of the TZ matrix and the number of \(y=U^{T}x\) compressed items for a given bucket on the compression algorithm. By increasing the number of columns, the acquisition time increases with no significant noise reduction, but with an increase in computational time. The increase of the peak size is relevant if the noise floor remains the same. From Table 5, the noise floor or bed can be inferred, increasing with increasing peak values which explains why the higher peaks reached by CS do not correspond with an increase in the SNR, in that sense the use of PODs are more relevant to obtain an optimal SNR. Another method to know if the algorithm enhances the acquisition is to compare both methods against the probability of detection.  
According to Figure 25a, the detection probability of the GNSS-SVD-C algorithm is similar to that of the FFT parallel code search algorithm with high SNR, which are both close to 1, and the false alarm probability is \(10^{-3}\). The detection probability of the FFT parallel code se  
Figure 24: PRN 16, channel 6, Dataset 3. Observe how the raw discriminator is saturate. The software discarded the PRN after processing.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Columns** & **Peak Size** & **Noise Floor Power** & **Mean Detection Time** & **SNR** \\ \hline
5 & \(3.49\times 10^{8}\) & \(3.30\times 10^{14}\) & 2.65 & 22.60 \\
10 & \(1.00\times 10^{9}\) & \(3.29\times 10^{15}\) & 2.70 & 25.48 \\
20 & \(4.55\times 10^{9}\) & \(5.93\times 10^{16}\) & 2.73 & 23.43 \\
30 & \(1.11\times 10^{10}\) & \(3.88\times 10^{17}\) & 2.70 & 24.39 \\
40 & \(2.08\times 10^{10}\) & \(1.37\times 10^{18}\) & 2.70 & 29.31 \\
50 & \(3.27\times 10^{10}\) & \(3.10\times 10^{18}\) & 2.79 & 22.95 \\
80 & \(8.73\times 10^{10}\) & \(1.77\times 10^{19}\) & 2.76 & 22.37 \\
150 & \(3.09\times 10^{11}\) & \(2.73\times 10^{20}\) & 4.31 & 27.82 \\
300 & \(1.15\times 10^{12}\) & \(4.42\times 10^{21}\) & 14.97 & 28.10 \\
350 & \(1.62\times 10^{12}\) & \(9.04\times 10^{21}\) & 144.09 & 23.89 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mean detection time and SNR, Dataset 1.  
rapidly along the decreasing SNR, while that of GNSS-SVD-C algorithm with R = 0.30 just reduces slowly. In addition, the detection probability of the FFT parallel code search algorithm deteriorates with the low SNR, and that of the GNSS-SVD-C also reduces rapidly, but lower than GNSS-SVD-C. Both algorithms perform the same above 20 dB with R = 0.30 for GNSS-SVD-C. FFT parallel code search algorithm performs better than GNSS-SVD-C when R = 0.90.  
Similar results are achieved in Figure 25b for the GPS and Giove satellites, the detection probability of the GNSS-SVD-C algorithm with R = 0.30 is 100% on the 20 dB SNR, close to the FFT parallel code search algorithm. FFT parallel code search algorithm performs better than GNSS-SVD-C when R = 0.90. According to Figure 26a,b the detection probability of the GNSS-SVD-C algorithm increases when the SNR increases and produces the best results for compression ratios from 0.10 to 0.30.