## 2 Mathematical Model  
According to earlier work [13], the sum of the sine and cosine samples of an incoming continuous signal can be approximated by Equation (1) in a complex form. \[\left\langle IQ\right\rangle_{\text{cos/sin}}\approx\int_{0}^{T}e^{i\left(2 \pi\cdot\Delta f\cdot t+\Delta\rho\right)}dt=\frac{\sin(\pi\cdot\Delta f\cdot T )}{\pi\cdot\Delta f\cdot T}e^{i\Delta\rho} \tag{1}\]  
Here, \(T\) is the coherent integration time of the accumulation-and-dump filter, \(\Delta\Phi\) is the phase error in radian units and \(\Delta f\) is the frequency error in Hz units. If the coherent integration time is limited to a code sequence period (or a single pulse period), a redefinition of Equation (1) for aperiodic pulsed navigation signals is straightforward, because the integration time is only affected by the duty ratio of the pulse. Therefore, the accumulation-and-dump output model of the incoming signal can be rewritten as Equation (2). \[\left\langle IQ\right\rangle_{\text{aperiodic}}=\frac{\sin(\pi\cdot\Delta f \cdot DR\cdot T_{\text{seq}})}{\pi\cdot\Delta f\cdot DR\cdot T_{\text{seq}}} \cdot\sqrt{2\frac{C}{N_{0}}\cdot DR\cdot T_{\text{seq}}}\cdot R(\tau)\cdot D \cdot e^{i\Delta\rho}+\eta \tag{2}\]  
Here, \(T_{seq}\) is the coherent integration time for a code sequence period, \(DR\) is the duty ratio of the pulse, \(C/N_{0}\) is the carrier to noise density ratio, \(R(\tau)\) is an auto-correlation function with a code phase error of \(\tau\), \(D\) is the data bit sign and \(\eta\) denotes the noise in a complex form. The first term in Equation (2) explains why the null-to-null space of the signal acquisition result in the frequency domain changes when a pulse is applied to an incoming code sequence, as presented in Figure 1. Figure 1a shows the 2D correlation outputs when a coherent integration time of 1 ms is used, as in a GPS navigation signal acquisition, while Figure 1b presents the pulsed signal acquisition result, for which the duty ratio is set to 0.1. Figure 1 shows that the null-to-null frequency space is increased in a manner inversely proportional to the duty ratio \(DR\), which is identical to the first term of Equation (2). Figure 1: Null-to-null space extension due to a single pulse. The second term of Equation (2) implies partial correlation effects on the power due to the pulsing scheme acting on a code sequence. Figure 2 shows that the post-correlation power is reduced by a pulse with a duty ratio of 0.1; the second term of Equation (2) explains why in a mathematical expression. Reduced power itself is not critical to a ground-based local navigation system, because it can be resolved by a systematic design, such as transmission power control. The important issue in Figure 2 is the change of the cross-correlation property. As shown in Figure 2, the cross-correlation separation is reduced from 24 to 25 dB to approximately 10 dB by pulsing. Assuming a Gaussian approximation, the upper bound of the partial correlation (_i.e._, the cross-correlation separation bound) of pulsed C/A Gold codes can be estimated, as shown in Figure 3. If the window size is set to 102-103 chips (\(DR=0.1\)), the upper bound approaches \(-10\) dB or more, as simulated in Figure 2.  
A new spreading code design to obtain good correlation properties is out of the scope of this paper. Therefore, a coherent integration time over a single code sequence period is unavoidable to meet the  
Figure 3: Cross-correlation separation of the pulsed C/A code. Figure 2: Partial correlation effect on the power: (**a**) duty ratio of 1.0; (**b**) duty ratio of 0.1.  
sensitivity requirements of navigation sensors in general. In this case, we must utilize the periodic pulse train model. Here, we define a rectangular pulse train signal model with Equations (3) and (4):  
\[p_{0}(t)= \left\{ \begin{aligned} & 1,&&\left|t\right|\leq DR \cdot T_{seq}\ /\ 2\\ & 0,&& DR\cdot T_{seq}\ /\ 2<\left|t\right|\leq T_{seq}\ /\ 2 \end{aligned} \right.