A method for locating radio frequency sources based on aerial drones

Through joint detection and frequency domain analysis of multiple drones, the problems of low positioning accuracy of radio frequency source and large environmental impact in the prior art are solved, and a high-precision and flexible radio frequency source positioning method is realized.

CN119689380BActive Publication Date: 2025-05-02CHENGDU RUIYUAN YUNQI TECH CO LTD
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Patent Information

Application Number
CN202510207303.9
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-02-25
Publication Date
2025-05-02
Estimated Expiration
2045-02-25

AI Technical Summary

Technical Problem

The existing radio frequency source positioning methods based on drones have problems such as low positioning accuracy and great influence from the environment. Especially in complex terrain, signals are easily blocked and positioning accuracy and efficiency are affected.

Method used

By adopting the joint detection method of multiple drones, the flight altitude, flight speed, current position and detected radio frequency signals of each drone are obtained by deploying at least three drones with positioning devices and radio frequency reception functions. Then, the radio frequency signals detected by each drone are analyzed and Doppler frequency shift correction are performed, the horizontal distance between each radio frequency source and each drone is calculated, the distance azimuth equation system is established, and the generalized inverse matrix calculation method is solved to obtain the positions of each radio frequency source.

Benefits of technology

It realizes high-precision radio frequency source positioning that is not affected by complex terrain environments and drone flight attitudes, and can detect multiple radio frequency sources simultaneously to meet the needs of various application scenarios.

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Abstract

The present invention discloses a method for locating a radio frequency source based on an aerial drone, which relates to the field of radio, including: deploying at least three drones equipped with a positioning device and a radio frequency receiving function, obtaining the flight altitude, flight speed, current position and detected radio frequency signals; obtaining the frequency value and amplitude of different sub-signals; obtaining the radio amplitude radiated from each radio frequency source to each drone; obtaining the horizontal distance from each radio frequency source to each drone; establishing a distance-azimuth equation group, and solving it through a generalized inverse matrix calculation method to obtain the position of each radio frequency source. The present invention makes full use of the low-cost and high-flexibility characteristics of drones. Through the joint detection of multiple drones, it is not only not restricted by complex terrain environments, but also can detect multiple radio frequency sources at the same time, and is not affected by the flight posture of the drones, meeting the needs of various application scenarios.
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Description

Technical Field

[0001] The invention relates to the field of radio, and in particular to a method for locating a radio frequency source based on an aerial unmanned aerial vehicle. Background Art

[0002] The positioning of radio frequency sources is of great significance. In the military field, it can be used to locate enemy communication equipment, radars, etc. In the civilian field, it can be used to find illegal signal sources that interfere with communications, determine the best location for mobile base stations, etc.

[0003] Traditional methods for locating radio frequency sources mainly include ground fixed station positioning and vehicle-mounted mobile positioning. Ground fixed station positioning is greatly restricted by the geographical environment, has monitoring blind spots, and has high construction and maintenance costs. Although vehicle-mounted mobile positioning has improved flexibility, the signal is easily blocked in complex terrain (such as mountainous areas and urban high-rise areas), affecting positioning accuracy and efficiency.

[0004] With the rapid development of drone technology, the use of drones to locate radio frequency sources has become a research hotspot. However, existing drone-based positioning methods still have some problems, such as low positioning accuracy and being greatly affected by the drone's posture. Therefore, a new method for locating radio frequency sources based on aerial drones is needed, which is not restricted by the drone's posture and improves positioning accuracy. Summary of the invention

[0005] In view of the above-mentioned deficiencies in the prior art, the present invention provides a radio frequency source positioning method based on an aerial drone, which solves the problems of low positioning accuracy and great environmental influence of the existing positioning methods.

