
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

70846
10.1038/s41598-024-70846-0
Article
LEO navigation observables extraction using CLOCFC network
Wang Zhisen 1
Lu Hu sdkmsdn@qq.com

12
Bian Zhiang 1
1 https://ror.org/00seraz22 grid.440645.7 0000 0004 1800 072X Information and Navigation School, Air Force Engineering University, Xi’an, 710077 China
2 Key Laboratory of Smart Earth, Beijing, 100029 China
4 9 2024
4 9 2024
2024
14 2057816 6 2024
21 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In case of mitigate the reliance of aviation users on the Global Navigation Satellite System (GNSS) in an increasingly interference-prone environment, utilizing opportunistic signals from Low-Earth Orbit (LEO) for navigation and positioning is an alternative approach. However, LEO satellite SOPs are not intended for navigation. Therefore, it is necessary to design methods to extract navigation observables from these signals. In this paper, we proposed a lightweight deep learning model with a two-branch structure called CLOCFC, designed to extract navigation observables. Furthermore, we have established a low Earth orbit satellite signal dataset by using ORBCOMM constellation signals as the input to the model and Doppler frequency as the label for the model. The results show that CLOCFC, as a lightweight model, demonstrates a significantly faster convergence rate and higher accuracy in navigation observables extraction compared to other models (ResNet, Swin Transformer, and Clo Transformer). In CLOCFC, we introduce the CFC module, a kind of Liquid Neural Network, to enhance the information acquisition capability through the spatiotemporal information in the data sequence. Finally, we have also conducted extensive experiments with the Doppler shift extraction of LEO satellites as an example, under various noise and resolution conditions, demonstrating the superiority of the CLOCFC.

Keywords

Signals of opportunity
Low earth orbit satellite communication
Instantaneous Doppler positioning
Lightweight network
CFC network
Subject terms

Aerospace engineering
Electrical and electronic engineering
Key Laboratory of Smart EarthKF2023ZD01-05 Lu Hu issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Conducting research on Signal of Opportunity (SOPs) positioning technology provides an effective way to address the problem of navigation and positioning when the Global Navigation Satellite System (GNSS) is not available. Additionally, compared to land-based SOPs, LEO-based SOPs have the advantages of global coverage, wide available frequency bands, strong electromagnetic interference resistance, no need for system construction, and low cost, making them more suitable as signal sources for positioning1,2.

However, LEO satellite SOPs are not designed for navigation purposes. Consequently, a primary challenge must be addressed to utilize their signals for navigation: Design specialized methods capable of extracting navigation observables from these downlink signals3.

Over the past five years, the application of Doppler positioning with Low Earth Orbit (LEO) satellites has garnered significant attention in the research community. In these studies, the research conducted by Kassas and his team is the most comprehensive. They have not only focused on positioning for various LEO constellations1,4,5 but also integrated other methods such as INS for localization3,6. In addition, Qin has made many significant contributions to the research on multi-constellation fusion positioning7,8. Wei et al.9 introduced an algorithm for the precise estimation of LEO satellite downlink Doppler frequency, utilizing both explicit and implicit pilots within the received signal. Ferre et al.10 delved into the suitability of existing LEO systems and the optimization steps required to construct novel LEO-PNT constellations aimed at enhancing positioning performance. Farhangian et al.11 concentrated on the design of a multi-constellation software-defined receiver for Doppler positioning with LEO satellites. Minetto et al.12 investigated a cognitive particle filter for collaborative Differential Global Navigation Satellite System (DGNSS) positioning, emphasizing the significance of processing pseudorange and Doppler measurements for improved accuracy. Chen et al.13 analyzed the signal characteristics of LEO satellites, including power spectral density and Doppler shift, for potential use in navigation augmentation systems. Jardak et al.14 explored the potential of LEO satellite-based opportunistic navigation for high dynamic applications, particularly when integrated with inertial measurement units. Farhangian et al.15 presented a high-order pseudorange rate measurement model for multi-constellation LEO/INS integration, focusing on the integration of inertial navigation systems with LEO signals of opportunity. Zhao et al.16 proposed a Doppler differential positioning framework using signals of opportunity from LEO satellites, particularly Iridium satellites, and assessed the influence of baselines on differential Doppler positioning. Kozhaya et al.17 introduced a novel blind spectral approach for blind beacon estimation, Doppler tracking, and opportunistic positioning with unknown LEO satellite signals, underscoring the use of Kalman filters for Doppler tracking and the convergence properties of blind beacon estimation frameworks. Collectively, these studies underscore the burgeoning interest in leveraging signals of opportunity from LEO satellites for Doppler positioning and navigation applications. However, these traditional methods for obtaining Doppler frequency are susceptible to noise interference, and they also require substantial computational resources during positioning, which greatly reduces the efficiency of data processing.

