Distributed MIMO radar AOA target positioning method based on kernel recursive least square

Abstract

The invention discloses a distributed MIMO radar AOA target positioning method based on kernel recursive least squares, and the method comprises the steps: carrying out the nonlinear mapping of AOA measurement data obtained by a radar to a high-dimensional RKHS based on a kernel method, converting a nonlinear target position estimation problem into a linear parameter estimation problem in the RKHS, carrying out the online learning through employing the obtained AOA measurement data, and carrying out the positioning of a target position. And outputting a position estimation result of the target on line to complete target positioning. According to the method, effective target positioning precision can be provided under the condition that measurement error prior information is unknown, target positioning precision higher than that of an existing method can be obtained under the condition that measurement errors are large, and the problem that timeliness still needs to be guaranteed under the condition that the requirement for radar target positioning precision is high is solved.

Description

Technical Field

[0001] This invention belongs to the field of radar target localization technology, specifically relating to a distributed MIMO radar AOA target localization method based on kernel recursive least squares. Background Technology

[0002] Target localization is a fundamental function of distributed multiple-input multiple-output (MIMO) radar. Localization methods are categorized into direct localization and indirect localization. For direct localization, the radar directly processes the received signal to obtain the target's position information. For indirect localization, the radar first measures parameters from the received signal, such as time difference of arrival (TDOA) or bistatic distance (BR), Doppler shift (DS), and direction of arrival (AOA), and then uses these measurements to estimate the target's position. Comparatively, direct localization methods can achieve higher accuracy at low signal-to-noise ratios than indirect localization, but require significantly higher computational and spatial resources. Indirect localization methods can obtain closed-form solutions for the target position, while indirect localization can only obtain non-closed-form solutions.

[0003] Most existing target localization methods are model-based, requiring statistical models of radar measurements to establish likelihood functions and design estimation algorithms. Furthermore, the design of these algorithms typically also utilizes the covariance matrix information of the radar measurements. In practical applications, when the radar measurement model is unknown or the radar operates in complex interference environments, model-based target localization methods often fail to achieve the expected localization accuracy. In recent years, data-driven deep learning-based radar target detection and recognition methods have received widespread attention and research. However, deep learning methods are difficult to apply to online applications because model training requires large amounts of data, and the training time is often insufficient to meet timeliness requirements.

[0004] Therefore, there is an urgent need to design an effective data-driven radar target localization method that meets the accuracy requirements of target localization while also ensuring timeliness. In fact, target localization is a typical nonlinear estimation problem. Because kernel methods have the ability to effectively approximate various nonlinear functions, they are widely used in nonlinear regression and classification tasks. Kernel methods can be effectively used for some online tasks, and thus various online kernel learning algorithms have been proposed for applications such as nonlinear system identification and nonlinear channel equalization. Among them, the kernel least-mean-square (KLMS) algorithm and the kernel recursive least square (KRLS) algorithm are the most classic online kernel learning algorithms. The paper "The kernel least-mean-square algorithm, IEEE Trans. Signal Process., vol. 56, no. 2, pp. 543-554, Feb. 2008" combines kernel tricks with the least-mean-square algorithm to provide a sample-by-sample update scheme for adaptive filters in the regenerating kernel Hilbert space, and proves that this scheme has good definition in the regenerating kernel Hilbert space under limited training data. The paper "The kernel-recursive least-squares algorithm, IEEE Trans. Signal Process., vol. 52, no. 8, pp. 2275-2285, Aug. 2004" proposes a nonlinear version of the recursive least squares method, performing linear regression in a high-dimensional feature space induced by the Mercer kernel. This can be used to recursively construct the minimum mean square error solution for nonlinear least squares problems. Compared to the previous method, the KRLS algorithm utilizes more data information to update the weights, thus exhibiting significantly faster convergence and better optimization performance.

[0005] In summary, applying online kernel learning methods to radar target localization has the potential to effectively improve target localization performance and timeliness. This research innovatively introduces online kernel learning methods into distributed MIMO radar target localization applications and provides a new research approach for radar target localization, possessing certain practical and theoretical research value. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention proposes a distributed MIMO radar AOA target localization method based on kernel recursive least squares, which solves the problem of ensuring timeliness even when radar target localization accuracy requirements are high.

