# Deep Learning of GNSS Acquisition
## 3 Deep Learning Method for GNSS Acquisition
### Noncoherent Integration through Fusion of Classifiers  
Coherent integration of long code sequences can be implemented in computing the CAF map, \(\mathcal{C}_{i}(\cdot,\cdot)\), in the usual manner. In implementing noncoherent integrations, an alternative is to fuse the multiple probability ratio maps resulting from processing CAF images through the NN architecture described earlier in Section 3.2. We denote by \(K\in\mathbb{Z}^{+}\) the total number of noncoherent integrations. This section discusses the data fusion of such multiple classifiers. It is known that increasing integration time (both coherently and noncoherently) improves the overall detection performance of the acquisition process, this same rational holds in the case of the data-driven classifier proposed here, whereby noncoherent integrations (i.e., fusion of multiple classifier solutions) improves the reliability of the so-called probability maps (i.e., by attenuating falsely detected peaks or enhancing locations where actual signals reside). When processing noncoherent snapshots of data, a set of \(K\) CAF maps is computed. In the standard approach, this would correspond to full CAF maps \(\mathbf{Z}_{i,k}\) with \(k=1,\ldots,K\). In the subimage approach, the result is a different subimage for every integration period, \(\mathbf{Z}_{i,k}^{(m,n)}\). In order to combine the class probabilities of the \(K\) classifiers (which are assumed conditionally independent given their own data), we use Bayes' rule to derive an optimal fusion rule. For an arbitrary \(\{m,n\}\) pair, the optimal Bayes detector based on the \(K\) noncoherent integrations is  
\[\mathcal{T}(\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})=\frac{p (\mathcal{H}_{1}|\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})}{p (\mathcal{H}_{0}|\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})} \gtrapprox 1\, \tag{20}\]  
where, by using the conditional independence assumption of the \(K\) snapshots, we obtain  
\[p(\mathcal{H}_{1}|\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})=\frac{\mathbb{P}(\mathcal{H}_{1})\prod_{k=1}^{K}p(\mathbf{Z}_{i, k}^{(m,n)}|\mathcal{H}_{1})}{p(\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})}\] \[\propto\left[\prod_{k^{\prime}=1}^{K-1}\mathbb{P}(\mathcal{H}_{1 })\right]^{-1}\prod_{k=1}^{K}p(\mathcal{H}_{1}|\mathbf{Z}_{i,k}^{(m,n)}) \tag{21}\]  
and  
\[p(\mathcal{H}_{0}|\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{ (m,n)})=\frac{\mathbb{P}(\mathcal{H}_{0})\prod_{k=1}^{K}p(\mathbf{Z}_{i,k}^{(m, n)}|\mathcal{H}_{0})}{p(\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})}\] \[\propto\left[\prod_{k^{\prime}=1}^{K-1}\mathbb{P}(\mathcal{H}_{0 })\right]^{-1}\prod_{k=1}^{K}p(\mathcal{H}_{0}|\mathbf{Z}_{i,k}^{(m,n)}) \tag{22}\]  
which explicitly contain the binary class probabilities of the \(K\) classifiers: \(p(\mathcal{H}_{0}|\mathbf{Z}_{i,k}^{(m,n)})\) and \(p(\mathcal{H}_{1}|\mathbf{Z}_{i,k}^{(m,n)})\). The statistical test can then be formulated as  
\[\mathcal{T}(\mathbf{Z}_{i,1}^{(m,n)},\ldots,\mathbf{Z}_{i,K}^{(m,n)})=\prod_{k =1}^{K}\frac{p(\mathcal{H}_{1}|\mathbf{Z}_{i,k}^{(m,n)})}{p(\mathcal{H}_{0}| \mathbf{Z}_{i,k}^{(m,n)})}\overset{\mathcal{H}_{1}}{\gtrless}\left(\frac{ \mathbb{P}(\mathcal{H}_{1})}{\mathbb{P}(\mathcal{H}_{0})}\right)^{K-1}\triangleq\gamma \tag{23}\]  
such that the decision threshold becomes \(\gamma=1\) when \(\mathbb{P}(\mathcal{H}_{0})=\mathbb{P}(\mathcal{H}_{1})\). It can be observed that the optimal fusion rule is to multiply the \(K\) binary class probabilities (similar to what was shown in [54]) resulting from the \(K\) noncoherent integrations processed by the NN classifier. The role of the decision threshold is relevant, as is discussed later, in establishing the \(P_{d}\) and \(P_{fa}\) of the overall classifier. A reasonable choice is to assume that both hypotheses are equally probable, such that \(\gamma=1\).