[0006] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0007] A method for locating a radio frequency source based on an aerial drone comprises the following steps:

[0008] S1. Deploy at least three drones equipped with positioning devices and radio frequency receiving functions to cover the radio frequency source detection area, and simultaneously obtain the flight altitude, flight speed, current location and detected radio frequency signals of each drone;

[0009] S2. Perform frequency domain analysis and Doppler shift correction on the radio frequency signals detected by each drone to obtain the frequency values ​​and amplitudes of different sub-signals detected by each drone;

[0010] S3, treating sub-signals with the same frequency value as being emitted by the same RF source, sorting out the frequency values ​​and amplitudes of all sub-signals measured by all drones, and obtaining the radio amplitude radiated from each RF source to each drone;

[0011] S4. Calculate the horizontal distance from each RF source to each UAV according to the flight altitude of each UAV and the radio amplitude radiated from each RF source to each UAV;

[0012] S5. According to the current position of each UAV and the horizontal distance from each RF source to each UAV, a distance-azimuth equation group is established, and the distance-azimuth equation group is solved by a generalized inverse matrix calculation method to obtain the position of each RF source.

[0013] The beneficial effects of the present invention are as follows: the present invention fully utilizes the low cost and high flexibility of UAVs. Through the joint detection of multiple UAVs, it is not only not restricted by complex terrain environments, but also can detect multiple radio frequency sources at the same time without being affected by the flight posture of the UAVs, thus meeting the needs of various application scenarios.

[0014] Furthermore, the method of performing frequency domain analysis on the radio frequency signals detected by each drone in S2 comprises the following steps:

[0015] A1. Sample and quantize the radio frequency signals detected by each drone at a frequency twice the highest frequency of the radio frequency source to be detected through a radio frequency analog-to-digital converter to obtain a radio frequency signal sequence detected by each drone;

[0016] A2. Perform FFT on the RF signal sequence detected by each drone to obtain the frequency domain complex sequence of each drone signal;

[0017] A3. Cut off the first half of the frequency domain complex sequence of each drone signal, take each element in the first half of the amplitude-frequency sequence of each drone signal as a frequency point, take the modulus operation value of the element as the amplitude of the frequency point, multiply the element number by the sampling quantization frequency of the RF analog-to-digital converter and then divide it by the sampling quantization length of the RF analog-to-digital converter as the frequency value of the frequency point.

[0018] Furthermore, the expression for Doppler frequency shift correction in S2 is:

[0019] ,

[0020] Among them, f is the corrected frequency value, f0 is the frequency value before correction, c is the speed of light, γ is the speed scaling multiplier, and v is the flight speed of the drone.

[0021] The beneficial effects of the above further scheme are: whether in the military or civilian fields, radio frequency communication is to modulate the information to be transmitted in a radio frequency carrier of a preset frequency to form a radio frequency signal, so the frequency value can be regarded as the identity tag of the radio frequency source to be detected. As an identity tag of the radio frequency source, the frequency value does not need to be accurately calculated, but the influence of the different speeds of each drone must be excluded. The present invention calculates the frequency based on FFT fast Fourier transform, and its resolution has met the needs of identity recognition; and uses Doppler frequency shift correction to eliminate the influence of the different speeds of each drone, which can accurately distinguish different radio frequency sources in the radio frequency source detection area and locate them at the same time.

[0022] Furthermore, the S4 comprises the following sub-steps:

[0023] S41, calculating the three-dimensional space distance from each radio frequency source to each drone according to the radio amplitude radiated from each radio frequency source to each drone;

[0024] S42. Calculate the horizontal distance from each radio frequency source to each drone according to the flight altitude of each drone and the three-dimensional space distance radiated from each radio frequency source to each drone.

[0025] Furthermore, the expression of S41 is:

[0026] ,

[0027] in, is the three-dimensional distance from the i-th RF source to the j-th UAV, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, u j,i is the radio amplitude from the i-th RF source to the j-th UAV, i is the RF source number, and j is the UAV number.

[0028] Furthermore, the signal power distance coupling factor of the i-th radio frequency source and the j-th drone is calibrated by the following steps:

[0029] B1, drive the jth UAV to move a short distance, and record the change in the radio amplitude from the i-th RF source to the j-th UAV;

[0030] B2. Calculate the signal power distance coupling factor of the jth UAV from the i-th RF source according to the moving distance of the jth UAV and the change in the radio amplitude from the i-th RF source to the j-th UAV through the following formula:

[0031] ,

[0032] Among them, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, Δd jis the moving distance of the jth UAV, Δu j,i is the change in radio amplitude from the i-th RF source to the j-th UAV.

