1 Introduction

Global Navigation Satellite System (GNSS) can provide global navigation and positioning services and is widely used in people's daily travel. To meet the growing demand for navigation and positioning, GNSS is often combined with other technologies and applied across various industries of social activities (Petritoli et al., 2014; Hinüber et al., 2017; Zhang et al., 2017). However, GNSS reliability and accuracy can be compromised in GNSS-denied environments such as urban canyons and indoor areas (Chen et al., 2022; Lu et al., 2021). To address this issue, several solutions based on signals of opportunity have been proposed, including Bluetooth (Grinyak et al., 2022), Wi-Fi (Li et al., 2018), ultra-wideband (UWB) (Huang et al., 2023), DTV (Jiao et al., 2023), 4G-LTE (Liu et al., 2023), 5G (Chen et al., 2022; Ruan et al., 2023) and their integration with other sensors for navigation (Liu et al., 2022). As an alternative solution, pseudolite systems offer flexible navigation capabilities and can enhance the performance of GNSS services in scenarios such as structural health monitoring (Shen et al., 2019) and urban canyons positioning (Montillet et al., 2009). Pseudolite systems can also provide stand-alone positioning services in GNSS-denied environments. These systems broadcast signals similar to GNSS, enabling commercial GNSS receivers to receive this service through hardware upgrades (Fujii et al., 2016; Sakamoto et al., 2011), or even without modification (Gan et al., 2019a, 2019b).

In recent years, indoor positioning has become one of the active fields in the research of pseudolite systems. Li et al. (2019) constructed a distributed pseudolite system in the \({7\times 10\mathrm{ m}}^{2}\) indoor scene, utilizing Universal Software Radio Peripheral (USRP) to receive pseudolite baseband signals, and obtained GNSS raw observations through signal post-processing. High-precision pseudorange observations are used for approximate coordinate initialization. Finally, they achieved high-precision indoor positioning by employing real-time kinematic (RTK) technology to resolve carrier phase ambiguities. However, the effective moving range was limited to approximately 1 m. However, the pseudorange accuracy output by commercial GNSS receivers cannot meet the requirements for approximate coordinate initialization. Therefore, Gan et al. (2022) adopted the strategy of UWB instead of pseudorange.

Gan et al., (2019a, 2019b) designed a novel array pseudolite system, which can realize indoor positioning in a \({20\times 24.5\mathrm{ m}}^{2}\) larger indoor scene (Sheng et al., 2022). They obtained the raw observation through the ublox receiver, and aimed to provide positioning services through smartphones (Gan et al., 2019a, 2019b). However, this array antenna led to poor DOP at the edge of the area, which impacted positioning accuracy. The actual moving area was about \({15\times 15\mathrm{ m}}^{2}\). Based on the above-mentioned array pseudolite system, some scholars have employed the pseudolites’ Channel State Information (CSI) fingerprint matching scheme, which is a common technique for indoor positioning. The pseudolites’ CSI includes the carrier phase and carrier-to-noise (C/N0) and so on. In the initial stage, a fingerprint dataset was established by obtaining measured CSI using a self-developed receiver and simulating it using wireless InSite ray tracing software (Huang et al., 2022). A variational auto-encoder (VAE) was adapted to establish a corresponding model, which can achieve meter-level positioning accuracy in the \({12\times 18\mathrm{ m}}^{2}\) test environment (Huang et al., 2022). Although the final positioning solution involved a smartphone, the acquisition method of the raw observations still needs to be connected to the smartphone through a pseudolite antenna (Huang et al., 2019) or a Bluetooth connection between the receiver and the smartphone (Huang et al., 2022), which increases the burden on the user.

Thanks to Google’s open access to GNSS raw observations for Android 7.0 and above operating systems in 2016 (European GNSS Agency et al. Agency and EGA 2017), developers can obtain GNSS raw observations through Android development, including pseudorange, carrier phase, Doppler, and C/N0, etc. However, smartphones typically use small omnidirectional linearly polarized antennas and low-cost, low-power GNSS chips, which can significantly impact on the quality of mobile phone GNSS signals (Li & Geng et al., 2019; Liu et al., 2019; Paziewski et al., 2019). This impact is often manifested as low signal strength, large pseudorange multipath errors, carrier phase observation poor value continuity, cycle-slip and half-wavelength cycle-slips, ambiguity does not have integer characteristics, etc. Some researchers have verified for the first time that GNSS observations based on smartphone antennas (Pesyna et al., 2014) can obtain centimeter-level high-precision positioning solutions through experiments. Gao et al. (2021) systematically studied the feasibility of fixing the ambiguity of the raw GNSS observations of the smartphone, and realized the phase ambiguity fixing for the first time by connecting the smartphone with a high-precision antenna, and obtained centimeter-level positioning results.

