Frequency diversity co-prime array angle and distance estimation method

Abstract

The invention is suitable for the technical field of array signal processing, and provides a frequency diversity co-prime array angle and distance estimation method, which comprises the following steps: designing a supervised hole completion network to establish a mapping model from a hole-containing virtual signal to a complete virtual array signal, and the calculation complexity in the subsequent space smoothing process is reduced by designing a matrix reconstruction method, and finally, a target sensing result is output by combining a MUSIC algorithm. The frequency diversity co-prime array receives signals as input, angle and distance information of each target can be output after iterative training, and estimation accuracy and robustness are improved.

Description

Technical Field

[0001] The present application relates to the field of array signal processing, and in particular to a method for estimating angle and distance of a frequency diversity coprime array. Background Art

[0002] In the field of wireless sensing, high-precision sensing capabilities within limited resource constraints are crucial. Frequency-diversity coprime arrays leverage the dual advantages of coprime sparse placement in the spatial domain and coprime frequency offset diversity in the frequency domain to multiply the degrees of freedom within the constraints of limited array element size and bandwidth, providing a system-level solution for precise positioning of devices at an unprecedented scale and density.

[0003] However, the inherent constraints of the array structure imposed by coprime design lead to the virtual domain hole problem. Current research methods have modeled the signal matrix filling problem as an atomic norm minimization model and solved it via convex relaxation. However, this approach introduces performance losses due to convex relaxation, and its target perception accuracy needs to be improved. Furthermore, it relies heavily on a precise array manifold, resulting in performance degradation under conditions of array mismatch. Summary of the Invention

[0004] An embodiment of the present application provides a frequency diversity coprime array angle and distance estimation method, which can solve the problems of low accuracy and robustness caused by modeling the signal matrix filling problem as an atomic norm minimization model and solving it through convex relaxation.

[0005] In a first aspect, an embodiment of the present application provides a method for estimating angle and distance of a frequency diversity coprime array, including:

[0006] Step 1: Design a frequency diversity coprime array structure;

[0007] Step 2: obtaining a received signal of the array;

[0008] Step 3: construct a covariance matrix based on the received signal, and perform vectorization on the covariance matrix to generate a virtual array received signal model containing holes;

[0009] Step 4: Create a data set based on the virtual domain signal received by the virtual array receiving signal model containing holes, wherein the data set includes labels of the virtual domain signal and the complete virtual array receiving signal;

[0010] Step 5: Predict the virtual domain signal using a preset supervised hole completion network to obtain a predicted value of the complete virtual array received signal; calculate the loss function between the predicted value and the true value, and perform multiple rounds of backpropagation and parameter update based on the loss function to obtain a trained hole completion network;

[0011] Step 6: Input the received signal of the array obtained in step 2 into the trained hole completion network, and output the completed continuous virtual array signal;

[0012] Step 7: Performing matrix reconstruction spatial smoothing processing on the completed continuous virtual array signal to generate a covariance matrix of extended degrees of freedom;

[0013] Step 8: Based on the covariance matrix of the extended degrees of freedom, a 2D-MUSIC algorithm is used to implement joint estimation of target space-frequency parameters.

[0014] In a possible implementation of the first aspect, step 1 specifically includes:

[0015] A joint receiving array with dual sparse uniform subarrays is used to receive the signal from the transmitter:

[0016] In the spatial dimension, the first subarray is set to contain M array elements with an array element spacing of Nd; the second subarray is set to contain N array elements with an array element spacing of Md, where (M, N) is a coprime parameter pair satisfying M < N, d is the array element unit spacing, and the two subarrays share the same first array element. The total number of array elements in the joint array is M + N - 1. The array element arrangement of the joint array is expressed as follows using a coprime parameter set:

[0017] L=L1∪L2, L1={nM,0≤n≤N-1}, L2={mN,0≤m≤M-1}

[0018] Where L1 and L2 are two mutually prime parameter subsets;

[0019] The array element physical position set is:

[0020] D=Ld

[0021] In the frequency domain, the frequency offset set of the frequency diversity signal has the same coprime structure as the set of array element physical positions, that is, the frequency set is:

[0022] F=f0+LΔf

[0023] Where f0 is the center reference frequency of the array and Δf is the unit frequency deviation.

