Enhancing Ranging Precision in OFDM-Based LEO Navigation: Signal Design and Receiver Implementation

Ranging Accuracy

To evaluate the ranging accuracy potential, the auto-correlation function (ACF) is first adopted. The sharpness of the ACF peak serves as a key indicator for achievable ranging resolution, where narrower ACF peaks correspond to higher precision in time-delay (κ) estimation (Liu et al., 2014). As shown in Figure 3(a), OFDM signals exhibit evident sensitivity to the SCS configuration. Increasing the SCS from 60 kHz to 120 kHz reduces the ACF peak width significantly. This substantial narrowing demonstrates the potential for enhanced ranging accuracy through SCS optimization. Notably, the OFDM(120) configuration achieves ACF sharpness comparable to that of BOC(10,5), a signal modulation widely recognized for high-precision satellite navigation applications.

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Figure 3

Performance of ranging accuracy for different modulation schemes (a) ACFs of different modulation schemes, (b) Gabor bandwidth as a function of front-end bandwidth.

In addition to ACF analysis, we employ the Gabor bandwidth to further quantify the signal characteristics related to ranging accuracy. The Gabor bandwidth quantifies the frequency spread of a signal’s energy (Amoroso, 1980), which is expressed as follows:

$\Delta \hspace{0.17em}{f}_{\mathrm{Gabor}}=\sqrt{{\int }_{-B_r/2}^{B_r/2}{f}^2\hspace{0.17em}G\hspace{0.17em}\left(f\right)df}$ 5

where G(f) is the normalized power spectral density (PSD) of the signal and B_r denotes the front-end bandwidth of the receiver.

A higher Gabor bandwidth corresponds to a broader frequency spread, which enhances code-tracking precision and reduces uncertainty in the TOA estimation of κ. As illustrated in Figure 3(b), the Gabor bandwidth of OFDM signals is directly influenced by their SCS when the number of subcarriers is fixed. Specifically, the Gabor bandwidth for OFDM(60) exceeds that of all other modulation schemes except BOC(10,5) and OFDM(120). The Gabor bandwidth for OFDM(120) surpasses that of BOC(10,5) when the front-end bandwidth exceeds 33 MHz. This finding demonstrates that OFDM signals with a larger SCS and consequently a wider signal bandwidth achieve a higher Gabor bandwidth, which translates to superior code-tracking performance. With a sufficient SCS, OFDM modulations can surpass all other modulation schemes evaluated in this study.

Power Efficiency and Bandwidth Requirement

The PSD and power containment percentages are plotted in Figures 4(a) and 4(b), respectively, to evaluate the power efficiency and determine the bandwidth requirements for different modulations. Figure 4(a) shows that the OFDM signal has a flat main lobe, which includes the majority of its energy, whereas the PSD of the other modulations exhibits pronounced peaks and valleys. This flat main lobe indicates that OFDM is a power-efficient modulation scheme, as it concentrates its energy uniformly across the frequency range. However, as the main lobe of the OFDM signal becomes wider, the signal strength per unit frequency decreases, necessitating a more sensitive receiver to process the wideband OFDM signal.

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Figure 4

Power efficiency and bandwidth requirement for different modulation schemes (a) PSD of the listed modulations, (b) Power containment of the listed modulations.

To complement the analysis of power efficiency, Figure 4(b) provides insights into the specific receiver bandwidth requirements for different modulations. Typically, the front-end bandwidth B_r of the receiver should be equal to at least the 90% power bandwidth β_90% to ensure efficient reception (Betz, 2001). According to Figure 4(b), the β_90% of the OFDM signal is broader than that of the BPSK(1) and BOC(1,1) signals and relatively similar to that of BPSK(10) and BOC(10,5), depending on the specific values of SCS and subcarrier number M. Thereby, while OFDM is power-efficient, a wider bandwidth is required for reception compared with narrower modulations to fully utilize the potential of OFDM. Collectively, these results demonstrate that OFDM not only optimizes the energy distribution but also requires careful receiver design to handle its wider frequency characteristics.