[0007] The technical solution adopted in this invention is: a distributed MIMO radar AOA target localization method based on kernel recursive least squares, the specific steps of which are as follows:

[0008] S1. Establish a distributed MIMO radar AOA target localization measurement model. Use the kernel method to nonlinearly map the radar-obtained AOA measurement data to the regenerating kernel Hilbert space RKHS, which will transform the nonlinear estimation problem in the original space into a linear estimation problem in RKHS.

[0009] S2. Based on step S1, by minimizing the Euclidean norm of the cumulative position estimation error up to the current time, an AOA target localization algorithm KRLS-AOA based on kernel recursive least squares is proposed to solve the linear estimation problem, output the target position estimate, and complete the target localization.

[0010] Furthermore, step S1 is specifically as follows:

[0011] S11, Radar AOA Measurement Model;

[0012] Set the distributed MIMO radar target localization scenario as follows One transmitting antenna and The receiving antennas are spatially diverse, among which the first... The positions of the transmitting antennas are as follows: , No. The positions of the receiving antennas are as follows: , Indicates transpose. The target's position. The radar is unknown.

[0013] All transmitting antennas transmit orthogonal waveforms. Let the transmitting antennas... The transmitted waveform is The carrier frequency is In the case of coherent reception, the receiving antenna... Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows:

[0014] (1);

[0015] in, Indicates time, Indicates an unknown amplitude. Indicates the signal from the transmitting antenna The transmitted signal is reflected by the target to the receiving antenna. Time delay, This represents zero-mean Gaussian white noise.

[0016] In the case of noncoherent reception, the receiving antenna Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows:

[0017] (2);

[0018] in, This indicates an unknown amplitude.

[0019] The receiving antenna measures the azimuth and elevation angles (AOA) of the target echo and transmits this information to the fusion center. The fusion center then uses all the AOA information to estimate the target's position. The target's position relative to the receiving antenna is then determined. The azimuth and elevation angles are defined as follows: and The expression is as follows:

[0020] (3);

[0021] (4);

[0022] Regarding the receiving antenna The obtained AOA measurement The expression is as follows:

[0023] (5);

[0024] in, Indicates receiving antenna The true value of AOA. express The measurement error vector follows a zero-mean Gaussian distribution, and its covariance matrix is... , and Indicates the target relative to the receiving antenna The variance of the azimuth and elevation angles.

[0025] Then the AOA measurement of all receiving antennas The expression is as follows:

[0026] (6);

[0027] in, This represents the true AOA value for all receiving antennas. Let the vectors of all AOA measurement errors follow a zero-mean Gaussian distribution, and the covariance matrix be . .

[0028] S12. Target localization optimization model based on kernel method;

[0029] First, establish AOA measurement. Regarding the target location The nonlinear relationship expression is as follows:

[0030] (7);

[0031] in, It is a nonlinear function. Representing the set of real numbers; the target localization problem is then transformed into a problem involving nonlinear functions. The approximation problem.

[0032] Given that the approximate location of the target is known, the input-output data pair sequence is obtained through simulation or actual measurement. .

[0033] in, Indicates the first One input data, Indicates the corresponding output label, Indicates the quantity of data.

[0034] Then use the data Approximating nonlinear functions based on kernel methods To obtain an estimate of the target position, based on the representation theorem and nonlinear functions. The parameterized expression is as follows:

[0035] (8);

[0036] in, Represents the coefficient. Indicates Mercer core Induced mapping, which takes input data from the original space Transition to the regenerating nucleus Hilbert space Unknown function From the weight matrix Parametric representation, i.e. Furthermore, a Gaussian kernel is used, defined as follows:

[0037] (9);

[0038] in, Indicates the independent variable. Indicates kernel parameters.

[0039] In the In the input data, the forward prediction error The expression is defined as follows:

[0040] (10);

[0041] in, .

