Non-Gaussian signal DOA estimation method based on three-level nested array vectorization DFT-Taylor algorithm
Through the vectorized DFT-Taylor algorithm of three-level nested arrays, the sparse array model of the fourth-order cumulative matrix is used to solve the problem of the degradation of the angle estimation performance of sparse arrays in non-Gaussian signal DOA estimation, and high efficiency and low complexity Angle estimation, suitable for large-scale arrays.
Patent Information
- Application Number
- CN202411424032.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-12
- Publication Date
- 2025-01-03
AI Technical Summary
In the non-Gaussian signal DOA estimation, the method based on the second-order cumulative amount ignores the virtual array elements, resulting in a degradation of the angular estimation performance, and lacks the design of physical sparse arrays based on the fourth-order cumulative amount.
A vectorized DFT-Taylor algorithm based on three-level nested arrays is proposed. By constructing a sparse array model with three-layer nested relationships, calculating the fourth-order cumulative quantity matrix of the received signal, vectorizing and sorting, and obtaining continuous virtual array elements , combined with DFT and Taylor expansion, achieve efficient angle estimation.
It improves the accuracy and freedom of angle estimation, reduces the computational complexity, is suitable for large-scale arrays, avoids the loss of freedom caused by spatial smoothing, and has better performance than traditional array structures.
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Abstract
Description
Technical Field
[0001] The invention relates to the technical field of array signal processing, in particular to a non-Gaussian signal DOA estimation method based on a three-level nested array vectorized DFT-Taylor algorithm. Background Art
[0002] Direction of Arrival (DOA) estimation is one of the basic problems in array signal processing and plays an important role in various fields such as radar systems, acoustics, navigation and wireless communications.
[0003] In the related research of DOA estimation, it is generally assumed that the signal source is a Gaussian signal. The second-order cumulant of the signal can be used to obtain the probability density function containing all the information in the signal source. However, in practical applications, non-Gaussian signals are usually encountered. Non-Gaussian signals are defined as the general term for signals that do not satisfy the Gaussian distribution. Their first-order and second-order statistics cannot fully describe the statistical characteristics of the signal. In this case, the use of high-order cumulants can not only obtain higher performance than second-order statistics, but also solve many problems that second-order statistics cannot solve. In a symmetric random process, the third-order cumulant of the signal is zero. Therefore, in the parameter estimation for non-Gaussian signals, the fourth-order cumulant of the signal is usually processed. Compared with the second-order cumulant, the fourth-order cumulant is not only blind to Gaussian noise but also can expand the array elements. Therefore, when the fourth-order cumulant is used for parameter estimation, a more accurate received signal matrix can be obtained, thereby obtaining a more accurate estimation result.
[0004] Compared with the traditional uniform linear array (ULA), the sparse array can obtain less mutual coupling and higher degrees of freedom (DOF). Due to its superior array performance, DOA estimation methods based on sparse arrays have been continuously proposed. This type of method can be roughly divided into two types, one is the deambiguation method and the other is the vectorization (virtualization) method. The former method is to obtain the estimated value by using the classical DOA estimation algorithm for the two sub-arrays respectively, and then use the corresponding array properties of the sparse array to compare and remove the ambiguous angle estimation value to complete the DOA estimation. The advantage of this method is that the principle is simple, and the angle estimation value can be obtained directly based on the sub-array using the classical DOA estimation method, but the operation of processing the sub-array separately will reduce the degree of freedom. The latter method is to perform vectorization operation on the covariance matrix of the received signal, and then use the reconstructed covariance matrix to realize DOA estimation. Since the vectorized matrix is a non-full-rank single-snapshot matrix, its subsequent processing can be restored to full rank through the spatial smoothing (SS) process and then the angle estimation value can be obtained by using the classic DOA estimation method. Typical representatives are SS-MUSIC algorithm, SS-ESPRIT algorithm and SS-PM algorithm, etc., or the vectorized covariance matrix of the single snapshot can be directly processed to obtain the DOA estimation value. Typical representatives are discrete Fourier transform (DFT), compressed sensing (CS) and other algorithms. The vectorized DFT algorithm involves two angle estimates. The initial estimation value can be obtained by searching the spectrum peak of the received signal vector after discrete Fourier transform, and the precise estimation result can be obtained based on the phase rotation compensation error. The algorithm can be directly used for single-snapshot virtual signals, without the need for decorrelation process, which improves the degree of freedom.
[0005] The current array structure design for sparse arrays focuses on the second-order cumulants, which have excellent performance when performing DOA estimation based on the second-order cumulants. However, when applied to vectorized DOA estimation for non-Gaussian signals, the fourth-order difference joint array of the physical array is obtained by performing another difference operation on the basis of the second-order difference joint array of the physical array. This processing method will ignore the virtual array elements in the second-order difference joint array, thereby generating more redundant array elements, which ultimately leads to a decrease in angle estimation performance. Therefore, when performing angle estimation for non-Gaussian signals, it is very necessary to design a physical sparse array based on the fourth-order cumulant. Summary of the invention
[0006] The object of the present invention is to provide a non-Gaussian signal DOA estimation method based on a three-level nested array vectorized DFT-Taylor algorithm with low computational complexity and reliable results.
[0007] The technical solution to achieve the purpose of the present invention is: a non-Gaussian signal DOA estimation method based on a three-level nested array vectorized DFT-Taylor algorithm, comprising the following steps:
[0008] Step 1: Establish a DOA estimation model for non-Gaussian signals;
[0009] Step 2: construct a sparse array model with three-layer nested relationships, which is a three-level nested array;
[0010] Step 3: Calculate the fourth-order cumulant matrix C of the received signal x(t) 4,x , and for C 4,x Perform vectorization, sorting and remove redundancy to obtain vector z;
[0011] Step 4: intercept vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K;
[0012] Step 5: construct a phase rotation matrix Φ(η);
[0013] Step 6: Search the phase offset in the range (-π / T, π / T) to obtain the optimal phase offset η k , get the angle estimate after phase rotation DFT precise estimation
[0014] Step 7: Perform error compensation based on Taylor expansion and combine the angle estimate with DFT precision estimation and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k .