Anti-Interference Capacity

Anti-interference capacity is a critical factor for potential navigation signals. The multipath error envelope (MEE), particularly within the context of a two-ray signal model, serves as a critical criteria for assessing the impact of multipath on tracking performance and ranging accuracy. The MEE is given as follows (Liu et al., 2014):

$ϵ\approx \frac{\pm a\hspace{0.17em}{\int }_{-B_r/2}^{B_r/2}G\hspace{0.17em}\left(f\right)\hspace{0.17em}\sin \hspace{0.17em}\left(2\hspace{0.17em}\pi \hspace{0.17em}f\hspace{0.17em}t\right)\hspace{0.17em}\sin \hspace{0.17em}\left(\pi \hspace{0.17em}f\hspace{0.17em}d\right)df}{2\hspace{0.17em}\pi \hspace{0.17em}{\int }_{-B_r/2}^{B_r/2}f\hspace{0.17em}G\hspace{0.17em}\left(f\right)\hspace{0.17em}\sin \hspace{0.17em}\left(\pi \hspace{0.17em}f\hspace{0.17em}d\right)\hspace{0.17em}\left[1\pm a\hspace{0.17em}\cos \hspace{0.17em}\left(2\hspace{0.17em}\pi \hspace{0.17em}f\hspace{0.17em}t\right)\right]df}$ 6

where a is the multipath-to-direct signal amplitude ratio and the receiver correlator spacing is denoted by d. The symbols “+” and “–” represent the phase relationship between the multipath signal and the direct line-of-sight signal. “+” indicates that the signals are in phase (0°), causing constructive interference, whereas “–” indicates that they are 180° out of phase, causing destructive interference.

We compute the MEE for a front-end bandwidth of 35 MHz. As shown in Figure 5(a), the OFDM(120) signal exhibits minimal multipath errors compared with other modulation schemes when a = 0 1. and d = 0 05. This result highlights the potential of OFDM modulation for maintaining precise tracking when facing multipath interference.

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Figure 5

Performance of anti-interference properties for different modulation schemes (a) MEE with a prefiltering bandwidth of 35 MHz, (b) SSC values for different modulation schemes.

The spectral separation coefficient (SSC) is employed to assess the interference level when a navigation signal candidate, sharing the same frequency band with other signals, is received. The SSC is defined as follows (Liu et al., 2014):

$k_{i\hspace{0.17em}d}={\int }_{-\infty }^{\infty }{G}_i\hspace{0.17em}\left(f\right)\hspace{0.17em}{G}_d\hspace{0.17em}\left(f\right)df$ 7

where G_i(f) and G_d(f) are the normalized PSD of the signals considered as the interference and desired signals, respectively. When both the interference and desired signals employ the same modulation scheme, G_i(f) and G_d(f) are identical. In such cases, k_id serves as a measure of self-interference SSC. Otherwise, k_id measures the cross-interference SSC.

Figure 5(b) shows the SSC values for cross-interference between example signals, displayed off the diagonals, with self-interference values indicated along the diagonals of the plot. For cross-interference SSC, it is evident that both the OFDM and BOC(10,5) signals exhibit relatively low SSC values among the various modulation schemes, indicating a lower likelihood of interference. For self-interference SSC, OFDM signals demonstrate superior resistance to intersystem interference, whereas BOC(10,5) signals are comparatively more vulnerable in cases of matched spectrum interference. Considering the significant advantages of OFDM in maintaining higher accuracy in timing output and offering superior interference resistance, we have selected OFDM as our preferred modulation scheme for LEO navigation signals.

Integer CFO

The m-sequence exhibits a defining characteristic under integer CFOs: when the frequency shift exceeds one SCS, i.e., when an integer CFO occurs, the correlation peak disappears, leaving only noise, as shown in Figure 6(b). This phenomenon can be mathematically represented as follows (Kaplan & Hegarty, 2017):

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Figure 6

Correlation peak detection results for the ZC sequence and m-sequence (a) Correlation peak of the ZC sequence, (b) Correlation peak of the m-sequence. For both panels, the colorbar denotes the normalized correlation peak amplitude.