In order to address the limitations of the above methods, the application of deep learning techniques based on computer vision for extracting Doppler frequency has been gradually explored. Ngo et al.18 introduced a cutting-edge deep learning approach for the prediction of Doppler shifts in mobile communication systems. Their methodology employed a hybrid model, combining convolutional neural networks (CNNs) with long short-term memory (LSTM) networks, to significantly enhance prediction accuracy. The versatile application of deep learning in sensory systems was further highlighted by Lu et al.19 who developed a deep learning framework tailored for the acquisition of weak signals. Their focus was on accurately determining the absolute position of correlation peak pixels through the use of CNNs. Additionally, Ngo et al.20 expanded the scope of deep learning in signal processing, particularly in predicting channel profiles, Doppler shifts, and signal-to-noise ratios within wireless communication systems. Their approach innovatively incorporated input diversity and binary prediction strategies, aiming to boost prediction accuracy while concurrently reducing computational complexity. However, these models are not well-suited for mobile devices due to their large model size and high number of floating-point operations (FLOPs). And their performance degrades significantly when directly scaled down to a mobile-friendly size. Moreover, considering the characteristics of the spectrum, in order to extract Doppler frequency more efficiently, the model needs to possess both global and local awareness simultaneously.

To solve these challenges, we introduce CLOCFC, a lightweight model that offers both sequence data inference capabilities and enhanced global and local perception. CLOCFC employs a two-branch structure, with each branch dedicated to perceiving global and local information, respectively. We drew inspiration from the Cloformer21 model and refined the integration method of local and global information, thereby achieving enhanced model performance. Therefore, the combination of these elements allows our model to simultaneously handle both local and global patterns, resulting in improved detection of Doppler frequencies in input data. Since LEO satellites continuously change their position and velocity over time, there is a spatiotemporal relationship between their Doppler frequencies. Consequently, we represent this dynamic using continuous differential equations, which is what we refer to as the CFC network. Therefore, the combination of these blocks allows our model to simultaneously capture global and local information, as well as the correlations among sequential data, thereby enhancing the detection of Doppler frequencies.

The contributions and novelties are summarized as follows:In CLOCFC, we utilize the CFC network, which represents the spatiotemporal correlations between sequential data through a set of continuous differential equations, and the outcomes are derived by obtaining the closed-form solutions of these differential equations. Therefore, our model is capable of rapidly processing sequential data. Furthermore, it employs a distinct method to integrate global and local information, achieving superior results.

We created a dataset for LEO satellite signals. Initially, we derived the Keplerian parameters of the LEO satellite from TLE files and predicted the satellite's orbit. Following this, we generate the satellite's transmitted signal, doppler shift, and propagation delay. Ultimately, the spectrogram is utilized as the input for the model.

We applied CLOCFC to the measurement of Doppler frequency. Concurrently, we validated the performance of the proposed model on the dataset and compared it with mainstream models. Notably, with the same order of magnitude parameters, CLOCFC outperforms other models.

The structure of this paper is as follows: in "Data" section, We present the related work on opportunistic signal positioning and deep learning techniques based on computer vision. In "Method" section, We describe the process of creating the dataset. Next, the structure and roles of each module in our model are presented in "Experiment" section. The evaluation of the experimental results and the analysis and discussion of their reasons are addressed in "Results and discussion" section. Finally, our conclusions are provided in "Conclusion" section.

Data

The experimental data was obtained by establishing a satellite-to-ground scenario. We generate satellite scenarios to model and visualize satellites in orbit and to perform additional analyses such as computing access with ground stations, latency, and Doppler shift. Additionally, to make the results more closely resemble real-world conditions, we used the HPOP (High Precision Orbit Propagator) algorithm as the orbit propagator.

As shown in Fig. 1, the generation of LEO satellite simulation signals is roughly divided into three parts: calculating the LEO satellite parameters, computing the latency and Doppler shift between the satellite and the ground station, and generating the spectrogram.Fig. 1 The process of generating satellite signals and their spectrograms. Fc represents carrier frequency.