[0015] Furthermore, the DOA estimation model for non-Gaussian signals is established as described in step 1, as follows:
[0016] First set (·) T 、(·) H 、(·) -1 and(·) * They represent transpose, conjugate transpose, inversion, and conjugate operations respectively; bold capital letters represent matrices, and bold lowercase letters represent vectors; the symbol cum(·) represents fourth-order cumulant calculation, and E(·) represents expectation calculation. represents the Kronecker product, vec(·) represents matrix vectorization, and angle(·) represents the phase angle of a complex number;
[0017] Assume that there are K narrowband far-field incoherent signals in space, and the one-dimensional wave arrival direction is θ k , k=1,2,...,K, the signal is incident on a linear array consisting of P elements, and the position set is V={v p d,p=1,2,...,P}, assuming that the received signal model of the array is expressed as:
[0018] x(t)=A(θ)s(t)+n(t)
[0019] Where x(t) is the signal receiving matrix of P elements in the array under L snapshots, 1≤t≤L, A(θ)=[a(θ1),a(θ2),...,a(θ K )]∈C P×K Represents the direction matrix, direction vector s(t)=[s1(t),s2(t),...,s K (t)] T represents a non-Gaussian signal source with a mean of 0; n(t) = [n1(t), n2(t), ..., n P (t)] T It is represented by a mean of 0 and a variance of s. 2 Additive Gaussian white noise; λ represents half wavelength, d represents the array element spacing;
[0020] For a zero-mean random process, given a random variable x k1 ,x k2 , For integers 1≤k1,k2,k3,k4≤P, the fourth-order cumulant is expressed as:
[0021]
[0022] The fourth-order signal accumulation for the entire hypothetical array received signal is expressed as:
[0023]
[0024] After calculating the fourth-order cumulant of the received signal, the corresponding element position of the fourth-order difference virtual array is expressed as:
[0025] Γ 4-dc ={(v p1 -v p2 )-(v p3 -v p4 )}={(v p1 +v p4 )-(v p2 +v p3 )}
[0026]
[0027] Among them, the integer value is 1≤p1,p2,p3,p4≤P; according to the above formula, the fourth-order differential virtual array of the physical array can be obtained by adding and subtracting the position numbers of any two array elements, that is, the second-order sum-difference joint array and the fourth-order differential virtual array can be transformed into each other.
[0028] Furthermore, the sparse array model with three-layer nested relationship constructed in step 2 is a three-level nested array, which is specifically as follows:
[0029] The nested array is composed of two linear arrays, and the array elements are distributed as follows:
[0030] V NA ={n1d,0≤n1≤N1-1}∪{[n2(N1+1)-1]d,1≤n2≤N2},n1∈Z,n2∈Z
[0031] The array element spacing is d = λ / 2, which is numerically equal to the half wavelength of the incident signal. The second-order differential virtual array corresponding to the array is a continuous array, and the array element position set is
[0032] The physical sensor locations of the three-level nested array THL-NA are as follows:
[0033] V THL-NA =V1∪V2∪V3
[0034] V1={n1d,1≤n1≤N1},n1∈Z
[0035] V2={n2(N1+1)d,1≤n2≤N2},n2∈Z
[0036] V3={n3(N2+1)(N1+1)d,1≤n3≤N3},n3∈Z
[0037] The array has a total of N = N1 + N2 + N3 array elements;
[0038] First, the second-order and virtual array element positions of the array are obtained, ranging from 2d to 2N3(N2+1)(N1+1)d, and the number of array element positions is not continuous;
[0039] Divide the second-order matrix into several parts, and express the positions of the array elements of each part in the form of a set:
[0040]
[0041] S1={n1d,2≤n1≤(N1+1)(N2+1)}
[0042] S2=(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1}
[0043] S3=2(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1}
[0044]
[0045]
[0046] The second-order sum matrix of the three-level nested matrix is divided into N3+1 parts, where the set S1 is continuous and the set is a nested matrix with sub-matrices (N1, N2), and the set is a three-level nested array with sub-arrays (N1, N2, N3) respectively; according to the continuous characteristic of the difference matrix of the nested array, the element position range of the second-order sum and difference joint array of the array is obtained, that is, the second-order sum and difference joint array of the three-level nested array is a continuous array, and the position set of the virtual array is
[0047]
[0048] From this, the closed-form solution of the continuous degrees of freedom of the three-level nested matrix is deduced as
[0049] cDOF=2[2N3(N2+1)(N1+1)-2]+1=4N3(N2+1)(N1+1)-3
[0050] Let M b =2N3(N2+1)(N1+1)-2, that is, the number of array element positions of the virtual array is in the interval [-M b ,M b ] is continuous, and the number of continuous array elements is T = 2M b +1.
[0051] Furthermore, the fourth-order cumulant matrix C of the received signal x(t) is calculated in step 3 4,x , and for C 4,x Vectorize, sort and remove redundancy to obtain vector z, as follows:
[0052] 1) First, solve the fourth-order cumulant matrix C of the received signal 4,x , and vectorize it:
[0053]
[0054] in
[0055]
[0056] B(θ) is the array manifold expanded using the fourth-order cumulant b(θ k ) is a(θ k ) The expanded array manifold; the fourth-order cumulant matrix C is vectorized 4,x To process:
[0057] z vec =vec(C 4,x )=vec[B(θ)C s B H (θ)]=Λ(θ)p
[0058] in
[0059]
[0060] and represents the fourth-order cumulant of the k-th source vector, where represents the fourth-order cumulant of the kth source;
[0061] Vectorized C 4,x The corresponding direction vector matrix Λ(θ) is expanded from the two-level differential array A(θ). According to the closed-form solution of continuous degrees of freedom derived in step 2, the set of element positions of the fourth-order differential array is expressed as
[0062]
[0063] The array elements of the fourth-order virtual array within the array element position set of the fourth-order differential array are completely continuous;
[0064] 2) For z vec Sorting and removing redundancy yields vector z:
[0065] according to The position of the fourth-order virtual array element is z vec Sort and remove redundant elements to construct a new vector z
[0066] z=Λ1(θ)p
[0067] Among them, Λ1(θ) is the matrix of Λ(θ) after removing redundancy and sorting; the direction matrix Λ1(θ) of the difference array is
[0068]
[0069] and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.