[0042] Finally, the target position is estimated at the current time by minimizing a cost function. Input-output data If it is available, then minimize the cumulative squared error. As the cost function, its expression is as follows:

[0043] (11);

[0044] in, Represents the regularization factor. Describing the Frobenius norm, .

[0045] Furthermore, step S2 is specifically as follows:

[0046] First, the feature mapping vector Simple representation At the current moment The feature input-output data is represented as Then, all feature input vectors and output vectors are represented as follows:

[0047] (12);

[0048] (13);

[0049] The cost function (11) can then be rewritten as follows:

[0050] (14);

[0051] in, Represents the trace of a matrix.

[0052] Seeking again about The derivative of this expression is as follows:

[0053] (15);

[0054] Then, based on the method for identifying the gradient matrix of a real matrix scalar function, we obtain... about gradient matrix The expression is as follows:

[0055] (16);

[0056] make The weight matrix is ​​obtained in the th The expression for the next iteration is as follows:

[0057] (17);

[0058] in, Represents the zero matrix. Represents the identity matrix. Represents the inverse of a matrix;

[0059] Then, using the matrix inversion lemma, we obtain the following expression:

[0060] (18);

[0061] The expression can then be obtained as follows:

[0062] (19);

[0063] Among them, the intermediate variable matrix .

[0064] Again The resulting expression is as follows:

[0065] (20);

[0066] in, Using the lemma of finding the inverse of a block matrix, we obtain the following expression:

[0067] (twenty one);

[0068] in, , ;

[0069] The resulting expression is as follows:

[0070] (twenty two);

[0071] in, The forward prediction error is expressed as follows:

[0072] (twenty three);

[0073] Finally, AOA measurement was given. The target location estimation expression is as follows:

[0074] (twenty four);

[0075] in, Representation matrix The OK.

[0076] The beneficial effects of this invention are as follows: The method of this invention is based on a kernel method, which nonlinearly maps the AOA measurement data acquired by radar to a high-dimensional RKHS. The nonlinear target position estimation problem is transformed into a linear parameter estimation problem in the RKHS. Then, the acquired AOA measurement data is used for online learning to output the target position estimation result online, thus completing the target localization. The method of this invention can provide effective target localization accuracy when the prior information of measurement error is unknown, and can achieve higher target localization accuracy than existing methods when the measurement error is large. It solves the problem of ensuring timeliness even when radar target localization accuracy requirements are high. Attached Figure Description

[0077] Figure 1 This is a flowchart of a distributed MIMO radar AOA target localization method based on kernel recursive least squares according to the present invention.

[0078] Figure 2 This is a schematic diagram of a distributed MIMO radar target localization scenario in an embodiment of the present invention.

[0079] Figure 3 This is a simulation scene diagram in an embodiment of the present invention.

[0080] Figure 4 Different kernel parameters in the embodiments of the present invention RMSE test curve of KRLS-AOA algorithm under certain conditions.

[0081] Figure 5 Different regularization parameters in the embodiments of the present invention RMSE test curve of KRLS-AOA algorithm under certain conditions.

[0082] Figure 6 The graphs show the RMSE test curves of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms in the embodiments of the present invention.

[0083] Figure 7 The graphs show the steady-state RMSE of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms in this embodiment of the invention as a function of the standard deviation of azimuth measurement.

[0084] Figure 8 The graphs show the steady-state RMSE of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms in the embodiments of the present invention as a function of the standard deviation of pitch angle measurement. Detailed Implementation

[0085] The method of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0086] like Figure 1The flowchart of a distributed MIMO radar AOA target localization method based on kernel recursive least squares according to the present invention is shown below. The specific steps are as follows:

[0087] S1. Establish a distributed MIMO radar AOA target localization measurement model. Use the kernel method to nonlinearly map the radar-obtained angle of arrival (AOA) measurement data to the regenerating kernel Hilbert space (RKHS), which will transform the nonlinear estimation problem in the original space into a linear estimation problem in the RKHS.

[0088] S2. Based on step S1, by minimizing the Euclidean norm of the cumulative position estimation error up to the current time, an AOA target localization algorithm KRLS-AOA based on kernel recursive least squares (KRLS) is proposed to solve the linear estimation problem, output the target position estimate, and complete the target localization.

