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

39237594
71620
10.1038/s41598-024-71620-y
Article
Enhancing the correlation between azimuth and Doppler frequency for speed estimation of nearby tangential targets
Li Wenjie 12
Luan Yuchen luanyc@aircas.ac.cn

1
Chai Zicheng 12
Chen Longyong 1
1 grid.9227.e 0000000119573309 National Key Laboratory of Microwave Imaging, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, 100190 China
2 https://ror.org/05qbk4x57 grid.410726.6 0000 0004 1797 8419 School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing, 100049 China
5 9 2024
5 9 2024
2024
14 2075926 1 2024
29 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/.
With the development of intelligent transportation systems, traffic supervision radar with wide coverage plays a crucial role in traffic management and vehicle-road coordination. The correlation between Doppler frequency and azimuth has been widely validated in wide coverage traffic supervision radar for high-precision velocity measurement. However, angular glint and noise of the nearby targets lead to a decrease in correlation between the azimuth and Doppler frequency, which negatively impacts the accuracy of velocity estimation. Currently, adopting separate filtering strategies for target azimuth and Doppler frequency has limited performance in enhancing correlation. This paper presents a joint observation model for azimuth and Doppler frequency to achieve the extraction of interrelated components from subspaces, which improves the accuracy of velocity measurement. The effectiveness of this approach is validated using data obtained from X-band and Ku-band sensors.

Subject terms

Electrical and electronic engineering
Characterization and analytical techniques
National Key R&D Program of China2022YFB3901601 Chen Longyong issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

The vehicle detector plays a crucial role in making traffic-related decisions, including traffic management and vehicle-road cooperation applications1–3. Its basic functions include traffic flow detection and speed measurement, which must be executed accurately and consistently under various lighting and weather conditions. Moreover, the installation method should not damage the road surface. As a result, microwave radar technology has become more prevalent than other systems, such as loop detectors, video detectors4,5, and infrared detectors6.

The use of side-mounted radar systems has become increasingly popular in recent years due to their capability to accurately detect and track vehicles in various traffic scenarios7,8. These systems are typically installed on lamp posts and are more easily deployed in urban road or highway settings compared to top-mounted installations9–11. However, in long-distance scenarios, where the radar line of sight cannot be perpendicular to the road7, the radar beam covers a wide range along the road direction. This can result in multiple vehicles entering the radar beam simultaneously, leading to the misidentification of adjacent vehicles in the same lane as a single entity12.

Long-distance coverage can be achieved when the radar’s line of sight is perpendicular to the road. Additionally, installing the radar perpendicularly to the road on long-distance lanes can prevent multiple vehicles from entering the beam simultaneously. This feature has attracted significant attention from researchers13–17. In the application scenario of side-mounted traffic radar, it has been pointed out that the target cannot be considered an ideal point. Angular glint must be considered18, and the optimal fixed-gain filter (OFGF) is used to smooth the noise of the Doppler frequency15. The correlation between the azimuth and Doppler frequency is utilized to achieve high-precision velocity measurement under the premise of zero Doppler center16. In reality, the correlation between azimuth and Doppler frequency is influenced by multiple factors. Due to the large size of the vehicle and its proximity to the antenna, the scattering center of the vehicle changes during driving, which is called angular glint. Additionally, due to the use of dual antenna reception, the scattering center of the car observed by the two antennas also changes, which is called baseline decoherence. In such situations, the correlation between azimuth and Doppler frequency decreases significantly. Seislet17,19 and OFGF15,20 are proposed for smoothing noise in Doppler frequency in traffic radar. However, they are unable to alleviate the the decline in correlation because the filtering process of azimuth and Doppler frequency are carried out separately.