[0070] Furthermore, in step 4, the vector z is intercepted and the range of continuous virtual array elements is obtained as [-Mb ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT The details are as follows:
[0071] The normalized DFT matrix is defined as
[0072]
[0073] Where, T = 2M b +1 indicates the element range [-M b ,M b ], the total number of elements of the continuous fourth-order virtual array within the matrix F with coordinates (p, q) is
[0074] The kth non-Gaussian source corresponds to the continuous range [-M b ,M b The direction vector of ] is expressed as a(θ k )∈C T×1 , multiply the F matrix on the left, and rewrite the direction vector to get
[0075]
[0076] The qth element and 1≤q≤T is represented by
[0077]
[0078] When Tsinθ k / 2=q k When is an integer, There is only one qth=q k elements are not 0, and the rest are 0;
[0079] Source angle θ k Initial estimate of pass The positions of the largest K elements in are determined. Assuming that the vectorized received signal after DFT transformation is
[0080] y ini =Fz1
[0081] The horizontal coordinates corresponding to the K largest peaks of the spatial spectrum are marked as The initial estimate of DOA is
[0082]
[0083] Furthermore, the phase rotation matrix Φ(η) is constructed as follows in step 5:
[0084] The phase rotation matrix is defined as
[0085]
[0086] Wherein, η∈(-π / T,π / T) is the phase offset, and T is the total number of virtual array elements in the virtual array.
[0087] Furthermore, in step 6, the phase offset is searched in the range (-π / T, π / T) to obtain the optimal phase offset η k , get the angle estimate after phase rotation DFT precise estimation The details are as follows:
[0088] Direction vector The phase shift is superimposed
[0089]
[0090] The qth element is represented by
[0091]
[0092] Rewrite
[0093]
[0094] Among them, 1≤q≤T; when Tsinθ k When / 2 is an integer, the phase shift η is 0; when Tsinθ k When / 2 is not an integer, the value of η is searched in (-π / T,π / T) so that Tsinθ k / 2+Tη k / (2π) is an integer;
[0095] Phase shift η k Obtained by searching in (-π / T,π / T)
[0096]
[0097] in, is the matrix F The computational complexity of this part is O(GKT), and G is the number of quantized grids in the region [-π / T,π / T];
[0098] The precise estimate after phase rotation is
[0099]
[0100] Furthermore, the error compensation is performed based on Taylor expansion in step 7, and the angle estimation value accurately estimated by DFT is combined with the error compensation based on Taylor expansion. and error compensation δ k, calculate the final angle estimate θ of the DFT-Taylor algorithm k , as follows:
[0101] The direction vector a(θ k )exist Expand, yes
[0102]
[0103] in,
[0104] Ignoring the second-order derivative and above, and ignoring the influence of noise, we have
[0105]
[0106] Among them, the direction vector Error matrix Δ = diag(δ1,δ2,...,δ K ), w = Δp;
[0107] Perform matrix transformation on the formula to get
[0108]
[0109] By performing variant derivation on the formula, we can obtain
[0110] Δ=w· / p
[0111] The final DOA estimate is
[0112]
[0113] A non-Gaussian signal DOA estimation system based on a three-level nested array vectorized DFT-Taylor algorithm, the system is used to implement the non-Gaussian signal DOA estimation method based on the three-level nested array vectorized DFT-Taylor algorithm, the system includes a DOA estimation model construction module, a sparse array model construction module, a vector z calculation module, a DFT initial estimation value calculation module, a phase rotation matrix construction module, a DFT precise estimation angle estimation value calculation module, and a final angle estimation value calculation module, wherein:
[0114] DOA estimation model building module, used to establish a DOA estimation model for non-Gaussian signals;
[0115] The sparse array model building module is used to construct a sparse array model with three-layer nesting relationship, which is a three-level nested array;
[0116] Vector z calculation module, used to calculate the fourth-order cumulant matrix C of the received signal x(t)4,x , and for C 4,x Perform vectorization, sorting and remove redundancy to obtain vector z;
[0117] The DFT initial estimate calculation module is used to intercept the vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K;
[0118] A phase rotation matrix building module, used to construct a phase rotation matrix Φ(η);
[0119] The angle estimation value calculation module of DFT precise estimation searches for the optimal phase offset η in the range (-π / T, π / T) k , get the angle estimate after phase rotation DFT precise estimation
[0120] The final angle estimation calculation module performs error compensation based on Taylor expansion and combines the angle estimation value accurately estimated by DFT and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k .
[0121] A mobile terminal comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the program, the non-Gaussian signal DOA estimation method based on a three-level nested array vectorized DFT-Taylor algorithm is implemented.
[0122] Compared with the prior art inventions, the present invention has the following significant advantages:
[0123] (1) The array of the present invention utilizes the continuous characteristics of nested array difference arrays. The proposed array can derive a provably completely continuous fourth-order virtual array element position, that is, the array has a closed-form solution, and the array has strong scalability and usability;
[0124] (2) The array THL-NA designed by the present invention has a large degree of freedom, and compared with the newly proposed sparse array SAFE-CPA, the traditional two-level nested array TL-NA, the four-level nested array FL-NA, and the traditional augmented coprime array ACA, it has greater continuous degrees of freedom;
[0125] (3) When the array THL-NA designed by the present invention is compared with the above-mentioned four array structures SAFE-CPA, TL-NA, FL-NA and ACA with the same number of array elements and other conditions, the angle estimation performance of the proposed array is better than that of the other array structures in terms of root mean square error performance;
[0126] (4) Expanding the virtual array aperture by combining sparse array with vectorization;
[0127] (5) It is applicable to DOA estimation for large-scale arrays with a large number of array elements or virtual array elements. The algorithm can be directly applied to the virtual signal of a single snapshot without the need for spatial smoothing, thus avoiding the loss of degrees of freedom caused by spatial smoothing and improving the spatial degrees of freedom of the array.
