# A Fine Fast Acquisition Scheme for a Communication and Navigation Fusion System
## 2 System Model and Acquisition Method
### Acquisition Method  
#### 2.2.1 CAF Evaluation  
CAF evaluation is the stage where the CAF is generated and processed. The radio frequency front end of CNFS positioning receiver firstly receives CNFS positioning signal and converts it to an intermediate frequency (IF) signal. The analog-to-digital converter (ADC) module is then utilized to realize the sampling of IF signal. Therefore, the received signal in discrete time domain can be expressed in the complex form as  
\[\mathrm{r(n)}=\sum_{i=1}^{\mathrm{N_{BS}}}\mathrm{A^{(i)}d^{(i)}(nT_{s})c^{(i)} (nT_{s})e^{i(2\pi(tf_{\mathrm{IF}}+f_{\mathrm{d}}^{(i)})nT_{s}+\varphi_{0}^{(i) })}+v(nT_{s})} \tag{3}\]  
Figure 1: Structure of FSBS. Figure 2: CNFS network topology. where \(\rm{N_{BS}}\) is the total number of FSBS signal received, \(\rm{T_{s}}\) denotes the sampling duration, \(\rm{A}\) is the amplitude of the signal, \(\rm{c(nT_{s})}\) is the PRN code utilized in CNFS, \(\rm{d(nT_{s})}\) is the navigation message, \(\rm{f_{ff}}\) is the intermediate frequency, \(\rm{f_{d}}\) is the Doppler frequency, \(\rm{\varphi_{0}}\) is the initial phase of the carrier and \(\rm{v(nT_{s})}\) is the additive white Gaussian noise (AWGN) with one-sided power spectral density (PSD) of \(\rm{N_{0}}\). Generally, CAF evaluation techniques used inside the receivers are based on the matched filter method or cyclic correlation method in time domain, or fast Fourier transform (FFT) method in frequency domain [33]. They give the same (or nearly the same) results and the choice of these methods mainly depends on the implementation of receiver hardware and software. Figure 3 gives the structure of match filter method. The received signal \(\rm{r(n)}\) is multiplied by two orthogonal sinusoids at different frequency and two branches of signal are generated, which are called in-phase and quadrature branches. The two signals are then multiplied by local replica code at different code ID or different delay. Coherent and non-coherent integration are used to improve the SNR of processed signal. For the convenience of analysis, only one FSBS signal is considered. The FFT based parallel acquisition approach in time domain uses FFT and Invert FFT (IFFT) to realize the correlation between received and local PRN code. PMF-FFT method combines the partial match filter (PMF) and FFT module, whose structure is shown in Figure 4. The integration result \(\rm{P_{MF}(\tau_{e},f_{e})}\) and \(\rm{P_{PMF-FFT}(\tau_{e},f_{e})}\) can be expressed as Equations (4) and (5). \[\rm{P_{MF}(\tau_{e},f_{e})}=\frac{1}{N_{nc}}\sum_{i=1}^{N_{nc}}a^{2}sin\,c^{2} (f_{e}T_{coh})R^{2}(\tau_{e})+v_{p}(n) \tag{4}\]  
\[\rm{P_{PMF-FFT}(\tau_{e},f_{e})}=\frac{1}{N_{nc}}\sum_{i=1}^{N_{nc}}a^{2}\! \left(\frac{\sin(\pi f_{e}PT_{s})\sin(\pi f_{e}NPT_{s}-\pi kN/F)}{\sin(\pi f_{ e}T_{s})\sin(\pi f_{e}PT_{s}-\pi k/F)}\right)^{2}\!R^{2}(\tau_{e})+v_{p}(n) \tag{5}\]  
\[\rm{R(\tau_{e})}=\left\{ \begin{array}{cc}1-|\tau_{e}|/T_{c},&|\tau_{e}|<T_{ c}\\ 0,&\rm{others}\end{array} \right. \tag{6}\]  
\[\rm{f_{e}=f_{IF}+f_{d}-f_{f},\ \tau_{e}=\tau-\tau_{f}} \tag{7}\]  
where \(\tau_{e}\) is the difference between received signal delay \(\tau\) and local replica code delay \(\tau_{l}\), \(\rm{f_{e}}\) is the difference between received signal carrier frequency \(\rm{f_{IF}+f_{d}}\) and local generated sinusoids frequency \(\rm{f_{f}}\), \(\rm{N_{nc}}\) is the number of data summed in non-coherent integration (NCT) calculation, a is the signal amplitude. \(\rm{T_{coh}}\) is the coherent integration time (CIT), \(\rm{v_{p}(n)}\) is noise component with power \(\rm{\sigma_{n}^{2}}\) after integration process, \(\rm{P}\) is the length of partial match filter, \(\rm{N}\) is the number of partial match filter, \(\rm{F}\) is the length of FFT operation, and \(\rm{k}\) indicates the index of FFT output. \(\rm{R(\tau_{e})}\) is the coherent integration result of code in received signal with local replica code, \(\rm{T_{c}}\) is the chip duration of the PRN code.