In conclusion, existing traffic radars already use the correlation between Doppler frequency and azimuth angle for speed measurement operations. However, they overlook the decline in correlation caused by noise, such as baseline decoherence and angular glint of nearby targets. In terms of signal processing, this paper points out that common filtering methods are ineffective against these noise effects and proposes a joint observation model for azimuth and Doppler frequency under the terms of such situations. The model allows for the extraction of interrelated components between azimuth and Doppler frequency from subspaces using two-dimensional rotation invariance property. This leads to interference suppression and a significant enhancement in the correlation between azimuth and Doppler frequency under low signal-to-noise ratio (SNR) conditions. Linear fitting between azimuth and Doppler frequency is used to further improve the accuracy of velocity measurement.

Methods

The correlation between azimuth and Doppler frequency

Under the condition of side view observation, the Doppler history of the ideal point is as Eq. (1).1 fd=2v2λRη

In which, η indicates slow time, λ represents the wavelength, R represents the distance from the scattering center to the radar, and v represents the velocity. However, this method faces significant limitations in practical applications due to angular glint. Due to not being considered as a point target, the main scattering center of the vehicle continuously changes with its movement in the antenna beam. As shown in Fig. 1a, the front half of the car is the main scattering center when the vehicle enters the beam, but it changes to the back of the car when the vehicle leaves the beam. To solve the problem, the azimuth and Doppler frequency of the target can be measured separately, and the velocity of the target can be determined through their relationship. At a certain slow time η, the Doppler frequency of the target can be represented by the Eq. (2).2 fd(η)=2vλsinθ(η)

Fig. 1 Azimuth and Doppler measurements model: (a) the explanation of angular glint; (b) the geometric model for rotational invariance applications.

In which, the azimuth of the scattering center θ(η) can be measured using a dual-channel antenna, as shown in Eq. (3).3 Δφ(η)=2π·Δxλsinθ(η)

where, Δφ represents the phase difference between signal of two channels, and Δx indicates the distance between the two receiving antennas. From Eq. (2) and (3), it can be seen that the speed of the vehicle can be completely obtained from the azimuth of the scattering center of the target and the Doppler frequency.4 v=fd(η)Δφ(η)π

Enhancing the correlation between azimuth and Doppler frequency via 2D rotational invariance

The following derivation demonstrates the principle of using rotational invariance to further enhance the correlation between azimuth and Doppler frequency. The geometric model for the side-mounted traffic radar is shown in Fig. 1b. In the coordinate defined by O, the radar is installed on the Z-axis with an altitude of H, and the target moves against the Y-direction with a constant velocity. The range between the phase center and the target at position A is R0. θ, and R(η) respectively represent the azimuth angle and the distance when the target at position B. Suppose that the linear frequency modulated (LFM) signal is transmitted, that targets with different range exhibit different frequencies after de-chirp21. Fast Fourier Transform (FFT) is used to recover the range profile, and the result can be given as5 sp(η)=χt-2R(η)c×exp4jπfcR(η)c

where, t and fc represent fast time and frequency. c and j represent the speed of light and the imaginary unit, respectively. The amplitude envelope of sp(η) is denoted by χ(·). To describe the two-dimensional rotation invariance of azimuth and Doppler Frequency, it is necessary to approximate the echo signal in a limited number of pulses. Considering the movement of the target from point B to point B′, as shown in Fig. 1b, the relationship of Eq. (6) can be constructed.6 R02+l+vη2=R2ηR02+l2=R0′2

By simplifying Eq. (6), the relationship between the distance R(η) and the starting distance R0′ can be expressed as7 R(η)=R0′2+2vη(vη+l)-v2η2

As shown in Fig. 1b, it is clear that vη+l=R0tanθ(η). Therefore, Eq. (7) can be further simplified to Eq. (8).8 R(η)=R0′×1+2vηR0tanθ(η)-v2η2R0′

Since both η and α are small, R0≈R0′cosθη , Eq. (8) can be expressed as a Taylor series, and the higher-order terms are small and negligible, therefore9 R(η)≈R0′+vηsinθ(η)

Combining Eq. (9) and (5), we have10 s(r,η)=χt-2R(η)c×exp4jπfcR0′c×exp4jπfcvηsinθ(η)c