Calculate the velocity and position of LEO satellites

To calculate the position and velocity of a LEO satellite, it is first necessary to obtain its Keplerian elements. The Keplerian elements, which define the orbits of Low Earth Orbit (LEO) satellites, are available to the public via the North American Aerospace Defense Command (NORAD). These elements are updated daily in the Two-Line Element (TLE) datasets. Utilizing TLEs in conjunction with orbit propagation algorithms, such as the High-Precision Orbit Propagator (HPOP), enables the computation of satellite positions and velocities. As shown in Fig. 2, we used the TLEs data of the Iridium, Iridium Next, and ORBCOMM constellations from November 11, 2023, and employed the HPOP to calculate the satellite orbits.Fig. 2 The trajectory of the ORBCOMM constellation for December 11, 2023. Map data: Google Earth 7.3.6 (https://www.google.cn/intl/zh-CN_ALL/earth/about/versions/).

Calculate the latency and Doppler shift

Firstly, we establish a satellite-to-ground simulation scenario with a start time of November 25, 2023, at 00:00:00 UTC, and a stop time 1 day later. We set the simulation's sample time interval to 60 s. Then, we add Xi'an as the ground station (GS), and specify its latitude and longitude. Secondly, we conduct an access analysis between the LEO satellites and the ground station to determine when the ground station is visible to the satellites. Figure 3 visualizes the satellites that can be observed at a particular moment. This analysis enables the visualization of which satellites are visible from the ground station with time. Finally, we calculate the latency and Doppler shift based on the outcomes of the access analysis.Fig. 3 An example of accessible LEO satellites. We set the minimum elevation angle to 10.

Generate spectrogram

The ORBCOMM satellite downlink utilizes the frequency band of 137–138 MHz, which includes a total of 13 channels. We have selected the eleventh channel as fc. The ORBCOMM satellite’s downlink utilizes SDPSK (Symmetrical Differential Phase Shift Keying) modulation, operating at a data rate of 4800 bits per second. Then, we generated the received signal yi with time offset and frequency offset according to Eq. (1).1 yit=aist-τiej2πft-τi+nit

In Eq. (1), i represents the LEO satellites number, and t denotes the sampling time. ai represents the attenuation coefficient of the signal transmitted by the i-th satellite, τ represents the time delay of the signal transmitted by the i-th satellite, f represents the Doppler shift of the signal transmitted by the i-th receiver (assuming that each receiver downconverts the received signal), C is the speed of signal propagation, and n represents the noise in the signal of the i-th satellite, assumed to be complex Gaussian noise and independent of each other. Finally, we plot the spectrogram in grayscale based on the obtained signals. An example of a spectrogram is shown in Fig. 4.Fig. 4 An example of a spectrogram.

By following the steps outlined above, we can obtain the spectrogram of the ORBCOMM signal. The corresponding Doppler frequency can be calculated based on the position and velocity of the satellite and the ground station. The satellite’s position and velocity can be determined from the published two-line element sets. The ground station’s location can be set as required. The spectrogram serves as the input to the model, while the Doppler frequency is the label for the model. Divide the above data into a training dataset and a validation dataset in the proportions of 80% and 20%, respectively.

Method

Overall architecture

In this section, we provide a comprehensive overview of the proposed CLOCFC. Firstly, we introduce the fundamental structure of the CLOCFC. Secondly, we explain its key modules such as convolution stem, Clo-block, ConvFFN, and CFC block. As depicted in Fig. 5. Initially, the input image is processed by the convolutional stem to generate patches. This stem is constructed from a series of four convolutional layers, each applied with stride values of 4, 4, 1, and 1, respectively. The first two convolutional layers are utilized for the purpose of patch embedding. Consequently, the patches are processed through four sequential stages of Clo blocks and ConvFFNs, which extract features at various levels of the hierarchy. In the final stage, global average pooling is employed, followed by a fully connected layer, to produce the predictions.Fig. 5 The illustration of CLOCFC. CLOCFC consists of three main components: the Clo-block, ConvFFN, and the CFC block, with the specific details of the CFC block depicted in Fig. 6.

