Wideband lfm signal doa estimation method based on variable scale discrete fresnel transform

Abstract

The application discloses an LFM non-stationary signal DOA estimation method based on a variable scale discrete Fresnel transform, and the DOA estimation of the non-stationary LFM signal can be realized by combining the traditional high-resolution MUSIC angle estimation algorithm with the novel variable scale discrete Fresnel transform. The method can realize high-resolution angle estimation of the non-stationary LFM signal by rotating the time-frequency surface, converting the signals received by all array elements into the Chirp domain, and then constructing the Chirp domain steering vector. The Chirp domain steering vector is only related to the incident angle and is not related to time, so that the high-resolution angle estimation of the non-stationary LFM signal can be realized. Compared with the wideband LFM signal based on the fractional Fourier transform, the method has lower calculation complexity.

Description

Technical Field

[0001] This invention belongs to the field of radar DOA estimation signal processing, specifically a broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform (SDFnT). Background Technology

[0002] Direction of Arrival (DOA) estimation is an important research direction in radar array direction finding technology. Related research includes: J. Capon's minimum variance distortion-free response beamforming method (Capon method) proposed in 1969; J.P. Burg's Maximum Entropy Spectrum Estimation (MESE) algorithm proposed in 1976; Roschmit's multiple signal classification algorithm (MUSIC) proposed in 1986; and Paulraj et al.'s Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) algorithm proposed in 1986. While these traditional algorithms can achieve high-resolution estimation of the transmit and receive angles, they are all based on narrowband stationary signal models, assuming that the array model's steering vector is only related to the target angle and independent of time. Therefore, these methods cannot reveal the trend of each frequency division in the signal changing over time.

[0003] Linear frequency modulated (LFM) signals are an important class of non-stationary signals, widely used in radar, sonar, communications, and other fields. DOA estimation for LFM signals has long been a difficult and important problem in array signal processing, and a hot research topic. Gershman's method, through interpolation in the spatial time-frequency distribution matrix (STFD), has achieved DOA estimation for broadband LFM signals. However, its main drawbacks are high computational cost and model bias; furthermore, this type of nonlinear time-frequency analysis method is susceptible to interference from cross-terms when selecting time-frequency points, leading to decreased estimation accuracy. Short-time Fourier transform (STFT) establishes the spatial frequency distribution, avoiding cross-term interference, but the window selection in this method significantly impacts estimation accuracy, making it difficult to choose a suitable window when multiple different targets exist. Fractional Fourier transform (FrFT) for broadband DOA estimation is currently a widely used method. This method belongs to linear time-frequency analysis, has no cross-term interference, and offers high accuracy, but suffers from high computational complexity and complex hardware implementation. Summary of the Invention

[0004] The purpose of this invention is to solve the problems existing in the prior art.

[0005] The technical solution to achieve the purpose of this invention is: a broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform, comprising the following steps:

[0006] Step 1: Select one element from the radar receiving array as the reference element, perform a variable-scale Fresnel transformation on it, and record the position of the spectral peak.

[0007] Step 2: Perform variable-scale Fresnel transformation on the received signals on all array elements to obtain the spectral peak position and peak point data of the k-th incident signal on the m-th array element.

[0008] Step 3: Construct the correlation matrix in the SDFnT domain based on the peak point data;

[0009] Step 4: Perform eigenvalue decomposition on the correlation matrix in the SDFnT domain to obtain the signal subspace matrix, and construct the spectral function;

[0010] Step 5: Traverse angle θ and perform spectral peak search on the spectral function to achieve incident angle estimation for K LFM signals.

[0011] Preferably, step 1, which involves selecting an array element as a reference element from the radar receiving array and performing a variable-scale Fresnel transformation on it, includes the following specific steps:

[0012] Step 1-1: When the radar transmits a broadband LFM signal and the array signal model satisfies the conditions that the source is far-field, the number of sources is less than the number of receiving array elements, and the receiving array elements are uniform linear array elements, select one array element as the reference array element with zero time delay, and transmit K targets, including K target angles.