Denote the received data vectors of the two channels as x=[x1,x2,...,xM]Tand y=[y1,y2,...,yM]T, respectively, where the n represents the discrete slow time. The radar installation is depicted in Fig. 1b, with both antenna patterns oriented in the same direction, perpendicular to the road. By substituting Eq. (2) into Eq. (10) and converting it to discrete form, x(n) can be expressed as11 x(n)=exp4jπfcvn·PRIsinθnc=exp2jπfd·n·PRI

Therefore, the Doppler frequency fd in Eq. (2) of the target can be obtained from the phase difference between adjacent pulses in the same channel, as shown in Eq. (12).12 αn=xn+1·xn∗=exp2jπfd·PRI

The target’s angle can be obtained from the phase difference between two channels at the same time, as shown in Eq. (13), corresponding to the complex form of Eq. (3).13 βn=xn·yn∗=exp2jπfcΔxsinθc

where, θ is azimuth angle, PRI represents the cycle of pulse repetition. To characterize the correlation between Doppler frequency and azimuth , the echoes of two channels at different times are expressed in matrix Sn as14 Sn=Aσk+N=xnxn-1...xn-k+1ynyn-1...yn-k+1xn+1xn...xn-kyn+1yn...yn-k

15 A=αn0αn1⋯αnkαn0αn1⋯αnkαn0αn1⋯αnkαn0αn1⋯αnk⊙βn0βn0⋯βn0βn1βn1⋯βn1βn0βn0⋯βn0βn1βn1⋯βn1

16 σk=diagσ1,⋯,σk

where, σk represents scattering coefficients of equivalent scattering centers at k slow times, and ⊙ represents Hadamard product. N represents Gaussian white noise matrix. Furthermore, the sub-matrix of A can be exploited to analyze its rotation invariance. Combined with Eq. (14-16), the following characteristics of Doppler frequency and azimuth can be obtained,17 JyαnA=αnJxαnAJyβnA=βnJxβnA

The property of matrix A expressed in Eq. (17) is called two-dimensional rotational invariance. Where, Jx(αn) and Jy(αn) are used to obtain a sub-matrix of A that describes the rotational invariance of αn and Jx(βn) and Jy(βn) are used to obtain a sub-matrix of A that describes the rotational invariance of βn. The specific form can be expressed as Eq. (18).18 Jx(αn)=10⊗I2Jy(αn)=01⊗I2Jx(βn)=I2⊗10Jy(βn)=I2⊗01

where, I2 represents second-order unit matrix and ⊗ represents Kronecker product. To use the two-dimensional rotation invariant property of matrix A to jointly enhance the correlation between the azimuth and Doppler of the target under terms of glint conditions, this paper consider extracting interrelated components by subspace. First note that the rank of A is only 4, since this is the number of rows of Sn. We compute the SVD of Sn, i.e. Sn=UsnΣVsn, where Usn has 4 columns, spanning the column space of Sn. Thus, the linear transformation of matrix A represented by Eq. (17) also holds for Usn. Thus, least squares22–24 is employed.19 αn^=JxαUsn†·JyαnUsnβn^=JxβUsn†·JyβnUsn

Fig. 2 The process of the proposed method.

Then, the acquired Doppler frequency arg(α^n) and azimuth arg(β^n) are linearly fitted. The speed of the extended target can be estimated by the slope of the fitting. Where arg(·) represents phase taking operation and † represents the pseudo-inverse of a matrix. The proposed solution process is shown in the Fig. 2.

In the signal processing stage, an adaptive filter is applied to suppress the stationary clutter25. The detected vehicle data x(n) and y(n) are then combined and subjected to a SVD operation. The signals x(n) and y(n) are acquired from dual channels, with the antenna pattern of each channel oriented perpendicularly to the road. Subsequently, 2D-ESPRIT least squares is utilized to extract azimuth and Doppler frequency-related components, which are linearly fitted. The slope of the linear fitting result can represent the speed of the target, as shown in Eq. (4).