[0024] Optionally, in another possible implementation of the first aspect, step 2 specifically includes:

[0025] Set K azimuth angles to θ k , the distance is r k , where k = 1,…,K, for a far-field incoherent target, the i-th array element receiving the q-th frequency signal is expressed as:

[0026]

[0027] where x k (t) represents the complex envelope of the signal, e is a natural constant, is the imaginary unit, f q =f0+l q Δf is the qth frequency, d i =l i d is the position of the i-th receiving element; l i , l q They represent the array element position and frequency offset index respectively, c is the speed of light, π is the circumference, λ q is the wavelength corresponding to the qth frequency, n i,q (t) is additive white Gaussian noise;

[0028] Represent the multidimensional received signal in vector form:

[0029] y i,q By stacking according to i,q=1,…,M+N-1, the received signal vector of the array is obtained as follows:

[0030]

[0031] x(t)=[x1(t),…,x K (t)] T

[0032] H d,f =[h d,f (θ1,r1),…,h d,f (θ K ,r K )]

[0033]

[0034] Where x(t) is the received signal vector, H d,f is the array manifold matrix, h d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the spatial steering vector h d (θ k ) and the frequency domain steering vector h f (r k ) is obtained through Kronecker product operation; T is the matrix transpose symbol.

[0035] Optionally, in another possible implementation of the first aspect, step 3 specifically includes:

[0036] The covariance matrix of the received signal is constructed as:

[0037]

[0038] Among them, E is the statistical significance expectation, I is the identity matrix, represents the noise power, R x is the covariance matrix of the received signal vector x(t). Under non-coherent conditions, the signal covariance matrix is ​​represented by a diagonal matrix of K target powers, that is,

[0039] Covariance matrix R y Perform vectorization processing to obtain the original virtual domain signal:

[0040]

[0041] in, represents the array manifold corresponding to the virtual array; represents a column vector consisting of K target powers; i = vec(I) represents a column vector obtained by vectorizing the unit matrix I;

[0042] For the reconstructed virtual array, its element spatial position and frequency offset can be obtained by the difference array D diff and F diff Decide:

[0043] D diff ={(l i -l j )d|l i ,l j ∈L,i,j=1,2,…,M+N-1}

[0044] F diff ={(l i -l j )Δf|l i ,l j ∈L,i,j=1,2,…,M+N-1}

[0045] Eliminate D diff and F diff The repeated elements in the original virtual domain signal y are selected accordingly. v The elements in get the virtual signal vector

[0046] For virtual signal vector Execute the zero-fill initialization strategy and The hole position elements of are interpolated to 0 to obtain the initialized virtual signal vector:

[0047]

[0048] Among them L I =[-M(N-1),M(N-1)] is a complete integer index set, Ldiff Represents the valid virtual array element index set, matrix element The virtual array element corresponding to the frequency offset jΔf and the array element position ld.

[0049] Optionally, in another possible implementation of the first aspect, step 4 specifically includes:

[0050] Under noise-free conditions, the perfect virtual array received signal, i.e., the ideal signal, is modeled as:

[0051]

[0052] A d,f =[a d,f (θ1,r1),…,a d,f (θ K ,r K )]

[0053]

[0054] Among them, A d,f is the array manifold corresponding to the complete virtual array, a d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the space-domain steering vector a d (θ k ) and the frequency domain steering vector a f (r k ) is obtained by Kronecker product operation;

[0055] Will initialize the virtual signal vector With the ideal signal Reshape into complex-valued tensors with side length 2M(N-1)+1 and

[0056] according to and Belonging dataset, generate dataset

[0057] Optionally, in another possible implementation of the first aspect, step 5 specifically includes:

[0058] The dataset D is fed into a preset convolutional neural network for training to construct a hole completion network. Each complex-valued tensor in the dataset is split into real and imaginary parts.

[0059] The hole completion network consists of an input layer, an output layer, and multiple hidden layers, and each network layer includes multiple neural units. The convolution process of each network layer is used as a function process:

[0060]

[0061] in represents the convolution kernel, V l i represents the lth element of the i-th feature map, f(·) is the activation function, is the i-th element of the complex matrix after the real and imaginary parts of the input are split, and b is the bias term;

[0062] Use the mean square error function as the objective optimization function for the hole completion task:

[0063]

[0064] Where ‖·‖2 is the Euclidean norm, and the network estimate Y is calculated. cnn With label The loss function is used, and the Adam optimizer is used to adjust the parameters during the back-propagation process. After training for the preset rounds, the model is saved to obtain the trained hole completion network model.

[0065] Optionally, in another possible implementation of the first aspect, step 6 specifically includes:

[0066] The received signal of the array obtained in step 2 is derived as an initialized virtual signal vector and input into the trained hole completion network model for hole completion to obtain the completed continuous virtual array signal Y cnn .

