A phase-coded pulse radar signal direction finding method

By employing a phase-coded pulse radar signal direction finding method, and utilizing FIFO buffering, STFT transformation, and multi-baseline interferometer algorithms, the problem of detecting and identifying weak target signals in complex signal environments was solved. This method achieves high-precision signal parameter estimation and direction finding, providing accurate guidance information for passive detection systems.

CN115494497BActive Publication Date: 2025-10-24SHANGHAI RADIO EQUIP RES INST
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Patent Information

Application Number
CN202211207983.7
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2022-09-30
Publication Date
2025-10-24
Estimated Expiration
2042-09-30

AI Technical Summary

Technical Problem

In complex signal environments, effectively suppressing noise and extracting weak target signals has become a challenge for passive detection systems. In particular, when faced with unknown target information in a wide frequency, high-frequency, and strong noise background, existing technologies struggle to accurately detect and identify phase-coded pulse radar signals.

Method used

The phase-coded pulse radar signal direction finding method is adopted. Through FIFO buffering, square and fourth power processing, STFT transformation, adaptive threshold detection, multiphase filtering combined with short-time Fourier transform and multi-baseline interferometer direction finding algorithm, it can detect and identify weak and complex modulated signals and obtain high-precision frequency and angle information.

Benefits of technology

It improves the sensitivity and accuracy of signal detection, provides high-gain signal processing, and can accurately determine direction under low probability of intercept. It is suitable for UAV-borne passive radar detection systems and has strong engineering application value.

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Abstract

The application discloses a phase encoding pulse radar signal direction finding method, which is used for two-phase encoding and four-phase encoding signals, and comprises the following steps: firstly, a flat method or a four-time method is used to pre-process a received signal; then, energy accumulation is carried out on frequency domain data of the received signal; and finally, a pulse starting time and a pulse ending time are obtained through adaptive threshold detection processing; intrapulse sampling data of an original sampling wideband phase encoding pulse signal corresponding to the starting time and the ending time are sent into a channelization detection processing module based on a multi-phase filter combined with a short-time Fourier transform, so that signal amplitude and phase parameters and high-precision frequency information corresponding to a single code element are obtained; and finally, target angle information is obtained based on an optimal ambiguous multi-baseline interferometer direction finding algorithm. The application can detect and identify weak and complex modulation signals, and can provide more accurate and reliable guiding information for a passive detection system.
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Description

TECHNICAL FIELD

[0001] The present application relates to the field of non-cooperative radar signal and information processing, in particular to a phase-coded pulse radar signal direction finding method. BACKGROUND

[0002] In the field of electronic detection application, both sides always try to use various advanced technical means to counterbalance the opponent, and a scene of confrontation practice promotes the prosperity and progress of the technical field. For the active detection system, by using the advantage of cooperative signal processing, not only can the flexible signal form be used in the sending end to ensure that the radar is not accurately detected by the opponent, but also the matching processing method can be used in the receiving end to improve the signal-to-noise ratio of the echo signal, so as to achieve the purpose of effectively extracting target information; for the defense system, not only unknown target information in the frequency wide opening, high density pulse stream and strong noise background, but also the signal type and signal modulation characteristics are more and more complex and changeable. How to suppress noise and extract effective target signal from complex signal environment has been a big problem for researchers. SUMMARY

[0003] The purpose of the present application is to provide a phase-coded pulse radar signal direction finding method, which can detect and identify weak complex modulated signals and provide more accurate and reliable guidance information for passive detection systems.

[0004] In order to achieve the above purpose, the present application provides a phase-coded pulse radar signal direction finding method, comprising the following steps:

[0005] The original sampling data is copied into two identical paths, one of which is processed through FIFO buffer, and the other is processed through square and fourth power processing. The data processed through square and fourth power processing is subjected to STFT transformation, and the data subjected to STFT transformation is subjected to energy accumulation. The start and end time of the pulse signal is automatically obtained through the adaptive threshold detection technology. According to the start and end time of the pulse signal, the data corresponding to the start and end time of the pulse signal is intercepted from the original sampling data subjected to FIFO buffer, and the intercepted data is sent into the channelization detection processing module based on multi-phase filtering combined with short-time Fourier transform to obtain the signal amplitude and phase parameters and high-precision frequency information corresponding to a single symbol. According to the obtained frequency and phase data, combined with the position information of each antenna, the optimal fuzzy-based multi-baseline interferometer direction finding algorithm is used to obtain the angular position of the target.

