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Sky–ground wave signal separation in enhanced Loran based on Levenberg–Marquart algorithm

2021; Institution of Engineering and Technology; Volume: 16; Issue: 1 Linguagem: Inglês

10.1049/rsn2.12036

1751-8792

Zhenzhu Zhao, Jiangfan Liu, Jinsheng Zhang, Xiaoli Xi,

Direction-of-Arrival Estimation Techniques

IET Radar, Sonar & NavigationVolume 16, Issue 1 p. 1-8 ORIGINAL RESEARCH PAPEROpen Access Sky–ground wave signal separation in enhanced Loran based on Levenberg–Marquart algorithm Zhenzhu Zhao, Zhenzhu Zhao Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorJiangfan Liu, Jiangfan Liu orcid.org/0000-0002-1960-4571 Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorJinsheng Zhang, Jinsheng Zhang Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorXiaoli Xi, Corresponding Author Xiaoli Xi xixiaoli@xaut.edu.cn orcid.org/0000-0003-4349-1308 Department of Electrical Engineering, Xi'an University of Technology, Xi'an, China Correspondence Xiaoli Xi, Department of Electrical Engineering, Xi'an University of Technology, 5 South Jinhua Road, Xi'an 710048, China. Email: xixiaoli@xaut.edu.cnSearch for more papers by this author Zhenzhu Zhao, Zhenzhu Zhao Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorJiangfan Liu, Jiangfan Liu orcid.org/0000-0002-1960-4571 Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorJinsheng Zhang, Jinsheng Zhang Department of Electrical Engineering, Xi'an University of Technology, Xi'an, ChinaSearch for more papers by this authorXiaoli Xi, Corresponding Author Xiaoli Xi xixiaoli@xaut.edu.cn orcid.org/0000-0003-4349-1308 Department of Electrical Engineering, Xi'an University of Technology, Xi'an, China Correspondence Xiaoli Xi, Department of Electrical Engineering, Xi'an University of Technology, 5 South Jinhua Road, Xi'an 710048, China. Email: xixiaoli@xaut.edu.cnSearch for more papers by this author First published: 19 October 2021 https://doi.org/10.1049/rsn2.12036AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract This study proposes a high-precision algorithm used in enhanced Loran sky–ground wave separation based on the Levenberg–Marquart algorithm. The simulation results show that, compared with the existing algorithms, the algorithm proposed in this study has higher accuracy, especially in the case of a low signal-to-noise ratio. Furthermore, when the skywave of enhanced Loran has at least a 35 μs time delay compared with the groundwave, the performance of the algorithm is significantly better than that of other algorithms. Analysis of an actual signal showed that the average correlation coefficient was greater than 99% and that the signal was successfully separated. 1 INTRODUCTION In recent years, some sudden global navigation satellite system (GNSS) interference events have caused huge military and economic losses [1-3]. There is an urgent need to develop an alternative position, navigation, and timing system to use in case of global positioning system loss or degradation. The results of a series of studies have led to a consensus that the Loran-C navigation system can be used as a GNSS backup system [4-7]. Loran-C is a terrestrial, low-frequency, hyperbolic radio navigation system with strong anti-interference performance and high reliability [8]. After signal acquisition and processing, the maximum interference of the enhanced Loran signal is its homologous skywave interference [9]. Technologies that distinguish between the groundwave and skywave components are of key importance. Considerable work has been carried out by many researchers on the algorithm for separation of sky–ground waves. The most classic algorithm is the inverse fast Fourier transform (IFFT) algorithm, which uses the principle of spectral division to separate the signal [10, 11]. Because the division in the frequency domain generates noise across the entire frequency band, it is necessary to perform a filtering process using a window function. This reduces the resolution of the algorithm. Algorithms based on current signal spectrum analysis have since been developed and have been used to estimate Loran skywave time delay. Examples of these algorithms are the auto-regressive moving average algorithm [12], the multiple signal classification algorithm [13], and the estimation of signal parameters by rotational invariance techniques (ESPRIT) algorithm [14, 15]. These algorithms use a parameterised method to perform spectral analysis of the spectral division results to obtain the Loran skywave time delay. However, because the fast Fourier transform (FFT) of a standard signal may have a value of zero (as a divisor), some uncertain results may be obtained. The hybrid weighted Fourier transform and relaxation (Hybrid-WRELAX) algorithm [16] and the extended invariance principle WRELAX (EXIP-WRELAX) algorithm [9] were developed for application to the estimation of the Loran skywave time delay to overcome this problem. These algorithms convert the time delay estimation problem into a non-linear least squares fitting problem and search for the global minimum, thereby greatly improving the accuracy of the solution [9, 16]. Nevertheless, the simulation results have shown that the performance of these algorithms is poorer at low signal-to-noise ratio (SNR) [9, 16]. Because these algorithms are both frequency-domain solution methods, the number of calculations involved is positively correlated to the calculation accuracy [9, 16]. Furthermore, when the number of multipaths increases, the calculation process becomes more complicated. To address these problems, we explored other algorithms. The Levenberg–Marquart (L–M) algorithm is effective in solving problems involving non-linear function parameter fitting [17-19]. The L–M algorithm not only has the global optimisation feature of the Gauss–Newton method but also reduces the requirements for initial values of parameters [20]. Furthermore, the L–M algorithm has high adaptability to different formulas [20]. The sky–ground wave signal is a combination of multiple multipath signals with different amplitudes and time delays in the time domain. When the signal is mathematically modelled in the time domain, the problem of sky–ground wave separation is transformed into a problem of parameter estimation for a known formula. As the number of multipaths increases, only the size of the matrix used in the calculation needs to be increased, and as a result, the calculation process remains unaffected. In this study, the L–M algorithm was applied to sky–ground wave separation. Because the received enhanced Loran signal is a synthetic wave of groundwaves and skywaves with different amplitudes and time delays, a non-linear model for the signal was established in the time domain. The IFFT method was then used to preprocess the parameters in the model to determine the number of received signals and the initial values of the parameters. Finally, the L–M algorithm was used to fit the parameters in the model to achieve the purpose of separating signals. Compared with the IFFT, Hybrid-WRELAX, and EXIP-WRELAX algorithms, the proposed algorithm achieves higher accuracy, especially at low SNR. Even when the time delay of the skywave of enhanced Loran is at least 35 μs relative to the groundwave, the performance of the algorithm proposed in this study is still good. Analysis of an actual signal showed that the average correlation coefficient was greater than 99% and that the signal was successfully separated. 2 RECEIVED SIGNAL MODEL AND IFFT METHOD 2.1 Received signal model The signal model used to study the sky–ground wave separation method assumes that the skywave and groundwave have the same shape as the standard signal but have different time delays and amplitudes. In the time domain, the received enhanced Loran signal can be expressed as follows: y ( t ) = ∑ l = 1 L a l x ( t − τ l ) + e ( t ) (1)where x ( t ) is the standard enhanced Loran signal, e ( t ) represents all interference and noise, and a l and τ l represent the amplitudes and time delays of the received signals, respectively. The subscript l indicates one of the Lth signals. The FFT of Equation (1) can be expressed as follows: Y ( f ) = X ( f ) ∑ l = 1 L a l e j 2 π f τ l + E ( f ) (2)where Y ( f ) , X ( f ) , and E ( f ) are the FFT of y ( t ) , x ( t ) , and e ( t ) , respectively. Equations (1) and (2) represent the time-domain and frequency-domain forms of the received signal, respectively. The simulated signal used in this study, shown in Figure 1, is a combination of a groundwave, one-hop skywave, and noise. FIGURE 1Open in figure viewerPowerPoint Time-domain and frequency-domain diagrams of the signal. The groundwave time delay is 77 μs, the skywave time delay is 133 μs, the skywave-to-groundwave ratio (SGR) is 12 dB, and the signal-to-noise ratio is 15 dB. (a) Time-domain signal; (b) Frequency-domain signal 2.2 IFFT method It has been shown in [9] that the IFFT of the ratio of the received composite signal spectrum to the standard enhanced Loran pulse spectrum generates a pulse at the arrival time of the skywave or the groundwave component. Analytically, this is expressed as follows: F − 1 { Y ( f ) X ( f ) } = ∑ l = 1 L a l δ ( t − τ l ) + F − 1 { E ( f ) X ( f ) } (3)where F − 1 represents the IFFT. Referring to Equation (3), the impulse function of the time delay τ l can be obtained in the time domain. Because the bandwidth of the enhanced Loran signal is only 20 kHz, the division operation greatly amplifies the noise component