[0089] In this embodiment, step S1 is specifically as follows:

[0090] S11, Radar AOA Measurement Model;

[0091] Setting up a distributed MIMO radar target localization scenario, such as Figure 2 As shown, One transmitting antenna and The receiving antennas are spatially diverse, among which the first... The positions of the transmitting antennas are as follows: , No. The positions of the receiving antennas are as follows: , Indicates transpose. The target's position. The radar is unknown.

[0092] All transmitting antennas transmit orthogonal waveforms. Let the transmitting antennas... The transmitted waveform is The carrier frequency is In the case of coherent reception, the receiving antenna... Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows:

[0093] (1);

[0094] in, Indicates time, Indicates an unknown amplitude. Indicates the signal from the transmitting antenna The transmitted signal is reflected by the target to the receiving antenna. Time delay, This represents zero-mean Gaussian white noise.

[0095] In the case of noncoherent reception, the receiving antenna Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows:

[0096] (2);

[0097] in, This indicates an unknown amplitude.

[0098] The receiving antenna measures the azimuth and elevation angles (collectively referred to as AOA) of the target echo and transmits this information to the fusion center. The fusion center then uses all the AOA information to estimate the target's position. The target's position relative to the receiving antenna... The azimuth and elevation angles are defined as follows: and The expression is as follows:

[0099] (3);

[0100] (4);

[0101] In practical applications, there are errors in the measurement of AOA; therefore, regarding the receiving antenna... The obtained AOA measurement The expression is as follows:

[0102] (5);

[0103] in, Indicates receiving antenna The true value of AOA. express The measurement error vector follows a zero-mean Gaussian distribution, and its covariance matrix is... , and Indicates the target relative to the receiving antenna The variance of the azimuth and elevation angles.

[0104] Then the AOA measurement of all receiving antennas The expression is as follows:

[0105] (6);

[0106] in, This represents the true AOA value for all receiving antennas. Let the vectors of all AOA measurement errors follow a zero-mean Gaussian distribution, and the covariance matrix be . .

[0107] S12. Target localization optimization model based on kernel method;

[0108] The main task of target localization is to use AOA measurement with error. To estimate the target's location As can be seen from equations (3) and (4), AOA measurement It's about the target location. The nonlinear function. Most existing methods transform equations (3) and (4) into functions relating to position. The pseudolinear function is then solved using the least squares method or its improved form. The localization performance of existing target localization methods is severely limited by the intensity of measurement noise, and prior information about the measurement noise is required in the algorithm design.

[0109] To address this, this embodiment proposes a data-driven nonlinear estimation method based on the online kernel method. This method does not require prior information about the measurement noise and can achieve the expected positioning performance even under harsh noise conditions.

[0110] First, establish AOA measurement. Regarding the target location The nonlinear relationship expression is as follows:

[0111] (7);

[0112] in, It is a nonlinear function. Let represent the set of real numbers. Therefore, the target localization problem can be transformed into a problem involving a nonlinear function. The approximation (or fitting) problem.

[0113] Given that the approximate location of the target is known (e.g., it is known to be within a general spatial range), the input-output data pair sequence is obtained through simulation or actual measurement. .

[0114] in, Indicates the first One input data, This indicates the corresponding output (label). Indicates the quantity of data.

[0115] Since kernel methods can effectively approximate nonlinear functions online, they can utilize data... And based on the kernel method, approximate the nonlinear function. This allows for the estimation of the target's location. Based on the representation theorem, nonlinear functions... The parameterized expression is as follows:

[0116] (8);

[0117] in, Represents the coefficient. Indicates Mercer core Induced mapping, which takes input data from the original space Transition to the regenerating nucleus Hilbert space Unknown function From the weight matrix Parametric representation, i.e. .

[0118] Because the Gaussian kernel has a wide approximation capability, it is used, and its definition is as follows:

[0119] (9);

[0120] in, Indicates the independent variable. Indicates kernel parameters.

[0121] For practical target localization applications, the data is acquired online, in the... In each input data (iteration), the forward prediction error The expression is defined as follows:

[0122] (10);

[0123] in, .use Alternative Because in the first When the data arrives It is unknown.