$\left|R_{I,m}\hspace{0.17em}\left(\Delta \hspace{0.17em}{f}_I,\Delta \hspace{0.17em}\kappa \right)\right|=\frac{1}{N}\hspace{0.17em}\left|\sum _{n=0}^{N-1}{c}_1\hspace{0.17em}\left(n+\kappa \right)\hspace{0.17em}{c}_2^{\ast }\hspace{0.17em}\left(n+{\kappa }^{\prime }\right)\right|=\{\begin{array}{cc}\left|\text{sinc}\hspace{0.17em}\left(\Delta \hspace{0.17em}{f}_I\right)\right|,& \text{if}{\kappa }^{\prime }=\kappa \\ \frac{1}{N},& \text{if}{\kappa }^{\prime }\ne \kappa \end{array}$ 8

where $\Delta \hspace{0.17em}{f}_I=\frac{f_I}{S\hspace{0.17em}C\hspace{0.17em}S}\ge 1$ is the normalized integer CFO, κ′ is the estimated TOA, $\Delta \hspace{0.17em}\kappa =\kappa -{\kappa }^{\prime }$ is the timing shift, f_I denotes the true integer CFO, and $c_2\hspace{0.17em}\left(n\right)=c_1\hspace{0.17em}\left(n\right)\cdot e^{\frac{j\hspace{0.17em}2\hspace{0.17em}\pi \hspace{0.17em}{f}_I\hspace{0.17em}n}{N}}$, with c₁(n) being an m-sequence of length N, generated by a maximal length linear feedback shift register.

The susceptibility of the m-sequence to integer CFOs is clearly demonstrated; there is no correlation peak when such CFO conditions exist in systems based on m-sequence SSs. Interestingly, this vulnerability to integer CFOs proves advantageous for our ranging applications. This advantage arises because, after performing a two-dimensional search within specific frequency ranges and successfully detecting an m-sequence, we ensure that the integer CFO is accurately compensated using the ML estimate of the TOA (Hua et al., 2014):

${\kappa }^{\prime }=\arg \hspace{0.17em}m\hspace{0.17em}a\hspace{0.17em}x\hspace{0.17em}{\left|R_{I,m}\hspace{0.17em}\left(\Delta \hspace{0.17em}{f}_I,\Delta \hspace{0.17em}\kappa \right)\right|}^2$ 9

Furthermore, residual fractional CFOs do not significantly impact the TOA κ′ estimation of the m-sequence, as will be elaborated upon in the following section. Therefore, if the m-sequence is successfully detected, the estimated κ′ should align precisely with the actual delay, remaining unaffected by any CFOs.

In contrast, the ZC sequence behaves differently under conditions of integer CFOs. As illustrated in Figure 6(a), the correlation peaks of the ZC sequence persist even in the presence of an integer CFO between the ZC sequence and its local replicas. When subject to an integer CFO, the correlation result of the ZC sequence, omitting noise, is mathematically represented as follows (Gul et al., 2012):

$\left|R_{I,z\hspace{0.17em}c}\hspace{0.17em}\left(\Delta \hspace{0.17em}{f}_I,\Delta \hspace{0.17em}\kappa \right))\right|=\alpha \hspace{0.17em}\left|\sum _{n=0}^{N-1}{e}^{j\hspace{0.17em}2\hspace{0.17em}\pi \hspace{0.17em}\frac{\left(\mu \hspace{0.17em}\left({\kappa }^{\prime }-N_g-\kappa \right)+f_I\right)\hspace{0.17em}n}{N}}\right|,\kappa \le {\kappa }^{\prime }\le \kappa +N_g$ 10

where μ is the root of the ZC sequence, which is received with a delay of κ samples. N_g and N represent the lengths of the CP and SS, respectively.

It can be inferred from Equation (10) that the time delay κ of the ZC sequence is estimated as follows:

${\kappa }^{\prime }=\arg \hspace{0.17em}max\hspace{0.17em}{\left|R_{I,z\hspace{0.17em}c}\hspace{0.17em}\left(\Delta \hspace{0.17em}{f}_I,\Delta \hspace{0.17em}\kappa \right)\right|}^2=\left\{{\kappa }^{\prime }|\mu \hspace{0.17em}\left({\kappa }^{\prime }-N_g-\kappa \right)+f_I=p\hspace{0.17em}N\right\}$ 11

where p is any integer. From Equation (11), it is evident that when an integer CFO f_I exists, a peak can still be observed if $\mu \hspace{0.17em}\left({\kappa }^{\prime }-N_g-\kappa \right)+f_I=p\hspace{0.17em}N$. This phenomena is referred to as frequency offset ambiguity (Chandramouli et al., 2019). Consequently, during the step of ZC sequence detection, the receiver cannot confirm whether it is affected by an integer CFO, which may increase the BER in data recovery. More critically, an integer CFO can induce a timing shift in the TOA estimation of κ′, leading to frequency offset ambiguity. This introduces uncertainty in SS detection, making it difficult to determine whether the estimated TOA κ′ is influenced by a potential integer CFO.