[0067] Optionally, in another possible implementation of the first aspect, step 7 specifically includes:

[0068] According to the completed continuous virtual array signal Y cnn , perform spatial smoothing to obtain the covariance matrix of the extended degrees of freedom:

[0069]

[0070] F p,q =vec(P p Y cnn Q q )

[0071] P p =[0 (U+1)×(-p) I (U+1)×(U+1) 0 (U+1)×(U+p) ]∈{0,1} (U+1)×(2U+1)

[0072]

[0073] U=M(N-1)

[0074] P, Q represents the selection matrix, which is used to select Y under the guidance of variables p, q. cnn Neutron matrix, where p∈[-U,0],q∈[0,U], 0 represents the zero matrix, I represents the identity matrix, and U represents the maximum index of the virtual array.

[0075] Optionally, in another possible implementation of the first aspect, step 8 specifically includes:

[0076] Covariance matrix for the expanded degrees of freedom Perform eigendecomposition and extract the noise subspace U N , construct 2D-MUSIC spectrum function:

[0077]

[0078] in, Represent the estimated angle and distance respectively, represents the continuous virtual array space-frequency joint steering vector.

[0079] This application provides a method for estimating the angle and distance of a frequency-diversified coprime array. This method employs a supervised hole completion network to establish a mapping model from a virtual signal containing holes to a complete virtual array signal. A matrix reconstruction method is then designed to reduce the computational complexity of the subsequent spatial smoothing process. Finally, the method combines the MUSIC algorithm to output target perception results. This method uses the received signal of a frequency-diversified coprime array as input and, after iterative training, can output angle and distance information for each target, improving the accuracy and robustness of the estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following briefly introduces the drawings required for use in the embodiments or descriptions of the prior art. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0081] Figure 1 A flowchart of a method for estimating angle and distance of a frequency-diversity coprime array provided for an embodiment of the present application;

[0082] Figure 2 A schematic diagram of the array structure of a frequency diversity coprime array provided in an embodiment of the present application;

[0083] Figure 3 A virtual array diagram of a frequency diversity coprime array provided in an embodiment of the present application;

[0084] Figure 4This is a graph of SNR-angle RMSE in a single-target embodiment provided in an embodiment of the present application;

[0085] Figure 5 This is a graph of SNR-distance RMSE in a single-target embodiment provided in an embodiment of the present application;

[0086] Figure 6 This is a snapshot number-angle RMSE curve graph for a single target embodiment provided in an embodiment of the present application;

[0087] Figure 7 This is a snapshot number-distance RMSE curve graph for a single target embodiment provided in an embodiment of the present application;

[0088] Figure 8 This is a graph of SNR-angle RMSE in a multi-objective embodiment provided in an embodiment of the present application;

[0089] Figure 9 This is a graph of SNR-distance RMSE in a multi-target embodiment provided in an embodiment of the present application;

[0090] Figure 10 The array position error-angle RMSE histogram provided in the embodiment of the present application;

[0091] Figure 11 This is a frequency accuracy error-distance RMSE histogram provided in an embodiment of the present application. DETAILED DESCRIPTION

[0092] In the following description, specific details such as specific system structures and techniques are provided for purposes of illustration rather than limitation to facilitate a thorough understanding of the embodiments of the present application. However, it will be apparent to those skilled in the art that the present application may be implemented in other embodiments without these specific details. In other cases, detailed descriptions of well-known systems, devices, circuits, and methods are omitted to avoid obscuring the description of the present application with unnecessary detail.

[0093] It should be understood that when used in the present specification and the appended claims, the term "comprising" indicates the presence of described features, integers, steps, operations, elements and / or components, but does not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components and / or collections thereof.

[0094] It will also be understood that the term "and / or" used in this specification and the appended claims refers to and includes any and all possible combinations of one or more of the associated listed items.

[0095] As used in this specification and the appended claims, the term "if" can be interpreted as "when" or "upon" or "in response to determining" or "in response to detecting," depending on the context. Similarly, the phrase "if it is determined" or "if [described condition or event] is detected" can be interpreted as meaning "upon determination" or "in response to determining" or "upon detection of [described condition or event]" or "in response to detecting [described condition or event]," depending on the context.

[0096] In addition, in the description of the present application specification and the appended claims, the terms "first", "second", "third", etc. are only used to distinguish the descriptions and cannot be understood as indicating or implying relative importance.

