A fast gridless direction of arrival estimation method for wideband linear frequency modulation signals
Patent Information
- Application Number
- CN202410102027.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-01-24
- Publication Date
- 2026-10-30
- Estimated Expiration
- 2044-01-24
AI Technical Summary
[0004]经过分数阶傅里叶变换后单个脉冲宽带线性调频信号的接收信号被转换为单个快拍形式，并且接收脉冲的数量是有限的，因此在分数阶域，接收信号可用的快照很少，这限制了基于子空间的算法的使用
[0048]在分数阶傅里叶变换域，线性调频信号表现为少快照特性，限制了子空间算法的性能，而已有的基于稀疏恢复的线性调频信号DOA估计方法计算复杂度高且受到离散字带来网格失配的影响，本发明利用分数阶傅里叶变换构建关于线性调频信号的导向向量时不变的阵列接受模型，并基于最小化原子范数构建稀疏恢复优化问题。该优化问题需要转化半正定规划问题求解，进一步针对半正定规划问题的求解效率，本发明基于快速内点法构建了快速优化框架，从而在实现更高分辨率和更高估计精度的同时，有效提升了算法的计算效率。
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of direction-of-arrival (DOA) estimation technology, and particularly relates to a fast gridless DOA estimation method suitable for broadband linear frequency modulated (LFM) signals. Background Technology
[0002] Direction-of-arrival (DOA) estimation is of great significance in the field of array signal processing. With the development of detection technology, linear frequency modulated (LFM) signals, due to their large time-width product, resistance to multipath propagation, and resistance to Doppler attenuation, have been widely used in sonar, radar, mobile communications, and other fields. Therefore, how to perform high-resolution ODA estimation for broadband LFM signals has become a research hotspot. Traditional broadband ODA methods are mainly divided into two categories: subspace methods and sparse recovery methods. Subspace methods, such as the incoherent signal subspace method and the coherent signal subspace method, estimate ODA based on the sampling covariance of the signal, which requires a high number of snapshots. To enhance the resolution of broadband signals with a finite number of snapshots, broadband ODA methods based on sparse recovery have been proposed. However, these algorithms are based on the idea of subband partitioning and estimate the ODA by combining the sparse recovery results of signals from each subband, leading to considerable computational complexity. At the same time, the inherent non-stationarity of LFM signals also poses a challenge to traditional broadband ODA methods.
[0003] In recent years, various direction-of-arrival (DOA) estimation methods based on time-frequency analysis tools have provided new approaches for DOA estimation of broadband non-stationary signals. Among them, the fractional Fourier transform (FFT) has attracted considerable attention due to its excellent time-frequency convergence properties, which decouple the array response vector and frequency variations while reducing interference from cross terms. For the DOA estimation problem of linear frequency modulated (LFM) signals, various algorithms based on the FFT have been proposed, such as the MUSIC algorithm based on the FFT (MUSIC-FRFT) and the sparse recovery algorithm using the L1 norm as a sparsity constraint (L1-FRFT).
[0004] After fractional Fourier transform, the received signal of a single-pulse broadband linear frequency modulated signal is converted into a single snapshot. Since the number of received pulses is finite, there are very few usable snapshots of the received signal in the fractional domain, which limits the use of subspace-based algorithms. Although L1-norm-based sparse recovery algorithms have been proposed to address this problem, they are still limited by computational complexity and mesh mismatch. Summary of the Invention
[0005] The purpose of this invention is to overcome the shortcomings of existing technologies and propose a fast, gridless direction-of-arrival estimation method suitable for broadband linear frequency modulated signals.
[0006] To achieve the above objectives, this invention proposes a fast, gridless direction-of-arrival estimation method suitable for broadband linear frequency modulated signals, comprising:
[0007] Step 1) Based on the fractional Fourier transform, convert the steering vector of the broadband linear frequency modulated signal about the uniform linear matrix into a time-invariant form;
[0008] Step 2) Combining the sparsity of the angle domain, we construct a sparse recovery optimization problem by using the continuous atomic norm of the dictionary as a sparsity constraint, and then transform the solution of the sparse recovery optimization problem into a semidefinite programming problem.