outside the frequency band. The solution is to add a window function in the frequency domain. Direct application of Equation (3) to estimate the skywave time delay requires a high SNR. To illustrate, Figure 2 shows the frequency-domain diagram of the magnitude of Y ( f ) / X ( f ) from 0 to 500 MHz and the time-domain diagram after IFFT, using the signal in Figure 1. The results show that the wave components are lost in the noise. Figure 3 shows the frequency-domain diagram of the magnitude of Y ( f ) / X ( f ) from 0 to 500 MHz after application of a window 50 kHz wide and the time-domain diagram after IFFT. The wave components can be obtained through peak detection. FIGURE 2Open in figure viewerPowerPoint Results of application of the inverse fast Fourier transform (IFFT) method without the window function after spectrum division. (a) Frequency-domain diagram of Y ( f ) / X ( f ) ; (b) Time-domain diagram after IFFT FIGURE 3Open in figure viewerPowerPoint Results for the inverse fast Fourier transform (IFFT) method with the window function applied after spectrum division. (a) Frequency-domain diagram of Y ( f ) / X ( f ) after window function application; (b) Time-domain diagram after IFFT Although this method reduces the resolution of the wave, the number of standard signals and meaningful initial values of the time delay parameters can be obtained simply and quickly. 3 LEVENBERG–MARQUART ALGORITHM This method evaluates whether the best approximation is achieved by iteratively calculating the sum of the squared residuals. The algorithm not only has the global optimisation feature of the Gauss–Newton method but also reduces the requirements for initial values of parameters [20]. The IFFT method was used in this study for preprocessing to determine the function model and initial parameter values and avoid the result converging to the local minimum. The objective function of the L–M method is expressed as follows [19]: χ 2 ( a ) = 1 N ∑ i [ y i − f ( x i ; a ) ] 2 (4)where χ 2 is the sum of the squares of the residual errors, y i is the actual value, N is the number of y, a = [ a 1 , a 2 , … a p ] , a is the parameter value to be determined, x i = [ x 1 i , x 2 i , … x h i ] , h is the number of x i , h = 1 in this study, and p is the number of parameters. According to the principle of least squares regression, when solving for the values of the parameters of a non-linear function, it is necessary to minimise the sum of the squares of the residual errors, that is, y = min χ 2 ( a ) (5) Multiple iterations of the L–M method yield values for the parameters that approach the optimal parameters. The iteration process can be described as follows: a min = a c u r + D − 1 [ − ∇ χ 2 ( a c u r ) ] (6)where D = H + λ I , I is the identity matrix, H is the Hessian matrix, and λ is the damping parameter. Because the H matrix is a function of multidimensional variables, its second-order partial derivative matrix can be expressed as follows: H ( χ 2 ) = [ ∂ 2 χ 2 ∂ a 1 2 ∂ 2 χ 2 ∂ a 1 ∂ a 2 ⋯ ∂ 2 χ 2 ∂ a 1 ∂ a p ∂ 2 χ 2 ∂ a 2 ∂ a 1 ∂ 2 χ 2 ∂ a 2 2 ⋯ ∂ 2 χ 2 ∂ a 2 ∂ a p ⋮ ⋮ ⋱ ⋮ ∂ 2 χ 2 ∂ a p ∂ a 1 ∂ 2 χ 2 ∂ a p ∂ a 2 ⋯ ∂ 2 χ 2 ∂ a p 2 ] (7) In the Hessian matrix, ∂ χ 2 ∂ a k = − 2 N ∑ i = 1 N [ y i − f ( x i ; a ) ] ∂ f ( x i ; a ) ∂ a k k = 1,2 … p (8) ∂ 2 χ 2 ∂ a k ∂ a l = 2 N ∑ i = 1 N [ ∂ f ( x i ; a ) ∂ a k ∂ f ( x i ; a ) ∂ a l − [ y i − f ( x i ; a ) ] ∂ 2 f ( x i ; a ) ∂ a k ∂ a l ] k = 1,2 … p l = 1,2 … p (9) 4 EXPERIMENTAL SIMULATION AND RESULTS The performance of the algorithm presented in this study is illustrated herein by the results of a series of experiments consisting of two parts, using the simulated signal and the actual measurement signal. 4.1 Simulation data The simulation experiment was repeatable. Signals were generated in the same manner each time and are shown in Figure 1. The time delay of the groundwave was 77 μs, and that of the skywave was 133 μs. The SNR was 15 dB, and the SGR was 12 dB. The signal sampling frequency was 1 MHz for 1000 μs. The convergence condition was set to <0.01. Because the IFFT method is a classic method and the Hybrid-WRELAX and EXIP-WRELAX algorithms are currently the most accurate algorithms available, we selected these three methods for comparison. We performed multiple simulations and recorded the time delay error results. Figure 4 shows the time delay error results for the four algorithms for the same conditions. Under these simulation conditions, the average number of iterations for the L–M method was 20. FIGURE 4Open in figure viewerPowerPoint Results for the time delay error (for a time delay between the skywave and groundwave of 56 μs and signal-to-noise ratio = 15 dB). (a) Comparison of all four methods; (b) Comparison of three methods; (c) Comparison of