# A Fine Fast Acquisition Scheme for a Communication and Navigation Fusion System
## 2 System Model and Acquisition Method
### Acquisition Method
#### 2.2.2 Signal Detection  
Once the CAF is evaluated, the positioning receiver can make the decision regarding the presence of FSBS and give the coarse estimation of \(\mathrm{f_{d}}\) and \(\mathrm{\tau}\). The MAX/TC criterion is widely used in DSSS-CDMA system to realize a trade-off between detection performance, hardware complexity, and acquisition time. The whole search space is divided into several blocks, and each block includes \(\mathrm{B_{f}}\times\mathrm{B_{c}}\) cells. The blocks are tested one-by-one while the cells in a block are tested in parallel. Different CAF evaluation method described in Section 2.2.1 can realize different block size based on different hardware resources. If the maximum CAF value in a block exceeds the pre-defined threshold, the successful detection is declared. Otherwise, it moves to the next block.  
The CAF value in each cell is also called detection variable. The decision of a cell is made based on two hypotheses: H0 and H1. H0-cell indicates the wrong estimation of \(\mathrm{f_{d}}\) and \(\mathrm{\tau}\) while H1-cell indicates the correct estimation. Block that contains H1-cell is called H1-block while others are called H0-blocks.  
The statistical characteristics of detection variable in a cell are determined by CAF evaluation method and detection method. The match filter method with NCT and power detection is taken as an example hereinafter. Assuming that the random variable in H0-cell is X and the random variable in H1-cell is Y. The probability density function (PDF) of both random variables can be expressed as [13]  
\[\mathrm{f_{X}}(\mathrm{x})=\frac{\mathrm{x}^{\mathrm{N_{nc}-1}}}{\left(2\sigma _{\mathrm{n}}^{2}\right)^{\mathrm{N_{nc}}}\mathrm{\Gamma}(\mathrm{N_{nc}})} \mathrm{e}^{-\frac{\mathrm{x}}{2\sigma_{\mathrm{n}}^{2}}} \tag{10}\]  
\[\mathrm{f_{Y}}(\mathrm{y})=\frac{1}{2\sigma_{\mathrm{n}}^{2}}\mathrm{e}^{- \frac{\mathrm{y}+\mathrm{N_{nc}}\mathrm{a}^{2}}{2\sigma_{\mathrm{n}}^{2}}} \Bigg{(}\frac{\mathrm{y}\sigma_{\mathrm{n}}^{2}}{\mathrm{N_{n}a}^{2}}\Bigg{)}^ {\frac{\mathrm{N_{bg}-1}}{2}}\mathrm{I_{0}}\Bigg{(}\frac{\mathrm{a}\sqrt{ \mathrm{N_{nc}}\mathrm{y}}}{\sigma_{\mathrm{n}}^{2}}\Bigg{)} \tag{11}\]  
where \(\mathrm{\Gamma}(\cdot)\) donates the gamma function, \(\mathrm{I_{0}}(\cdot)\) is the modified Bessel function of the first kind and zero order. For H1-block, there are three disjoint events: correct detection \(\mathrm{P_{Bd}}\), missed detection \(\mathrm{P_{Bm}}\) and error (false alarm in H1-block) \(\mathrm{P_{Be}}\). For H0-block, there are two disjoint events: correct rejection \(\mathrm{P_{Bcr}}\), false alarm \(\mathrm{P_{Bfa}}\). The probability of them can be expressed as [14]  
\[\mathrm{P_{Bd}}=\int_{\mathrm{th}}^{\infty}\mathrm{f_{Y}}(\mathrm{y})\Bigg{[} \int_{0}^{\mathrm{y}}\mathrm{f_{X}}(\mathrm{x})\mathrm{d}\mathrm{x}\Bigg{]}^{ \mathrm{B_{B}}\mathrm{c}}\mathrm{d}\mathrm{y} \tag{12}\]  
\[\mathrm{P_{Bm}}=\int_{0}^{\mathrm{th}}\mathrm{f_{Y}}(\mathrm{y})\mathrm{d} \mathrm{u}\Bigg{[}\int_{0}^{\mathrm{th}}\mathrm{f_{X}}(\mathrm{x})\mathrm{d} \mathrm{x}\Bigg{]}^{\mathrm{B_{B}}\mathrm{c}-1} \tag{13}\]  
\[\mathrm{P_{Be}}=1-\mathrm{P_{Bd}}-\mathrm{P_{Bm}} \tag{14}\]  
\[\mathrm{P_{Bcr}}=\left[\int_{0}^{\mathrm{th}}\mathrm{f_{X}}(\mathrm{x})\mathrm{ d}\mathrm{x}\right]^{\mathrm{B_{f}}\mathrm{B_{c}}} \tag{15}\]  
\[\mathrm{P_{Bfa}}=1-\mathrm{P_{Bcr}} \tag{16}\]  
where th is the detection threshold. Different \(\mathrm{B_{f}}\mathrm{B_{c}}\) and th lead to different performance, hardware complexity, and acquisition time. Fast acquisition requires a high probability of correct detection and low PFA in H1 block and H0 block. However, the fluctuation of background noise often causes the changes in these probabilities, which may increase the acquisition time.