Results

System descriptions

This paper deploys the Ku-band and X-band dual channel traffic radars on the side of a road to obtain data in three scenarios, as shown in Fig. 3. Both X-band and Ku band radars transmitt FMCW signals and are equipped with one transmitter and two receivers. The de-chirp reference signals of two receiving channels are coupled from the transmitting channel to ensure the coherence of the radar. By adjusting its pitch angle to an appropriate value, full lane coverage can be achieved. All radars operate on a pulse repetition cycle of 2 kHz.Fig. 3 Experimental scenarios: (a) scenario for data acquisition on highways; (b) scenario for data acquisition on busy street; (c) data acquisition scenario for closer targets.

The working parameters of the radar in the above three scenarios are shown in Table 1. In the first scenario, the radar is mounted on the side of a highway, simulating the most common use case of traffic radar. In the second scenario, the radar is placed on the side of a busy street, evaluating the effectiveness of the proposed method in dense traffic conditions. In the third scenario, the radar is set at a low height and closer to the vehicles, verifying the effectiveness of the proposed algorithm under severe angular glint effects. Table 1 Radar parameters in different scenarios.

Scenario	Frequency	Antenna spacing	Height	
Scenario 1	14.6 GHz	30 cm	6 m	
Scenario 2	14.6 GHz	25 cm	4 m	
Scenario 3	9.6 GHz	7.4 cm	1.5 m	

Different vehicles

In this section, the variations in Doppler and azimuth characteristics of different vehicle types, as well as the performance of the proposed and comparison algorithms on different vehicle types, are compared. Data for trucks and cars are obtained in scenario 1, while data for bicycles are collected in scenario 2.

In Fig. 4a, the short-time Fourier transform (STFT) of the full-time signal shows that a bicycle is represented as a single frequency point at a specific slow time. This strongly supports the observation that the bicycle’s small size results in minimal Doppler bandwidth. Additionally, the low Doppler frequency in Fig. 4a indicates that the bicycle is moving at a slow speed. Moreover, the zero Doppler ridge in the figures represents ground reflection clutter. It is necessary to eliminate this part through clutter suppression techniques, as it almost always accompanies the bicycle signal. In Fig. 4b, the disappearance of the zero Doppler ridge demonstrates the effectiveness of clutter suppression. In Fig. 4c, e and g, the compact size of bicycles makes them less susceptible to glint noise and results in nearly linear changes in the Doppler frequency. In Fig. 4c and g, both the mean filter and the proposed method effectively remove noise, with the filtered curves closely matching the pre-filtered curves. Wherein, Fig. 4d and f show a stronger correlation between Doppler frequency and azimuth. In Fig. 4e, PFGF causes significant deviation from the pre-filtered results. Because of the excessive deviation from the original phase curve, the Doppler frequency and azimuth angle no longer exhibit a linear correlation, as shown in Fig. 4f. This indicates that OFGF is not suitable for situations with strong interference.Fig. 4 Enhancing the correlation between azimuth and Doppler frequency (bicycle, range = 20 m): (a) STFT results before clutter suppression; (b) STFT results after clutter suppression; comparison of phase history smoothing between mean filtering (c), OFGF (e), and the proposed method (g); comparison of Doppler phase and azimuth correlation between mean filtering (d), OFGF (f), and the proposed method (h).

Fig. 5 Enhancing the correlation between azimuth and Doppler drequency (car, range = 15 m): (a) STFT results before clutter suppression; (b) STFT results after clutter suppression; comparison of phase history smoothing between mean filtering (c), OFGF (e), and the proposed method (g); comparison of Doppler phase and azimuth correlation between mean filtering (d), OFGF (f), and the proposed method (h).

Fig. 6 Enhancing the correlation between azimuth and Doppler drequency (truck, range = 10 m): (a) STFT results before clutter suppression; (b) STFT results after clutter suppression; comparison of phase history smoothing between mean filtering (c), OFGF (e), and the proposed method (g); comparison of Doppler phase and azimuth correlation between mean filtering (d), OFGF (f), and the proposed method (h).