Clo block

Due to the characteristics of the input data, relying solely on either local or global perception does not yield satisfactory results. Therefore, to extract more comprehensive features, it is necessary for CLOCFC to separately compute and integrate both global and local information. Clo block, is a lightweight block that leverages context-aware local enhancement.

Clo blocks play a crucial role in CLOCFC. Each of these blocks consists of two primary parts: a local branch and a global branch. Additionally, to enhance the feature representation of the input data, instead of splitting the channels as in the original method21, we input all channels into both the global branch and the local branch separately.

Global branch

In the global branch, as shown in Fig. 5, we first downsample K and V by average pool layer, and then, we apply a nonlinear transformation to derive the Q (query), K (key), and V (value) matrices2 K,V=FCPoolXinQ=FCXin

After that, we proceed with the standard attention mechanism, operating on the Q, K, and V matrices to extract the low-frequency global information:3 Xglobal=Attention(Q,K,V)

Local branch

In this section, we introduce the AttnConv in detail, which has sufficient capacity to capture high-frequency local information. AttnConv is a vital component that empowers our model to deliver superior performance. It integrates several standard attention mechanisms. Specifically, within the AttnConv module, we initiate the process by applying a nonlinear transformation to derive the Q (query), K (key), and V (value) matrices, which is the same as the conventional attention method.4 Q,K,V=ConvXinput

In Eq. (2), X represents the input to the AttnConv module. Conv refers to the convolutional layer. Following the nonlinear transformation, we initiate the local feature aggregation process on V using shared weights. Subsequently, leveraging the transformed V along with Q and K, we perform the context-aware local enhancement.

Local feature aggregation

As illustrated in Fig. 5, we apply a basic depth-wise convolution (DWconv) to Q, K and V for the purposes of local feature aggregation. The weights of DWconv are globally shared:5 QS,KS,VS=DWconvQ,K,V

Context-aware local enhancement

In Eq. (5), we combine Q and K to generate context-aware weights by using DWconvs. Subsequently, we calculate the Hadamard product of the Q (query) and K (key) matrices. Then, we apply a sequence of transformations to this product to generate context-aware weights that range between − 1 and 1. These weights are utilized to enhance the local features. The entire procedure can be summarized as follows:6 Attnt=FCSwishFCQS⊙KSAttn=TanhAttntdXlocal=Attn⊙VS

Fusion with global branch

Unlike the original Cloformer, we employ a straightforward method to merge the outputs from the local branch and the global branch. Specifically, we add the two outputs along the channel dimension. Experiments demonstrate that this simple modification can significantly enhance the model's accuracy. In addition, we apply a fully connected (FC) layer to the channel dimension:7 XFusion=AddXlocal,XglobalXout=FCXFusion

ConvFFN

To integrate local information into the Feedforward Network (FFN) process, we have substituted the standard FFN with a ConvFFN. The key difference between the ConvFFN and the standard FFN is that the ConvFFN applies a depthwise convolution (DWconv) operation after the GELU activation, which allows it to aggregate local information effectively. The presence of DWconv in ConvFFN enables direct downsampling within the module, obviating the need for a separate PatchMerge module.

As shown in Fig. 5, The CLOCFC architecture utilizes two variants of ConvFFN. The first is the in-stage ConvFFN, which incorporates a direct skip connection. The second is the ConvFFN which connects two stages.

CFC block

In fact, we can only receive signals from low Earth orbit (LEO) satellites at specific elevation angles. Therefore, the appearance and disappearance of LEO satellite signals is a continuous process, and this characteristic is also reflected in the collected data. In other words, there is temporal information between consecutive data generated by the same satellite. To capture the interdependencies between sequence data, we apply a CFC network, which can perform auto-regressive modellings based on an entire sequence of observations. In addition, CFC networks constructed by ODEs (Ordinary Differential Equations) are expressive models useful in modeling data with complex dynamics.

CFC models

Leveraging the scalar closed-form solution, Hasani22 distils liquid time-constant network (LTC) into a model that can be trained at scale. Formally, the hidden states x(t), which consist of D hidden units at each time step t, can be obtained as:8 x(t)=σ(-f(x,I;θf)t)⊙g(x,I;θg)+1-σ-f(x,I;θf)t⊙hx,I;θh

where σ and (1-σ) play a gating role which can resolve the potential gradient issues. It signifies an m-dimensional input vector at each specific time step t. Additionally, f(.), g(.) and h(.) are three neural networks parametrized by θ=Wlxm×D,WxxD×D,bxD. ⊙ is the Hadamard product.