[0013] Steps 1-2: Perform SDFnT transformation on the data received by the reference array element, and form an energy-concentrated spectral peak in the SDFnT domain at the scaling factor β.

[0014] Steps 1-3 involve performing a spectral peak search on the transformed signal to estimate the number of targets and the positions of their spectral peaks. and peak point data β k This represents the scale factor for the SDFnT transform of the k-th incident LFM signal.

[0015] Preferably, the scaling factor β is specifically:

[0016]

[0017] Where L is the number of fast sampling points in the time domain, tb is the time domain sampling interval, and the spectral peak position is: f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal.

[0018] Preferably, in step 2, a scaling factor of β is applied to the received signals on all array elements using SDFnT to obtain the spectral peak position of the k-th incident signal on the m-th array element. and peak point data The specific process includes:

[0019] Step 2-1: Obtain the kth incident signal received on the m-th array element based on the kth incident LFM signal.

[0020] The expression for the k-th incident LFM signal is:

[0021]

[0022] f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal;

[0023] The incident signal received by the m-th array element is:

[0024]

[0025] in The delay of the kth incident signal on the mth receiving element;

[0026] The SDFnT transformation matrix is ​​represented as follows:

[0027]

[0028] Where m and n represent the time-domain sampling points and frequency-domain sampling points after discretization, respectively, N is the number of subcarriers, and mod2 represents the remainder;

[0029] When N is even, substituting the LFM expression into the SDFnT transform matrix, we obtain the SDFnT expression for the k-th LFM signal as follows:

[0030]

[0031] In the formula, f0 is the initial frequency of the k-th LFM signal, and A is the initial amplitude of the k-th LFM signal;

[0032] Step 2-2, for Perform SDFnT transform to obtain S k m The rotation angle β and the peak position u of (β,u):

[0033]

[0034]

[0035] Where tb is the time-domain sampling interval, and L is the number of snapshots. μ is the scale factor for the SDFnT transform of the k-th incident signal on the m-th array element. k and f k Let be the frequency modulation slope and initial frequency of the k-th incident signal, respectively. The x-coordinate position of the peak value of the kth incident signal after performing SDFnT transformation on the mth array element;

[0036] The relationship between the spectral peak position of the m-th receiving element and the spectral peak position of the reference element is as follows:

[0037]

[0038] Steps 2-3: Compare the SDFnT of the reference element with the SDFnT of the m-th element, and simplify to obtain:

[0039]

[0040] Wherein, the direction vector A of the k-th LFM incident signal in the SDFnT domain is... k The expression is:

[0041]

[0042]

[0043] This vector is only related to the delay term. In the far-field condition, the direction of light (DOA) depends only on the angle of the k-th incident signal and is time-invariant. Using this data model instead of the traditional array data model allows the time-varying direction matrix of the LFM signal to be transformed into a fixed direction matrix, thus enabling DOA estimation using traditional high-resolution algorithms.

[0044] Preferably, step 3 involves constructing the correlation matrix R in the SDFnT domain, specifically including:

[0045] The received data model for the array signal is determined as follows:

[0046] X(t) = AS(t) + N(t)

[0047] Where X(t) is M×1 dimensional snapshot data, A is M×K dimensional steering vector, S(t) is K×1 incident signal, K is the number of incident signals, and N(t) is M×1 dimensional noise data.

[0048] X(t) = [x1(t), x2(t), ..., x M (t)] T

[0049] A = [a(θ1), a(θ2), ..., a(θ)] K )] T

[0050] S(t) = [s1(t), s2(t), ..., s K (t)] T

[0051] N(t) = [n1(t), n2(t), ..., n M (t)] T

[0052] Where, x m (t) represents the received data of the m-th array element. For the array manifold of the k-th signal, the phase term in the steering vector θ k Let s be the angle of the incident signal. k (t) represents the k-th incident signal, n m (t) represents the noise signal received by the m-th array element;

[0053] The covariance matrix of the array output signal is expressed as:

[0054] R = E[XX] H ] = AE[SS H A H +E[NN H ] = AR S A H +R N

[0055] Where R S and R N Let X be the signal covariance matrix and the noise covariance matrix, and let X be the SDFnT domain received signal data matrix.