[0124] To estimate the target location, a cost function needs to be minimized. At the current time... Input-output data It is available (previously) The accumulated value at time n is the first (a data pair), therefore, we consider minimizing the cumulative squared error. As the cost function (least squares), its expression is as follows:

[0125] (11);

[0126] The regularization term is introduced to prevent overfitting (because the weight matrix...). (The dimensions are very high) Represents the regularization factor. Describing the Frobenius norm, The cost function can be solved based on the representation theorem and kernel trick, and its solution lies in the space spanned by past feature mapping vectors.

[0127] In this embodiment, step S2 is specifically as follows:

[0128] First, for simplicity, we will represent the feature mapping vector. Represented as At the current moment The feature input-output data is represented as The input and output vectors of all features are represented as follows:

[0129] (12);

[0130] (13);

[0131] Therefore, the cost function (11) can be rewritten as the following expression:

[0132] (14);

[0133] in, Represents the trace of a matrix.

[0134] beg about The derivative of this expression is as follows:

[0135] (15);

[0136] Then, based on the method for identifying the gradient matrix of a real matrix scalar function, we obtain... about gradient matrix The expression is as follows:

[0137] (16);

[0138] make The weight matrix is ​​obtained in the th The expression for the next iteration is as follows:

[0139] (17);

[0140] in, Represents the zero matrix. Represents the identity matrix. This represents the inverse of a matrix.

[0141] According to the lemma of matrix inversion, the expression is as follows:

[0142] (18);

[0143] Therefore, the expression can be obtained as follows:

[0144] (19);

[0145] Among them, the intermediate variable matrix .

[0146] Again The resulting expression is as follows:

[0147] (20);

[0148] in, Using the lemma of finding the inverse of a block matrix, we can obtain the following expression:

[0149] (twenty one);

[0150] in, , .

[0151] Therefore, the expression can be obtained as follows:

[0152] (twenty two);

[0153] in, The forward prediction error is expressed as follows:

[0154] (twenty three);

[0155] Given AOA measurement The target location estimation expression is as follows:

[0156] (twenty four);

[0157] in, Representation matrix The OK.

[0158] In this embodiment, the method of the present invention further includes step S3, which involves setting the simulation scenario and parameters, then verifying the effectiveness of the KRLS-AOA algorithm under different parameters, and finally comparing the effectiveness and superiority of the KRLS-AOA algorithm with other typical algorithms in terms of algorithm convergence performance and target localization accuracy under low signal-to-noise ratio conditions through simulation, as detailed below:

[0159] S31, Simulation Settings;

[0160] This embodiment sets the simulation scenario as follows: Figure 3As shown, the distributed MIMO radar consists of one transmitting station and five receiving stations, used for target localization. The target is located within the spatial region [0km~2000km, -1000km~1000km, 4800km~5200km]. Through digital simulation, input-output data is generated within the defined region, with 2000 data points used for training and 400 data points used for testing. Key simulation parameters are shown in Table 1.

[0161] Table 1

[0162]

[0163] The root mean square error (RMSE) is used to evaluate target localization performance, and its definition is as follows:

[0164] (25);

[0165] in, Indicates the first The target location data for the second training iteration. During training and testing, Indicates the first Target location data, Indicates the quantity of data.

[0166] S32. Validity verification;

[0167] This embodiment uses the average of 200 Monte Carlo experiments to obtain the simulation results. The KRLS-AOA algorithm has two free parameters: the kernel parameter and the... and regularization factor Therefore, the first step is to verify different kernel parameters. and different regularization factors The impact of this condition on algorithm performance. In the simulation of this embodiment, the standard deviation of the azimuth and elevation angle measurements is set to... and ( ).

[0168] Different kernel parameters Simulation results under the value condition are as follows Figure 4 As shown. From Figure 4 As can be seen, the kernel parameters The value of the kernel parameter affects the RMSE convergence speed and steady-state performance of the KRLS-AOA algorithm, but there is no obvious pattern. In practical applications, the kernel parameter is selected experimentally. The value of . Furthermore, existing research has proposed numerous methods for selecting kernel parameters, such as cross-validation, which can be used for kernel parameter selection. The choice.