[0009] Step 3) Construct a fast optimization framework based on the fast interior point method to solve the semidefinite programming problem, thereby recovering the covariance matrix of fractional domain few snapshot data;
[0010] Step 4) Obtain the direction of arrival estimation results without mesh partitioning through van der Monte decomposition.
[0011] Preferably, the uniform linear array in step 1) comprises M array elements with an element spacing of d, and receives K signals from a direction θ. k The far-field broadband linear frequency modulated signal, where k = 1, 2, ..., K; the received signal x of the i-th array element. i (t) is:
[0012]
[0013] in, Let τ be the k-th wideband linear frequency modulated signal, where t represents the time domain. k,i = (i-1)dsinθ k / c represents the time delay of the i-th array element relative to the reference array element with respect to the k-th signal, f k and μ k Let n be the initial frequency and modulation frequency of the k-th wideband linear frequency modulated signal, c represent the signal propagation speed, and n be the frequency of the k-th signal. i (t) represents isotropic uniform white noise.
[0014] Preferably, step 1) includes:
[0015] Perform a discrete fractional Fourier transform on the received signal from the array, and the single pulse sk(t) is expressed as f k The sampling frequency yields a discrete signal s of length N. k (n), where n is the number of sampling points, L is the total number of received pulses, and the array receives the signal X in the fractional-order domain. F for:
[0016]
[0017] Among them, A FFor array manifold vector matrix, a f (θ k )=[1,b(k,2),...,b(k,i),...,b(k,M)] T Let T be the array manifold vector, and T denote the transpose. For the i-th element, b(k,i) with respect to the k-th signal is... and m k,1 =f k Nsinα k,1 / f s The coordinates of the reference array element when the received signal reaches its maximum value in the fractional-order domain are given. S represents the peak value matrix of the received signal in the fractional-order domain of the reference array element. k For the peak value of the k-th signal, For Gaussian white noise, the covariance matrix R0 of the signal can be written in the form T(u) using the Hermit Toplitz operator T:
[0018]
[0019] Among them, u=[u0,u1,…,u M-1 ], H represents conjugate transpose.
[0020] Preferably, step 2) includes:
[0021] Introducing the atomic norm as a sparsity constraint, we define the atomic set. The atomic norm of the array in the fractional-order domain. for:
[0022]
[0023] in, c k =||S k ||2, A(θ k ,ψ k ) = a f (θ k )ψ k ,
[0024] Solving the sparse recovery problem based on minimizing the atomic norm is transformed into solving the following positive definite programming problem:
[0025]
[0026]
[0027]
[0028] Where φ is the index of some array elements. For partial observations, X is the received signal of the array element in the fractional-order domain φ. F X is the variable to be restored. Fφ Let φ be the variable to be recovered for the array element in φ. Considering the existence of observation noise, the error between the recovered variable and the observation is controlled by η. W and u are variables to be solved.
[0029] Preferably, step 3) includes:
[0030] The iteration termination condition is set, and the following optimization problem is obtained iteratively. Numerical solution:
[0031]
[0032]
[0033] Let the solution of the (i-1)th iteration be Then the solution for the i-th iteration is Where p i t represents the search direction, β represents the step size, and t represents the search direction. i Let be the path parameters for the i-th iteration. It is a positive semi-definite Hermitian-Toeplitz cone.
[0034] The solutions for the other original variables are obtained from the following formula:
[0035]
[0036]
[0037] in, W i The path parameter t i X below F The solution for W. Let h = T be the Hermitian Toplitz matrix. * (I), where I is the identity matrix, and the elements of matrix Γ are chosen as follows: x and j are the row and column indices of Γ, respectively, φ x Let x be the x-th element in φ.