[0056] Preferably, step 4 involves eigenvalue decomposition of the covariance matrix R to obtain the signal subspace matrix U. s And construct the spectral function f(θ), the specific process includes:

[0057] Perform eigenvalue decomposition on the covariance matrix R:

[0058] R = U S ∑ S U S H +U N ∑ N U NH

[0059] Among them, U S U represents the signal subspace, corresponding to the eigenvectors of the largest eigenvalues ​​obtained after eigenvalue decomposition of R. N Let represent the noise subspace, corresponding to the eigenvectors of the small eigenvalues ​​obtained after eigenvalue decomposition of R.

[0060] Preferably, step 5, which involves traversing angle θ and searching for the peaks of the f(θ) spectrum to estimate the incident angles of the K LFM signals, specifically includes:

[0061] The spectral estimation formula for the MUSIC algorithm is:

[0062]

[0063] According to this formula, by iterating through the angle θ, the pseudo-spectrum of the angle estimation by the MUSIC algorithm can be obtained, and high-resolution angle estimation of non-stationary LFM signals can be achieved by searching for spectral peaks.

[0064] Compared with existing technologies, the significant advantages of this invention are as follows: This invention belongs to the linear time-frequency transformation method, which has good energy focusing characteristics and high accuracy for non-stationary signals such as LFM; This invention combines a novel variable-scale discrete Fresnel transform with a high-resolution MUSIC estimation algorithm to achieve DOA estimation for non-stationary signals; This invention has lower computational complexity and higher estimation accuracy.

[0065] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0066] Figure 1 This is a flowchart of the broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform of the present invention.

[0067] Figure 2 This is an array signal receiving model.

[0068] Figure 3 A comparison of the computation time for DOA estimation using the traditional FrFT method and the SDFnT method used in this invention when traversing the number of snapshots K.

[0069] Figure 4 A comparison of the computation time for DOA estimation using the traditional FrFT method and the SDFnT method used in this invention when traversing rotation factors α and β.

[0070] Figure 5 Comparison of root mean square errors of DOA estimation using the FrFT and SDFnT methods when traversing SNR.

[0071] Figure 6MUSIC spatial spectrum for single-objective estimation using the SDFnT method.

[0072] Figure 7 MUSIC spatial spectrum for multi-objective estimation using the SDFnT method.

[0073] Figure 8 When performing multi-target estimation for the SDFnT method, the MUSIC spatial spectrum of the temporal snapshot number is traversed.

[0074] Figure 9 When performing DOA estimation for the SDFnT method, the MUSIC space spectrum of the array elements is traversed. Detailed Implementation

[0075] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0076] In one embodiment, combined Figure 1 A method for estimating the DOA of a broadband LFM signal based on the variable-scale discrete Fresnel transform, the method comprising the following steps:

[0077] Step 1: Select one element from the radar receiving array as the reference element, perform a variable-scale Fresnel transformation on it, and record the position of the spectral peak.

[0078] Step 2: Perform variable-scale Fresnel transformation on the received signals on all array elements to obtain the spectral peak position and peak point data of the k-th incident signal on the m-th array element.

[0079] Step 3: Construct the correlation matrix in the SDFnT domain based on the peak point data;

[0080] Step 4: Perform eigenvalue decomposition on the correlation matrix in the SDFnT domain to obtain the signal subspace matrix, and construct the spectral function;

[0081] Step 5: Traverse angle θ and perform spectral peak search on the spectral function to achieve incident angle estimation for K LFM signals.

