# Deep Learning of GNSS Acquisition
## 3 Deep Learning Method for GNSS Acquisition
### Data-Driven, Physics-Based Signal Acquisition  
This section formulates the probabilistic hypothesis test, which in this work is solved through a data-driven approach. More precisely, the proposed approach is also informed by the nominal model discussed in Section 2, whereby the CAF for a given satellite \(\mathcal{C}_{i}(\cdot,\cdot)\) is computed in order to extract the signal from the noise floor, thus enabling acquisition by the data-driven model. The intuition is that the _physics_ of the problem are accounted for (that is, the optimal solution using the CAF), while augmented with a data-driven model in the vein of [48]. The data fed to the NNs are the CAF's delay/Doppler map for the \(i\)-th satellite, which we denote with \(\mathbf{Z}_{i}\in\mathbb{R}^{n_{\tau}\times n_{f}}\) in the sequel. The proposed methodology works on a per-satellite basis. That is, the \(\{m,n\}\) element of the input matrix is defined as  
\[[\mathbf{Z}_{i}]_{m,n}=\left|\mathcal{C}_{i}([\tau]_{m},[\mathbf{f}_{d}]_{n}) \right|^{2}\,, \tag{13}\]  
where \(\tau\) and \(\mathbf{f}_{d}\) are vectors containing the computed delay and Doppler-shifts, respectively. We use the convention that \([\mathbf{a}]_{m}\) represents the \(m\)-th element in the vector, \(\mathbf{a}\), and that \([\mathbf{A}]_{m,n}\) provides a shortcut for the element of \(\mathbf{A}\) in the \(m\)-th row and \(n\)-th column. Figure 1: CAF evaluation at the delay/Doppler grid in the (**a**) absence and (**b**) presence of a signal with \(C/N_{0}=39\) dB-Hz. In the Bayesian sense, the information of the models is gathered in their a posteriori distribution after observing the data. An optimal (Bayesian) test between \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\) is given by the ratio,  
\[\frac{p(\mathcal{H}_{1}|\mathbf{Z}_{i})}{p(\mathcal{H}_{0}|\mathbf{Z}_{i})} \gtrapprox 1\, \tag{14}\]  
in which case we basically favor the model with the largest a posteriori probability. This can be further expanded in terms of the likelihood and a priori distributions as  
\[\frac{p(\mathbf{Z}_{i}|\mathcal{H}_{1})}{p(\mathbf{Z}_{i}|\mathcal{H}_{0})} \frac{\mathbb{P}(\mathcal{H}_{1})}{\mathbb{P}(\mathcal{H}_{0})}\gtrapprox 1\, \tag{15}\]  
where we readily identify that \(\mathbb{P}(\mathcal{H}_{i})\) denotes the a priori probability of the \(i\)-th hypothesis. In the absence of better priors, we may assume equally likely hypotheses \(\mathbb{P}(\mathcal{H}_{0})=\mathbb{P}(\mathcal{H}_{1})=1/2\). Otherwise, we might incorporate that information in the hypothesis test, resulting in the adjustment of a threshold \(\gamma\). The resulting test statistic is such that  
\[\mathcal{T}(\mathbf{Z}_{i})\triangleq\frac{p(\mathbf{Z}_{i}|\mathcal{H}_{1}) }{p(\mathbf{Z}_{i}|\mathcal{H}_{0})}\gtrapprox 1\, \tag{16}\]  
which would substitute the standard acquisition test defined in (9). Since the test statistic is a ratio of probabilities, we have that \(0<\mathcal{T}(\mathbf{Z}_{i})<\infty\). The trained NNs (explained below) are then providing the probabilities of each of the two hypotheses in (16). Therefore, the input data would be \(\mathbf{Z}_{i}\) and the output of the NN would be the estimated probability for the \(i\)-th satellite to be absent or present in the dataset \(\mathbf{y}\) used to build \(\mathbf{Z}_{i}\). If the test in (16) results in favor of \(\mathcal{H}_{1}\), then an estimate of the delay/Doppler for the \(i\)-th satellite is given by the arguments of the largest element in \(\mathbf{Z}_{i}\). That is,  
\[\{\hat{m},\hat{n}\}=\arg\max_{m,n}\ \mathcal{T}([\mathbf{Z}_{i}]_{m,n}) \tag{17}\]  
such that \(\hat{\tau}_{i}=[\boldsymbol{\tau}]_{i\hat{n}}\) and \(\hat{f}_{d,i}=[\mathbf{f}_{d}]_{\hat{n}}\). As a consequence, once a signal is detected at a specific delay/Doppler bin, that would become the coarse estimate for those parameters. NNs are models composed of neurons, which are information processing units for complex data processing.