# A Fine Fast Acquisition Scheme for a Communication and Navigation Fusion System
## 3 Fast Acquisition Method
### DD-MAX/TC-CACFAR Detection Method  
SSR method described above can dramatically reduce the acquisition time. However, the acquisition time is also influenced by detection performance. The decrease of probability of detection and increase of probability of false alarm cause long acquisition time. DD-MAX/TC-CACFAR detection method described in this section add adaptive threshold in MAX/TC criterion to realize stable detection performance when the background noise fluctuates. The search space is still divided into several blocks. Detection threshold is generated based on the detection variables in cells in the block. Therefore, the threshold is also a random variable. For the convenience of description, \(\mathrm{P}(\tau_{\mathrm{e}},\mathrm{f}_{\mathrm{e}})\) is rewritten as \(\mathrm{P}_{\mathrm{k}}\), where \(\mathrm{k}\) is the index of CAF value in one block. The threshold can be expressed as  
\[\mathrm{U}=\frac{\mathrm{T}}{\mathrm{B}_{\mathrm{f}}\mathrm{B}_{\mathrm{c}}-1 }\sum_{\mathrm{k}=1,\mathrm{k}\
eq\mathrm{k}_{\mathrm{m}}}^{\mathrm{B}_{ \mathrm{f}}\mathrm{B}_{\mathrm{c}}}\mathrm{P}_{\mathrm{k}} \tag{21}\]  
\[\mathrm{k}_{\mathrm{m}}=\max_{\mathrm{k}}(\mathrm{P}_{\mathrm{k}}),\mathrm{k} =1,2,\ldots,\mathrm{B}_{\mathrm{f}}\mathrm{B}_{\mathrm{c}} \tag{22}\]  
where \(\mathrm{T}\) donates the multiple parameter. Because the maximum variable is removed in the sums, all CAF values in Equation (21) can be seen as random variable \(\mathrm{X}\) in Equation (10). In the case, \(\mathrm{U}\) obeys Gamma distribution with shape parameter being \((\mathrm{B}_{\mathrm{f}}\mathrm{B}_{\mathrm{c}}-1)\mathrm{N}_{\mathrm{nc}}\) and scale parameter being \(\frac{2\mathrm{T}\sigma_{\mathrm{n}}^{2}}{\mathrm{B}_{\mathrm{f}}\mathrm{B}_{ \mathrm{c}}-1}\). If we set \(\mathrm{L}=\mathrm{B}_{\mathrm{f}}\mathrm{B}_{\mathrm{c}}-1\) for convenience, the PDF of \(\mathrm{U}\) can be expressed as  
\[\mathrm{f}_{\mathrm{U}}(\mathrm{u})==\frac{\left(\mathrm{L}\right)^{\mathrm{ LN}_{\mathrm{nc}}}}{\left(2\mathrm{T}\sigma_{\mathrm{n}}^{2}\right)^{\mathrm{ LN}_{\mathrm{nc}}}\Gamma(\mathrm{LN}_{\mathrm{nc}})}\mathrm{u}^{\mathrm{LN}_{ \mathrm{nc}}-1}\mathrm{e}^{-\mathrm{Lu}/2\mathrm{T}\sigma_{\mathrm{n}}^{2}} \tag{23}\]  
where \(\Gamma(\bullet)\) donates the gamma function. The five probabilities: probability of correct detection \(\mathrm{P}_{\mathrm{Bdi}}\), probability of missed detection \(\mathrm{P}_{\mathrm{Bmi}}\), probability of error (false alarm in H1-block) \(\mathrm{P}_{\mathrm{Bei}}\), probability of correct rejection \(\mathrm{P}_{\mathrm{Bcri}}\), and probability of false alarm \(\mathrm{P}_{\mathrm{Btai}}\) can be expressed as  
\[\mathrm{P}_{\mathrm{Bdi}} =\int_{0}^{\infty}\mathrm{f}_{\mathrm{Y}}(\mathrm{y})\left[\int_{ 0}^{\mathrm{y}}\mathrm{f}_{\mathrm{x}}(\mathrm{x})\mathrm{d}\mathrm{x}\right]^{ \mathrm{L}}\int_{0}^{\mathrm{y}}\mathrm{f}_{\mathrm{U}}(\mathrm{u})\mathrm{ d}\mathrm{d}\mathrm{y} \tag{24}\] \[\mathrm{P}_{\mathrm{Bmi}} =\int_{0}^{\infty}\mathrm{f}_{\mathrm{U}}(\mathrm{u})\left[\int_{ 0}^{\mathrm{u}}\mathrm{f}_{\mathrm{x}}(\mathrm{x})\mathrm{d}\mathrm{x}\right]^{ \mathrm{L}}\int_{0}^{\mathrm{u}}\mathrm{f}_{\mathrm{Y}}(\mathrm{y})\mathrm{ d}\mathrm{d}\mathrm{u}\] \[\mathrm{P}_{\mathrm{Bei}} =1-\mathrm{P}_{\mathrm{Bdi}}-\mathrm{P}_{\mathrm{Bmi}}\] \[\mathrm{P}_{\mathrm{Bcri}} =\int_{0}^{\infty}\mathrm{f}_{\mathrm{U}}(\mathrm{u})\left[\int_{ 0}^{\mathrm{u}}\mathrm{f}_{\mathrm{x}}(\mathrm{x})\mathrm{d}\mathrm{x}\right]^{ \mathrm{L}}\mathrm{d}\mathrm{u}\] \[\mathrm{P}_{\mathrm{Bfai}} =1-\mathrm{P}_{\mathrm{Bcri}}\]  
where different subscript \(\mathrm{i}\) donates different detection stage. \(\mathrm{i}=1\) represents first dwell coarse acquisition and \(\mathrm{i}=2\) represents second dwell fine acquisition. System only declares successful acquisition when both stages realize successful detection. Therefore, the detection performance of the acquisition process can be expressed as the equation  
\[