[0082] Furthermore, in one embodiment, step 1, which involves selecting an array element as a reference element in the radar receiving array, performing a variable-scale Fresnel transform on it, and recording the spectral peak position, specifically includes:

[0083] Step 1-1: When the radar transmits a broadband LFM signal and the array signal model satisfies the conditions that the source is far-field, the number of sources is less than the number of receiving array elements, and the receiving array elements are uniform linear array elements, select one array element as the reference array element with zero time delay, and transmit K targets, including K target angles.

[0084] Steps 1-2 involve performing a spectral peak search on the transformed signal to estimate the number of targets and the positions of their spectral peaks. and peak point data β k This represents the scale factor for the SDFnT transform of the k-th incident LFM signal.

[0085] The scaling factor β is specifically:

[0086]

[0087] Where L is the number of fast sampling points in the time domain, tb is the time domain sampling interval, and the spectral peak position is: f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal.

[0088] Furthermore, in one embodiment, step 1 involves performing an SDFnT with a scaling factor of β on the received signals across all array elements to obtain the spectral peak position of the k-th incident signal on the m-th array element. and peak point data The specific process includes:

[0089] Step 2-1, the expression for the k-th incident LFM signal is:

[0090]

[0091] f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal.

[0092] Therefore, the k-th incident signal received on the m-th array element is:

[0093]

[0094] in Just and It is related to time t, but not to time t. The delay of the k-th incident signal on the m-th receiving element is given.

[0095] The SDFnT transformation matrix is ​​represented as follows:

[0096]

[0097] Where m and n represent the time-domain and frequency-domain sampling points after discretization, respectively, N is the number of subcarriers, and mod2 represents the remainder. For ease of description, we only consider the case where N is even. Substituting the above LFM expression into the SDFnT transform matrix, we obtain the SDFnT expression for the k-th LFM signal as follows:

[0098]

[0099] Step 2-2, for Perform SDFnT transform to obtain S k m The rotation angle β and the peak position u of (β,u):

[0100]

[0101]

[0102] From the formula above, it can be seen that after SDFnT processing of different received array element signals, the scaling factor β remains unchanged, but the spectral peak position changes. The relationship between the spectral peak position of the m-th received array element and the spectral peak position of the reference array element is as follows:

[0103]

[0104] Where tb is the time-domain sampling interval, and L is the number of snapshots. μ is the scale factor for the SDFnT transform of the k-th incident signal on the m-th array element. k and f k Let be the frequency modulation slope and initial frequency of the k-th incident signal, respectively. Let x be the x-coordinate of the peak value of the k-th incident signal after performing SDFnT transformation on the m-th array element.

[0105] The relationship between the spectral peak position of the m-th receiving element and the spectral peak position of the reference element is as follows:

[0106]

[0107] Steps 2-3, comparing the SDFnT of the reference element and the SDFnT of the m-th element, and simplifying, yield the following:

[0108]

[0109] in The term is very small and can be ignored. The direction vector A of the k-th LFM incident signal in the SDFnT domain is... k The expression is:

[0110]

[0111]

[0112] This vector is only related to the delay term. In the far-field condition, the direction of light (DOA) depends only on the angle of the k-th incident signal and is time-invariant. Using this data model instead of the traditional array data model allows the time-varying direction matrix of the LFM signal to be transformed into a fixed direction matrix, thus enabling DOA estimation using traditional high-resolution algorithms.

[0113] Furthermore, in one embodiment, step 3 involves constructing the correlation matrix R of the SDFnT domain. k The specific process includes:

[0114] Step 3-1, the data model for receiving array signals is as follows:

[0115] X(t) = AS(t) + N(t)

[0116] Where X(t) is M×1 dimensional snapshot data, A is M×K dimensional steering vector, S(t) is K×1 incident signal, K is the number of incident signals, and N(t) is M×1 dimensional noise data.