In this study, we only use smartphones to replace pseudorange observations with a simple pseudolite C/N0 ranging model to achieve indoor positioning in a large scene. In Sect. 2, a distributed time-asynchronous indoor pseudolite system is introduced, which consists of a transmitter with an unmodified smartphone, and an observation model. The proposed model based on real indoor experimental scenarios, as well as the actual positioning situation, are shown in Sect. 3. In the final section, we summarize the contributions and future work of this study.

2 Distributed indoor pseudolite asynchronous system

2.1 Transmitter and smartphone receiver

Figure 1 illustrates the composition of the distributed indoor pseudolite asynchronous system. It consists of three parts: a dual-channel radio frequency (RF) signal transmitter, distributed RF antenna, and a multi-channel clock distribution accessory. Since the coverage of the indoor pseudolite system is much smaller than that of the outdoor pseudolite system, a dual-channel transmitter with the same clock source is used to simplify the process of time synchronization. Each channel transmits a signal with different spread spectrum codes, and the navigation message is modulated at 1575.42 MHz. It should be noted that the signals sent by the two channels of the same transmitter will not be completely time-synchronized due to the difference in hardware delay and the difference in the resistance between the cable and the transmitting antenna. However, as they come from the same clock source, their frequencies are consistent (Wang et al., 2019), which is enough for our system.

Fig. 1
figure 1

Composition of distributed indoor pseudolite system

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An unmodified smartphone Mi 8 is chosen as the receiver for indoor positioning, which is no different from the smartphone used in daily life. The Mi 8, released by Xiaomi in 2018, is renowned as the world's first dual-frequency GPS positioning smartphone. It is equipped with a BCM47755 chip and supports L1/L5 dual-frequency satellite signals (European GNSS Agency (EGA) 2018). It is also the most commonly used test receiver in most smartphone outdoor GNSS high-precision positioning research (Li et al., 2019; Liu et al., 2019; Paziewski et al., 2019). Through the Android software of the pseudolite indoor ubiquitous positioning system (PIUPS) developed by our team, the GNSS raw observation of the smartphone can be obtained.

2.2 Observation model

In classic received signal strength ranging, RSS is defined as the voltage measured by a receiver’s Receiver Signal Strength Indicator (RSSI) circuit, which corresponds to received power measured on a logarithmic scale. RSS measurements are usually modeled as (Fontanella et al., 2012; Lindström et al., 2007; Patwari et al., 2005):

$$P\left(\rho \right)={P}_{0}-10\alpha {log}_{10}\left(\frac{\rho }{{\rho }_{0}}\right)$$
(1)

where \(P\left(\rho \right)\) is the RSS measured at the distance \(\rho\) from the transmitter, \(\alpha\) is the path-loss exponent and \({P}_{0}\) is the power received at a short reference distance, \({\rho }_{0}\).

In this paper, the RSSI obtained by the smartphone is C/N0. Although the observed value of C/N0 can be influenced by the signal processing capability and signal bandwidth of the receiver itself, C/N0 can effectively reflect the distance or occlusion between the receiver and the transmitter when the same receiver is used to receive the same system signal (Li et al., 2022). In particular, Eq. (1) can be rewritten in terms of C/N0 measurements as (Borio et al., 2016):

$${\left(C/N0\right)}_{j}={K}_{j}-\alpha 10{log}_{10}\left({\rho }_{j}\right)$$
(2)

where the index \(j\) is introduced to denote C/N0 measurements from the \({j}^{th}\) transmitter and \({K}_{j}\) is a constant accounting for the power of the \({j}^{th}\) transmitted signal and for the reference distance \({\rho }_{0}\). Unless specified, the C/N0 will always be expressed in units of dB-Hz. In actual measurement, the C/N0 obtained by the smartphone is easily affected by multipath and body occlusion, which can be mitigated by Kalman filtering (KF).

3 Experiment designment and results

3.1 Experimental scene and model building

To evaluate the C/N0 of the pseudolite received by the smartphone, an underground garage with a length of about 45 m and a width of about 15 m is chosen as the experimental site. As shown in Fig. 2, this environment is a typical indoor scene, with complex topological relationships, uncertain parking numbers and locations, resulting in poor passage conditions.

