# Band-Pass Sampling in High-Order BOC Signal Acquisition
## 3 BOC Acquisition Based on Band-Pass Sampling
### Acquisition of Band-Pass Sampled BOC Signals  
Taking the velocity into consideration, the sampled signal can be expressed as:  
\[s_{BS}(nT_{s})=\,Aa_{1}\cdot c(nT_{s})\sin\bigl{[}2\pi\bigl{(}f_{sc}^{\prime} +f_{dsc}\bigr{)}nT_{s}\bigr{]}\cos\bigl{[}2\pi\bigl{(}f_{IF}^{\prime}+f_{d} \bigr{)}nT_{s}+\theta_{0}\bigr{]} \tag{16}\]  
where \(f_{d}\) is carrier Doppler; \(f_{dsc}\) is subcarrier Doppler; \(T_{s}\) is the sampling interval.  
Given \(v\) as the relative speed between the satellite and the receiver, the Doppler frequencies of the lower band and upper band are:  
\[\begin{array}{l}f_{dL}=-\frac{v}{c}(f_{0}-f_{sc})\\ f_{dU}=-\frac{v}{c}(f_{0}+f_{sc})\end{array}\]  
Then the Doppler frequencies of the carrier and subcarrier are:  
\[\begin{array}{l}f_{d}=\frac{f_{dL}+f_{dU}}{2}=-\frac{v}{c}f_{0}\\ f_{dsc}=\frac{f_{dU}-f_{dL}}{2}=-\frac{v}{c}f_{sc}\end{array}\]  
Equation (16) shows that the sampled signal is still a BOC signal, with subcarrier frequency \(f_{sc}^{\prime}\) and intermediate frequency \(f_{IF}^{\prime}\).  
In the acquisition process, the receiver must determine the carrier frequency offset, the subcarrier frequency offset and the time offset. Then the acquisition turns into a 3-dimensional searching process. However, since the ratio between \(f_{d}\) and \(f_{dsc}\) always equals the ratio between \(f_{0}\) and \(f_{sc}\); that is, \(f_{d}/f_{dsc}=f_{0}/f_{sc}\), the acquisition is actually a 2-dimensional searching process.  
An acquisition structure based on dual-loop is proposed in Figure 5.  
Figure 4: PSD (power spectrum density) and ACF (autocorrelation function) of the sample d BOC (binary offset carrier) signals. (**a**) PSD; (**b**) ACF.  
In Figure 5, \(k=f_{d}/f_{dsc}=f_{0}/f_{sc}\). The correlators' outputs are:  
\[\begin{array}{ll}II&=\frac{Aa_{1}}{N}\sum\limits_{n=0}^{N-1}s_{BS}(nT_{s})c(nT_ {s}-\varepsilon)\cos\Bigl{(}2\pi\hat{f}_{IE}nT_{s}+\hat{\theta}_{0}\Bigr{)}\sin \Bigl{(}2\pi\hat{f}_{sc}nT_{s}+\hat{\theta}_{sc}\Bigr{)}\\ &=\frac{Aa_{1}}{4N}\sum\limits_{n=0}^{N-1}c(nT_{s}-\varepsilon)\cos(2\pi \Delta f_{dsc}t+\Delta\theta_{sc})\cdot\cos(2\pi\Delta f_{d}t+\Delta\theta_{0}) \\ &\approx\widetilde{R}(\varepsilon,\Delta f_{dsc},\Delta f_{d})\cos[\pi\Delta f _{dsc}(N-1)T_{s}+\Delta\theta_{sc}]\cos[\pi\Delta f_{d}(N-1)T_{s}+\Delta \theta_{0}]+Z_{II}\\ IQ&\approx\widetilde{R}(\varepsilon,\Delta f_{dsc},\Delta f_{d})\sin[\pi \Delta f_{dsc}(N-1)T_{s}+\Delta\theta_{sc}]\cos[\pi\Delta f_{d}(N-1)T_{s}+ \Delta\theta_{0}]+Z_{IQ}\\ QI&\approx\widetilde{R}(\varepsilon,\Delta f_{dsc},\Delta f_{d})\cos[\pi \Delta f_{dsc}(N-1)T_{s}+\Delta\theta_{sc}]\sin[\pi\Delta f_{d}(N-1)T_{s}+ \Delta\theta_{0}]+Z_{QI}\\ QQ&\approx\widetilde{R}(\varepsilon,\Delta f_{dsc},\Delta f_{d})\sin[\pi \Delta f_{dsc}(N-1)T_{s}+\Delta\theta_{sc}]\sin[\pi\Delta f_{d}(N-1)T_{s}+ \Delta\theta_{0}]+Z_{QQ}\end{array} \tag{17}\]  
where \(\widetilde{R}(\varepsilon,\Delta f_{dsc},\Delta f_{d})=\frac{Aa_{1}}{4}R( \varepsilon)\frac{\sin(\pi\Delta f_{dsc}NT_{s})}{N\sin(\pi\Delta f_{dsc}T_{s} )}\frac{\sin(\pi\Delta f_{d}NT_{s})}{N\sin(\pi\Delta f_{dsc}T_{s})}\); \(R(\varepsilon)\) is the ACF of the PN-code; \(\varepsilon\) is the time offset; \(\hat{f}_{sc}\) and \(\hat{f}_{IF}\) are the frequencies of the subcarrier numerically controlled oscillator (NCO) and carrier NCO; \(\hat{\theta}_{sc}\) and \(\hat{\theta}_{0}\) are the phases of the subcarrier NCO and carrier NCO; \(\Delta f_{dsc}\) and \(\Delta f_{d}\) are the Doppler errors of the subcarrier and carrier; \(\Delta\theta_{sc}\) and \(\Delta\theta_{0}\) are the phase errors of the subcarrier and carrier; \(Z_{II}\), \(Z_{IQ}\), \(Z_{QI}\), \(Z_{QQ}\) are independent zero-mean Gaussian variables.  
Then the decision statistic is given by:  
\[X=II^{2}+IQ^{2}+QI^{2}+QQ^{2} \tag{18}\]