[0128] Method (6) uses DFT changes to obtain an initial estimate, then completes the DFT precise estimate with fewer grid searches, and finally uses the Taylor expansion algorithm to compensate for the error in the phase-rotated DFT precise estimate to obtain the final angle estimate. The above three steps all have low computational complexity. BRIEF DESCRIPTION OF THE DRAWINGS
[0129] Figure 1 It is a schematic diagram of the three-level nested array structure.
[0130] Figure 2 It is the physical array of the three-level nested array, the second-order sum array, and the fourth-order difference array element position diagram.
[0131] Figure 3 This is a schematic diagram of the positions and weights of the second-order virtual array elements of the three-level nested array.
[0132] Figure 4 It is a schematic diagram of the positions and weights of the fourth-order virtual array elements of the three-level nested array.
[0133] Figure 5 There is no leakage peak in the direct DFT algorithm (θ k =arcsin(60 / 253)) graph.
[0134] Figure 6 There is a leaked peak in the direct DFT algorithm (θ k =60°) figure.
[0135] Figure 7 This is the estimation result diagram of the vectorized DFT-Taylor algorithm.
[0136] Figure 8 It is a comparison chart of the estimation performance of the vectorized DFT-Taylor algorithm under different snapshot numbers.
[0137] Fig. 9This is a comparison chart of the computational complexity of different algorithms under different numbers of snapshots. DETAILED DESCRIPTION
[0138] The present invention provides a vectorized DFT-Taylor algorithm for non-Gaussian signals based on a three-level nested array (THL-NA), which is characterized by specifically comprising the following steps:
[0139] Step 1: Establish a DOA estimation model for non-Gaussian signals;
[0140] Step 2: construct a sparse array model with three-layer nested relationships, which is a three-level nested array;
[0141] Step 3: Calculate the fourth-order cumulant matrix C of the received signal x(t) 4,x , use the formula to vectorize it, sort it and remove the redundancy to obtain the vector z;
[0142] Step 4: intercept vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K;
[0143] Step 5: construct a phase rotation matrix Φ(η);
[0144] Step 6: Search the phase offset in the range (-π / T, π / T) to obtain the optimal phase offset η k , get the angle estimate after phase rotation DFT precise estimation And k=1,2,...,K.
[0145] Step 7: Error compensation based on Taylor expansion, combined with DFT precise estimation and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k , and k=1,2,...,K.
[0146] Symbols: In the present invention, (·) T ,(·) H (·) -1 and(·) * They are transpose, conjugate transpose, inversion and conjugate operations respectively. Bold capital letters represent matrices, bold lowercase letters represent vectors, the symbol cum(·) represents fourth-order cumulant calculation, E(·) represents expectation calculation, represents the Kronecker product, vec(·) represents matrix vectorization, and angle(·) represents the phase angle of a complex number.
[0147] The specific implementation of the present invention is as follows:
[0148] Step 1: Establish a DOA estimation model for non-Gaussian signals:
[0149] Assume that there are K narrowband far-field incoherent signals in space, and their one-dimensional wave arrival direction is θ k , k=1,2,...,K, the signal is incident on a linear array consisting of P elements, whose position set is V={v p d,p=1,2,...,P}, assuming that the received signal model of the array can be expressed as:
[0150] x(t)=A(θ)s(t)+n(t)
[0151] Where x(t) is the signal receiving matrix of P elements in the array under L snapshots, 1≤t≤L, A(θ)=[a(θ1),a(θ2),...,a(θ K )]∈C P×K Represents the direction matrix, direction vector s(t)=[s1(t),s2(t),...,s K (t)] T represents a non-Gaussian signal source with a mean of 0, n(t) = [n1(t), n2(t), ..., n P (t)] T It is represented by a mean of 0 and a variance of s. 2 Additive Gaussian white noise. λ represents half wavelength and d represents the array element spacing.
[0152] For a zero-mean random process, given a random variable x k1 ,x k2 , For integers 1≤k1,k2,k3,k4≤P, the fourth-order cumulant is expressed as:
[0153]
[0154] Applying this definition to the entire hypothetical array, the fourth-order signal accumulant of the received signal for the entire hypothetical array is expressed as:
[0155]
[0156] After calculating the fourth-order cumulant of the received signal, the corresponding element position of the fourth-order difference virtual array can be expressed as:
[0157] Γ 4-dc ={(vp1 -v p2 )-(v p3 -v p4 )}={(v p1 +v p4 )-(v p2 +v p3 )}
[0158]
[0159] The integer value is 1≤p1,p2,p3,p4≤P. According to the above formula, the fourth-order differential virtual array of the physical array can be obtained by adding and subtracting the position numbers of any two array elements, that is, the second-order sum-difference joint array and the fourth-order differential virtual array can be transformed into each other. Based on the above definition, the subsequent steps will discuss and analyze the arrangement of the fourth-order differential virtual array elements of the proposed three-level nested array.
