# Variance-Triggered Two-Step GPS Acquisition
## 3 Variance-Triggered Two-Step Acquisition
### The Coarse Search Step  
The proposed strategy computes partial correlation values for all cells of the search space using a group of interleaved Subsequent samples in such a way that one sample per chip is used at a time. Although computationally equivalent to when all samples are used, this scheme is considerably different from the conventional acquisition, which calculates the energy of each cell at a time using all samples and visiting each cell only once per phase and frequency pair. The proposed scheme allows for a reduction in the number of samples needed to acquire the signal, as detailed in Section 3.1.1. Another strategy that reduces computational complexity is applied to the frequency search, reducing the number of frequency bins in the coarse search. Consequently, the bandwidth of each bin is enlarged, reducing accuracy in this step, which is reestablished in the FS step. The rate of reduction in the number of bins is a design parameter of the receiver and should depend mainly on its application domain. The details of this dimensionality reduction strategy are presented in Section 3.1.2. The correlation values \(R_{C_{f,\hat{g}}}\) of the cells with probing estimated frequency \(\hat{f}\) and probing estimated phase \(\hat{\phi}\) in the coarse search are incrementally computed using 1 millisecond of the signal, i.e., with \(L=1\), as follows. \[R_{C_{f,\hat{g}}}^{d+1} = R_{C_{f,\hat{g}}}^{d}+\sum_{ \begin{subarray}{c}n=d\\ \text{steps of }P\end{subarray} }^{N-1}\left(C_{a}\left(\text{mod}\left(\frac{n}{P}+ \hat{\phi},1023\right)\right)S(n)\cos\left(2\pi\frac{\hat{f}}{f_{s}}n\right) \right)^{2}\;+\] \[\left(C_{a}\left(\text{mod}\left(\frac{n}{P}+\hat{\phi},1023 \right)\right)S(n)\sin\left(2\pi\frac{\hat{f}}{f_{s}}n\right)\right)^{2}\;,\] \[\forall\hat{f} \in \left\{f_{c}+\delta_{\text{CS}}\left(x-\frac{N_{b}^{\text{CS}}-1 }{2}\right)\mid x\in\mathbb{N},x\;<N_{b}^{\text{CS}}\right\},\] \[\forall\hat{\phi} \in \left\{y\mid y\in\mathbb{N},y\;<1023\right\},\] \[R_{C_{f,\hat{g}}}^{-1} = 0,\]  
where \(S\) is the vector with \(N\) samples of the incoming signal, \(P=\frac{N}{1023}\) is the number of samples per chip, \(d\) is the displacement of the interleaved samples in the range \([0,P-1]\), \(f_{s}\) is the sampling frequency, \(f_{c}\) is the central frequency used to transmit the signal, \(N_{b}^{\text{CS}}\) is the number of frequency bins of the CS step, which should be an odd number to include \(f_{c}\) in the middle of the range, and \(\delta_{\text{CS}}\) is the bandwidth of each of those bins. The vector \(C_{a}\) contains the sequence of the 1023 chips for the given satellite. The code \(C_{a}\) is shifted by the probing phase \(\hat{\phi}\), wrapped around over the 1023 th chip,and then multiplied by the incoming signal \(S\) modulated in phase and in quadrature. The in-phase and quadrature multiplication are then each squared and then accumulated. With the proposed strategy, a coarse correlation peak rises gradually around the searched cell upon each partial increment of Equation (2). The pseudo code of the CS procedure is listed in Algorithm 1.