Rather than training the three neural network instances separately, we construct them to share initial layers, creating a common backbone from which they diverge into their respective functions. This architectural choice enables our model to develop shared representations across these instances. Consequently, it accelerates and enhances the stability of the learning process.

The CFC architecture is illustrated in Fig. 6.Fig. 6 CFC block architecture.

Experiment

Experimental settings

The experiment was carried out on the UBUNTU 22.04 operating system, utilizing an NVIDIA 4080 graphics card equipped with 16 GB of memory. We randomly divided all data into training and validation sets with respective proportions of 80% and 20%. The training set was utilized for model training, while hyperparameter tuning and model selection were carried out using the validation set. The hyperparameters presented in this paper are those that have been optimized. We evaluated several supervised deep learning models known for their substantial contributions to navigation observables extraction, namely ResNet23, Swin Transformer24, and CloFormer21.

In our experimental setup, we maintained consistent training and evaluation strategies across all models. Additionally, in our experiments, each sequence data contains 10–16 images. The total number of training epochs was determined to be 300. For the loss function, we opted for the mean squared error (MSE) loss function integrated within PyTorch. As the optimization algorithm, we selected AdamW, which, while similar to the Adam optimizer, offers improved generalization capabilities. Furthermore, to refine the model training process, we implemented a dynamic learning rate schedule. The learning rate was initially set to a relatively high value of 0.001 to facilitate rapid convergence to a favorable solution. As training advanced, we progressively reduced the learning rate to achieve more precise and accurate fine-tuning of the model parameters in the vicinity of the optimal solution. This approach aimed to mitigate excessive parameter fluctuations during the later stages of training, thereby fostering a more stable and reliable training process.

Results and discussion

We intend to evaluate the experimental outcomes of our approach in comparison to other methods by examining both visual and metric-based criteria.

Performance evaluation

Comparison of performance under different noise conditions

We report the navigation observables extraction results in Table 1. The results show that our models outperform previous models when they have comparable model sizes and FLOPs even under different noise conditions. As shown in Table 1, under the condition of SNR = −5 dB, the CLOCFC model attains a Mean Absolute Error (MAE) of 0.000296531, while only utilizing 14.7 million parameters and 75.4 G FLOPs, outperforming Res18, SwinTransformer, and Cloformer by margins of 45.56%, 97.41%, and 19.63%, respectively. Additionally, the CLOCFC operates with approximately one-fourteenth the computational load, in terms of FLOPs, when compared to Resnet34. Under different noise conditions, our CLOCFC model exhibits better performance than the Cloformer, but only with an increase of 12% in parameters and 18% in FLOPs. All the above results verify CLOCFC’s superiority. Table 1 Comparison of performance under different noise conditions.

Model	Params(M)	FLOPs(G)	MAE(1e-4)	
SNR = -5	SNR = 5	SNR = 15	SNR = 25	
SwinTransformer	67.107073	519.22	245.789	245.445	244.957	244.958	
Res18	14.051702	684.096	11.6828	5.05401	5.25621	3.74715	
Cloformer	12.929729	63.814	7.9085	5.96624	5.34761	3.35058	
CLOCFC	15.751702	75.418	6.3557	3.92089	3.87133	2.96531	
Res34	25.286593	1410.427	8.3573	5.38394	3.97562	3.35804	
The FLOPs are measured at a resolution of 640 × 1920. The labels in the dataset represent the frequency values of the spectrograms, and the labels have been normalized. For every increase of 0.0001 in MAE, the average error increases by 0.653 Hz.

Comparison of performance at different resolutions

The higher the pixel resolution of the input image and the larger its size, the more computational resources are required. In order to reduce model power consumption and enable its application on mobile devices, the model often needs to operate at lower resolutions. In Table 2, we present a comparative analysis of our proposed CLOCFC model against prior Vision Transformer (ViT) and Convolutional Neural Network (CNN) architectures under different resolution conditions. Table 2 Comparison of performance at different resolutions.