three methods based on the absolute value of the error. EXIP-WRELAX, extended invariance principle WRELAX; Hybrid-WRELAX, hybrid weighted Fourier transform and relaxation; L–M, Levenberg–Marquart As Figure 4a shows, the time delay error of the IFFT algorithm was larger than that of the other three algorithms, resulting in differences between them that cannot be observed. Therefore, the differences are shown separately in Figure 4b,c. The L–M algorithm has an error variance of ≤0.1 μs, and that of the other two algorithms is ≤0.2 μs. Figure 5 shows a comparison of the average estimation errors of the four algorithms for SNR values from 0 to 20 dB SNR. The L–M algorithm performed better, especially at SNR = 0 dB, and the accuracy was greatly improved. Figure 6 shows the relevant simulation results for SNR = 0 dB. The error range of the L–M method was significantly smaller than that of the other two methods. In addition, according to the results shown in Figure 5, the higher the SNR is, the higher is the accuracy. When the SNR is sufficiently large, the accuracy improvement is very small. FIGURE 5Open in figure viewerPowerPoint Comparison of estimation errors of four algorithms under different signal-to-noise ratio. EXIP-WRELAX, extended invariance principle WRELAX; Hybrid-WRELAX, hybrid weighted Fourier transform and relaxation; IFFT, inverse fast Fourier transform; L–M, Levenberg–Marquart FIGURE 6Open in figure viewerPowerPoint Results for the time delay error (for a time delay between the skywave and the groundwave of 56 μs and signal-to-noise ratio = 0 dB) (taking the absolute value of the error). EXIP-WRELAX, extended invariance principle WRELAX; Hybrid-WRELAX, hybrid weighted Fourier transform and relaxation; L–M, Levenberg–Marquart According to estimates, the minimum time delay between the groundwave and skywave should be approximately 35 μs [8]. We therefore conducted simulation experiments and analysed the performance of the algorithms for this case. The time delay of the skywave was 70 μs, and that of the groundwave was 35 μs. The other simulation conditions are the same as those for the case illustrated in Figure 4. The results are shown in Figure 7. The performance of the L–M algorithm was still good, with an error within 0.5 μs accounting for 95%, whereas the hybrid-WRELAX and EXIP-WRELAX algorithms did not perform well. FIGURE 7Open in figure viewerPowerPoint Results for the time delay error (for a time delay between the skywave and the groundwave of 35 μs and signal-to-noise ratio = 15 dB). EXIP-WRELAX, extended invariance principle WRELAX; Hybrid-WRELAX, hybrid weighted Fourier transform and relaxation; L–M, Levenberg–Marquart 4.2 Actual data To further verify the actual application effect of the algorithm, the L–M algorithm was used to analyse actual data. A diagram of the signal acquisition process is shown in Figure 8. The whip antenna transmits the received signal to the analogue circuit processing module, which amplifies and filters the analogue signal. The purpose of the analogue amplifier circuit is to adjust the amplitude of the signal to an appropriate amplitude to meet the input requirements of the analogue-to-digital converter (ADC), and its gain is 20 dB. The analogue bandpass filter has a passband in the range of 60–140 kHz and a stopband attenuation of −40 dB and is mainly used to filter out the power frequency interference. After being amplified and filtered, the analogue signal is converted into a digital signal by a high-speed ADC and sent to a field-programmable gate array (FPGA) for processing. The FPGA performs bandpass filtering and resampling of the signal. The bandpass filtering range used is 90–110 kHz to filter out-of-band interference. The purpose of resampling is to reduce the signal sampling rate and reduce the pressure of post-stage signal processing. The filtered and resampled signal is uploaded to the host computer via a universal serial bus 3.0 connection for storage. FIGURE 8Open in figure viewerPowerPoint Signal acquisition process. ADC, analogue-to-digital converter; FPGA, field-programmable gate array Because the experimental location was nearest to the Pucheng station and the other stations were farther away, the signal from the Pucheng station was the strongest and most stable. The enhanced Loran signal transmitted from the Pucheng station in China as the main station was taken on 16 May 2020 at 3:00 PM, in a sunny weather condition. In this environment, the wave changes were relatively smooth, and the test results were stable. Therefore, it was easy to confirm the accuracy of the test method. The measured pulse train of the enhanced Loran master station and the single Loran signal are shown in Figure 9. First, the IFFT method was used to analyse