[0097] References to "one embodiment" or "some embodiments" in this specification mean that a particular feature, structure, or characteristic described in conjunction with that embodiment is included in one or more embodiments of the present application. Thus, phrases such as "in one embodiment," "in some embodiments," "in other embodiments," and "in other embodiments" appearing in various places in this specification do not necessarily refer to the same embodiment, but rather mean "one or more but not all embodiments," unless otherwise specifically emphasized. The terms "including," "comprising," "having," and variations thereof all mean "including but not limited to," unless otherwise specifically emphasized.

[0098] A frequency diversity coprime array angle and distance estimation method provided by the present application is described in detail below with reference to the accompanying drawings.

[0099] Figure 1 A flow chart of a frequency diversity coprime array angle and distance estimation method provided in an embodiment of the present application is shown.

[0100] S1, design frequency diversity coprime array structure;

[0101] Furthermore, in the embodiment of the present application, the above S1 includes:

[0102] A joint receiving array with dual sparse uniform subarrays is used to receive the signal from the transmitter:

[0103] In the spatial dimension, the first subarray is set to contain M array elements with an array element spacing of Nd; the second subarray is set to contain N array elements with an array element spacing of Md, where (M, N) is a coprime parameter pair satisfying M < N, d is the array element unit spacing, and the two subarrays share the same first array element. The total number of array elements in the joint array is M + N - 1. The array element arrangement of the joint array is expressed as follows using a coprime parameter set:

[0104] L=L1∪L2, L1={nM,0≤n≤N-1}, L2={mN,0≤m≤M-1}

[0105] Where L1 and L2 are two mutually prime parameter subsets;

[0106] The array element physical position set is:

[0107] D=Ld

[0108] In the frequency domain, the frequency offset set of the frequency diversity signal has the same coprime structure as the set of array element physical positions, that is, the frequency set is:

[0109] F=f0+LΔf

[0110] Where f0 is the center reference frequency of the array and Δf is the unit frequency deviation.

[0111] Preferably, in one embodiment, the coprime number combination M=3, N=5, the center frequency f0=10GHz, the unit frequency deviation Δf=30KHz, and the array element spacing d=λ / 2 are taken. Under this parameter configuration condition, the frequency diversity coprime array structure formed is as follows: Figure 2 shown.

[0112] S2, obtaining a received signal of the array;

[0113] Furthermore, in the embodiment of the present application, the above S2 includes:

[0114] Set K azimuth angles to θ k , the distance is r k , where k = 1,…,K, for a far-field incoherent target, the i-th array element receiving the q-th frequency signal is expressed as:

[0115]

[0116] where x k (t) represents the complex envelope of the signal, e is a natural constant, is the imaginary unit, f q =f0+l q Δf is the qth frequency, d i =l i d is the position of the i-th receiving element; l i , l q They represent the array element position and frequency offset index respectively, c is the speed of light, π is the circumference, λ q is the wavelength corresponding to the qth frequency, n i,q (t) is additive white Gaussian noise;

[0117] Represent the multidimensional received signal in vector form:

[0118] yi,q By stacking according to i,q=1,…,M+N-1, the received signal vector of the array is obtained as follows:

[0119]

[0120] x(t)=[x1(t),…,x K (t)] T

[0121] H d,f =[h d,f (θ1,r1),…,h d,f (θ K ,r K )]

[0122]

[0123] Where x(t) is the received signal vector, H d,f is the array manifold matrix, h d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the spatial steering vector h d (θ k ) and the frequency domain steering vector h f (r k ) is obtained through Kronecker product operation; T is the matrix transpose symbol.

[0124] S3. Constructing a covariance matrix according to the received signal, and performing vectorization operation on the covariance matrix to generate a virtual array received signal model containing holes;

[0125] Furthermore, in the embodiment of the present application, the above S3 includes:

[0126] The covariance matrix of the received signal is constructed as:

[0127]

[0128] Among them, E is the statistical significance expectation, I is the identity matrix, represents the noise power, R x is the covariance matrix of the received signal vector x(t). Under non-coherent conditions, the signal covariance matrix is ​​represented by a diagonal matrix of K target powers, that is,

[0129] It should be noted that in the actual system, R y T snapshots are often used for estimation.