Model	Params(M)	FLOPs(G)	MAE(1e-4)	
256*256	640*640	384*1152	640*1920	
SwinTransformer	67.107073	170.386	245.016	244.572	244.637	244.452	
Res18	14.051702	36.463	8.38518	5.23753	5.01875	4.09413	
Cloformer	12.929729	3.190	4.62905	4.42346	4.50977	3.46148	
CLOCFC	15.751702	4.023	2.86085	3.00988	3.09517	2.8328	
Res34	25.286593	75.223	8.3099	3.99075	3.90637	3.727	
The FLOPs are measured at resolution 256 × 256. The labels in the dataset represent the frequency values of the spectrograms, and the labels have been normalized. For every increase of 0.0001 in MAE, the average error increases by 0.653 Hz.

As shown in Table 2, under the condition of a resolution of 256*256, the CLOCFC model achieved an MAE of 0.000286085 while using only 14.7 million parameters and 4.023 G FLOPs. Its performance surpasses that of Res18, SwinTransformer, and Cloformer by 65.88%, 98.88%, and 38.19%, respectively.

Time-consuming analysis

In Fig. 7, we have recorded the training time and convergence rate of different methods. The results indicate that the CLOCFC model outperforms previous models in both training duration and convergence speed. Compared to other methods, the CLOCFC model we proposed has a faster convergence rate and is more stable after convergence. At the same time, its performance on both the training set and the validation set is superior to other models.Fig. 7 Comparison of convergence rates of high-performance models.

The reason for the results shown in Fig. 7 may lie in the fact that convolutional layers act similarly to high-pass filters, focusing on high-frequency local information, while Multi-Head Attention (MHA) is akin to a low-pass filter, focusing on low-frequency global information25. The Swin Transformer may not capture local information well, making it difficult to converge to a high level of accuracy. The ResNet34, although it only uses convolutional layers, still manages to obtain some global information due to the stacking of multiple convolutional layers, allowing it to converge to a higher level of accuracy but lacking stability. Compared to the original CloFormer, our proposed CLOCFC better integrates local and global information, achieving faster convergence and more stable results. This is also why CLOCFC still achieves better results than other methods under different noise and resolution conditions.

Ablation study

In this section, we decouple the Clo block into several modules and perform comprehensive ablation studies. We aim to validate the contribution of each component of CLOCFC to the extraction of navigation observations.

Global branch

As depicted in Table 3, we conduct a comparison of the model’s performance when it employs only the global branch of the complete CLOCFC. The complete CLOCFC outperforms using only the global branch by 52.36%. Compared to vanilla attention, although the global branch enhances the extraction of global information by reducing FLOPs, it still fails to capture high-frequency local details. Table 3 Ablation of CLOCFC. The FLOPs are measured at resolution 256 × 256.

Model	Params(M)	FLOPs(G)	MAE(1e-4)	
train	valid	
Only global branch	11.532417	2.877	5.321	5.516	
Only local branch	13.470337	3.566	5.959	6.005	
Global branch + local branch	15.708481	3.927	4.861	4.372	
CLOCFC	15.751702	4.023	2.416	2.861	

Local branch

As depicted in Table 3, we conduct a comparison of the model’s performance when it employs only the local branch of the complete CLOCFC. The complete CLOCFC outperforms using only the local branch by 48.13%. The role of the local branch is to capture high-frequency, local information. Although the computation of the Q (query) and K (key) matrices takes into account context-awareness, it still cannot substitute for the global branch's function in capturing low-frequency global information.

Fusion of global and local information.

In Fig. 5, we have improved the original Clo block by enabling both the global and local branches to process the complete data simultaneously, and we have also modified the method of fusing information from the global and local branches. The effect of this strategy is clear from Table 3, where the modification of the fusion method leads to a notable enhancement in the model's convergence rate, achieving a smaller mean error than the original case after 300 epochs of training. This improved performance can be attributed to the model's ability to obtain more comprehensive global and local information.

CFC block

As depicted in Table 3, we investigated the model's performance when combining the CFC block with both the global branch and the local branch. By adding a CFC block, the model can capture the interdependencies between sequence data, finally resulting in better performance. In fact, not only does the model's mean error decrease by 34.56%, but the number of epochs needed for convergence also decreases.

We recorded the data generated from the ablation experiment and plotted it in Fig. 8.Fig. 8 Ablation study results.