Similar to ZC sequences, chirp signals are also influenced by frequency offset ambiguity, commonly referred to as range–Doppler coupling in radar signal processing. This effect causes a mismatch during matched filtering, ultimately degrading range estimation (Richards et al., 2010). To suppress this coupling and increase ranging accuracy, two closely spaced up-chirp and down-chirp pulses can be utilized to estimate and compensate for the observed Doppler shift. A similar approach can be applied to ZC sequences to address frequency offset ambiguity. However, this method significantly increases the correlation computational load, increasing the processing complexity of the receiver.

In contrast to both ZC sequences and chirp signals, the m-sequence offers a distinct advantage in terms of integer CFO compensation. Recall that the m-sequence generates a unique and unambiguous peak in the correlation matrix, as shown in Figure 6(b). This property highlights the m-sequence’s ability to support integer CFO compensation through straightforward searching across two-dimensional Doppler-phase ranges, validating its superior suitability for LEO ranging missions.

When the received data are affected by fractional CFOs, the subcarriers lose orthogonality, leading to inter-carrier interference and an increased ranging error. To understand the mathematical factors influencing the timing properties of the two candidate SSs under a fractional CFO, we define the correlation function $\left|R_{F,\ast }\hspace{0.17em}\left(\Delta \hspace{0.17em}\lambda ,\Delta \hspace{0.17em}\kappa \right)\right|$ for each candidate in the presence of fractional CFOs, following the approach given by Hua et al. (2014):

$\left|R_{F,\ast }\hspace{0.17em}\left(\Delta \hspace{0.17em}\lambda ,\Delta \hspace{0.17em}\kappa \right)\right|=\{\begin{array}{cc}\left|\text{sinc}\hspace{0.17em}\left(\Delta \hspace{0.17em}\lambda -\Delta \hspace{0.17em}{\lambda }_{\Delta \hspace{0.17em}\kappa }^{†}\right)\right|,& \text{if}{R}_{F,\ast }=R_{F,z\hspace{0.17em}c}\\ \left|\text{sinc}\hspace{0.17em}\left(\Delta \hspace{0.17em}\lambda \right)\ast R_m\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)\right|,& \text{if}{R}_{F,\ast }=R_{F,m}\end{array}$ 12

where:

$\Delta \hspace{0.17em}{\lambda }_{\Delta \hspace{0.17em}\kappa }^{†}\triangleq \mu \hspace{0.17em}\Delta \hspace{0.17em}\kappa +l^{†}\hspace{0.17em}N,l^{†}=\arg \hspace{0.17em}\underset{l\in Z}{min}\hspace{0.17em}\left|\mu \hspace{0.17em}\Delta \hspace{0.17em}\kappa +l\hspace{0.17em}N\right|$13

and R_F,∗ denotes the fractional correlation value for different SS candidates under fractional CFOs. $\Delta \hspace{0.17em}\lambda =\frac{f_F}{\text{SCS}}<1$denotes the normalized fractional CFO, and f_F is the true fractional CFO value. The function $\left|R_m\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)\right|$ is defined as follows:

$\left|R_m\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)\right|=\{\begin{array}{cc}1& \text{if}\Delta \hspace{0.17em}\kappa =0\\ \frac{1}{N}& \text{if}\Delta \hspace{0.17em}\kappa \ne 0\end{array}$

As shown in Equations (12) and (13), unlike R_F,m, the value of R_F,zc depends on both the length N and root μ of the ZC sequence. In Equation (12),$\Delta \hspace{0.17em}{\lambda }_{\Delta \hspace{0.17em}\kappa }^{†}$ indicates the critical frequency offset for R_F,zc at a specific incorrect shift offset of $\Delta \hspace{0.17em}{\kappa }^{†}\ne 0$. At this incorrect shift offset $\Delta \hspace{0.17em}{\kappa }^{†}$, with a fixed μ, the critical timing offset can have a small absolute value, which leads to a large value of R_F,zc at a nonzero location $\Delta \hspace{0.17em}{\kappa }^{†}\ne 0$. This situation results in an undesirable increase from correct estimations when $\Delta \hspace{0.17em}\kappa =0$ in the correlation function, increasing the likelihood of incorrect timing estimation.