[0117] X(t) = [x1(t), x2(t), ..., x M (t)] T

[0118] A = [a(θ1), a(θ2), ..., a(θ)] K )] T

[0119] S(t) = [s1(t), s2(t), ..., s K (t)] T

[0120] N(t) = [n1(t), n2(t), ..., n M (t)] T

[0121] Where, x m (t) represents the received data of the m-th array element. For the array manifold of the Kth signal, the phase term in the steering vector θ k It depends on the angle of the incident signal, s k (t) represents the k-th incident signal, n m (t) represents the noise signal received by the m-th array element;

[0122] The covariance matrix of the array output signal is expressed as:

[0123] R = E[XX] H ] = AE[SS H A H +E[NN H ] = AR S A H +R N

[0124] Where R S and R N Let X be the signal covariance matrix and the noise covariance matrix, and let X be the SDFnT domain received signal data matrix.

[0125] Furthermore, in one embodiment, step 1 involves R... k Eigenvalue decomposition is performed to obtain the signal subspace matrix U. s And construct the spectral function f(θ), the specific process includes:

[0126] Perform eigenvalue decomposition on the covariance matrix R:

[0127] R = U S ∑ S U S H +U N ∑ N U N H

[0128] U S U represents the signal subspace, corresponding to the eigenvectors of the largest eigenvalues ​​obtained after eigenvalue decomposition of R. N Let represent the noise subspace, corresponding to the eigenvectors of the small eigenvalues ​​obtained after R eigendecomposition. Ideally, the noise subspace and the signal subspace are orthogonal, and their steering vectors are also orthogonal.

[0129] Furthermore, in one embodiment, step 5, which involves traversing angle θ and searching for the peaks of the f(θ) spectrum, can achieve the estimation of the incident angles of K LFM signals. The specific process includes:

[0130] Based on the above formula, the spectral estimation formula for the MUSIC algorithm is obtained as follows:

[0131]

[0132] According to this formula, by iterating through the angle θ, the pseudo spectrum of the angle estimation by the MUSIC algorithm can be obtained. High-resolution angle estimation of non-stationary LFM signals can be achieved by searching the spectral peaks. Moreover, this method can estimate the number of targets when the number of signal sources is unknown.

[0133] This invention proposes a novel variable-scale discrete Fresnel transform (DFR) method, which is combined with the traditional MUSIC high-resolution algorithm to achieve high-resolution DOA estimation for non-stationary signals such as broadband LFM. This not only further reduces computational complexity but also improves estimation accuracy.

[0134] This invention presents a novel variable-scale discrete Fresnel transform (DFR) method, and implements high-resolution DOA estimation for broadband LFM signals based on this method. This method adds a scaling factor to the Fresnel transform formula, achieving a time-frequency surface rotation effect similar to the FrFT transform. It exhibits excellent energy focusing characteristics for non-stationary signals like LFM, and compared to the traditional FrFT method, it not only further reduces computational complexity but also improves estimation accuracy.

[0135] In one embodiment, a broadband LFM signal DOA estimation system based on variable-scale discrete Fresnel transform (SDFnT) is provided, the system comprising:

[0136] The number of targets and peak information are obtained by performing SDFnT transformation on several non-stationary LFM target signals received by the reference array element of the radar receiver.

[0137] Perform SDFnT transformation on the other array element signals containing time delay information at the receiving end to obtain a time-invariant direction vector that is only related to the incident angle and independent of time.

[0138] Traditional high-resolution subspace algorithms, such as the MUSIC algorithm, are used to estimate the DOA of LFM-type signals, thereby estimating the number of sources and the angle of arrival.

[0139] As a specific example, the invention will be further verified and illustrated in one embodiment.

