# Band-Pass Sampling in High-Order BOC Signal Acquisition
## 3 BOC Acquisition Based on Band-Pass Sampling
### Band-Pass Sampling for High-Order BOC Signals  
Since the filtered high-order BOC signal can be treated as a dual-sideband signal, the band-pass sampling technique can be applied. In this study, the signal will be sampled at intermediate frequencies, and the intermediate frequency is an important parameter.  
If the intermediate frequency of BOC(\(m\),\(n\)) is \(f_{IF}\), then the center frequencies of the two sidebands are:  
\[\begin{array}{l}f_{1}=f_{IF}-f_{sc}\\ f_{2}=f_{IF}+f_{sc}\end{array}\]  
Define \(p_{1}=\text{fix}\Big{(}\frac{f_{1}}{f_{s}/2}\Big{)}\) and \(p_{2}=\text{fix}\Big{(}\frac{f_{2}}{f_{s}/2}\Big{)}\). To simplify the analysis, we choose the sampling frequency \(f_{s}\) to make sure \(p_{2}\) is even. Then the spectrum transition will be discussed for two cases.  
**Case a: \(p_{1}\) is odd.**  
In this case, the frequencies of the sampled signals are:  
\[\left\{\begin{array}{ll}f^{\prime}_{1}=(1+k_{1})f_{s}-f_{1}\\ f^{\prime}_{2}=f_{2}-k_{2}f_{s}\end{array}\right. \tag{10}\]  
where \(k_{1}=\text{fix}(f_{1}/f_{s})\) and \(k_{2}=\text{fix}(f_{2}/f_{s})\).  
The spectrum transition is depicted in Figure 2. Since the signal is real, there is always a negative spectrum.  
Figure 2: Spectrum transition where \(p_{1}\) is odd.  
As shown in Figure 2, the spectrum marked \(C^{\prime}\) is actually from \(C\), and it is an inverse version of \(A\). So, if we treat \(A^{\prime}\) and \(B^{\prime}\) as the target sampled signal, then \(C^{\prime}\) and \(D^{\prime}\) are the negative spectrum. For the sake of clarity, we will change the sign of \(f_{1}\), then (10) should be rewritten as:  
\[\left\{\begin{array}{l}f_{1}^{\prime}=f_{1}-(1+k_{1})f_{s}\\ f_{2}^{\prime}=f_{2}-k_{2}f_{s}\end{array}\right. \tag{11}\]  
**Case b: \(p_{1}\) is even.**  
In this case, the frequencies of the sampled signals are:  
\[\left\{\begin{array}{l}f_{1}^{\prime}=f_{1}-k_{1}f_{s}\\ f_{2}^{\prime}=f_{2}-k_{2}f_{s}\end{array}\right. \tag{12}\]  
The spectrum transition is depicted in Figure 3.  
In summary, the frequencies of the sampled signals are:  
\[\left\{\begin{array}{l}f_{1}^{\prime}=f_{1}-k_{1}f_{s}\\ f_{2}^{\prime}=f_{2}-k_{2}f_{s}\end{array}\right. \tag{13}\]  
where \(k_{2}=\text{fix}(f_{2}/f_{s})\) and:  
\[k_{1}=\left\{\begin{array}{l}\text{fix}(f_{1}/f_{s})+1\quad p_{1}\quad\text{ is odd}\\ \text{fix}(f_{1}/f_{s})\quad\quad p_{1}\quad\text{is even}\end{array}\right.\]  
Consequently, given the center frequencies of sampled sidebands, the sampling frequency and the intermediate frequency can be determined as follows:  
\[\left\{\begin{array}{l}f_{IF}^{\prime}-f_{sc}^{\prime}=f_{1}-k_{1}f_{s}\\ f_{IF}^{\prime}+f_{sc}^{\prime}=f_{2}-k_{2}f_{s}\\ f_{s}>2B\end{array}\right. \tag{14}\]  
where \(2f_{sc}^{\prime}\) is the distance between the two sidebands of the sampled signal; \(f_{IF}^{\prime}=\left(f_{1}^{\prime}+f_{2}^{\prime}\right)/2\) is the intermediate frequency of the sampled signal; \(B\) is the bandwidth of either sideband.  
Equation (14) can be simplified as:  
\[\left\{\begin{array}{l}f_{s}=2\frac{f_{sc}-f_{sc}^{\prime}}{k_{2}-k_{1}}\\ f_{s}>2B\\ f_{IF}=f_{IF}^{\prime}+\frac{k_{1}+k_{2}}{2}f_{s}\end{array}\right. \tag{15}\]  
The sampling frequency must also satisfy (8).  
Taking Galileo E1 PRS signal (BOC(15,2.5) modulated) as an example, Figure 4 illustrates the power spectrum density (PSD) and the ACF of the sampled signal (\(f_{sc}^{\prime}=2.5575\,\text{MHz}_{z}f_{IF}^{\prime}=0\)).  
Figure 3: Spectrum transition where \(p_{1}\) is even.  
The sampling frequency and the intermediate frequency are \(f_{s}\ =\ 12.7875\) MHz and \(f_{IF}\ =\ 25.575\) MHz, respectively.  
Figure 4 shows that the PSD and ACF of the sampled signal are similar to that of a BOC(\(n\),\(n\)) signal and the number of the remaining secondary peaks is reduced to 2, much less than that of BOC(15,2.5).