To evaluate the TOA estimation performance for both SS candidates under fractional CFOs, two key metrics have been introduced by Hua et al. (2014), i.e., the timing detection probability $P_{\Delta \hspace{0.17em}{\kappa }^{\star }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)$ and the timing error probability P_e. The timing detection probability $P_{\Delta \hspace{0.17em}{\kappa }^{\star }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)$ is expressed as follows:

$P_{\Delta {\kappa }^{*}}\left(\Delta \kappa \right)=\frac{1}{W}\sum _{\kappa \in H}P(\Delta {\kappa }^{*}=\Delta \kappa \mid \kappa )$14

where κ* denotes the superior timing offset estimation and κ represents the actual timing offset, with $\Delta \hspace{0.17em}{\kappa }^{\ast }\triangleq {\kappa }^{\ast }-\kappa$ defining their difference. The variable W refers to the length of the timing hypothesis window H. As demonstrated in Equation (14), $P_{\Delta \hspace{0.17em}{\kappa }^{\star }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)$ indicates the probability of different Δκ values corresponding to the superior detection. The timing error probability is given as follows:

$P_e=1-P_{\Delta \hspace{0.17em}{\kappa }^{\ast }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa =0\right)$15

where $P_{\Delta \hspace{0.17em}{\kappa }^{\ast }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa =0\right)$ quantifies the likelihood of the ML algorithm correctly outputting the actual TOA κ′ when κ′=κ between the transmitter and the receiver.

The value of $P\hspace{0.17em}\left(\Delta \hspace{0.17em}{\kappa }^{\ast }=\Delta \hspace{0.17em}\kappa \mid \kappa \in H\right)$ is derived to be positively correlated with $\left|R_{F,\ast }\hspace{0.17em}\left(\Delta \hspace{0.17em}\lambda ,\Delta \hspace{0.17em}\kappa \right)\right|$ (Hua et al., 2014). As a result, when the ZC sequence is used as the $\mathrm{SS}\hspace{0.17em},P_{\Delta \hspace{0.17em}{\kappa }^{\ast }}\hspace{0.17em}\left(\Delta \hspace{0.17em}{\kappa }^{†}\right)$ can also become unexpectedly large, as R_F,zc exhibits a large value at $\Delta \hspace{0.17em}{\kappa }^{†}\ne 0.$ Consequently, $P_{\Delta \hspace{0.17em}{\kappa }^{\ast }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa =0\right)$ becomes relatively small, leading to a high P_e value across various SNRs. Figures 7 and 8 illustrate $P_{\Delta \hspace{0.17em}{\kappa }^{\ast }}\hspace{0.17em}\left(\Delta \hspace{0.17em}\kappa \right)$ and P_e for the m-sequence and ZC sequence, respectively, for a varying ZC root μ, under a time uncertainty window of |H| = 16, with N = 839 and a fractional CFO of Δλ = 0.55.

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Figure 7

Probability of detection and timing error for the m-sequence and ZC sequence, with µ = 140 for the ZC sequence (a) Probability of detection for the m-sequence and ZC sequence for a fractional CFO of Δλ = 0.55, (b) Probability of timing error for the m-sequence and ZC sequence across various SNR levels.

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Figure 8

Probability of detection and timing error for the m-sequence and ZC sequence, with µ = 367 for the ZC sequence (a) Probability of detection for the m-sequence and ZC sequence for a fractional CFO of Δλ = 0.55, (b) Probability of timing error for the m-sequence and ZC sequence across various SNR levels.