[0033] The beneficial effect of the above further scheme is that the strength of the radio frequency receiving signal is related to the strength of the radio frequency transmitting signal and the transmitting and receiving distance. Although the radio frequency transmitting signal strength position depends on the high flexibility of the UAV, the present invention dynamically and flexibly calibrates the signal power distance coupling factor through the short-distance movement of the UAV, so that the distance calculation has a high accuracy, laying the foundation for high-precision positioning.

[0034] Furthermore, the expression of S42 is:

[0035] ,

[0036] Among them, d j,i is the horizontal distance from the i-th RF source to the j-th UAV, is the three-dimensional distance from the i-th RF source to the j-th UAV, h j is the flight altitude of the jth UAV.

[0037] The beneficial effect of the above further solution is: since the drone is an aerial unit, in order to calculate its horizontal distance from the radio frequency source, it is necessary to perform a projection correction on the three-dimensional space distance based on the principle of a right triangle.

[0038] Furthermore, in S5, according to the current position of each drone and the horizontal distance from each radio frequency source to each drone, the distance and azimuth equations are established as follows:

[0039] ,

[0040] Among them, x1 is the horizontal coordinate of the current position of the first drone, x2 is the horizontal coordinate of the current position of the second drone, and x R-1 is the horizontal coordinate of the current position of the R-1th UAV, x R is the horizontal coordinate of the current position of the Rth UAV, R is the total number of UAVs, y1 is the vertical coordinate of the current position of the first UAV, y2 is the vertical coordinate of the current position of the second UAV, and y R-1 is the ordinate of the current position of the R-1th UAV, y R is the ordinate of the current position of the Rth UAV, x(i) is the abscissa of the position of the i-th RF source, y(i) is the ordinate of the position of the i-th RF source, d 1,i is the horizontal distance from the ith RF source to the first UAV, d 2,i is the horizontal distance from the ith RF source to the second UAV, d R-1,i is the horizontal distance from the ith RF source to the R-1th UAV, d R,iis the horizontal distance from the i-th RF source to the R-th UAV, and i is the RF source number.

[0041] Furthermore, the S5 solves the distance-azimuth equations by a generalized inverse matrix calculation method according to the following formula to obtain the position of each radio frequency source:

[0042] ,

[0043] Where D(i) is the position vector of the i-th RF source, and its elements are ,

[0044] W is the multiplier matrix of the distance and azimuth equations, whose elements are ,

[0045] κ(i) is the bias vector of the distance and azimuth equations, and its elements are

[0046] .

[0047] The beneficial effect of the above further solution is that each UAV and the i-th radio frequency source have the following azimuth distance relationship between them:

[0048] ,

[0049] Subtract the last equation from each equation of the system of equations and factorize the quadratic term to obtain the distance-azimuth equation system. Solving the system of equations can obtain the position of the i-th radio frequency source. After the above modeling, the present invention uses the generalized inverse matrix calculation method in matrix theory to construct a feasible solution method for the system of equations model, and finally obtains the position of the i-th radio frequency source. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 A flowchart of a method for locating a radio frequency source based on an aerial drone is provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0051] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0052] like Figure 1 As shown, in one embodiment of the present invention, a method for locating a radio frequency source based on an aerial drone comprises the following steps:

[0053] S1. Deploy at least three drones equipped with positioning devices and radio frequency receiving functions to cover the radio frequency source detection area, and simultaneously obtain the flight altitude, flight speed, current location and detected radio frequency signals of each drone.

[0054] S2. Perform frequency domain analysis and Doppler shift correction on the radio frequency signals detected by each drone to obtain the frequency values ​​and amplitudes of different sub-signals detected by each drone.