[0160] Step 2: Construct a sparse array model with three layers of nested relationships, which is a three-level nested array:
[0161] The classic nested array consists of two linear arrays, and the array elements are distributed as follows:
[0162] V NA ={n1d,0≤n1≤N1-1}∪{[n2(N1+1)-1]d,1≤n2≤N2},n1∈Z,n2∈Z
[0163] The array element spacing is d = λ / 2, which is numerically equal to the half wavelength of the incident signal. The second-order differential virtual array corresponding to the array is a continuous array, and the array element position set is
[0164] The physical sensor location set of the three-level nested array (THL-NA) proposed in the present invention is as follows:
[0165] V THL-NA =V1∪V2∪V3
[0166] V1={n1d,1≤n1≤N1},n1∈Z
[0167] V2={n2(N1+1)d,1≤n2≤N2},n2∈Z
[0168] V3={n3(N2+1)(N1+1)d,1≤n3≤N3},n3∈Z
[0169] The schematic diagram of the array element position distribution is as follows Figure 1 As shown, the array has a total of N=N1+N2+N3 array elements.
[0170] First, the virtual array element position of the second-order sum array of the array can be obtained, which ranges from 2d to 2N3(N2+1)(N1+1)d, and the number of array element positions is not continuous. It can be observed that although the number of array element positions of the second-order sum array of the array is not continuous, there is a nested relationship between them. In order to observe more clearly, the sum array is divided into several parts, and the array element positions of each part are represented in the form of a set.
[0171]
[0172] S1={n1d,2≤n1≤(N1+1)(N2+1)}
[0173] S2=(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1}
[0174] S3=2(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1}
[0175]
[0176]
[0177] The second-order sum matrix of the three-level nested matrix is divided into N3+1 parts, where the set S1 is continuous and the set is a nested matrix with sub-matrices (N1, N2), and the set is a three-level nested array with sub-arrays (N1, N2, N3). According to the continuous characteristic of the difference matrix of the nested array, the element position range of the second-order sum and difference joint array of the array is obtained, that is, the second-order sum and difference joint array of the three-level nested array is a continuous array, and the position set of the virtual array is
[0178]
[0179] From this, we can deduce that the closed-form solution of the continuous degrees of freedom of the three-level nested matrix is
[0180] cDOF=2[2N3(N2+1)(N1+1)-2]+1=4N3(N2+1)(N1+1)-3
[0181] Let M b =2N3(N2+1)(N1+1)-2, that is, the number of array element positions of the virtual array is in the interval [-M b ,M b ] is continuous, and the number of continuous array elements is T = 2M b +1.
[0182] Step 3: Calculate the fourth-order cumulant matrix C of the received signal x(t)4,x , use the formula to vectorize it, sort it and remove redundancy to obtain vector z:
[0183] 1) First, solve the fourth-order cumulant matrix C of the received signal 4,x , and vectorize it:
[0184]
[0185] in
[0186]
[0187] B(θ) is the array manifold expanded using the fourth-order cumulant b(θ k ) is a(θ k ) is the expanded array manifold. To further improve the DOA estimation accuracy, a vectorization method is used to calculate the fourth-order cumulant matrix C 4,x To process:
[0188] z vec =vec(C 4,x )=vec[B(θ)C s B H (θ)]=Λ(θ)p
[0189] in
[0190]
[0191] and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.
[0192] Vectorized C 4,x The corresponding direction vector matrix Λ(θ) can be considered to be expanded from the two-level differential array A(θ) The number of array elements in the three sub-arrays of the three-level nested array is set to N1=3, N2=3, and N3=4. The position comparison diagram of several different array elements under this parameter configuration is as follows Figure 2 As shown; the second-order virtual array element position corresponding to the array manifold B(θ) is as follows Figure 3 As shown, the virtual array is continuous; the fourth-order virtual array element position corresponding to Λ(θ) is as follows Figure 4 As shown, the virtual array is continuous. According to the closed-form solution of continuous degrees of freedom derived in step 2, the array element position set of the fourth-order difference array can be expressed as
[0193]
[0194] The array elements of the fourth-order virtual array within the above position set are completely continuous.
[0195] 2) For z vec Sorting and removing redundancy yields vector z:
[0196] according to The position of the fourth-order virtual array element is z vec Sort and remove redundant elements to construct a new vector z
[0197] z=Λ1(θ)p
[0198] Among them, Λ1(θ) is the matrix of Λ(θ) after removing redundancy and sorting. Based on the above, we can know that the direction matrix Λ1(θ) of the differential array is
[0199]
[0200] and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.
[0201] Step 4: intercept vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT
[0202] The DFT algorithm is a DOA estimation method proposed for arrays with a large number of array elements or virtual array elements. In the parameter estimation of Gaussian signals, the vectorized single-snapshot covariance matrix can be processed and a high-precision estimate can be obtained. Considering the conclusion that the fourth-order virtual array of the three-level nested array is a uniform linear array, its array element elements are in the interval [-M b ,M b ] is continuous, the present invention proposes a vectorized DFT-Taylor algorithm to obtain the DOA estimation value of non-Gaussian signal. This estimation algorithm is an algorithm that does not require spatial smoothing. It completes the initial estimation by constructing a normalized DFT matrix, then uses phase rotation to achieve DFT precision estimation, and finally achieves precision estimation based on the estimation result by approximate expansion of the Taylor algorithm for error compensation, thereby obtaining a precision estimated angle value.
[0203] The normalized DFT matrix is defined as
[0204]
[0205] Where, T = 2M b +1 indicates the element range [-M b ,M b ], the total number of elements of the continuous fourth-order virtual array within the matrix F with coordinates (p, q) is The kth non-Gaussian source corresponds to the continuous range [-M b ,M b The direction vector of ] can be expressed as a(θ k )∈C T×1 , multiply it by the F matrix on the left, and rewrite the direction vector to get
[0206]
[0207] Its qth element and 1≤q≤T can be expressed as
[0208]
[0209] When Tsinθ k / 2=q k When is an integer, There is only one qth=q k elements are not 0, and the rest are 0, such as Figure 5 As shown, the DFT algorithm used is based on a three-level nested array with N1=3, N2=3, and N3=4. The total number of elements in the virtual array is T=253. At this time, θ k =arcsin(60 / 253). When Tsinθ k / 2=q k When it is not an integer, most of the elements in the formula are close to 0, except for the q=round{q k} elements, such as Figure 6 As shown, the total number of array elements T = 253, at this time θ k =60°.
