# Deep Learning of GNSS Acquisition
## 3 Deep Learning Method for GNSS Acquisition
### Model Structure
These subimages are separately fed to multiple (parallelized) NNs that provide the corresponding class probability conditional on  
Figure 2: Classification of signal (\(\mathcal{H}_{1}\)) or noise (\(\mathcal{H}_{0}\)) in CAFs as part of the proposed GNSS signal acquisition scheme. Particularly, a set of convolutional layers followed by fully connected layers provide the capabilities of deep learning from large datasets. a specific delay/Doppler hypothesis or bin. The concept is sketched in Figure 3, where the \(\{m,n\}\)-th subimage corresponds to the correct location of the delay/Doppler. The output of the DNN structure, labeled as \(K=1\) in the plot, is the probability ratio map derived by the Bayesian hypothesis test. The subimages are possibly overlapping regions of the full CAF map, which are centered at a specific delay/Doppler bin hypothesis. While this might increase the overall computational complexity, it enables direct parallelization of the task, which would counteract such complexity. Recall that indices \(m\) and \(n\) map to the corresponding delay \([\boldsymbol{\tau}]_{m}\) and Doppler \([\mathbf{f}_{d}]_{n}\) values, respectively. Therefore, the \(\{m,n\}\)-th subimage for the \(i\)-th satellite will be defined as  
\[\mathbf{Z}_{i}^{(m,n)}=[\mathbf{Z}_{i}]_{m+\delta_{m},n+\delta_{n}} \tag{18}\]  
where \(\delta_{m}=[-\Delta_{m},\ldots,0,\ldots,\Delta_{m}]\) and \(\delta_{n}=[-\Delta_{n},\ldots,0,\ldots,\Delta_{n}]\) for some positive integers \(\Delta_{m},\Delta_{n}\in\mathbb{Z}^{+}\), thus resulting in a su-image dimension of \((2\Delta_{m}+1)\times(2\Delta_{n}+1)\), which is much smaller than the original CAF dimension of \(n_{\tau}\times n_{f}\). Figure 4 provides an example of an arbitrary subimage \(\mathbf{Z}_{i}^{(m,n)}\). As a consequence of the splitting image approach, the statistical test in (14) is in reality implemented for each subimage, such that  
\[\mathcal{T}(\mathbf{Z}_{i}^{(m,n)})\triangleq\frac{p(\mathcal{H}_{1}|\mathbf{ Z}_{i}^{(m,n)})}{p(\mathcal{H}_{0}|\mathbf{Z}_{i}^{(m,n)})}\gtrappro{\mathcal{H}_{1}}{ \mathcal{H}_{0}}1 \tag{19}\]  
is computed for each \(\{m,n\}\) pair, resulting in a _probability ratio map_ (in contrast to the CAF map) for every test delay and Doppler value in \(\boldsymbol{\tau}\) and \(\mathbf{f}_{d}\). Recall that \(m=\{1,\ldots,n_{\tau}\}\) and \(n=\{1,\ldots,n_{f}\}\). It is worth noting that the probability ratio map may contain false peaks, as shown in Figure 3 under \(K=1\). To mitigate those potential false detections, Section 3.3 describes a methodology to fuse noncoherent integrations of \(K\) DNN outputs. The effect of those integrations is depicted in Figure 3 in the rightmost panel for \(K=6\) noncoherent integrations, where the signal probability is accentuated in the correct delay/Doppler bin, while false peaks arising from noise are attenuated in the fused probability ratio map. Figure 3: Proposed acquisition method, where the CAF map \(\mathbf{Z}_{i}\) is split into smaller subimages \(\mathbf{Z}_{i}^{(m,n)}\) which are fed to a bank of parallel DNN binary classifiers to produce probability ratio maps. To increase accuracy, several (\(K>1\)) probability ratio maps can be noncoherently fused, as shown in the rightmost plot.