[0055] The method for performing frequency domain analysis on the radio frequency signals detected by each drone includes the following steps:

[0056] A1. Sample and quantize the radio frequency signals detected by each drone at a frequency twice the highest frequency of the radio frequency source to be detected through a radio frequency analog-to-digital converter to obtain a radio frequency signal sequence detected by each drone;

[0057] A2. Perform FFT on the RF signal sequence detected by each drone to obtain the frequency domain complex sequence of each drone signal;

[0058] A3. Cut off the first half of the frequency domain complex sequence of each drone signal, take each element in the first half of the amplitude-frequency sequence of each drone signal as a frequency point, take the modulus operation value of the element as the amplitude of the frequency point, multiply the element number by the sampling quantization frequency of the RF analog-to-digital converter and then divide it by the sampling quantization length of the RF analog-to-digital converter as the frequency value of the frequency point.

[0059] The expression of Doppler shift correction is:

[0060] ,

[0061] Among them, f is the corrected frequency value, f0 is the frequency value before correction, c is the speed of light, γ is the speed scaling multiplier, and v is the flight speed of the drone.

[0062] Whether in the military or civilian fields, radio frequency communications modulate the information to be transmitted in a radio frequency carrier of a preset frequency to form a radio frequency signal. Therefore, the frequency value can be regarded as the identity tag of the radio frequency source to be detected. As an identity tag of the radio frequency source, the frequency value does not need to be accurately calculated, but the influence of the different speeds of each drone must be excluded. The present invention calculates the frequency based on FFT fast Fourier transform, and its resolution has met the needs of identity recognition; and uses Doppler frequency shift correction to eliminate the influence of the different speeds of each drone, which can accurately distinguish different radio frequency sources in the radio frequency source detection area and locate them at the same time.

[0063] In this embodiment, the value of the flight speed v of the drone has a positive and negative sign, and a positive direction needs to be preset; the value of the speed scaling multiplier γ is calibrated in advance through experiments, and the calibration accuracy is based on the ability to effectively distinguish different frequency RF sources participating in the experiment.

[0064] S3. Sub-signals with the same frequency value are regarded as being emitted by the same RF source, and the frequency values ​​and amplitudes of all sub-signals measured by all UAVs are sorted out to obtain the radio amplitude radiated from each RF source to each UAV.

[0065] S4. Calculate the horizontal distance from each RF source to each UAV according to the flight altitude of each UAV and the radio amplitude radiated from each RF source to each UAV.

[0066] S4 includes the following sub-steps:

[0067] S41. According to the radio amplitude radiated from each radio frequency source to each drone, the three-dimensional space distance radiated from each radio frequency source to each drone is calculated by the following formula:

[0068] ,

[0069] in, is the three-dimensional distance from the i-th RF source to the j-th UAV, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, u j,i is the radio amplitude from the i-th RF source to the j-th UAV, i is the RF source number, and j is the UAV number.

[0070] The signal power distance coupling factor of the jth UAV from the ith RF source ς j,i Calibrate through the following steps:

[0071] B1, drive the jth UAV to move a short distance, and record the change in the radio amplitude from the i-th RF source to the j-th UAV;

[0072] B2. Calculate the signal power distance coupling factor of the jth UAV from the i-th RF source according to the moving distance of the jth UAV and the change in the radio amplitude from the i-th RF source to the j-th UAV through the following formula:

[0073] ,

[0074] Among them, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, Δd j is the moving distance of the jth UAV, Δu j,i is the change in radio amplitude from the i-th RF source to the j-th UAV.

[0075] The strength of the radio frequency receiving signal is related to the strength of the radio frequency transmitting signal and the transmitting and receiving distance. Although the radio frequency transmitting signal strength position depends on the high flexibility of the UAV, the present invention dynamically and flexibly calibrates the signal power distance coupling factor through the short-distance movement of the UAV, so that the distance calculation has a high accuracy, laying the foundation for high-precision positioning.

[0076] S42. According to the flight altitude of each UAV and the three-dimensional space distance radiated from each RF source to each UAV, the horizontal distance from each RF source to each UAV is calculated by the following formula:

[0077] ,

[0078] Among them, d j,i is the horizontal distance from the i-th RF source to the j-th UAV, is the three-dimensional distance from the i-th RF source to the j-th UAV, h j is the flight altitude of the jth UAV.

