# Deep Learning of GNSS Acquisition
## 5 Results  
The proposed data-driven signal detection scheme was tested, and its performance was assessed through simulated data. The details of the model can be consulted in Section 3, while the training process is discussed in Section 4. While the training of the model was conducted using CAF images produced by 1 ms coherent integration times, the overall method was tested with and without noncoherent integration schemes. Particularly, when considering noncoherent integration times, \(K=6\) was considered. To assess the performance of the detection scheme, its ROC curves were empirically obtained through simulations and compared with the theoretical performance of standard methods (as reviewed in Section 2.2 or more in-depth in [10]). Figure 6 provides results for \(K=6\) noncoherent integration periods (dashed lines), as compared with the theoretical performance (solid lines) of standard methods (aiming at maximizing the CAF) with the coherent/noncoherent values. Results show that whereas at low \(C/N_{0}\) values the proposed method can barely achieve the state-of-the-art performance, it does remarkably well at larger \(C/N_{0}\) values. It is worth noting that the improved performance starts at \(C/N_{0}\) as low as 36 dB-Hz, which could be considered to be on the limit of the moderate-low range. An explanation is that for low \(C/N_{0}\), the DNN cannot extract the relevant features from the corresponding _subimage_, \(\mathbf{Z}_{i}^{(m,n)}\), but at higher \(C/N_{0}\) values, the relevant features can be extracted, and the classification task successfully performed with desirable \(P_{d}\) and \(P_{fa}\) rates. Surprisingly, the results in Figure 6 also show that the proposed data-driven scheme outperforms current performance bounds, suggesting it is leveraging additional information. This additional information comes from the prior that is embedded in the classifier through the seen training dataset. More precisely, whereas standard methods are based on the maximization of the CAF and identifying the associated bin, the proposed data-driven method exploits the correlation across neighboring bins to compute the class probability. That is, the classifier uses a subimage that contain a detail of the CAF that, under \(\mathcal{H}_{1}\), contains the relevant waveform of the CAF and its delay/Doppler correlated values. On the other side, it is worth mentioning that the performance of the scheme for \(K=1\) is substantially degraded compared with standard model-based schemes. This is explained by the low signal-to-noise ratio in this situation, as argued similarly earlier. A benefit of the Bayes test approach considered in this work is that the adjustment of the detection threshold \(\gamma\) in (19) (or the one in (23) when \(K>1\)) has a probabilistic interpretation: how much larger the posterior probability of \(\mathcal{H}_{1}\) has to be from \(\mathcal{H}_{0}\) to be accepted. A reasonable choice would be \(\gamma=1\), such that one picks the class with the largest posterior probability. Figure 6 shows the ROC results when such a choice is made for the detection threshold. According to the results, \(\gamma=1\) provides good results for \(C/N_{0}>36\) dB-Hz, with low false alarm and outstanding detection probabilities. For the sake of completeness, Figure 7 shows the false alarm and the detection probabilities corresponding to the ROC in Figure 6.  
Figure 6: ROC curves (detection versus false-alarm probability) for a recording with 1 ms coherent integration and \(K=6\) noncoherently processed blocks. Several relevant \(C/N_{0}\) values are shown. The detection performance of the proposed scheme (dashed lines) is compared with the theoretical performance of standard methods (solid lines). Figure 7: Probabilities for a 1 ms coherently integrated snapshot, \(K=6\) noncoherent processing, and a variety of \(C/N_{0}\) values.