[0006] According to the peak stability after STFT transformation, the coding type of the data is judged;

[0007] Only the signal after square or fourth power processing is a single frequency signal, and the peak value of STFT transformation can be a series of relatively stable values in amplitude and phase;

[0008] If the peak value of the squared STFT transform is stable, the original signal is determined as a two-phase coded signal.

[0009] If the peak value of the squared STFT transform is not stable, and the peak value of the fourth power STFT transform is stable, the original signal is determined as a four-phase coded signal.

[0010] The STFT transform is realized by using a 128-point FFT mode, and the sliding point number is also 128 points.

[0011] After the STFT transform, 128-channel amplitude-phase data are output in series. Since there is an image in the FFT transform, only the first 64 channels are selected to construct an amplitude / time graph, wherein the time resolution Δt is determined by the sampling rate and the single signal processing point number:

[0012] Δt = N / f s

[0013] Wherein, N is the FFT point number, f s is the sampling rate.

[0014] In the time-frequency graph after the STFT transform, it is determined in turn whether the energy of each channel signal at the same time exceeds the set threshold value after energy accumulation. If it exceeds, the channel with the maximum energy is selected as the signal detection reference channel, otherwise the energy of each channel signal at the next time is compared with the threshold.

[0015] The threshold generation mode is:

[0016] Thre = A*B+C

[0017] Wherein, Thre is the signal detection dynamic threshold; C is a fixed threshold, which is a binding component; B is an erosion component; A is an intra-channel adjustment component.

[0018] When signal detection is performed, the original data within a period of time is first intercepted, and it is uniformly divided into N1 segments. The minimum value in each segment is calculated in turn, and the maximum value of the minimum values in each segment is selected as the adjustment component A, and B=3.1625 is selected as the erosion component.

[0019] The method for obtaining signal amplitude-phase parameters and high-precision frequency information corresponding to a single code element comprises:

[0020] In order to obtain the amplitude-phase information of the signal, first, all the received channel amplitude-phase data are added and summed, and are smoothed by using the following formula:

[0021]

[0022] Wherein, x k represents the amplitude value of a certain channel at the current time, yk represents the filter output value; the detection algorithm contains 64 channels, so there are 64 groups of y k , the amplitude data of the nth channel is represented by ; when the signal amplitude value corresponding to the fourth time scale exceeds the threshold, the fourth time scale corresponding to the frequency spectrum amplitude and phase data is taken as the amplitude and phase of the signal;

[0023] The threshold detection is performed on the amplitude value of the nth channel , the upper index represents the channel number, and the lower index represents the time. The third threshold value is taken as the pulse arrival time TOA, and the pulse end time is recorded at the pulse falling edge. The pulse width PW is calculated;

[0024] When the amplitude value of the nth channel , the current channel number CH is recorded; when , the cross-channel flag is sent out at the same time; the frequency information of the signal is obtained by combining the channel number and the phase of the signal:

[0025]

[0026] Where N1 represents the channel number; represents the phase difference corresponding to a single time scale of the same channel.

[0027] The method for obtaining the angular position of the target by using the multi-baseline interferometer direction-finding algorithm based on optimal ambiguity includes:

[0028] Taking an eight-antenna interferometer as an example, the detailed steps are as follows:

[0029] Determine the angular search range;

[0030] The pitch angle is β ∈ [0, 60°], the azimuth angle is α ∈ [0, 360°], the search pitch angle search step is set to 0.25°, and the pitch angle search step is set to 1°. Then (α i ,β j ) is obtained, where i ∈ [0, 240], j ∈ [0, 360], and i ∈ Z, j ∈ Z; the measured phase difference (φ mn )' between each antenna at each position is recorded; ij ;

[0031] Iterate through all angles (α i ,β j ), and calculate the theoretical phase value (φ mn ) ij of any group of antennas at the corresponding angle according to the following formula;

[0032] (φ mn ) ij = 2π / λsinβ jcos α i [(x m -x n )sin β j sin α i +(y m -y n )cos α i ]

[0033] where λ is the wavelength of the received signal, α i is the azimuth angle of the target incidence, β j is the pitch angle of the target incidence, (x i ,y i ) is the horizontal and vertical coordinates of the antenna.