[0130] Covariance matrix R y Perform vectorization processing to obtain the original virtual domain signal:

[0131]

[0132] in, represents the array manifold corresponding to the virtual array; represents a column vector consisting of K target powers; i = vec(I) represents a column vector obtained by vectorizing the unit matrix I;

[0133] For the reconstructed virtual array, its element spatial position and frequency offset can be obtained by the difference array D diff and F diff Decide:

[0134] D diff ={(l i -l j )d|l i ,l j ∈L,i,j=1,2,…,M+N-1}

[0135] F diff ={(l i -l j )Δf|l i ,l j ∈L,i,j=1,2,…,M+N-1}

[0136] Eliminate D diff and F diff The repeated elements in the original virtual domain signal y are selected accordingly. v The elements in get the virtual signal vector

[0137] For virtual signal vector Execute the zero-fill initialization strategy and The hole position elements of are interpolated to 0 to obtain the initialized virtual signal vector:

[0138]

[0139] Among them L I =[-M(N-1),M(N-1)] is a complete integer index set, L diff Represents the valid virtual array element index set, matrix element The virtual array element corresponding to the frequency offset jΔf and the array element position ld.

[0140] According to the parameter configuration conditions of an optional embodiment of S1, the complete integer index set L I =[-12,12], valid virtual array element index set L diff={-12,-10,-9,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,9,10,12}. Matrix elements The virtual array element corresponding to the frequency offset jΔf and the array element position ld, under the condition of this parameter configuration, the virtual array structure of the frequency diversity coprime array is formed as follows: Figure 3 shown.

[0141] S4. Creating a data set based on the virtual domain signal received by the virtual array receiving signal model containing holes, where the data set includes labels of the virtual domain signal and the complete virtual array receiving signal;

[0142] Furthermore, in the embodiment of the present application, the above S4 includes:

[0143] Under noise-free conditions, the perfect virtual array received signal, i.e., the ideal signal, is modeled as:

[0144]

[0145] A d,f =[a d,f (θ1,r1),…,a d,f (θ K ,r K )]

[0146]

[0147] Among them, A d,f is the array manifold corresponding to the complete virtual array, a d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the space-domain steering vector a d (θ k ) and the frequency domain steering vector a f (r k ) is obtained by Kronecker product operation;

[0148] Will initialize the virtual signal vector With the ideal signal Reshape into complex-valued tensors with side length 2M(N-1)+1 and

[0149] according to and Belonging dataset, generate dataset

[0150] S5. Predict the virtual domain signal using a preset supervised hole completion network to obtain a predicted value of the complete virtual array received signal; calculate the loss function between the predicted value and the true value, and perform multiple rounds of backpropagation and parameter updates based on the loss function to obtain a trained hole completion network;

[0151] Furthermore, in the embodiment of the present application, the above S5 includes:

[0152] The dataset D is fed into a preset convolutional neural network for training to construct a hole completion network. Each complex-valued tensor in the dataset is split into real and imaginary parts.

[0153] The hole completion network consists of an input layer, an output layer, and multiple hidden layers, and each network layer includes multiple neural units. The convolution process of each network layer is used as a function process:

[0154]

[0155] in represents the convolution kernel, V l i represents the lth element of the i-th feature map, f(·) is the activation function, is the i-th element of the complex matrix after the real and imaginary parts of the input are split, and b is the bias term;

[0156] Use the mean square error function as the objective optimization function for the hole completion task:

[0157]

[0158] Where ‖·‖2 is the Euclidean norm, and the network estimate Y is calculated. cnn With label The loss function is used, and the Adam optimizer is used to adjust the parameters during the back-propagation process. After training for the preset rounds, the model is saved to obtain the trained hole completion network model.

[0159] It should be noted that in order to avoid complex number input, the complex matrix can be split into real and imaginary parts. Transformed into

[0160] As a possible implementation, we can build a hole completion network based on a convolutional neural network. Set the number of network layers to 4, with the number of convolution kernels in each layer being 128, 256, 64, and 2, respectively, and the kernel dimensions being 5 × 5. Set the activation function to the PReLU function.

[0161] S6. Input the received signal of the array obtained in S2 into the trained hole completion network, and output the completed continuous virtual array signal;

[0162] In the embodiment of the present application, the received signal of the array obtained in step 2 is derived as an initialized virtual signal vector and input into the trained hole completion network model for hole completion, and the completed continuous virtual array signal Y is obtained. cnn .

