# Energy Efficient GNSS Signal Acquisition Using Singular Value Decomposition (SVD)
## 5 Numerical Results
### Datasets  
The datasets were created using two signal records from the book \"A Software Defined GPS and Galileo Receiver\", the files: GPSdata-DiscreteComponents-fs38_192-if9_55.bin (collected at the University of Colorado, Boulder, CO, USA), GPS_and_GIOVE_A-NN-fs16_3676-if4_1304.bin (collected in Turin, Italy), and a third dataset Feb6.us.bin (collected in Randsburg, CA, USA). Interestingly enough, the algorithm was validated with real GNSS-recorded data under avionic conditions for the receiver. Dataset 3 contains GNSS data from a high-power rocket flight that captured GPS RF data for post-processing. A description of the data is presented now in Table 1, and the parameters necessary for processing the datasets are as follows:  
The C/A code repeats every millisecond, but the data packets are modulated by the C/A code every 50 bps, and there is a possibility of a bit transition every 20 ms; therefore, a 1 ms chunk of data is reliable for satellite acquisition and is widely adopted in practice [5]. To ensure good probability of successful acquisition 763,840 samples from Dataset 1 were divided into 10 segments of 76,384 samples, and each segment was segmented into 40 vectors, each of a length of 1910. Thus, we have \(m=1910\), \(n=40\) and \(t=76\),400. For illustration purposes, after an acquisition using the algorithm with 2.5% (\(\frac{1910}{76,400}\)) or compressed measurements R = 0.025 which means the Toeplitz matrix has N = 40 columns for Dataset 1, results showed the signal is acquired by the acquisition software, and results match the ones in the book \"A Software-Defined GPS and Galileo Receiver\". Regarding the validation of the algorithm, PRN 21 is present, since the SoftGNSS Version 2 software detects this satellite, it validates the developed algorithm. Additionally, the book indicates that the file already mentioned does not include PRN 19, the results from SoftGNSS does not detect it either, but detects all the PRN as stated in the book. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data Set** & **File Name/Reference** & **Sample Frequency (MHz)** & **Intermediate Frequency** & **Signed Character** & **Doppler Frequency Search** \\ \hline \multirow{2}{*}{1} & GPSdata-DiscreteComponents-fs38\_192-if9\_55.bin (\(\mu\)42) & 38.192 & 9.55 MHz & Bit8 & \(\pm 10\) kHz \\ \hline \multirow{2}{*}{2} & GPS,and_GIOVE_A-NN-fs16\_3676-if4\_1304.bin (\(\mu\)42) & 16.3676 & 4.1304 MHz & Bit8 & \(\pm 10\) kHz \\ \hline \multirow{2}{*}{3} & Feb6.us.bin/ [43] & 2.048 & 2210.53 Hz & uchar & \(\pm 10\) kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Datasets. Figure 12 shows the correlation outputs for a signal that was acquired using a regular method, which is a parallel code phase search, with a Doppler search step: 500 Hz and 2 ms data length (corresponding to 1 PRN) sampled at Fs = 38.192 MHz. The acquisition is successful when a satellite is visible, and one is provided with a coarse estimation of the carrier frequency of the GPS raw signal, as well as its code phase. In theory, only one dominant peak should be observed at the correct code phase-frequency bin combination. Peaks of smaller magnitude may coexist due to signal and noise interference [38]. Traditional methods for acquisition performance assume the satellite is acquired if a certain threshold is obtained. The computation of the metric is obtained by SoftGNSS by dividing the maximum peak coefficient by the second highest correlation peak in the same frequency bin, and that threshold was set to 2.5. The correlation peak is shown in Figure 13a,b, and by comparing both figures the peak size can be observed as larger by two orders of magnitude for the compressed sensing signal. Figure 13 reverberates what it was described before in Section 3.2, the most dominant coefficients are the only useful information, and it is representative of the signal's time delay and Doppler shift.