[0210] In summary, the source angle θ k Initial estimate of Can be The positions of the largest K elements in are determined. In addition, The number of larger elements in can also be used to determine the number of signal sources. In practical applications, the target parameter estimation value can be processed by sorting and removing the redundant vector z1. Assume that the vectorized received signal after DFT transformation is
[0211] y ini =Fz1
[0212] The horizontal coordinates corresponding to the K largest peaks of its spatial spectrum are marked as The initial estimate of DOA is
[0213]
[0214] Step 5: Construct the phase rotation matrix Φ(η):
[0215] It can be seen from the figure that when Tsinθ k / 2=q k When it is not an integer, the DOA estimate has a large deviation even in the absence of noise. In practice, most angles do not satisfy Tsinθ. k / 2=q k The condition that is an integer is met, so the DFT precision estimation is realized based on the phase rotation matrix. The phase rotation matrix is defined as
[0216]
[0217] Wherein, η∈(-π / T,π / T) is the phase offset, and T is the total number of virtual array elements in the virtual array.
[0218] Step 6: Search the phase offset in the range (-π / T, π / T) to obtain the optimal phase offset η k , get the angle estimate after phase rotation DFT precise estimation
[0219] Direction vector The superposition phase shift gives
[0220]
[0221] Its qth element can be expressed as
[0222]
[0223] For convenience, rewrite
[0224]
[0225] Among them, 1≤q≤T. When Tsinθ k When / 2 is an integer, the phase shift η is 0; when Tsinθ k When / 2 is not an integer, we can search for the value of η in (-π / T,π / T) so that Tsinθ k / 2+Tη k / (2π) is an integer. There will be only one non-zero element. Therefore, the introduction of phase offset can effectively solve Tsinθ k / 2=q k The problem of not being an integer is that the phase shift η k It can be obtained by searching in (-π / T,π / T)
[0226]
[0227] in, is the matrix F The computational complexity required for this part is O(GKT), and G is the number of quantized grids in the region [-π / T,π / T]. The exact estimate after phase rotation is
[0228]
[0229] Step 7: Perform error compensation based on Taylor expansion to obtain the final angle estimate:
[0230] As can be seen from the previous section, phase rotation requires rotating each signal source in turn, and the accuracy of its DOA estimation result depends on the number of searches in the region [-π / T,π / T]. When the number of searches is large, the complexity of the algorithm will also be relatively high. This section first uses a few searches to improve the accuracy of the initial estimate, completes the precise estimation of the DFT algorithm, and then uses the precise estimation algorithm based on Taylor approximation expansion, which can effectively improve the algorithm estimation accuracy.
[0231] The direction vector a(θ k )exist Expand, yes
[0232]
[0233] in, Ignoring the second-order derivative and above, and ignoring the influence of noise, we have
[0234]
[0235] Among them, the direction vector Error matrix Δ = diag(δ1,δ2,...,δ K ), w = Δp. By matrix transformation, we can get
[0236]
[0237] By deriving the formula in a different form, we can get
[0238] Δ=w· / p
[0239] The final DOA estimate is
[0240]
[0241] The present invention also provides a non-Gaussian signal DOA estimation system based on a three-level nested array vectorized DFT-Taylor algorithm, which is used to implement the non-Gaussian signal DOA estimation method based on the three-level nested array vectorized DFT-Taylor algorithm. The system includes a DOA estimation model construction module, a sparse array model construction module, a vector z calculation module, a DFT initial estimation value calculation module, a phase rotation matrix construction module, a DFT precise estimation angle estimation value calculation module, and a final angle estimation value calculation module, wherein:
[0242] DOA estimation model building module, used to establish a DOA estimation model for non-Gaussian signals;
[0243] The sparse array model building module is used to construct a sparse array model with three-layer nesting relationship, which is a three-level nested array;
[0244] Vector z calculation module, used to calculate the fourth-order cumulant matrix C of the received signal x(t) 4,x , and for C 4,x Perform vectorization, sorting and remove redundancy to obtain vector z;
[0245] The DFT initial estimate calculation module is used to intercept the vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K;
[0246] A phase rotation matrix building module, used to construct a phase rotation matrix Φ(η);
[0247] The angle estimation value calculation module of DFT precise estimation searches for the optimal phase offset η in the range (-π / T, π / T) k , get the angle estimate after phase rotation DFT precise estimation
[0248] The final angle estimation calculation module performs error compensation based on Taylor expansion and combines the angle estimation value accurately estimated by DFT and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k .
[0249] The present invention also provides a mobile terminal, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the non-Gaussian signal DOA estimation method based on the three-level nested array vectorized DFT-Taylor algorithm is implemented.
[0250] In summary, the present invention proposes a sparse array structure of a three-level nested array on the one hand, and on the other hand, uses a vectorized DFT-Tarloy algorithm based on the three-level nested array to perform DOA estimation in the incident scenario of a non-Gaussian signal.
[0251] Performance verification and analysis:
[0252] The following Monte Carlo simulation is used to verify the effectiveness and parameter estimation performance of the vectorized DFT-Taylor algorithm. RMSE is used to measure the parameter estimation performance and is defined as
[0253]
[0254] Among them, θ k represents the true incident angle of the kth source, is the angle θ in the i-th Monte Carlo simulation k , K represents the number of incident far-field narrowband uncorrelated zero-mean non-Gaussian sources.
[0255] a. Estimation lattices with different angle parameters
[0256] Figure 7 The figure shows the estimated dot pattern obtained by the vectorized DFT-Taylor algorithm proposed in the present invention when incident on a three-level nested array at different angles. Consider that there are K=7 unrelated narrowband signals incident on a three-level nested array with array configurations of N1=3, N2=3, and N3=4, and the azimuth angles of the signals are evenly distributed between 5° and 65° at intervals of 10°, and the number of snapshots is set to L=1000, the signal-to-noise ratio is 5dB, and the number of tests is set to 100 times. The red dotted line in the figure represents the actual angle value, and the black dot represents the angle estimation value obtained by the algorithm. It can be seen from the figure that the vectorized DFT-Taylor algorithm proposed in the present invention can achieve accurate DOA angle estimation, and the simulation verifies the effectiveness of the proposed algorithm.