Figures 5a and 6a show the STFT results for a car and a truck. Due to their considerable length, cars and trucks cannot be considered point targets. When such vehicles enter the beam, they exhibit positive Doppler components, and as more of the vehicle enters the beam, the Doppler bandwidth gradually increases. At this point, the zero Doppler ridge in Fig. 5a and 6a represents clutter-induced Doppler. When the vehicle fully occupies the beam, the Doppler bandwidth of the echo signal reaches its maximum. Additionally, because large vehicles block part of the ground reflection, the clutter component in the radar signal decreases. As the vehicle continues to move and starts to exit the radar coverage, the Doppler bandwidth of the radar signal gradually decreases, and the zero Doppler ridge becomes prominent again. This analysis indicates that when large vehicles enter or exit the beam, the echo signal also contains both the vehicle and strong ground clutter. Therefore, clutter suppression is necessary to purify the signal. In Fig. 5b and 6b, the disappearance of the zero Doppler ridge demonstrates the effectiveness of clutter suppression. In Fig. 5 and 6c, e, and g, the longer lengths of vehicles result in significant fluctuations in their Doppler phase and azimuth angle, differing from the bicycles in Fig. 4. Despite significant fluctuations in Doppler frequency and azimuth, the original scatter plots in Fig. 5 and 6d, f, and h show a degree of linear correlation. However, the correlation is diminished by noise. In Fig. 5 and 6d and h, both mean filtering and the proposed method effectively smooth noise, aligning with the pre-filtered data and significantly enhancing the correlation between Doppler frequency and azimuth angle. Moreover, the proposed method suppresses outliers better than the mean filter, leading to smoothed data with a stronger correlation, compared to mean filtering. However, the presence of strong singular values cause the OFGF to accumulate errors resulting in smoothed results that mismatch with the pre-smoothed data in Fig. 5 and 6e. As a result, the scatter plot in Fig. 5f and 6f no longer align with the original data.

Different scenarios

In this section, a more comprehensive statistical analysis of the performance of different algorithms across various scenarios is presented. Table 2 shows the coefficient of determination R2 of the Doppler frequency and azimuth for bicycles, cars, and trucks at different distances, indicating that the proposed method significantly outperforms the compared algorithms in enhancing the correlation between Doppler frequency and azimuth. R2 is the coefficient of determination, as shown in Eq. (20).20 R2=1-SSESST

where SSE represents the Sum of Squares Residual, indicating the fitting error of the regression model, and SST denotes the Sum of Squares Total, reflecting the degree of dispersion within the overall data. This formula evaluates the goodness of fit of the regression model to the data. The closer this value approaches 1, the greater the explanatory power of the model, thereby indicating a stronger predictive capability. Table 2 Coefficient of determination R2 of different algorithms.

Number	Scenario	Type	Range	Original	OFGF	Mean Filtering	Proposed Method	
1	Scenario 1	Car	15 m	0.4459	0.8223	0.8508	0.9027	
2	Scenario 1	Truck	10 m	0.5343	0.9626	0.8904	0.9868	
3	Scenario 2	Car	13 m	0.9526	0.9354	0.9958	0.9985	
4	Scenario 2	Car	11 m	0.7818	0.9575	0.9688	0.9996	
5	Scenario 2	Bicycle	9 m	0.8747	0.9254	0.9933	0.9997	
6	Scenario 2	Bicycle	8 m	0.9481	0.9245	0.9978	0.9995	
7	Scenario 2	Car	17m	0.6341	0.8233	0.9410	0.9710	
8	Scenario 2	Car	14m	0.6777	0.9363	0.9870	0.9996	
9	Scenario 2	Bicycle	20 m	0.9775	0.9963	0.9985	0.9993	
10	Scenario 3	Car	3 m	02195	0.7568	0.7018	0.9171	
11	Scenario 3	Car	3 m	0.2365	0.8426	0.7257	0.9080	