[0140] In this example, two LFM signals are selected for far-field incident illumination. Signal 1: initial frequency 200Hz, frequency modulation slope 200Hz / s, incident angle 20°; Signal 2: initial frequency 100Hz, frequency modulation slope 100Hz / s, incident angle 25°. The time-domain snapshot number is set to 256, the number of receiver array elements to 8, the signal-to-noise ratio to be -14:8, and 500 Monte Carlo experiments are performed. Table 1 compares the computation time used for angle estimation by the two different methods under different snapshot numbers. Table 2 shows the root mean square error (RMSE) of the two methods when the target angle is 20° and 25°.

[0141] Table 1 Comparison of computation time between the two methods

[0142]

[0143] As can be seen from Table 1, by comparing the computation time of the SDFnT method proposed in this invention and the FrFT-based method in combination with the high-resolution MUSIC algorithm for DOA estimation, it can be seen that the computation time of this method is less than that of the FrFT-based method, and it has lower computational complexity.

[0144] Table 2 shows the root mean square error of DOA estimation for both methods.

[0145]

[0146] As shown in Table 2, comparing the DOA estimation results of the SDFnT method proposed in this invention and the FrFT-based method combined with the high-resolution MUSIC algorithm, it can be seen that the calculation accuracy of this method is higher than that of the FrFT-based method. This is mainly because the FrFT method requires multiple calculations of trigonometric functions, which demands high accuracy, while the method of this invention avoids these calculations and maintains good performance even at low signal-to-noise ratios. The table also shows that when the signal-to-noise ratio is below -10dB, the DOA estimation performance drops significantly. This is because the low signal-to-noise ratio causes the peak signal after transformation to the Chirp domain to be completely submerged in noise.

[0147] In summary, the method of this invention can achieve DOA estimation for non-stationary LFM signals, and compared with existing methods, it has lower computational complexity and higher estimation accuracy.

[0148] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A method for estimating the DOA of a broadband LFM signal based on variable-scale discrete Fresnel transform, characterized in that, Includes the following steps: Step 1: Select one element from the radar receiving array as the reference element, perform a variable-scale Fresnel transformation on it, and record the position of the spectral peak. Step 2: Perform variable-scale Fresnel transformation on the received signals on all array elements to obtain the spectral peak position and peak point data of the k-th incident signal on the m-th array element. Step 3: Construct the correlation matrix in the SDFnT domain based on the peak point data; Step 4: Perform eigenvalue decomposition on the correlation matrix in the SDFnT domain to obtain the signal subspace matrix, and construct the spectral function; Step 5: Traverse angle θ and perform spectral peak search on the spectral function to achieve incident angle estimation for K LFM signals.

2. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 1, characterized in that, Step 1 involves selecting one element from the radar receiving array as the reference element and performing a variable-scale Fresnel transformation on it. The specific process includes: Step 1-1: When the radar transmits a broadband LFM signal and the array signal model satisfies the conditions that the source is far-field, the number of sources is less than the number of receiving array elements, and the receiving array elements are uniform linear array elements, select one array element as the reference array element with zero time delay, and transmit K targets, including K target angles. Steps 1-2: Perform SDFnT transformation on the data received by the reference array element, and form an energy-concentrated spectral peak in the SDFnT domain at the scaling factor β. Steps 1-3 involve performing a spectral peak search on the transformed signal to estimate the number of targets and the positions of their spectral peaks. and peak point data This represents the scale factor for the SDFnT transformation of the k-th incident LFM signal in the reference array element.

3. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 2, characterized in that, The scaling factor for the SDFnT transform of the k-th incident LFM signal of the reference array element is specifically: Where L is the number of fast sampling points in the time domain, tb is the time domain sampling interval, and the spectral peak position is: f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal.

4. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 3, characterized in that, Step 2: Perform SDFnT with a scaling factor of β on the received signals of all array elements to obtain the spectral peak position of the k-th incident signal on the m-th array element. and peak point data The specific process includes: Step 2-1: Obtain the k-th incident signal received on the m-th array element based on the k-th incident LFM signal. The expression for the k-th incident LFM signal is: f k Let μ be the initial frequency of the k-th incident signal. k The frequency modulation slope of the LFM signal; The incident signal received by the m-th array element is: in The delay of the kth incident signal on the mth receiving element; The SDFnT transformation matrix is ​​represented as follows: Where h and n represent the time-domain sampling points and frequency-domain sampling points after discretization, respectively, N is the number of subcarriers, and mod2 represents the remainder; When N is even, substituting the LFM expression into the SDFnT transform matrix, we obtain the SDFnT expression for the k-th LFM signal as follows: In the formula, f0 is the initial frequency of the k-th LFM signal, and A1 is the initial amplitude of the k-th LFM signal; Step 2-2, for Perform SDFnT transform to obtain Rotation angle β and spectral peak position u: Where tb is the time-domain sampling interval, and L is the number of snapshots. μ is the scale factor for the SDFnT transform of the k-th incident signal on the m-th array element. k and f k Let be the frequency modulation slope and initial frequency of the k-th incident signal, respectively. The x-coordinate position of the peak value of the kth incident signal after performing SDFnT transformation on the mth array element; The relationship between the spectral peak position of the m-th receiving element and the spectral peak position of the reference element is as follows: Steps 2-3: Compare the SDFnT of the reference element with the SDFnT of the m-th element, and simplify to obtain: Wherein, the direction vector A of the k-th LFM incident signal in the SDFnT domain is... k The expression is: This vector is only related to the delay term. Regarding the far-field condition, it depends only on the angle of the k-th incident signal and is time-invariant.

5. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 4, characterized in that, Step 3 involves constructing the correlation matrix R in the SDFnT field. The specific process includes: The received data model for the array signal is determined as follows: X(t) = AS(t) + N(t) Where X(t) is M×1 dimensional snapshot data, A is M×K dimensional steering vector, S(t) is K×1 incident signal, K is the number of incident signals, and N(t) is M×1 dimensional noise data. X(t)=[x1(t),x2(t),…,x M (t)] T A=[a(θ1),a(θ2),…,a(θ K )] T S(t)=[s1(t),s2(t),…,s K (t)] T N(t)=[n1(t),n2(t),…,n M (t)] T Where, x m (t) represents the received data of the m-th array element. k = 1, 2, ..., K is the array manifold of the k-th signal, and the phase term in the steering vector... θ k Let s be the angle of the incident signal. k (t) represents the k-th incident signal, n m (t) represents the noise signal received by the m-th array element; The covariance matrix of the array output signal is expressed as: R=E[XX H ]=AE[SS H ]A H +E[NN H ]=AR S A H +R N Where R S and R N Let X be the signal covariance matrix and the noise covariance matrix, and let X be the SDFnT domain received signal data matrix.

6. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 5, characterized in that, Step 4 involves eigenvalue decomposition of the covariance matrix R to obtain the signal subspace matrix U. s And construct the spectral function f(θ), the specific process includes: Perform eigenvalue decomposition on the covariance matrix R: R=U S ∑ S IN S H +U N ∑ N IN N H Among them, U S U represents the signal subspace, corresponding to the eigenvectors of the largest eigenvalues ​​obtained after eigenvalue decomposition of R. N Let represent the noise subspace, corresponding to the eigenvectors of the small eigenvalues ​​obtained after eigenvalue decomposition of R.

7. The broadband LFM signal DOA estimation method based on variable-scale discrete Fresnel transform according to claim 6, characterized in that, Step 5, which involves traversing angle θ and searching for peaks in the f(θ) spectrum to estimate the incident angles of K LFM signals, specifically includes: The spectral estimation formula for the MUSIC algorithm is: In the formula, U N Represents the noise subspace; According to this formula, by iterating through the angle θ, the pseudo-spectrum of the angle estimation by the MUSIC algorithm can be obtained, and high-resolution angle estimation of non-stationary LFM signals can be achieved by searching for spectral peaks.

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