[0169] Different regularization factors Simulation results under the value condition are as follows Figure 5 As shown. From Figure 5 As can be seen, the regularization factor The value of the regularization factor affects the RMSE convergence speed and steady-state performance of the KRLS-AOA algorithm. Generally, the regularization factor... The smaller the value of the regularization factor, the faster the convergence speed, but the worse the steady-state performance. Therefore, in practical applications, the regularization factor... It is usually taken as a smaller value, such as 0.01.

[0170] S33, Performance Comparison;

[0171] This embodiment considers various scenarios with different combinations of standard deviations for azimuth and elevation angle measurements, and selects the Least Mean Square (LMS-AOA) and Recursive Least Squares (RLS-AOA) positioning algorithms for performance comparison. Existing model-based positioning algorithms were not used for comparison because they require AOA measurement covariance information, which is unknown to the radar in the application scenarios considered in this embodiment.

[0172] In this embodiment, the standard deviation of the azimuth and elevation angle measurements is [value missing]. and ( Under the given conditions, the RMSE test curves of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms are as follows: Figure 6 As shown. From Figure 6 As can be seen, the KRLS-AOA algorithm has the fastest convergence speed and steady-state performance; the KRLS-AOA algorithm performs better than the RLS-AOA algorithm because the RLS-AOA algorithm is a linear estimation method; the RLS-AOA algorithm performs better than the LMS-AOA algorithm because the RLS-LMS algorithm only uses the current error information for weight update, while the RLS-AOA algorithm uses all the error information up to this point for weight update.

[0173] The standard deviation of the pitch angle measurement is ( The steady-state RMSE of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms varies with the standard deviation of azimuth measurements under different azimuth measurement standard deviations, as shown in the curves. Figure 7 As shown. From Figure 7 It can be observed that the larger the standard deviation of the azimuth measurement, the larger the steady-state RMSE of all algorithms; the KRLS-AOA algorithm has the best steady-state estimation performance, and this advantage is more obvious when the standard deviation of the azimuth measurement is larger.

[0174] The standard deviation of azimuth measurement is ( The steady-state RMSE of the LMS-AOA, RLS-AOA, and KRLS-AOA algorithms varies with the standard deviation of pitch angle measurements under different pitch angle measurement standard deviations, as shown in the curves below. Figure 8 As shown. From Figure 8 It is possible to obtain, such as Figure 7 The same conclusion holds. In particular, when the standard deviation of AOA measurements is large, the KRLS-AOA algorithm exhibits significantly better steady-state estimation performance than other algorithms.

[0175] In summary, digital simulations verify the effectiveness and superiority of the proposed method in terms of algorithm convergence performance and target positioning accuracy under low signal-to-noise ratio conditions. The simulation results show that the KRLS-AOA algorithm proposed in this invention can provide effective target positioning accuracy when the prior information of measurement error is unknown, and can achieve higher target positioning accuracy than existing methods when the measurement error is large. This solves the problem of ensuring timeliness when radar target positioning accuracy requirements are high.

[0176] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the implementation methods of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the present invention.

Claims

1. A distributed MIMO radar AOA target localization method based on kernel recursive least squares, the specific steps of which are as follows: S1. Establish a distributed MIMO radar AOA target localization measurement model. Use the kernel method to nonlinearly map the radar-obtained AOA measurement data to the regenerating kernel Hilbert space RKHS, which will transform the nonlinear estimation problem in the original space into a linear estimation problem in RKHS. S2. Based on step S1, by minimizing the Euclidean norm of the cumulative position estimation error up to the current time, an AOA target localization algorithm KRLS-AOA based on kernel recursive least squares is proposed to solve the linear estimation problem, output the target position estimate, and complete the target localization.

