## 2 Mathematical Model
\tag{3}\] \[p(t)=p_{0}(t\pm n\cdot T_{seq}),\ \ \ \ \ n=0,\pm 1,\pm 2,... \tag{4}\]  
Equations (3) and (4) correspond to the aperiodic and the periodic model, respectively. The series coefficients of Equation (3) are derived as follows:  
\[c_{n}\approx\frac{1}{T_{seq}}\int_{-T_{seq}^{\prime}/2}^{T_{seq}/2}p(t)\cdot e^ {-\mu\frac{2\pi}{T_{seq}}}dt=\frac{1}{n\pi}\sin(DR\cdot n\cdot\pi)=\frac{\sin( DR\cdot n\cdot\pi)}{DR\cdot n\pi} \tag{5}\]  
By inserting Equation (5) into the exponential Fourier series definition, the pulse train in Equation (4) can be rewritten as Equation (6). \[p(t)=DR\cdot\sum_{n\leftarrow-\infty}^{\infty}\sin c(n\cdot DR)\cdot e^{\mu \frac{2\pi}{T_{seq}}t} \tag{6}\]  
Equation (1) can be rewritten as Equation (7) by applying Equation (6), as follows:  
\[\left\langle IQ\right\rangle_{\text{co/sin}}^{periodic} =\int_{0}^{KT_{seq}}e^{/(2\pi\cdot M+\lambda\theta)}\cdot DR\cdot \sum_{n\leftarrow-\infty}^{\infty}\sin c(n\cdot DR)\cdot e^{\mu\frac{2\pi}{T_{ seq}}}dt \tag{7}\] \[=DR\cdot\left[\sum_{n\rightarrow-\infty}^{\infty}\sin c(n\cdot DR )\right]_{0}^{KT_{seq}}e^{/\left[2\pi(\Delta^{\prime}+\mu f_{seq})+\lambda \theta\right]}\cdot dt\right]\]  
Here, \(f_{seq}\) denotes the code sequence frequency in Hz (=1/\(T_{seq}\)). The integral term in Equation (7) resembles the form of Equation (1) exactly; thus, we can easily rearrange Equation (7) above into Equation (8):  
\[\left\langle IQ\right\rangle_{\text{co/sin}}^{periodic}=DR\cdot\left[\sum_{n \leftarrow-\infty}^{\infty}\sin c(n\cdot DR)\cdot\sin c\cdot\left((\Delta^{ \prime}+\mu f_{seq})\cdot K\cdot T_{seq}\right)\cdot e^{\mu\sigma}\right] \tag{8}\]  
Finally, the accumulation-and-dump output model of the periodic pulsed navigation signal can be defined as Equation (9) by applying Equation (8) as the first term of Equation (2). Because the \(DR\) term in Equation (6) will affect the noise density, as well, we normalized Equation (8) by \(DR\) to define Equation (9). \[\left\langle IQ\right\rangle_{periodic}= \left[\sum_{n\leftarrow-\infty}^{\infty}\sin c(n\cdot DR)\cdot \sin c((\Delta^{\prime}+n\cdot f_{seq})\cdot K\cdot T_{seq})\right] \tag{9}\] \[\cdot\sqrt{2\frac{C}{N_{0}}\cdot DR\cdot T_{seq}}\cdot R(\tau) \cdot D\cdot e^{\mu\frac{1}{\sigma}}+\eta\]  
Equation (9) indicates that the sinc function train repeats every \(f_{seq}\) Hz and that the overall shape on the frequency axis is limited by the sinc function presented in Equation (2). This is a type of aliasing phenomenon caused by additional periodic pulse samples, which may lead to a false lock during the signal acquisition process. In Figure 4, we showed the two-dimensional signal acquisition results as simulated using the FFT technique. A GPS C/A code-like sequence was used with a duty ratio of 0.1, and the length of the coherent integration time was two sequences. As derived by Equation (9), we note that there is a sinc function train under the wide sinc function in Figure 4. The false lock issue caused by the sinc function train (or the side lobes in the frequency domain) is a major concern in the pulsed signal processing. The various algorithms utilized in earlier work and the algorithm suggested in this paper to dissolve the false lock issue will be described in the next section. In Figure 5, we present the signal acquisition results of pulsed signal samples with a duty ratio of 0.1 while varying the number of coherent integrations \(K\). For the simulation, we used a Gold code, for which the code sequence length is 1023 chips. Figure 4: Side lobes due to periodic pulsed signals. \[c_{n}\approx\frac{1}{T_{seq}}\int_{-T_{seq}^{\prime}/2}^{T_{seq}/2}p(t)\cdot e^ {-\mu\frac{2\pi}{T_{seq}}}dt=\frac{1}{n\pi}\sin(DR\cdot n\cdot\pi)=\frac{\sin( DR\cdot n\cdot\pi)}{DR\cdot n\pi} \tag{5}\]0 Auto-correlation function (_N_ = the number of coherent integrations in sequence units). The upper left figure shows the signal acquisition results when the number of coherent integrations \(K\) is set to one (1 ms, 1023 chips).