[0038] Preferably, the iteration termination condition is determined by the duality gap ξ. i and relative dual gap ξ i / f(μ i )Sure, That is, ξ i <ε abs ξ i / f(μ i )<ε rel , where ε abs , ε rel Two set end thresholds;
[0039] The iterative update criterion is: Where σ represents the number of hyperparameters for updating the center path t, and the dual gap ξ i =f(μ i )-f LB f LB Update according to the following formula:
[0040] V i =τI
[0041] s i =τh
[0042]
[0043] Among them, V i s i , The original problem variables are W and X, respectively. F , The dual variables V, Y, s in hyperparameter t i The following solution, if Established in Represents the dual cone, for The dual cone.
[0044] Preferably, step 4) includes:
[0045] According to Vandermont's decomposition, Written as Where a(f′) k p is the array manifold vector obtained from the decomposition. k These are the coefficient values obtained from the decomposition;
[0046] With a f (θ k The frequency values were obtained through comparison. Where k = 1, ... K, the directions of arrival of K incoming wave signals can be obtained.
[0047] Compared with the prior art, the advantages of the present invention are:
[0048] In the fractional Fourier transform domain, linear frequency modulated (LFM) signals exhibit a rare snapshot characteristic, limiting the performance of subspace algorithms. Existing LFM signal DOA estimation methods based on sparse recovery suffer from high computational complexity and are affected by grid mismatch caused by discrete words. This invention utilizes the fractional Fourier transform to construct a time-invariant array receiving model for the steering vector of the LFM signal, and constructs a sparse recovery optimization problem based on minimizing the atomic norm. This optimization problem requires transformation into a positive semidefinite programming problem. To further address the efficiency of solving positive semidefinite programming problems, this invention constructs a fast optimization framework based on the fast interior-point method, thereby effectively improving the computational efficiency of the algorithm while achieving higher resolution and higher estimation accuracy. Attached Figure Description
[0049] Figure 1 This is a flowchart of the fast gridless direction-of-arrival estimation method for broadband linear frequency modulated signals according to the present invention;
[0050] Figure 2 These are the normalized spatial spectra of four existing algorithms and the method of this invention;
[0051] Figure 3 The root mean square errors of four existing algorithms and the method of this invention under different grids;
[0052] Figure 4 The computation time of four existing algorithms and the method of this invention is shown under different grid conditions. Detailed Implementation
[0053] This invention proposes a fast meshless direction-of-arrival (DOA) estimation method suitable for broadband linear frequency modulated (LFM) signals. This method utilizes fractional Fourier transform and atomic norm to model the DOA estimation problem of LFM signals as a sparse recovery problem in the fractional domain. Furthermore, this problem is transformed into a semidefinite programming problem, which is solved using a fast optimization framework based on the fast interior-point method. This recovers the covariance matrix of fractional domain data with few snapshots, and the DOA estimation result under meshless partitioning is obtained through VanderMont decomposition.
[0054] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.
[0055] Example
[0056] Embodiment 1 of this invention proposes a fast, meshless direction-of-arrival (DOA) estimation method suitable for broadband linear frequency modulated (LFM) signals. Based on fractional Fourier transform, the steering vector of the LFM signal with respect to a uniform linear matrix is converted into a time-invariant form. Taking advantage of the sparse nature of the angle domain, a sparse recovery optimization problem is constructed using the continuous atomic norm as a sparsity constraint. The solution to this optimization problem can be obtained by transforming the atomic norm minimization problem into a positive semidefinite programming problem and solving it using an optimization framework constructed based on the fast interior-point method. This invention recovers the signal covariance matrix from finite snapshot data in the fractional domain and obtains the DOA estimation result in a meshless manner through VanderMonte decomposition.
[0057] 1. Problem Description
[0058] Assume a uniform linear array with M elements receives K signals from direction θ. k For a far-field broadband linear frequency modulated signal, k = 1, 2, ..., K, with an array element spacing of d, the received signal of the i-th array element can be written as:
[0059]
[0060] in, For the k-th wideband linear frequency modulated signal, τ k,i = (i-1)dsinθ k / c represents the time delay of the i-th array element relative to the reference array element with respect to the k-th signal, f k and μ k Let n be the initial frequency and modulation frequency of the k-th wideband linear frequency modulated signal, c represent the signal propagation speed, and n be the frequency of the k-th signal. i (t) represents isotropic uniform white noise.