Conclusion

In this paper, we propose a recognition method for the Doppler frequency of non-cooperative LEO satellites, a lightweight visual transformer with sequence data processing capabilities, and a dual-branch structure (local and global), and we have generated a dataset for testing. Our proposed CLOCFC model demonstrates competitive accuracy when compared to models with equivalent FLOPs and size. Additionally, our improved Clo block more effectively utilizes input information. In this paper, taking the signals of Low Earth Orbit (LEO) as an example, we demonstrate that CLOCFC is a robust and lightweight method for Doppler frequency identification, outperforming many existing methods. However, as a general-purpose lightweight deep learning model, the application of CLOCFC in various tasks will be the focus of our next work.

Author contributions

Conception and design of study: Zhisen Wang, Hu Lu; Acquisition of data: Zhisen Wang; Analysis and/or interpretation of data: Zhisen Wang, Hu Lu; Drafing the manuscript: Zhisen Wang, Hu Lu, Zhiang Bian; Revising the manuscript critically for important intellectual content: Hu Lu; All authors reviewed the manuscript.

Funding

This work was supported by Key Laboratory of Smart Earth (No. KF2023ZD01-05).

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Morales, J., Khalife, J. & Kassas, Z. M. Simultaneous tracking of orbcomm LEO Satellites and inertial navigation system aiding using Doppler measurements. In 2019 IEEE 89th Vehicular Technology Conference (VTC2019-Spring) pp. 1–6 (2019). 10.1109/VTCSpring.2019.8746485.
2. Reid TGR Neish AM Walter T Enge PK Broadband LEO constellations for navigation Navig. J. Inst. Navig. 2018 65 205 220 10.1002/navi.234
Reid, T. G. R., Neish, A. M., Walter, T. & Enge, P. K. Broadband LEO constellations for navigation. Navig. J. Inst. Navig. 65, 205–220 (2018).10.1002/navi.234
3. Khalife, J. J. & Kassas, Z. M. Receiver design for Doppler positioning with LEO satellites. In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5506–5510 (2019). 10.1109/ICASSP.2019.8682554.
4. Morales, J. J., Khalife, J., Cruz, U. S. & Kassas, Z. M. Orbit modeling for simultaneous tracking and navigation using LEO satellite signals. In Miami, Florida, pp. 2090–2099 (2019). 10.33012/2019.17029.
5. Ardito, C. T., Morales, J. J., Khalife, J., Abdallah, Ali, A. & Kassas, Z. M. Performance evaluation of navigation using LEO satellite signals with periodically transmitted satellite positions, Reston, Virginia, pp. 306–318 (2019). 10.33012/2019.16743.
6. Orabi, M., Khalife, J. & Kassas, Z. M. Opportunistic navigation with Doppler measurements from iridium next and Orbcomm LEO satellites. In 2021 IEEE Aerospace Conference (50100), pp. 1–9 (2021). 10.1109/AERO50100.2021.9438454.
7. Tan Z Qin H Cong L Zhao C New method for positioning using IRIDIUM satellite signals of opportunity IEEE Access 2019 7 83412 83423 10.1109/ACCESS.2019.2924470
Tan, Z., Qin, H., Cong, L. & Zhao, C. New method for positioning using IRIDIUM satellite signals of opportunity. IEEE Access 7, 83412–83423 (2019).10.1109/ACCESS.2019.2924470
8. Wu N Qin H Zhao C Long-baseline differential Doppler positioning using space-based SOP based on BPVGMM IEEE Trans. Instrum. Meas. 2023 72 1 10 37323850
Wu, N., Qin, H. & Zhao, C. Long-baseline differential Doppler positioning using space-based SOP based on BPVGMM. IEEE Trans. Instrum. Meas. 72, 1–10 (2023).37323850
9. Wei Q Chen X Zhan YF Exploring implicit pilots for precise estimation of LEO satellite downlink Doppler frequency IEEE Commun. Lett. 2020 24 2270 2274 10.1109/LCOMM.2020.3003791
Wei, Q., Chen, X. & Zhan, Y. F. Exploring implicit pilots for precise estimation of LEO satellite downlink Doppler frequency. IEEE Commun. Lett. 24, 2270–2274 (2020).10.1109/LCOMM.2020.3003791
10. Morales-Ferre, R., Lohan, E. S., Falco, G. & Falletti, E. GDOP-based analysis of suitability of LEO constellations for future satellite-based positioning. In 2020 IEEE International Conference on Wireless for Space and Extreme Environments (WiSEE), pp. 147–152 (2020). 10.1109/WiSEE44079.2020.9262624.