Fig. 2
figure 2

Experimental environments and setup: (a) the distance between pseudolite antennas is 45 m, (b) the distance between pseudolite antennas is 26 m, (c) the distance between pseudolite antennas is 25 m

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Due to objective constraints, the initial evaluation of this experiment involved the use of only two pseudolite antennas. As shown in Fig. 2, the dual-channel transmitters were all linked by a 30-m cable, and the transmitting antennas were placed at distances of 45 m (a), 26 m (b) and 25 m (c) meters. Among them, 45 m is the maximum distance of the underground garage.

To construct the relationship between C/N0 observations and distance, we used smartphone to measure many static pseudolite C/N0 observations in scenarios of Fig. 2 (a) and (b). At the connection position of the two pseudolite antennas, each position 0.5 m apart will be measured for 30 s, and the output frequency is 1 Hz. According to the measured C/N0 and the distance between the receiver and the transmitter, the relevant model was successfully established, as shown in the Fig. 3.

Fig. 3
figure 3

Measurement and fitting relationship between distance and C/N0

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As shown in Fig. 3, there is a certain corresponding relationship between the measured C/N0 at different distances. The fitting relationship is obtained by fitting with Eq. (2). It should be stated that in the distance of 0–10 m, with the fitting relationship (blue line) as the dividing line, the number of the lower measurement relationship (red dots) is more than that of the upper ones, but the slope of fitting relationship (blue line) increases, mainly because the red points above cover each other. The numerical values output in the upper part are relatively stable and fixed, while the numerical values in the lower data are relatively messy. In addition, it can also be seen from the figure that the measured C/N0 at different distances has large fluctuations, which may be due to multipath effects, near-far effects, and the limited processing performance of the GNSS module in smartphones.

3.2 Result analysis

In the kinematic positioning experiment, two pseudolite transmitting antennas were placed in the middle of the underground garage, 25 m apart. The tester held the smartphone (Mi8) and moves back and forth at the connection position of the two transmitting antennas to obtain their C/N0. The true positions were obtained using a total station as a reference. The experiment is divided into four tests, of which the first two tests are from the one transmitting antenna to another transmitting antenna. The latter two tests followed the opposite path, as shown in Fig. 2(c) and Fig. 4.

Fig. 4
figure 4

X-axis kinematics experiment based on distributed dual pseudolite system

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Since there are only two pseudolite transmitting antennas, the Y and Z axes can be fixed, and the coordinate track of the X-axis can be calculated from the ranging result of C/N0. The X-axis about kinematic positioning trajectory comparison is shown in Fig. 5. The green line represents the reference, the black line depicts the X-axis positioning trajectory of the original C/N0, and the red line represents the X-axis positioning trajectory obtained by the Kalman filter C/N0 ranging.

Fig. 5
figure 5

Kinematic positioning X-axis effects comparison

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In Fig. 5, it is worth noting that for the 20th epoch of test1 and 2 and the starting and ending epochs of test3 and 4, the original results may be better than the filtered results. At these epochs, the smartphone is extremely close to one antenna and far away from the other antenna, causing serious near-far effects. The influence of the near-far effect cannot always be completely consistent with the filter configuration, so it may further increase the difference between the ranging results and the real situation at some special locations. Table 1 statistics the positioning accuracy of each test. The total RMSE of the X-axis positioning of the original C/N0 ranging is 5.18 m. The total RMSE of the X-axis positioning of the C/N0 ranging after Kalman filtering is 2.93 m.

Table 1 Kinematic positioning X-axis error statistics and comparison
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4 Conclusion

In this work, we have introduced a novel approach that utilizes unmodified smartphones to acquire pseudolite observations and develop a simple C/N0 ranging model for indoor positioning. In the initial evaluation, single-axis positioning of dual base stations was implemented in an indoor scene with a length of 45 m. In our indoor positioning system, wide-coverage and low-complexity are the primary research goals. One of the key advantages of our proposed method is that it does not require fingerprint matching or deep learning algorithms. This ensures that the method does not significantly increase power consumption on smartphones. The positioning accuracy of RMSE better than 3 m has been successfully achieved, and the C/N0 ranging is used to replace the pseudorange. This addresses the limitation of traditional pseudolite systems that cannot utilize pseudorange measurements for initial position estimation in indoor scenarios. For future work, we will consider using adaptive filtering to improve the filtering results and further enhance the ranging accuracy. Additionally, we plan to expand experimental area by establishing an indoor scene with multiple pseudolite base stations.