[0061] 2. Design of DOA estimation method
[0062] (1) Received signal model in the fractional domain
[0063] First, a discrete fractional Fourier transform is performed on the received signal from the array, and a single pulse s k (t) with f k The sampling frequency yields a discrete signal s of length N. k (n), where L is the total number of received pulses, the received signal of the fractional-order array can be written as:
[0064]
[0065] in, a f (θ k b(k,2), ..., b(k,M)] T For the i-th array element, b(k,i) with respect to the k-th signal is and m k,1 =f k Nsinα k,1 / f s The coordinates of the reference array element when the received signal reaches its maximum value in the fractional-order domain are given. The peak matrix representing the received signal of the reference array element in the fractional-order domain, with respect to the peak value S of the k-th signal. k for: It is still Gaussian white noise. The covariance matrix of the received signal can be written as: R0 is the signal covariance matrix, which has a Hermitian Toplitz structure and can be written in the form T(u): where u=[u0,u1,…,u M-1 Q is the noise covariance matrix.
[0066] (2) Establishing a sparse recovery problem based on atomic norm minimization
[0067] Furthermore, by introducing the atomic norm as a sparsity constraint, the received signal of the array in the fractional-order domain can be written as: c k =||S k ||2, A(θ k ,ψ k ) = a f (θ k )ψ k The atomic norm of this signal can be written as:
[0068]
[0069] Solving the sparse recovery problem based on atomic norm minimization can be transformed into solving the following semidefinite programming problem.
[0070]
[0071] Where φ is the index of some array elements. For partial observations, X is the received signal of the array element in the fractional-order domain φ. F X is the variable to be restored. Fφ φ is the variable to be recovered for the array elements, and W and u are variables to be solved. Equation (4) can be solved using the convex optimization toolbox, but this method has high computational complexity.
[0072] (3) Fast optimization framework based on fast interior point method
[0073] This invention uses a fast optimization framework based on the fast interior-point method to solve this semidefinite programming problem. First, it defines... Let u be a real-valued variable that satisfies the following mapping relationship: And define about The Hermitt-Toplitz operator is Its Hermitian Toplitz matrix First action The definition of the adjoint Hermitian Toplitz operator is also given for matrices. Its accompanying Hermitian Toplitz matrix is:
[0074]
[0075] Further, a primal-dual model is constructed, and a logarithmic determinant barrier function is introduced into the primal problem to ensure that the optimization variables fall within the feasible region during iterative optimization. The primal problem variables are then defined. Its dual variable is γ=(V,Y,s), and the model can be written as:
[0076]
[0077] Where t represents the path parameter. τ is a hyperparameter that needs to be set manually. This parameter plays a role in balancing the sparsity of the optimized variables and the effect of signal denoising.
[0078]
[0079]
[0080]
[0081] The KKT conditions for this optimization model can be written as:
[0082]
[0083] Where, μ t and γ t This represents the optimal solution for the original and dual variables under parameter t. According to the first condition of formula (6), we can obtain:
[0084]
[0085] According to the fourth condition of formula (6), we can obtain:
[0086]
[0087] in, W t , X under path parameter t F W The solution, Let h = T be the Hermitian Toplitz matrix.* (I) Select the elements of matrix Γ as x and j are the row and column indices of Γ, respectively, φ x Let x be the x-th element in φ, and I be the identity matrix; observing formula (8), we can find that... The analytical solution is difficult to obtain, but it can be obtained through the following optimization problem iteration. Numerical solution:
[0088]
[0089] in, A semidefinite Hermitian Toeplitz cone This optimization problem is a monotonically convex function, which can be solved iteratively using line search and gradient methods. Assume the solution for the (i-1)th iteration is... The solution for the i-th iteration can be written as Where the search direction p i It can be obtained through the finite memory BFGS method, and the step size β can be obtained through... We obtain that c is the parameter that needs to be set. According to formulas (7) and (8), the original variable... and dual variable γ i =(V i ,s i ,Y i ) can be accessed The iteration yields the result. According to Schuler's complement theorem, the second condition of formula (6) is satisfied upon initialization of the original variables. It can be guaranteed that the third condition can be determined by the properties of finite-length autocorrelation sequences.