2. The distributed MIMO radar AOA target localization method based on kernel recursive least squares according to claim 1, characterized in that, The specific steps of S1 are as follows: S11, Radar AOA Measurement Model; Set the distributed MIMO radar target localization scenario as follows One transmitting antenna and The receiving antennas are spatially diverse, among which the first... The positions of the transmitting antennas are as follows: , No. The positions of the receiving antennas are as follows: , Indicates transpose; the location of the target. Unknown to radar; All transmitting antennas transmit orthogonal waveforms. Let the transmitting antennas... The transmitted waveform is The carrier frequency is In the case of coherent reception, the receiving antenna Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows: （1）； in, Indicates time, Indicates an unknown amplitude. Indicates the signal from the transmitting antenna The transmitted signal is reflected by the target to the receiving antenna. Time delay, This represents zero-mean Gaussian white noise; In the case of noncoherent reception, the receiving antenna Received by the transmitting antenna Transmit signals reflected by the target The expression is as follows: （2）； in, Indicates an unknown amplitude; The receiving antenna measures the azimuth and elevation angles (AOA) of the target echo and transmits this information to the fusion center. The fusion center then uses all the AOA information to estimate the target's position; thus, the target's position relative to the receiving antenna... The azimuth and elevation angles are defined as follows: and The expression is as follows: （3）； （4）； Regarding the receiving antenna The obtained AOA measurement The expression is as follows: （5）； in, Indicates receiving antenna The true value of AOA. express The measurement error vector follows a zero-mean Gaussian distribution, and its covariance matrix is... , and Indicates the target relative to the receiving antenna The variance of the azimuth and elevation angles; Then the AOA measurement of all receiving antennas The expression is as follows: （6）； in, This represents the true AOA value for all receiving antennas. Let the vectors of all AOA measurement errors follow a zero-mean Gaussian distribution, and the covariance matrix be . ; S12. Target localization optimization model based on kernel method; First, establish AOA measurement. Regarding the target location The nonlinear relationship expression is as follows: （7）； in, It is a nonlinear function. Representing the set of real numbers; the target localization problem is then transformed into a problem involving nonlinear functions. The approximation problem; Given that the approximate location of the target is known, the input-output data pair sequence is obtained through simulation or actual measurement. ; in, Indicates the first One input data, Indicates the corresponding output label, Indicates the quantity of data; Then use the data Approximating nonlinear functions based on kernel methods To obtain an estimate of the target position, based on the representation theorem and nonlinear functions. The parameterized expression is as follows: （8）； in, Represents the coefficient. Indicates Mercer core Induced mapping, which takes input data from the original space Transition to the regenerating nucleus Hilbert space Unknown function From the weight matrix Parametric representation, i.e. Furthermore, a Gaussian kernel is used, and its definition expression is as follows: （9）； in, Indicates the independent variable. Indicates kernel parameters; In the In the input data, the forward prediction error The expression is defined as follows: （10）； in, ; Finally, the target position is estimated at the current time by minimizing a cost function. Input-output data If it is available, then minimize the cumulative squared error. As the cost function, its expression is as follows: （11）； in, Represents the regularization factor. Describing the Frobenius norm, .

3. The distributed MIMO radar AOA target localization method based on kernel recursive least squares according to claim 2, characterized in that, Step S2 is as follows: First, the feature mapping vector Simple representation At the current moment The feature input-output data is represented as Then, all feature input vectors and output vectors are represented as follows: （12）； （13）； The cost function (11) can then be rewritten as follows: （14）； in, Represents the trace of a matrix; Seeking again about The derivative of this expression is as follows: （15）； Then, based on the method for identifying the gradient matrix of a real matrix scalar function, we obtain... about gradient matrix The expression is as follows: （16）； make The weight matrix is ​​obtained in the th The expression for the next iteration is as follows: （17）； in, Represents the zero matrix. Represents the identity matrix. Represents the inverse of a matrix; Then, using the matrix inversion lemma, we obtain the following expression: （18）； The expression can then be obtained as follows: （19）； Among them, the intermediate variable matrix ; Again The resulting expression is as follows: （20）； in, Using the lemma of finding the inverse of a block matrix, we obtain the following expression: （21）； in, , ; The resulting expression is as follows: （22）； in, The forward prediction error is expressed as follows: （23）； Finally, AOA measurement was given. The target location estimation expression is as follows: （24）； in, Representation matrix The OK.

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