[0034] (k mn ) ij is obtained by formula and step S5.2 and (φ mn ) ij , where (k mn ) ij is the ambiguity number between the measured phase and the theoretical phase.

[0035] For any fixed i and j, m = 1…7; n = m+1…8, a set of (k mn ) ij can be obtained, and the sum s ij of the ambiguity errors of all baselines is calculated by the following formula:

[0036]

[0037] where round((k mn ) ij ) represents the integer closest to (k mn ) ij , and |·| represents the absolute value;

[0038] Record s i for all angles (α j ,β ij ), and find the angle (α ij ,β I ) corresponding to the minimum value of s J , which is the real angle (α I ,β J ) of the target, where I and J are the index numbers corresponding to the minimum value of s ij .

[0039] The present application has the advantages that: from the system point of view, the signal detection, identification, key parameter estimation, direction finding and other detailed detection steps of the low-interception-probability radar signal are given; higher processing gain is obtained through energy accumulation; the in-pulse sampling data is intercepted through high-gain detection; the frequency and phase parameters corresponding to a single symbol are obtained through the signal detection of the multi-phase filtering combined with the short-time Fourier transform, and finally the high-precision interferometer direction finding provides more accurate angle guidance information for the superior system; the present application occupies less resources in the hardware implementation process, and has strong engineering application value under the premise of guaranteeing the real-time requirement of the system. BRIEF DESCRIPTION OF DRAWINGS

[0040] Figure 1 is a working flow block diagram of a phase-coded pulse radar signal direction finding method provided by the present application.

[0041] Figure 2 is a signal detection schematic diagram of the present application. DETAILED DESCRIPTION

[0042] The preferred embodiments of the present application are specifically described below according to Figure 1 and Figure 2 .

[0043] For the active detection radar, the transmitted signal and the target echo signal are cooperative signals, and high accumulation gain can be obtained through the cooperative signal processing mode such as matched filtering and coherent accumulation, and then accurate target parameters can be obtained; but for the passive defense system, it is difficult to obtain high accumulation gain due to the difficulty in obtaining matched filtering parameters, and only the non-coherent accumulation and other non-cooperative signal processing modes can be used for signal detection. For the conventional radar signal, the system often uses the frequency, pulse width, time of arrival, amplitude, phase and angle of the target signal characteristics for classification and identification, and adopts certain classification criteria to eliminate noise and interference, so as to achieve the purpose of extracting target information. However, when the signal bandwidth is greater than the channel processing bandwidth, the traditional narrowband receiver will cause information loss. Developing a wideband radar channelized receiver is inevitable for wideband signal processing, because it has incomparable advantages over narrowband receivers in terms of target interception probability, signal processing gain, channel equalization and software reconfigurability. The conventional channelized detection scheme based on short-time Fourier transform (STFT) has an instantaneous processing bandwidth of up to 1GHz, but the signal bandwidth of a single channel processing is narrow, which cannot adapt to wideband radar signals such as linear frequency modulation signals and phase-coded signals. The channelized detection scheme based on the multi-phase filtering structure not only has a high instantaneous processing bandwidth, but also has a high single-channel processing bandwidth, which can effectively process the above-mentioned wideband signals, and the present application carries out the detection and processing of phase-coded signals on this basis.

[0044] As Figure 1As shown, the application provides a method for direction finding of phase-coded pulse radar signals, comprising the following steps:

[0045] Step S1, copying the original sampling data into two identical paths, one of which is processed through a FIFO buffer, and the other is processed by squaring and quartic.

[0046] Step S2, performing STFT transformation on the data processed by squaring and quartic, and judging the coding type of the data according to the peak stability of the STFT transformed data;

[0047] Only the signal after squaring or quartic is a single frequency signal, and the peak of the STFT transformation can be a series of relatively stable values;

[0048] If the peak data of the STFT transformation after squaring is stable, it is determined that the original signal is a two-phase coded signal;

[0049] If the peak data of the STFT transformation after squaring is not stable, and the peak data of the STFT transformation after quartic is stable, it is determined that the original signal is a four-phase coded signal;