The results in Table 2 show that bicycles, due to their smaller size and reduced impact from angle scintillation and baseline decorrelation noise, maintain a high correlation between Doppler frequency and azimuth angle even when using raw data fitting. However, for cars and trucks, the correlation is below 0.5 when fitting raw data. Vehicles 4, 5, and 6 are driving simultaneously in adjacent lanes, and after applying the proposed filtering method, the correlation between Doppler frequency and azimuth angle is significantly improved. This indicates that the proposed method performs well even under complex road conditions. In Scenario 3, the close proximity of the radar to test vehicles 11 and 12 leads to a significant decrease in the correlation between Doppler frequency and azimuth angle in the raw data, and the effectiveness of OFGF and mean filtering is limited. However, the proposed method still maintains a correlation above 0.9 between Doppler frequency and azimuth, further demonstrating its superiority.

Furthermore, the efficiency of the algorithm is evaluated. The test platform utilized a 12th Gen Intel(R) Core(TM) i9-12900H CPU, with Matlab R2022a serving as the testing software. Following the completion of 150 test runs, the average computation time for each method was calculated. The results are presented in Table 3. Table 3 Efficiency of different algorithms.

Pulse Number	Original	OFGF	Mean Filtering	Proposed Method	
1000	0.002015 s	0.005257 s	0.002541 s	0.032789 s	
2000	0.003987 s	0.022419 s	0.004816 s	0.092776 s	
3000	0.005853 s	0.027925 s	0.006827 s	0.179343 s	

The results presented in Table 3 demonstrate a notable reduction in the efficiency of the proposed method. Nevertheless, in certain applications requiring high-precision inverse synthetic aperture radar (ISAR) imaging26,27, accuracy supersedes efficiency in importance.

Statistical performance

This study demonstrates the relationship between velocity measurement error, determination coefficient R2, and SNR through 1,000 Monte Carlo experiments. The computer simulation experiment was conducted based on a target moving at a speed of 15 m/s and located 40 m from the radar. Meanwhile, Gaussian white noise was added to the echo signal.

In addition to the determination coefficient obtained through linear regression using original data and the proposed method, comparisons were made using mean filtering and median filtering methods since the form of the glint noise is similar to salt-and-pepper noise in images28. Based on the aforementioned experimental results, the OFGF exhibited performance degradation during the tracking of the Doppler history and azimuth angle of the prototype vehicle, due to the influence of the glint effect. Thus, it was excluded from the comparative analysis in this study. The experimental results are shown in Fig. 7a and b.Fig. 7 The impact of SNR. (a) R2 vs SNR; (b) error in speed measurement vs SNR.

The proposed algorithm significantly improves the R2 and speed measurement error compared to that of mean filtering and median filtering, especially under low SNR conditions. The speed measurement accuracy of the proposed algorithm can be maintained at 1.7 km/h under an SNR of 20 dB.

Conclusions

This work demonstrates a method for high-precision velocity measurement using the correlation between azimuth and Doppler history in traffic radar. The subspace algorithm for enhancing correlation was proposed to jointly estimate Doppler frequency and azimuth of the target, significantly improving the correlation between azimuth and Doppler frequency in the presence of angular glint. Even when SNR is below 20 dB, the speed measurement accuracy of the proposed method can be maintained at 1.7 km/h, which allows for high-precision speed measurement with a wide coverage range and provides technical support for the low-cost construction of Intelligent Transportation Systems.

Author contributions

Conceptualization, W.L. and Y.L.; methodology, W.L. and Y.L.; software, Z.C.; validation, W.L., L.C. and Y.L.; formal analysis, W.L.; investigation, Y.L.; resources and data curation, L.C.; writing original draft preparation, W.L. and Z.C.; writing review and editing, W.L.; supervision, L.C. and Y.L.; funding acquisition, L.C.; All authors have read and agreed to the published version of the manuscript.

Data availability

The datasets used and analysed during the current study available from the corresponding author on 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.
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