[0163] S7, performing matrix reconstruction spatial smoothing processing on the completed continuous virtual array signal to generate a covariance matrix of extended degrees of freedom;

[0164] Furthermore, in the embodiment of the present application, the above S7 includes:

[0165] According to the completed continuous virtual array signal Y cnn , perform spatial smoothing to obtain the covariance matrix of the extended degrees of freedom:

[0166]

[0167] F p,q =vec(P p Y cnn Q q )

[0168] P p =[0 (U+1)×(-p) I (U+1)×(U+1) 0 (U+1)×(U+p) ]∈{0,1} (U+1)×(2U+1)

[0169]

[0170] U=M(N-1)

[0171] P, Q represents the selection matrix, which is used to select Y under the guidance of variables p, q. cnn Neutron matrix, where p∈[-U,0],q∈[0,U], 0 represents the zero matrix, I represents the identity matrix, and U represents the maximum index of the virtual array.

[0172] S8, based on the covariance matrix of the extended degrees of freedom, combined with the 2D-MUSIC algorithm to achieve joint estimation of the target space-frequency parameters. Further, in the embodiment of the present application, the above S8 includes:

[0173] Covariance matrix for the expanded degrees of freedom Perform eigendecomposition and extract the noise subspace U N , construct 2D-MUSIC spectrum function:

[0174]

[0175] in, Represent the estimated angle and distance respectively, represents the continuous virtual array space-frequency joint steering vector.

[0176] This application proposes a method for estimating angles and distances for frequency-diversified coprime arrays. This method employs a supervised hole completion network to establish a mapping model from virtual signals containing holes to complete virtual array signals. Furthermore, a matrix reconstruction method is designed to reduce the computational complexity of the subsequent spatial smoothing process. Finally, the method combines this with the MUSIC algorithm to output target perception results. This method uses the received signal from a frequency-diversified coprime array as input and, after iterative training, can output angle and distance information for each target, improving both the accuracy and robustness of the estimation.

[0177] The above embodiment demonstrates the overall process of the frequency diversity coprime array angle and distance estimation method proposed in this application. The superiority of the algorithm's calculation effect is verified through specific implementation below.

[0178] 1. Experimental parameter settings

[0179] The method proposed in this invention is referred to as CASS-MUSIC. In order to evaluate the comprehensive performance of the CASS-MUSIC algorithm proposed in this chapter in detail, the theoretical performance boundary of CRB is first used as the algorithm evaluation standard. The basic spatial smoothing algorithm SST-MUSIC is used as the benchmark algorithm. SST-MUSIC only uses the continuous part of the virtual array and does not perform the hole filling task, and abandons the non-continuous part of the virtual array. Secondly, the DANM hole interpolation method for frequency-diversity coprime arrays is strictly reproduced to compare the effectiveness of the CASS-MUSIC algorithm in performing hole interpolation. Finally, in order to enhance the completeness of the comparative experiment, an end-to-end Re-Transformer deep learning architecture is added. This algorithm is a purely data-driven algorithm that integrates the secondary attention mechanism into the basic Transformer structure to perform a direct mapping from the original data to the estimated results.

[0180] 2. Single target estimation performance

[0181] The training data targets are randomly generated in the range of [-45°, 45°] and [2000m, 3000m]. The angle and distance of the random signal source in the test set follow the range of θ~N(30°, (1°) 2 ) and r~N(2500m,(10m) 2 ). First, Figure 4 and Figure 5 The RMSE curves for single-target DoA and range estimation are shown for a fixed snapshot count of 200 and a dynamic range of signal-to-noise ratio (SNR) from -15dB to 20dB. The experimental results show that CASS-MUSIC consistently maintains the lowest estimation error under varying SNR conditions. Figure 6 and Figure 7 The simulation results show the comparison of DoA and distance estimation errors with the number of snapshots under a fixed signal-to-noise ratio of 10dB. The simulation results further confirm the performance advantages of the CASS-MUSIC method proposed in this paper.

[0182] 3. Multi-objective estimation performance

[0183] Further considering the multi-target scenario, under the premise of ensuring that the target interval is greater than the inherent resolution threshold of the array system, each training sample is calculated by randomly generating 56 targets in the range of [-85°, 85°] and [250m, 4750m]. The targets in the test set are evenly distributed at 7 angles between [-85°, 85°] and 8 distances between [250m, 4750m], and the superposition variance is ((1°) 2 ,(5m) 2 ) is a Gaussian distribution. Figure 8 and Figure 9 The results show how the DoA and distance estimation errors vary with the signal-to-noise ratio (SNR) for a multi-source scenario with 200 snapshots. When the number of targets exceeds the degrees of freedom of the continuous portion of the virtual array, SST-MUSIC, which uses only the continuous portion, cannot distinguish the targets and therefore exhibits large estimation errors under all SNR conditions. CASS-MUSIC, on the other hand, consistently maintains the lowest RMSE: compared to Re-Transformer and DANM, it achieves an average accuracy improvement of 15.4dB and 13.6dB for DoA estimation across all SNR conditions; and an average accuracy improvement of 6.41dB and 6.54dB, respectively, for distance estimation.