[0079] Since the drone is an aerial unit, to calculate its horizontal distance from the radio frequency source, it is necessary to perform a projection correction on the three-dimensional space distance based on the principle of right triangles.

[0080] S5. According to the current position of each UAV and the horizontal distance from each RF source to each UAV, the following distance and azimuth equations are established:

[0081] ,

[0082] Among them, x1 is the horizontal coordinate of the current position of the first drone, x2 is the horizontal coordinate of the current position of the second drone, and x R-1 is the horizontal coordinate of the current position of the R-1th UAV, x R is the horizontal coordinate of the current position of the Rth UAV, R is the total number of UAVs, y1 is the vertical coordinate of the current position of the first UAV, y2 is the vertical coordinate of the current position of the second UAV, and y R-1 is the ordinate of the current position of the R-1th UAV, y R is the ordinate of the current position of the Rth UAV, x(i) is the abscissa of the position of the i-th RF source, y(i) is the ordinate of the position of the i-th RF source, d 1,i is the horizontal distance from the ith RF source to the first UAV, d 2,i is the horizontal distance from the ith RF source to the second UAV, d R-1,i is the horizontal distance from the ith RF source to the R-1th UAV, d R,i is the horizontal distance from the i-th RF source to the R-th UAV, where i is the RF source number;

[0083] And the distance and azimuth equations are solved by the generalized inverse matrix calculation method to obtain the position of each RF source:

[0084] ,

[0085] Where D(i) is the position vector of the i-th RF source, and its elements are ,

[0086] W is the multiplier matrix of the distance and azimuth equations, whose elements are ,

[0087] κ(i) is the bias vector of the distance and azimuth equations, and its elements are

[0088] .

[0089] It is worth noting that the azimuth distance relationship between each UAV and the i-th radio frequency source is as follows:

[0090] ,

[0091] Subtract the last equation from each equation of the system of equations and factorize the quadratic term to obtain the distance-azimuth equation system. Solving the system of equations can obtain the position of the i-th radio frequency source. After the above modeling, the present invention uses the generalized inverse matrix calculation method in matrix theory to construct a feasible solution method for the system of equations model, and finally obtains the position of the i-th radio frequency source.

[0092] In summary, the present invention makes full use of the low cost and high flexibility of drones. Through the joint detection of multiple drones, it is not only not restricted by complex terrain environments, but also can detect multiple radio frequency sources at the same time without being affected by the flight posture of the drones, thus meeting the needs of various application scenarios.

[0093] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for locating a radio frequency source based on an aerial drone, characterized in that: The following steps are involved: S1. Deploy at least three drones equipped with positioning devices and radio frequency receiving functions to cover the radio frequency source detection area, and simultaneously obtain the flight altitude, flight speed, current location and detected radio frequency signals of each drone; S2. Perform frequency domain analysis and Doppler shift correction on the radio frequency signals detected by each drone to obtain the frequency values ​​and amplitudes of different sub-signals detected by each drone; S3, treating sub-signals with the same frequency value as being emitted by the same RF source, sorting out the frequency values ​​and amplitudes of all sub-signals measured by all drones, and obtaining the radio amplitude radiated from each RF source to each drone; S4. Calculate the horizontal distance from each RF source to each UAV according to the flight altitude of each UAV and the radio amplitude radiated from each RF source to each UAV; S5. According to the current position of each UAV and the horizontal distance from each RF source to each UAV, a distance-azimuth equation group is established, and the distance-azimuth equation group is solved by a generalized inverse matrix calculation method to obtain the position of each RF source.

2. The method for locating a radio frequency source based on an aerial drone according to claim 1, characterized in that: The method for performing frequency domain analysis on the radio frequency signals detected by each drone in S2 comprises the following steps: A1. Sample and quantize the radio frequency signals detected by each drone at a frequency twice the highest frequency of the radio frequency source to be detected through a radio frequency analog-to-digital converter to obtain a radio frequency signal sequence detected by each drone; A2. Perform FFT on the RF signal sequence detected by each drone to obtain the frequency domain complex sequence of each drone signal; A3. Cut off the first half of the frequency domain complex sequence of each drone signal, take each element in the first half of the amplitude-frequency sequence of each drone signal as a frequency point, take the modulus operation value of the element as the amplitude of the frequency point, multiply the element number by the sampling quantization frequency of the RF analog-to-digital converter and then divide it by the sampling quantization length of the RF analog-to-digital converter as the frequency value of the frequency point.