[0090] During the iteration process, the parameter t needs to be updated, and the update criterion is as follows: Where the dual gap ξ i =f(μ i )-f LB If the third condition of the KKT conditions is true, then update. The iteration termination condition can be determined by the dual gap and the relative dual gap, i.e., ξ i <ε abs ξ i / f(μ i )<ε rel ε abs , ε rel To set the end threshold.
[0091] In the overall optimization framework, the computational complexity of inverting the Hermit Toplitz matrix can be reduced using the Levinson-Durbin algorithm, and the computational complexity of multiplying the Toplitz matrix with a vector can be further reduced using FFT. For inverting the non-Hermit Toplitz matrix in the first equation of formula (8), the computational complexity can be reduced using the conjugate gradient method.
[0092] (4) Van der Monte decomposition
[0093] Once obtained, the frequency value can be derived through VanderMont decomposition. Thus, the directions of arrival of K incoming wave signals can be obtained.
[0094] 3. Performance Analysis:
[0095] Monte Carlo simulation was used to analyze the direction-of-arrival estimation performance of the ANM-FastIPM-FRFT proposed in this invention, and it was compared with CBF-FRFT based on conventional beamforming, existing algorithms MUSIC-FRFT and L1-FRFT, and ANM-FRFT, which uses a convex optimization toolbox to solve semidefinite programming problems. To ensure the reliability of the simulation results, the number of simulations was set to 100. The following simulation scenario was considered: multiple targets exist within the observation angle range; the center frequency of the linear frequency modulated (LFM) signal is 20 kHz, the bandwidth is 2 kHz, and the pulse width is 100 ms; a uniform linear array with 10 elements and an element spacing of 0.03 m is used to receive the LFM signal with a pulse width of 1.
[0096] Figure 2 Normalized spatial spectra of existing CBF-FRFT, MUSIC-FRFT, L1-FRFT, ANM-FRFT algorithms and the ANM-FastIPM-FRFT method of this invention are presented at a signal-to-noise ratio of 10dB and an arrival direction of [-20.5°, -10.5°, 20°]. Figure 3 and Figure 4 The root mean square error and computation time of four existing algorithms and the proposed method under different grid conditions are presented for a signal-to-noise ratio of 10 dB and an incoming wave direction of [-20.2°, -9.2°]. It can be seen that the resolution of the proposed method is significantly higher than that of existing algorithms, and the estimation accuracy of the proposed algorithm is not affected by grid division. Under fine grid division, the computational efficiency is significantly improved compared to the L1-FRFT and ANM-FRFT algorithms.
[0097] Innovation points:
[0098] 1) This invention establishes an array receiving model for the time-invariant steering vector of a broadband linear frequency modulated signal in the fractional-order domain;
[0099] 2) This invention introduces the atomic norm as a sparsity constraint, constructs a sparse recovery model in the fractional domain, and transforms it into a semidefinite programming problem for solution. By estimating the direction of arrival of the linear frequency modulated signal in a meshless manner, the resolution and estimation accuracy are significantly improved.
[0100] 3) This invention constructs a fast optimization framework based on the fast interior point method, and solves the semidefinite programming problem according to this optimization framework, thereby improving the computational efficiency of sparse recovery optimization solution;
[0101] 4) This invention proposes a new fast gridless direction-of-arrival estimation method suitable for broadband linear frequency modulated signals, which effectively improves the computational efficiency of the algorithm while achieving higher resolution and higher estimation accuracy.
[0102] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.