[0184] 4. Robust performance against array errors

[0185] Under the condition of fixed SNR=10dB, array position offset and frequency accuracy error are introduced to simulate array mismatch in actual environment. φ represents the degree of deviation of array position error and frequency accuracy error respectively. Figure 10 and Figure 11 The results show that the SST-MUSIC and DANM algorithms, which rely entirely on the ideal array model assumption, experience significant deterioration in positioning error as the error ratio increases. This is due to their sensitivity to array manifold mismatch. In contrast, the Re-Transformer and CASS-MUSIC algorithms, which incorporate data-driven capabilities, demonstrate stronger environmental adaptability. While CASS-MUSIC's performance slightly degrades due to the impact of errors, it still maintains a relatively optimal RMSE metric. These experimental results demonstrate the excellent robustness of the proposed method.

[0186] It should be understood that the size of the serial numbers of the steps in the above embodiments does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of this application.

[0187] The above-described embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present application, and should all be included in the scope of protection of the present application.

Claims

1. A method for estimating angle and distance of a frequency-diversity coprime array, characterized in that: The following steps are involved: Step 1: Design a frequency diversity coprime array structure; Step 2: obtaining a received signal of the array; Step 3: construct a covariance matrix based on the received signal, and perform vectorization on the covariance matrix to generate a virtual array received signal model containing holes; Step 4: Create a data set based on the virtual domain signal received by the virtual array receiving signal model containing holes, wherein the data set includes labels of the virtual domain signal and the complete virtual array receiving signal; Step 5: predicting the virtual domain signal through a preset supervised hole completion network to obtain a predicted value of the complete virtual array received signal; Calculate the loss function between the predicted value and the true value, and perform multiple rounds of backpropagation and parameter updates based on the loss function to obtain a trained hole completion network; Step 6: Input the received signal of the array obtained in step 2 into the trained hole completion network, and output the completed continuous virtual array signal; Step 7: Performing matrix reconstruction spatial smoothing processing on the completed continuous virtual array signal to generate a covariance matrix of extended degrees of freedom; Step 8: Based on the covariance matrix of the extended degrees of freedom, a 2D-MUSIC algorithm is used to implement joint estimation of target space-frequency parameters.

2. The method according to claim 1, characterized in that Step 1 specifically includes: A joint receiving array with dual sparse uniform subarrays is used to receive the signal from the transmitter: In the spatial dimension, the first subarray is set to contain M array elements with an array element spacing of Nd; the second subarray is set to contain N array elements with an array element spacing of Md, where (M, N) is a coprime parameter pair satisfying M < N, d is the array element unit spacing, and the two subarrays share the same first array element. The total number of array elements in the joint array is M + N - 1. The array element arrangement of the joint array is expressed as follows using a coprime parameter set: L=L1∪L2, L1={nM,0≤n≤N-1}, L2={mN,0≤m≤M-1} Where L1 and L2 are two mutually prime parameter subsets; The array element physical position set is: D=Ld In the frequency domain, the frequency offset set of the frequency diversity signal has the same coprime structure as the set of array element physical positions, that is, the frequency set is: F=f0+LΔf Where f0 is the center reference frequency of the array and Δf is the unit frequency deviation.

3. The method according to claim 2, characterized in that Step 2 specifically includes: Set K azimuth angles to θ k , the distance is r k , where k = 1,…,K, for a far-field incoherent target, the i-th array element receiving the q-th frequency signal is expressed as: where x k (t) represents the complex envelope of the signal, e is a natural constant, is the imaginary unit, f q =f0+l q Δf is the qth frequency, d i =l i d is the position of the i-th receiving element; l i , l q They represent the array element position and frequency offset index respectively, c is the speed of light, π is the circumference, λ q is the wavelength corresponding to the qth frequency, n i,q (t) is additive white Gaussian noise; Represent the multidimensional received signal in vector form: The y i,q By stacking according to i,q=1,…,M+N-1, the received signal vector of the array is obtained as follows: H d,f =[h d,f (θ1,r1),…,h d,f (θ K ,r K )] Where x(t) is the received signal vector, H d,f is the array manifold matrix, h d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the spatial steering vector h d (θ k ) and the frequency domain steering vector h f (r k ) is obtained through Kronecker product operation; T is the matrix transpose symbol.