[0050] Selecting a 128-point FFT mode for implementation, the sliding point number is also 128 points (no overlap), and after STFT transformation, 128 channel amplitude and phase data are output in series. Since the FFT transformation has an image, only the first 64 channels can be selected to construct an amplitude / time graph, wherein the time resolution is determined by the sampling rate and the single signal processing point number;

[0051] The calculation formula is:

[0052] Δt = N / f s

[0053] Where N is the FFT point number, f s is the sampling rate;

[0054] Step S3, performing energy accumulation on the data after STFT transformation, thereby smoothing the peak data, and then automatically obtaining the start and end time of the pulse signal through an adaptive threshold detection technology to improve the detection sensitivity of the low probability of intercept radar;

[0055] In the time-frequency graph after STFT transformation, it is determined in turn whether the energy of each channel signal at the same time exceeds the set threshold value after energy accumulation. If it exceeds, the channel with the maximum energy is selected as the signal detection reference channel, otherwise the energy of each channel signal at the next time is compared with the threshold;

[0056] Figure 2 is a schematic diagram of signal detection, wherein each vertical line represents the spectrum value of the signal, and the interval between the vertical lines is the time accuracy Δt;

[0057] The threshold is realized based on an adaptive manner, has strong anti-noise and target parameter change ability, and is generated in the following manner:

[0058] Thre=A*B+C

[0059] Wherein, Thre is a signal detection dynamic threshold; C is a binding component, which is a fixed threshold; B is an erosion component; A is an intra-channel adjustment component;

[0060] When signal detection is performed, first, the original data within a period of time is intercepted, and is evenly divided into N1 segments, the minimum value within each segment is calculated in turn, and the maximum value among the minimum values within each segment is selected as the adjustment component A. Here, B=3.1625 is selected as the erosion component, because 20log10(3.1625)=10, representing that the signal energy exceeds the fixed threshold by 10 dB;

[0061] Step S4, according to the start and end time of the pulse signal obtained in step S3, the data corresponding to the start and end time of the pulse signal is intercepted from the original sampling data through the FIFO buffer in step S1, and the intercepted data is sent into a channelization detection processing module based on polyphase filtering combined with short-time Fourier transform, to obtain signal amplitude and phase parameters and high-precision frequency information corresponding to a single code element;

[0062] Only when the 128-point FFT data is all the data of a code element, the FFT spectrum peak is a single-frequency signal spectrum peak, otherwise the signal energy is dispersed in a wide frequency spectrum range, and for subsequent direction finding, only the amplitude and phase data and frequency data within a code element are correctly selected, the analysis result is effective;

[0063] In order to obtain the amplitude and phase information of the signal, first, all the received channel amplitude and phase data are added and summed, and are smoothed by using the following formula:

[0064]

[0065] Wherein, x k represents the amplitude value of a certain channel at the current time, y k represents the filter output value; The detection algorithm contains 64 channels in total, and therefore there are 64 groups of y k , the amplitude data of the nth channel is represented by x ; In the time-frequency diagram shown in Figure 2 , when the signal amplitude values corresponding to three consecutive time scales of a certain channel exceed the threshold, the frequency spectrum amplitude and phase data corresponding to the fourth time scale are taken as the amplitude and phase of the signal;

[0066] The amplitude value of the nth channel x (upper index represents channel number, lower index represents time) do threshold detection, take the third threshold value as the pulse arrival time TOA, record the pulse end time at the pulse falling edge, calculate the pulse width PW of each pulse;

[0067] When the n-th channel amplitude value , record the current channel number CH; when , send out the cross-channel flag at the same time; combined with the channel number and the phase of the signal, the frequency information of the signal can be obtained, and the calculation formula is as follows:

[0068]

[0069] Wherein, N1 represents the channel number; represents the phase difference corresponding to the single time scale of the same channel;

[0070] Through the above method, the pulse description word information PDW of a single radar signal is obtained, and the PDW information of multiple signals arriving at the same time can be obtained by detecting different channels;

[0071] Step S5, using the frequency and phase data obtained in step S4, combined with the position information of each antenna, using the multi-baseline interferometer direction finding algorithm based on optimal ambiguity, the angular position of the target is obtained, and accurate guidance information is provided for the upper system;

[0072] Taking an eight-antenna interferometer as an example, the detailed steps are as follows:

[0073] Step S5.1, determine the angle search range;