the signal. The results are shown in Figure 10. There are five main enhanced Loran signals, so the time-domain function model can be expressed as follows: y ( t ) = ∑ l = 1 5 a l x ( t − τ l ) (10) FIGURE 9Open in figure viewerPowerPoint Actual signal. (a) Pulse train; (b) Single signal FIGURE 10Open in figure viewerPowerPoint Results for the inverse fast Fourier transform (IFFT) algorithm, Figure 9b The results of the parameter fitting using the L–M algorithm are shown in Table 1, and the correlation coefficient obtained was more than 99%. The signal was separated successfully, as the results are shown in Figure 11. In practice, signal propagation through the Earth's surface or ionospheric reflection will cause dispersion, resulting in signal distortion. However, the analysis results indicate that the correlation coefficient of the fitting result can reach 99% when only the amplitude and time delay are considered. We think this is because the bandwidth of the enhanced Loran signal is only 20 kHz and because the signal energy is relatively concentrated, so the dispersion is small and can be ignored. TABLE 1. Fitting results for L–M algorithm Parameters Fitting results Amplitude Time delay Multipath 1 52.6288826 35.5730441 μs Multipath 2 7.40948602 91.4399731 μs Multipath 3 −8.5262195 141.423505 μs Multipath 4 3.58738472 206.036894 μs Multipath 5 −5.0915786 288.491786 μs Correlation coefficient 0.993990 Abbreviation: L–M, Levenberg–Marquart. FIGURE 11Open in figure viewerPowerPoint Result of sky–ground wave separation using Levenberg–Marquartalgorithm According to the results shown in Figure 10 and Table 1, there are five main enhanced Loran signals. Because the distance between the measurement point and the transmission station is approximately 92 km, the earliest arrival and the strongest signal within this distance is undoubtedly the groundwave signal. The amplitudes of the other four signals are significantly smaller than that of the groundwave signal, which can be considered the one-hop skywave and multipath. The multipath may come from the multilayer reflection of the ionosphere or reflection from the surrounding high-rise buildings. The other Loran stations are farther away, and the signal is weaker. In this study, we mainly explored the method for separating the signal in the time domain; the specific source of the signal was not examined. The signals for the entire pulse train were analysed, and the parameter values obtained for the ground wave signal are shown in Table 2. The results are consistent with the characteristics of the signal in the enhanced Loran. Since the phase is uniformly set to 0 rad during the fitting process, negative amplitude values will appear. We can also observe a modulation phenomenon of ±1 or 0 μs . TABLE 2. Fitting results for groundwave in a pulse train Parameter Fitting results Amplitude Signal 1 −52.33464853 36.63145999 μs Signal 2 −52.32097227 36.67132160 μs Signal 3 52.55986085 35.57973174 μs Signal 4 52.6288826 35.57304410 μs Signal 5 −52.22477246 37.55929700 μs Signal 6 52.48553795 37.55036201 μs Signal 7 −52.6246686 36.54921456 μs Signal 8 52.58591216 36.54690676 μs Signal 9 −52.46185704 36.57369682 μs Average correlation coefficient 0.995336804 5 CONCLUSIONS In this study, a L–M optimal approximation algorithm was used for enhanced Loran sky–ground wave separation. The simulation results obtained showed that, compared with the IFFT, Hybrid-WRELAX, and EXIP-WRELAX algorithms, the L–M algorithm achieves higher accuracy, especially at low SNR values. Furthermore, when the enhanced Loran skywave time delay was at least 35 μs with respect to the groundwave, the performance of the algorithm was significantly better than that of the Hybrid-WRELAX and EXIP-WRELAX algorithms. Analysis of an actual signal showed that the proposed algorithm yielded an average correlation coefficient greater than 99% and that the signal was successfully separated. ACKNOWLEDGEMENTS This work was supported in part by the National Natural Science Foundation of China under Grant 61771389 and Grant 61701398, and in part by Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, China under Grant 19JS050. CONFLICT OF INTEREST We declared that we have no conflicts of interest to this work. PERMISSION TO REPRODUCE MATERIALS FROM OTHER SOURCES None. Open Research DATA AVAILABILITY STATEMENT Data available on request from the authors. REFERENCES 1Russia suspected of jamming GPS signal in Finland. https://www.bbc.com/news/world-urope-46178940. Accessed 12 Nov 2018 Google Scholar 2South Korea issues warning over suspected North Korean GPS disruption. https://www.gpsworld.com/south-korea-issueswarning-over-suspected-north-korean-gps-disruption. 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