Figure 7 shows the case in which the chosen value of μ degrades the TOA estimation performance, specifically when μ =140. In Figure 7(a), $\Delta {\lambda }_{\Delta \kappa }^{†}=\pm 1$ and ±2 for timing offsets of $\Delta {\kappa }^{†}=\pm 6$ and ±12, respectively. The small value of $\Delta {\lambda }_{\Delta \kappa }^{†}=\pm 1$ leads to relatively large values of $\left|R_{zc}(\Delta \lambda ,\Delta \kappa )\right|$, resulting in multiple spurious peaks at $∆{\kappa }^{†}\pm 6$ and ±12, in addition to $∆\kappa =0$. This result suggests a higher likelihood of erroneous detection at these offsets, which could mislead the receiver’s estimations of TOA κ′. In contrast, the m-sequence exhibits a single prominent peak only at $∆\kappa =0$ indicating more accurate κ′ estimations. Figure 7(b) supports these observations by showing that the m-sequence exhibits a sharp decrease in P_e as the SNR increases; conversely, the ZC sequence shows a more gradual decline, hindered by the four unexpected spurious peaks. These distinct patterns in P_e between the m-sequence and the ZC sequence persist across various fractional CFOs, $∆\lambda \ne 0$.

Figure 8 explores improvements in κ′ estimation performance for the ZC sequence, obtained by strategically selecting the root value as μ=367. In this case, the smallest $\Delta {\lambda }_{\Delta \kappa }^{†}=\pm 52\gg 1$ When $\Delta {\kappa }^{†}\pm 7$, which allows little effect on $\left|\begin{array}{c}{R}_{ZC}\left(∆\lambda ,∆\kappa \right)\end{array}\right|$. The results in Figures 8(a) and 8(b) indicate that both the detection probability $P_{∆\kappa *}\left(∆\kappa \right)$ and the error probability P_e of the ZC sequence now mirror those of the m-sequence, showcasing high detection accuracy and rapid declines in P_e, indicating a minimal degradation in timing performance.

These findings emphasize that while the ZC sequence can outperform the m-sequence in ranging capabilities under ideal conditions with no CFOs in random roots and may slightly surpass the m-sequence with fractional CFOs if the root is carefully selected, it has significant limitations. In contrast, the m-sequence offers a much more reliable performance owing to its regular structure, which ensures precise κ′ estimations without the need for selecting specific internal parameters, even in the presence of fractional CFOs. Moreover, the inherent property of the m-sequence allows it to compensate for the integer CFO through straightforward two-dimensional searching. In contrast, the ZC sequence struggles with frequency offset ambiguity, where integer CFOs induce timing shifts that impair κ′ estimation accuracy. Considering these strengths, the m-sequence emerges as the superior choice for our LEO ranging tasks, providing robustness and simplicity without the complications associated with parameter tuning or ambiguity from CFOs.

Numerical Simulations

The simulations in this section directly compare the ranging performance of ZC sequences and m-sequences under challenging LEO navigation conditions characterized by large dynamic CFO variations and low SNRs, using the proposed DCA-NOLRT-based receiver for a fair assessment. Each simulation condition was executed 20 times to ensure statistical reliability. Evaluations employ a modified non-terrestrial network tapped delay line-B (NTN-TDL-B) channel model (3GPP, 2018).While the standard NTN-TDL-B model fixes the satellite elevation at 50°,we extend the elevation angle range from 0° to 90° to simulate dynamic CFO behavior induced by elevation angle changes of the satellite. Within this modified NTN-TDL-B simulated environment, signals experience impairments commonly encountered in LEO navigation, such as delay spread causing temporal dispersion and CFOs from satellite motion. Detailed parameters are provided in Table 2.

The results in Figures 13(a) and 13(b) illustrate the ZC sequence performance across the full CFO shift range derived from satellite dynamics in Table 2. Figure 13(a) demonstrates that without mitigation, CFOs exceeding the SCS induce frequency ambiguity. Although signal detection is successful, such ambiguity introduces the timing shift analyzed in Section 3.1, resulting in step-like ranging errors. Furthermore, applying a CFO mitigation method (Van de Beek et al., 1997) proves largely ineffective or even counterproductive under these low-SNR impaired conditions, as shown in Figure 13(b). This adverse effect is due to inaccurate CFO estimation and compensation during the mitigation procedure, which introduces larger CFO errors than those without mitigation. As a result, even when the actual CFO falls below half the SCS (within the applicable range of the CFO mitigation method), CFO estimations exhibit destabilizing fluctuations that increase the ranging error magnitude. These findings highlight the challenges of traditional CFO estimation and mitigation strategies under harsh conditions of LEO navigation, especially when th SNR is low and channel impairments are considered.