Claims
1. A fast, gridless direction-of-arrival estimation method suitable for broadband linear frequency modulated signals, comprising: Step 1) Based on the fractional Fourier transform, convert the steering vector of the broadband linear frequency modulated signal about the uniform linear matrix into a time-invariant form; Step 2) Combining the sparsity of the angle domain, we construct a sparse recovery optimization problem by using the continuous atomic norm of the dictionary as a sparsity constraint, and transform the solution of the sparse recovery optimization problem into a semidefinite programming problem. Step 3) Construct a fast optimization framework based on the fast interior point method to solve the semidefinite programming problem, thereby recovering the covariance matrix of fractional domain few snapshot data; Step 4) Obtain the direction-of-arrival estimation results without mesh partitioning through van der Monte gravimetric decomposition; Step 2) includes: Introducing the atomic norm as a sparsity constraint, we define the atomic set. , representing the atomic norm of the received signal in the fractional-order domain. for: in, , , , , For array manifold vectors, This refers to the uniform linear array receiving in step 1). One from the direction is The far-field broadband linear frequency modulated signal, in which ; For the reason about the The peak value of the signal; Solving the sparse recovery problem based on minimizing the atomic norm is transformed into solving the following positive definite programming problem: in, This is an index for some array elements. For partial observations, it is a fractional domain. The signal received by the central array element For variables to be restored, for The variables to be recovered for the array elements are determined by considering the presence of observation noise. Controlling the error between the recovered variable and the observation, and All are variables to be solved; The covariance matrix of the signal via the Hermitt-Pulitz operator The form in which it is written; Step 3) includes: The iteration termination condition is set, and the following optimization problem is obtained iteratively. Numerical solution: Setting the first i The solution for the -1st iteration is Then the first i The solution of the next iteration is ,in For the search direction, Step size, For the first i Path parameters for the next iteration It is a positive semi-definite Hermitian-Toeplitz cone. , ; The solutions for the other original variables are obtained from the following formula: in, , Path parameters Below , The solution, , , For Hermitian Toplitz matrix, , Choose the matrix as the identity matrix. The elements are , x and j are respectively row and column indexes, for The Middle x Each element.
2. The fast gridless direction-of-arrival estimation method for broadband linear frequency modulated signals according to claim 1, characterized in that, The uniform linear array in step 1) comprises M array elements, with an element spacing of [missing information]. d ,take over One from the direction is The far-field broadband linear frequency modulated signal, in which ;No. The received signal of each array element for: in, For the first A broadband linear frequency modulated signal. t Representing the time domain, For the first Each array element relative to the reference array element regarding the first... The time delay of each signal and For the first The initial frequency and modulation frequency of a broadband linear frequency modulated signal. Represents the speed of signal propagation. It is an isotropic, uniform white noise.
3. The fast gridless direction-of-arrival estimation method for broadband linear frequency modulated signals according to claim 2, characterized in that, Step 1) includes: Perform a discrete fractional Fourier transform on the received signal from the array, and obtain a single pulse. by The sampling frequency yields a length of discrete signals , n Where L is the number of sampling points and L is the total number of received pulses, the array receives signals in the fractional-order domain. for: in, For array manifold vector matrix, , Let T be the array manifold vector, and T denote the transpose. For the th Regarding the first array element a signal for , and The coordinates of the reference array element when the received signal reaches its maximum value in the fractional-order domain are given. The peak value matrix representing the received signal of the reference array element in the fractional-order domain. For the reason about the The peak value of the signal. , Given Gaussian white noise, the covariance matrix of the signal is... via the Hermitt-Pulitz operator Written Format: in, H represents the conjugate transpose.
4. The fast gridless direction-of-arrival estimation method for broadband linear frequency modulated signals according to claim 3, characterized in that, The iteration termination condition is achieved through the duality gap. and relative duality gap Sure, ,Right now , ,in, , Two set end thresholds; The iterative update criterion is: ,in, Indicates updating the central path hyperparameters, dual gap , Update according to the following formula: in, , , These are the original problem variables. dual variables In hyperparameters The following solution, if Established ,in Represents the dual cone, for The dual cone.
5. The fast gridless direction-of-arrival estimation method for broadband linear frequency modulated signals according to claim 4, characterized in that, Step 4) includes: According to Vandermont's decomposition, Written as ,in The array manifold vectors obtained from the decomposition. These are the coefficient values obtained from the decomposition; and The frequency values were obtained through comparison. ,in Thus, the directions of arrival of K incoming wave signals can be obtained.