[0074] The pitch angle is β∈[0, 60°], the azimuth angle is α∈[0, 360°], the search pitch angle search step is set to 0.25°, and the pitch angle search step is set to 1°, then (α i ,β j ) is obtained, wherein i∈[0, 240], j∈[0, 360], and i∈Z, j∈Z; record the measured phase difference (φ mn ) ij between each antenna at each position;

[0075] Step S5.2, traverse all angles (α i ,β j ), calculate the theoretical phase value (φ mn ) ij of any group of antennas at the corresponding angle according to the following formula;

[0076] (φ mn ) ij =2π / λsinβ j cosα i [(xm -x n )sinβ j sinα i +(y m -y n )cosα i ]

[0077] Where λ is the wavelength of the received signal, α i is the azimuth angle of the target incident, β j is the pitch angle of the target incident, (x i ,y i ) are the horizontal and vertical coordinates of the antenna;

[0078] Step S5.3, by formula And the result from step S5.2 and (φ mn ) ij Get (k mn ) ij , where (k mn ) ij is the fuzzy number between the measured phase and the theoretical phase.

[0079] Step S5.4: For any fixed i and j, traverse m=1…7; n=m+1…8 to obtain a set of (k mn ) ij , calculate the sum of the blur errors of all baselines s using the following formula ij :

[0080]

[0081] Among them, round((k mn ) ij ) represents the same as (k mn ) ij The nearest integer, |·| means taking the absolute value;

[0082] Record all angles (α i ,β j ) ij , and find out s ij The angle corresponding to the minimum value (α I ,β J ), which is the true angle of the target (α I ,β J ), where I and J are s ij The index number corresponding to the minimum value.

[0083] Step S6: reporting the above detection, identification and parameter estimation results to the upper level system for its decision-making.

[0084] The application gives the detection, identification and parameter processing steps of two-phase coding, four-phase coding pulse radar signals from the perspective of system, can detect and identify weak complex modulation signals, and utilize the frequency and phase parameter information in the symbol to carry out multi-baseline interferometer direction finding. The application is suitable for unmanned aerial passive radar detection system, and can be realized by embedded DSP / FPGA / GPU software. The application can adapt to different bandwidth, different modulation and coding mode and other complex signal environment, has strong search tracking and direction finding capability for phase coding pulse radar signals, provides more accurate and reliable guiding information for passive detection system, and has high engineering application value.

[0085] It should be noted that in the embodiments of the present application, the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the embodiments, and do not indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as a limitation of the present application. In addition, the terms "first", "second", "third" are only for the purpose of description, and cannot be understood as indicating or implying relative importance.

[0086] Although the content of the present application has been described in detail by the above preferred embodiments, it should be recognized that the above description should not be considered as a limitation of the present application. After reading the above content, various modifications and alternatives of the present application will be obvious to those skilled in the art. Therefore, the protection scope of the present application should be defined by the appended claims.

Claims

1. A method of direction finding of phase-coded pulsed radar signals, characterized in that, It comprises the following steps: The original sampling data is copied into two paths, one of which is processed by FIFO cache, and the other is processed by square and fourth power. The data processed by square and fourth power is subjected to STFT transformation, and the start and end time of the pulse signal is automatically obtained by using an adaptive threshold detection technology. According to the start and end time of the pulse signal, the data corresponding to the start and end time of the pulse signal is cut from the original sampling data subjected to FIFO cache, and the cut data is sent to a channelization detection processing module based on multi-phase filtering combined with short-time Fourier transform to obtain signal amplitude and phase parameters and high-precision frequency information corresponding to a single code element. According to the obtained frequency and phase data, the angle position of the target is obtained by using a multi-baseline interferometer direction-finding algorithm based on optimal ambiguity and combined with the position information of each antenna.

2. The method of direction finding of phase-coded pulse radar signals as claimed in claim 1, characterized in that, The coding type of the data is determined according to the peak stability after STFT transformation; Only the signal processed by square or fourth power is a single-frequency signal, and the peak of STFT transformation can be a series of relatively stable values; If the peak data of STFT transformation after square is stable, it is determined that the original signal is a two-phase coded signal; If the peak data of STFT transformation after square is unstable, and the peak data of STFT transformation after fourth power is stable, it is determined that the original signal is a four-phase coded signal.