11. Farhangian F Landry R Multi-constellation software-defined receiver for Doppler positioning with LEO satellites Sensors 2020 20 5866 10.3390/s20205866 33081355
Farhangian, F. & Landry, R. Multi-constellation software-defined receiver for Doppler positioning with LEO satellites. Sensors 20, 5866 (2020).33081355 10.3390/s20205866
12. Minetto A Gurrieri A Dovis F A cognitive particle filter for collaborative DGNSS positioning IEEE Access 2020 8 194765 194779 10.1109/ACCESS.2020.3033626
Minetto, A., Gurrieri, A. & Dovis, F. A cognitive particle filter for collaborative DGNSS positioning. IEEE Access 8, 194765–194779 (2020).10.1109/ACCESS.2020.3033626
13. Chen L Signal acquisition of Luojia-1A low earth orbit navigation augmentation system with software defined receiver Geo-Spatial Inf. Sci. 2022 25 47 62 10.1080/10095020.2021.1964386
Chen, L. et al. Signal acquisition of Luojia-1A low earth orbit navigation augmentation system with software defined receiver. Geo-Spatial Inf. Sci. 25, 47–62 (2022).10.1080/10095020.2021.1964386
14. Jardak N Jault Q The potential of LEO satellite-based opportunistic navigation for high dynamic applications Sensors 2022 22 2541 10.3390/s22072541 35408156
Jardak, N. & Jault, Q. The potential of LEO satellite-based opportunistic navigation for high dynamic applications. Sensors 22, 2541 (2022).35408156 10.3390/s22072541
15. Farhangian F Landry R High-order pseudorange rate measurement model for multi-constellation LEO/INS integration: case of Iridium-NEXT, Orbcomm, and Globalstar Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 2023 237 925 939 10.1177/09544100221113123
Farhangian, F. & Landry, R. High-order pseudorange rate measurement model for multi-constellation LEO/INS integration: case of Iridium-NEXT, Orbcomm, and Globalstar. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 237, 925–939 (2023).10.1177/09544100221113123
16. Zhao C Qin H Wu N Wang D Analysis of baseline impact on differential Doppler positioning and performance improvement method for LEO opportunistic navigation IEEE Trans. Instrum. Meas. 2023 72 1 10 37323850
Zhao, C., Qin, H., Wu, N. & Wang, D. Analysis of baseline impact on differential Doppler positioning and performance improvement method for LEO opportunistic navigation. IEEE Trans. Instrum. Meas. 72, 1–10 (2023).37323850
17. Kozhaya, S., Kanj, H. & Kassas, Z. M. Multi-constellation blind beacon estimation, Doppler tracking, and opportunistic positioning with oneweb, Starlink, Iridium NEXT, and Orbcomm LEO Satellites. In 2023 IEEE/ION Position, Location and Navigation Symposium (PLANS), pp. 1184–1195 (2023). 10.1109/PLANS53410.2023.10139969.
18. Ngo, T., Kelley, B. T. & Rad, P. Deep learning based prediction of doppler shift for mobile communications. In 2021 Telecoms Conference (ConfTELE), pp. 1–6 (2021). 10.1109/ConfTELE50222.2021.9435519.
19. Lu, B., Chen, W. & Qu, C. Deep learning for weak signal acquisition: correlation peak pixel absolute position using deep CNNs. In 2021 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), pp. 1–6 (2021). 10.1109/ICSPCC52875.2021.9564619.
20. Ngo, T., Kelley, B. & Paul, R. Deep learning for signal processing with predictions of channel profile, Doppler shift and signal-to-noise ratio. Preprint at 10.36227/techrxiv.14787810.v1 (2021).
21. Fan, Q., Huang, H., Guan, J. & He, R. Rethinking local perception in lightweight vision transformer. Preprint at https://arxiv.org/abs/2303.17803 (2023).
22. Hasani R Closed-form continuous-time neural networks Nat. Mach. Intell. 2022 4 992 1003 10.1038/s42256-022-00556-7
Hasani, R. et al. Closed-form continuous-time neural networks. Nat. Mach. Intell. 4, 992–1003 (2022).10.1038/s42256-022-00556-7
23. He, K., Zhang, X., Ren, S. & Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016).
24. Liu, Z. et al. Swin transformer: Hierarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 10012–10022 (2021).
25. Park, N. & Kim, S. How Do Vision Transformers Work? Preprint at 10.48550/arXiv.2202.06709 (2022)