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Figure 13

Ranging errors of the ZC sequence under NTN-TDL-B channel conditions for various SCS values The ZC sequences are truncated from 511 to 276 chips with root µ=20. (a) Results obtained without the CFO compensation method, (b) Results obtained with the CFO compensation method.

In contrast to the ineffective CFO mitigation strategies illustrated in Figure 13, Figure 14 presents ranging outcomes for the m-sequence obtained through a two-dimensional frequency-code phase search. Because m-sequences suffer no frequency ambiguity, no timing shift is induced when the correlation peak is detected. Consequently, across the full CFO shift range indicated on the x-axis, step-like ranging errors are effectively mitigated, and a satisfactory ranging accuracy is guaranteed. These results demonstrate the inherent resilience of the m-sequence against CFO effects, allowing accurate ranging estimation despite challenging conditions under LEO navigation scenarios.

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Figure 14

Ranging errors of the m-sequence under NTN-TDL-B channel conditions for various SCS values

In Figure 14, the remaining fluctuations in ranging errors are primarily attributed to delay spread within the modified NTN-TDL-B model. Additionally, similar small fluctuations in ranging errors are observed in Figure 13(a) when the CFO is below the SCS, which are also caused by the simulated multipath delay spread. This consistent behavior under the modified NTN-TDL-B conditions indicates that the observed errors are primarily caused by multipath effects inherent in the channel model. Such multipath-induced errors can be effectively suppressed by employing a larger SCS. An increased SCS enhances the time resolution and signal bandwidth (Abrudan et al., 2013), thereby improving the ability to discriminate between line-of-sight and multipath signals. As a result, error fluctuations caused by multipath interference are significantly reduced.

Furthermore, for ZC sequences, larger SCS values significantly reduce frequency ambiguity-induced ranging errors over wider CFO ranges, as shown in Figure 13. This advantage explains the preference for wider SCSs in LEO navigation systems. However, when the available bandwidth is limited, increasing the SCS reduces the number of available subcarriers, thereby impairing data transmission efficiency. This inherent trade-off between transmission efficiency and CFO resistance represents a common challenge for LEO systems. To balance these competing factors, an SCS of 60 kHz is selected for our designed LEO signal configuration.

Terrestrial Ranging Experiments

In this section, we validate the ranging accuracy of the proposed LEO navigation framework through a terrestrial experiment. The experimental setup, illustrated in Figure 15, consists of a custom-designed navigation transmitter (Tx) and a receiver (Rx) based on an Ettus Universal Software Radio Peripheral (USRP) X310. The receiver is controlled by the USRP hardware driver software on a host PC, which stores received signal samples for offline postprocessing. The positions of the transmitter and receiver are accurately surveyed using a u-blox EVK-F9P receiver. The transmitter is located at 31.0269865°N, 121.4400866°E, with an elevation of 18 m, and the receiver is located at 31.0273907°N, 121.4404318°E, with an elevation of 23 m. The ground-truth distance calculated between the transmitter and receiver is 61.30 m.

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Figure 15

Experimental hardware used in ground tests

To emulate challenging LEO signal conditions, a 1-W navigation signal is attenuated by 60 dB before transmission to simulate path loss. Furthermore, an 80-kHz CFO is manually introduced offline, with a changing rate of 100 Hz/s, to mimic the dynamic Doppler shifts typical in LEO navigation scenarios. These configurations replicate low-SNR and time-varying CFO conditions within a controlled ground environment.

We calculate the range between the transmitter and receiver every 6 s of each epoch. As shown in Figure 16, the ranging errors across different estimation epochs are presented, with a root mean square error of 3.34 m, validating the effectiveness of the proposed LEO navigation framework under harsh conditions. It is important to clarify that in the ground experiments, we were unable to fully replicate the delay spread and other channel impairments characteristic of the NTN-TDL-B channel. This limitation is the primary reason why the ranging errors observed in the ground experiments are generally smaller than those in Figure 14.

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Figure 16

Ranging performance of the proposed LEO navigation framework in ground experiments

By combining the ranging performance results from both numerical simulations and ground tests, the reliability of our proposed LEO navigation signal and receiver framework is demonstrated. The system achieves satisfactory ranging accuracy in simulations and even better performance in controlled ground experiments, highlighting its robustness under challenging conditions of LEO navigation.