3. The method of direction finding of phase-coded pulse radar signals as claimed in claim 2, characterized in that, The STFT transformation is realized by using a 128-point FFT mode, and the sliding point number is also 128 points. After STFT transformation, 128-channel amplitude and phase data are output in series. Since there is mirror image in FFT transformation, only the first 64 channels are selected to construct an amplitude / time graph, wherein the time resolution Δt is determined by the sampling rate and the single signal processing point number: Δt = N / f s where N is the FFT point number, f s is the sampling rate.

4. The method of direction finding of phase-coded pulse radar signals as claimed in claim 3, characterized in that, In the time-frequency graph after STFT transformation, it is determined whether the energy of each channel signal at the same time exceeds the set threshold value after energy accumulation. If yes, the channel with the maximum energy is selected as the signal detection reference channel, otherwise, the energy of each channel signal at the next time is compared with the threshold. The threshold generation mode is: Thre=A*B+C Wherein, Thre is the signal detection dynamic threshold; C is a fixed threshold; B is an erosion component; A is an intra-channel adjustment component; When performing signal detection, the original data within a period of time is first intercepted, and it is uniformly divided into N1 segments. The maximum value of the minimum values in each segment is selected as the adjustment component A, and B=3.1625 is selected as the erosion component.

5. The method of direction finding of phase-coded pulse radar signals as claimed in claim 4, characterized in that, The method for obtaining signal amplitude and phase parameters and high-precision frequency information corresponding to a single code element comprises: In order to obtain the amplitude and phase information of the signal, all the received channel amplitude and phase data are first added and summed, and smoothed by using the following formula: wherein, x k represents the amplitude value of a certain channel at the current time, y k represents the filtered output value; the detection algorithm contains 64 channels, thus there are 64 groups of y k , the amplitude data of the nth channel is represented by ; when the signal amplitude value corresponding to three consecutive time scales of a certain channel exceeds the threshold, the amplitude and phase data of the fourth time scale corresponding to the spectrum are taken as the amplitude and phase of the signal. n-th channel amplitude value Threshold detection, the upper index represents the channel number, the lower index represents the time, and the third threshold value is taken as the pulse arrival time TOA. The pulse end time is recorded at the pulse falling edge, and the pulse width PW is calculated. When the n-th channel amplitude value is recorded, the current channel number CH is recorded; when the cross-channel flag is sent out simultaneously; in combination with the channel number and the phase of the signal, the frequency information of the signal is determined: N1 indicates a channel number; φ indicates a phase difference corresponding to a single time scale of a co-channel.

6. The method of direction finding of phase-coded pulse radar signals as claimed in claim 4, characterized in that, The method for obtaining the angle position of the target by using an eight-antenna interferometer direction-finding algorithm based on optimal ambiguity comprises: Determine the angle search range; The pitch angle is taken as β ∈ [0, 60°], the azimuth angle is taken as α ∈ [0, 360°], the search pitch angle search step is set as 0.25°, the pitch angle search step is set as 1°, and (α i , β j ) is obtained, wherein i ∈ [0, 240], j ∈ [0, 360], and i ∈ Z and j ∈ Z; the phase difference (φ mn )' between each antenna is recorded at each position ij . Traverse all angles (α i ,β j ), calculate the theoretical phase value (φ) of any set of antennas at the corresponding angle according to the following formula mn ) ij ; (φ mn ) ij = 2π / λ sin β j cos α i [(x m - x n ) sin β j sin α i + (y m - y n ) cos α i ] where λ is the wavelength of the received signal, a i is the azimuth angle of the target incidence, β j is the elevation angle of the target incidence, (x i , y i ) are the horizontal and vertical coordinates of the antenna. By the formula and (φ mn ) ij and (φ mn ) ij (k mn ) ij where (k mn ) ij is the ambiguity number between the measured phase and the theoretical phase. For any fixed i and j, a set of (k mn ) ij The sum s of the blurring errors of all the baselines is calculated using the following formula ij : where round((k mn ) ij ) denotes the integer closest to (k mn ) ij and |·| denotes the absolute value. Record s ij at all angles (α ij , β I ) and find the angles (α J , β I ) corresponding to the minimum value of s J , which is the real angle (α ij , β It comprises the following steps: ) of the target, where I, J are the index numbers corresponding to the minimum value of s .

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