3. The method for locating a radio frequency source based on an aerial drone according to claim 1, characterized in that: The expression of Doppler shift correction in S2 is: , Among them, f is the corrected frequency value, f0 is the frequency value before correction, c is the speed of light, γ is the speed scaling multiplier, and v is the flight speed of the drone.

4. The method for locating a radio frequency source based on an aerial drone according to claim 1, characterized in that: The S4 comprises the following sub-steps: S41, calculating the three-dimensional space distance from each radio frequency source to each drone according to the radio amplitude radiated from each radio frequency source to each drone; S42. Calculate the horizontal distance from each radio frequency source to each drone according to the flight altitude of each drone and the three-dimensional space distance radiated from each radio frequency source to each drone.

5. The method for locating a radio frequency source based on an aerial drone according to claim 4, characterized in that: The expression of S41 is: , in, is the three-dimensional distance from the i-th RF source to the j-th UAV, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, u j,i is the radio amplitude from the i-th RF source to the j-th UAV, i is the RF source number, and j is the UAV number.

6. The method for locating a radio frequency source based on an aerial drone according to claim 5, characterized in that: The signal power distance coupling factor of the jth UAV of the i-th radio frequency source is calibrated by the following steps: B1, drive the jth UAV to move a short distance, and record the change in the radio amplitude from the i-th RF source to the j-th UAV; B2. Calculate the signal power distance coupling factor of the jth UAV from the i-th RF source according to the moving distance of the jth UAV and the change in the radio amplitude from the i-th RF source to the j-th UAV through the following formula: , Among them, ς j,i is the signal power distance coupling factor of the jth UAV from the ith RF source, Δd j is the moving distance of the jth UAV, Δu j,i is the change in radio amplitude from the i-th RF source to the j-th UAV.

7. The method for locating a radio frequency source based on an aerial drone according to claim 4, characterized in that: The expression of S42 is: , Among them, d j,i is the horizontal distance from the i-th RF source to the j-th UAV, is the three-dimensional distance from the i-th RF source to the j-th UAV, h j is the flight altitude of the jth UAV.

8. The method for locating a radio frequency source based on an aerial drone according to claim 1, characterized in that: In S5, according to the current position of each drone and the horizontal distance from each radio frequency source to each drone, the distance and azimuth equations are established as follows: , Among them, x1 is the horizontal coordinate of the current position of the first drone, x2 is the horizontal coordinate of the current position of the second drone, and x R-1 is the horizontal coordinate of the current position of the R-1th UAV, x R is the horizontal coordinate of the current position of the Rth UAV, R is the total number of UAVs, y1 is the vertical coordinate of the current position of the first UAV, y2 is the vertical coordinate of the current position of the second UAV, and y R-1 is the ordinate of the current position of the R-1th UAV, y R is the ordinate of the current position of the Rth UAV, x(i) is the abscissa of the position of the i-th RF source, y(i) is the ordinate of the position of the i-th RF source, d 1,i is the horizontal distance from the ith RF source to the first UAV, d 2,i is the horizontal distance from the ith RF source to the second UAV, d R-1,i is the horizontal distance from the ith RF source to the R-1th UAV, d R,i is the horizontal distance from the i-th RF source to the R-th UAV, and i is the RF source number.

9. The method for locating a radio frequency source based on an aerial drone according to claim 8, characterized in that: The S5 solves the distance-azimuth equations by the generalized inverse matrix calculation method according to the following formula to obtain the position of each radio frequency source: , Where D(i) is the position vector of the i-th RF source, and its elements are , W is the multiplier matrix of the distance and azimuth equations, whose elements are , κ(i) is the bias vector of the distance and azimuth equations, and its elements are 。

Citation Information

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