4. The method according to claim 3, characterized in that Step 3 specifically includes: The covariance matrix of the received signal is constructed as: Among them, E is the statistical significance expectation, I is the identity matrix, represents the noise power, R x is the covariance matrix of the received signal vector x(t), and the signal covariance matrix under non-coherent conditions is represented by a diagonal matrix of K target powers, that is, Covariance matrix R y Perform vectorization processing to obtain the original virtual domain signal: in, represents the array manifold corresponding to the virtual array; represents a column vector consisting of K target powers; i = vec(I) represents a column vector obtained by vectorizing the unit matrix I; For the reconstructed virtual array, its element spatial position and frequency offset can be obtained by the difference array D diff and F diff Decide: D diff ={(l i -l j )d∣l i ,l j ∈L,i,j=1,2,…,M+N-1} F diff ={(l i -l j )Δf∣l i ,l j ∈L,i,j=1,2,…,M+N-1} Eliminate D diff and F diff The repeated elements in the original virtual domain signal y are selected accordingly. v The elements in get the virtual signal vector For virtual signal vector Execute the zero-fill initialization strategy and The hole position elements of are interpolated to 0 to obtain the initialized virtual signal vector: Among them L I =[-M(N-1),M(N-1)] is a complete integer index set, L diff Represents the valid virtual array element index set, matrix element The virtual array element corresponding to the frequency offset jΔf and the array element position ld.

5. The method according to claim 4, characterized in that Step 4 specifically includes: Under noise-free conditions, the perfect virtual array received signal, i.e., the ideal signal, is modeled as: A d,f =[a d,f (θ1,r1),…,a d,f (i K ,r K )] Among them, A d,f is the array manifold corresponding to the complete virtual array, a d,f (θ k ,r k ) is the space-frequency joint steering vector corresponding to the kth target, and the space-domain steering vector a d (θ k ) and the frequency domain steering vector a f (r k ) is obtained by Kronecker product operation; Will initialize the virtual signal vector With the ideal signal Reshape into complex-valued tensors with side length 2M(N-1)+1 and according to and Belonging dataset, generate dataset 6. The method according to claim 5, characterized in that Step 5 specifically includes: The dataset D is fed into a preset convolutional neural network for training to construct a hole completion network, where each complex-valued tensor in the dataset is split into a real part and an imaginary part; The hole completion network includes an input layer, an output layer and multiple hidden layers, and each network layer includes multiple neural units. The convolution process of each network layer is used as a function process: in represents the convolution kernel, V l i represents the lth element of the i-th feature map, f(·) is the activation function, is the i-th element of the complex matrix after the real and imaginary parts of the input are split, and b is the bias term; Use the mean square error function as the objective optimization function for the hole completion task: Where ‖·‖2 is the Euclidean norm, and the network estimate Y is calculated. cnn With label The loss function is used, and the Adam optimizer is used to adjust the parameters during the back-propagation process. After training for the preset rounds, the model is saved to obtain the trained hole completion network model.

7. The method according to claim 6, characterized in that Step 6 specifically includes: The received signal of the array obtained in step 2 is derived as an initialized virtual signal vector and input into the trained hole completion network model for hole completion to obtain the completed continuous virtual array signal Y cnn .

8. The method according to claim 7, characterized in that Step 7 specifically includes: According to the completed continuous virtual array signal Y cnn , perform spatial smoothing to obtain the covariance matrix of the extended degrees of freedom: F p,q =vec(P p Y cnn Q q ) P p =[0 (U+1)×(-p) I (U+1)×(U+1) 0 (U+1)×(U+p) ]∈{0,1} (U+1)×(2U+1) U=M(N-1) P, Q represents the selection matrix, which is used to select Y under the guidance of variables p, q. cnn Neutron matrix, where p∈[-U,0],q∈[0,U], 0 represents the zero matrix, I represents the identity matrix, and U represents the maximum index of the virtual array.

9. The method according to claim 8, characterized in that Step 8 specifically includes: Covariance matrix for the expanded degrees of freedom Perform eigendecomposition and extract the noise subspace U N , construct 2D-MUSIC spectrum function: in, Represent the estimated angle and distance respectively, represents the continuous virtual array space-frequency joint steering vector.

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