[0257] b. Estimation performance comparison under different snapshot numbers
[0258] Figure 8 The figure shows the parameter estimation performance comparison of the vectorized DFT-Taylor algorithm under different snapshot numbers, where K=2 non-Gaussian sources with far-field narrowband uncorrelated are incident at angles θ=[10° 40°] into a three-level nested array with array configurations of N1=3, N2=3, and N3=4. The increase in the number of snapshots, that is, the increase in sampled data, means that the vectorized fourth-order cumulant matrix is more accurate. It can be seen from the figure that the angle estimation performance of the algorithm improves with the increase in the number of snapshots and with the increase in the signal-to-noise ratio.
[0259] Computational complexity analysis and comparison:
[0260] The computational complexity of the vectorized DFT-Taylor algorithm is analyzed below, and the computational complexity is evaluated using the number of complex multiplications.
[0261] Assume that the number of array elements of the three sub-arrays of the three-level nested array is N1, N2, and N3, the number of signal sources is K, the number of snapshots is L, the total number of array elements is N, and N = N1 + N2 + N3, the number of continuous virtual array elements generated by the second-order sum and difference virtual array of the array is T, and T = 2M b +1,M b =2N3(N2+1)(N1+1)-2.
[0262] The computational complexity of vectorization algorithms mainly includes: calculating the fourth-order cumulant matrix of the received signal requires O{LN 4}, the vectorized received signal after DFT transformation is calculated as O{(2M b +1) 2}, DFT accurate estimation of phase rotation process phase offset search requires O{G(2M b +1)K}, and G is the number of grid searches. The number of searches in the vectorized DFT-Taylor algorithm is only a few times, so it is set to G1 = 5. The number of searches in the DFT fine estimation affects the algorithm estimation accuracy, so it is set to G2 = 1000. The complexity of using the Taylor approximate expansion algorithm for fine estimation is O{(8K 2 +2K)(M b +1)}, then the computational complexity required for the vectorized DFT-Taylor algorithm is O{LN 4 +(2M b +1) 2 +G(2M b +1)K+(8K 2 +2K)(M b +1)}; the computational complexity of the vectorized DFT precision estimation algorithm is O{LN 4 +(2M b +1) 2 +G2(2M b +1)K}, the computational complexity of the vectorized DFT initial estimation algorithm is O{LN 4 +(2M b +1) 2}, and M b =2N3(N2+1)(N1+1)-2. Table 1 lists the corresponding computational complexity of the three algorithms.
[0263] Table 1 Computational complexity of three algorithms in the vectorized DFT-Taylor algorithm
[0264]
[0265] Fig. 9 The figure shows the computational complexity comparison of the vectorized DFT precise estimation algorithm and the vectorized DFT-Taylor algorithm under different snapshot numbers, where K=2 far-field narrowband uncorrelated non-Gaussian sources are incident at an angle θ=[10° 40°] into a three-level nested array with an array configuration of N1=3, N2=3, and N3=4.
[0266] As can be seen from the figure, the computational complexity of the algorithm increases with the increase of the number of snapshots, and the complexity of the vectorized DFT precise estimation algorithm is slightly greater than the complexity of the vectorized DFT-Taylor algorithm.
[0267] In summary, the advantages of the present invention are: 1. The array of the present invention utilizes the continuous characteristics of the nested array difference array. The proposed array can derive a provably completely continuous fourth-order virtual array element position, that is, the array has a closed-form solution, and the array has strong scalability and availability; 2. The array THL-NA designed by the present invention has a large degree of freedom, and compared with the newly proposed sparse array SAFE-CPA, the traditional two-level nested array TL-NA, the four-level nested array FL-NA, and the traditional augmented coprime array ACA, it has a larger degree of continuous freedom; 3. The array THL-NA designed by the present invention is superior to the above-mentioned four array structures SAFE-CPA, TL-NA, FL-NA and ACA in terms of root mean square error performance when the number of array elements and other conditions are the same. From the comparison above, the angle estimation performance of the array proposed in the present invention is better than other array structures; 4. The proposed algorithm adopts a sparse array combined with vectorization to expand the virtual array aperture; 5. The proposed algorithm is suitable for large-scale arrays with a large number of array elements or virtual array elements for DOA estimation. The algorithm can be directly used for single-snap virtual signals without the need for spatial smoothing, thus avoiding the loss of degrees of freedom caused by spatial smoothing and thus improving the spatial degrees of freedom of the array; 6. The proposed algorithm uses DFT changes to obtain an initial estimate, and then completes the DFT precise estimation with a smaller number of grid searches. Finally, the Taylor expansion algorithm is used to compensate for the error on the result of the phase-rotated DFT precise estimation to obtain the final angle estimate. The above three steps all have low computational complexity.
Claims
1. A non-Gaussian signal DOA estimation method based on a three-level nested array vectorized DFT-Taylor algorithm, characterized in that: The following steps are involved: Step 1: Establish a DOA estimation model for non-Gaussian signals; Step 2: construct a sparse array model with three-layer nested relationships, which is a three-level nested array; Step 3: Calculate the fourth-order cumulant matrix C of the received signal x(t) 4,x , and for C 4,x Perform vectorization, sorting and remove redundancy to obtain vector z; Step 4: intercept vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K; Step 5: construct a phase rotation matrix Φ(η); Step 6: Search the phase offset in the range (-π / T, π / T) to obtain the optimal phase offset η k , get the angle estimate after phase rotation DFT precise estimation Step 7: Perform error compensation based on Taylor expansion and combine the angle estimate with DFT precision estimation and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k .
2. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 1 is characterized in that: Step 1 establishes a DOA estimation model for non-Gaussian signals, as follows: First set (·) T 、(·) H 、(·) -1 and(·) * They represent transpose, conjugate transpose, inversion, and conjugate operations respectively; bold capital letters represent matrices, and bold lowercase letters represent vectors; the symbol cum(·) represents fourth-order cumulant calculation, and E(·) represents expectation calculation. represents the Kronecker product, vec(·) represents matrix vectorization, and angle(·) represents the phase angle of a complex number; Assume that there are K narrowband far-field incoherent signals in space, and the one-dimensional wave arrival direction is θ k , k=1,2,...,K, the signal is incident on a linear array consisting of P elements, and the position set is V={v p d,p=1,2,...,P}, assuming that the received signal model of the array is expressed as: x(t)=A(θ)s(t)+n(t) Where x(t) is the signal receiving matrix of P elements in the array under L snapshots, 1≤t≤L, A(θ)=[a(θ1),a(θ2),...,a(θ K )]∈C P×K Represents the direction matrix, direction vector s(t)=[s1(t),s2(t),...,s K (t)] T represents a non-Gaussian signal source with a mean of 0; n(t) = [n1(t), n2(t), ..., n P (t)] T It is represented by a mean of 0 and a variance of s. 2 Additive Gaussian white noise; λ represents half wavelength, d represents the array element spacing; For a zero-mean random process, given a random variable x k1 ,x k2 , For integers 1≤k1,k2,k3,k4≤P, the fourth-order cumulant is expressed as: The fourth-order signal accumulation for the entire hypothetical array received signal is expressed as: After calculating the fourth-order cumulant of the received signal, the corresponding element position of the fourth-order difference virtual array is expressed as: Γ 4-dc ={(v p1 -v p2 )-(v p3 -v p4 )}={(v p1 +v p4 )-(v p2 +v p3 )} Among them, the integer value is 1≤p1,p2,p3,p4≤P; according to the above formula, the fourth-order differential virtual array of the physical array can be obtained by adding and subtracting the position numbers of any two array elements, that is, the second-order sum-difference joint array and the fourth-order differential virtual array can be transformed into each other.
3. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 2 is characterized in that: The sparse array model with three-layer nested relationship constructed in step 2 is a three-level nested array, which is as follows: The nested array is composed of two linear arrays, and the array elements are distributed as follows: IN NA ={n1d,0≤n1≤N1-1}∪{[n2(N1+1)-1]d,1≤n2≤N2},n1∈Z,n2∈Z The array element spacing is d = λ / 2, which is numerically equal to the half wavelength of the incident signal. The second-order differential virtual array corresponding to the array is a continuous array, and the array element position set is The physical sensor locations of the three-level nested array THL-NA are as follows: In THL-NA =V1∪V2∪V3 V1={n1d,1≤n1≤N1},n1∈Z V2={n2(N1+1)d,1≤n2≤N2},n2∈Z V3={n3(N2+1)(N1+1)d,1≤n3≤N3},n3∈Z The array has a total of N = N1 + N2 + N3 array elements; First, the second-order and virtual array element positions of the array are obtained, ranging from 2d to 2N3(N2+1)(N1+1)d, and the number of array element positions is not continuous; Divide the second-order matrix into several parts, and express the positions of the array elements of each part in the form of a set: S1={n1d,2≤n1≤(N1+1)(N2+1)} S2=(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1} S3=2(N1+1)(N2+1)d+{n1d,1≤n1≤N1}∪{n2(N1+1)d,1≤n2≤N2+1} The second-order sum matrix of the three-level nested matrix is divided into N3+1 parts, where the set S1 is continuous and the set is a nested matrix with sub-matrices (N1, N2), and the set is a three-level nested array with sub-arrays (N1, N2, N3) respectively; according to the continuous characteristic of the difference matrix of the nested array, the element position range of the second-order sum and difference joint array of the array is obtained, that is, the second-order sum and difference joint array of the three-level nested array is a continuous array, and the position set of the virtual array is From this, the closed-form solution of the continuous degrees of freedom of the three-level nested matrix is deduced as cDOF=2[2N3(N2+1)(N1+1)-2]+1=4N3(N2+1)(N1+1)-3 Let M b =2N3(N2+1)(N1+1)-2, that is, the number of array element positions of the virtual array is in the interval [-M b ,M b ] is continuous, and the number of continuous array elements is T = 2M b +1.
4. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 3 is characterized in that: Step 3 calculates the fourth-order cumulant matrix C of the received signal x(t) 4,x , and for C 4,x Vectorize, sort and remove redundancy to obtain vector z, as follows: 1) First, solve the fourth-order cumulant matrix C of the received signal 4,x , and vectorize it: in B(θ) is the array manifold expanded using the fourth-order cumulant b(θ k ) is a(θ k ) The expanded array manifold; the fourth-order cumulant matrix C is vectorized 4,x To process: z vec =vec(C 4,x )=vec[B(θ)C s B H (θ)]=Λ(θ)p in and represents the fourth-order cumulant of the k-th source vector, where represents the fourth-order cumulant of the kth source; Vectorized C 4,x The corresponding direction vector matrix Λ(θ) is expanded from the two-level differential array A(θ). According to the closed-form solution of continuous degrees of freedom derived in step 2, the set of array element positions of the fourth-order differential array is expressed as The array elements of the fourth-order virtual array within the array element position set of the fourth-order differential array are completely continuous; 2) For z vec Sorting and removing redundancy yields vector z: according to The position of the fourth-order virtual array element is z vec Sort and remove redundant elements to construct a new vector z z=Λ1(θ)p Among them, Λ1(θ) is the matrix of Λ(θ) after removing redundancy and sorting; the direction matrix Λ1(θ) of the difference array is and represents the fourth-order cumulant of the k-th source vector, where Represents the fourth-order cumulant of the kth source.
5. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 4 is characterized in that: Step 4 intercepts the vector z and obtains the continuous virtual array element range of [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT The details are as follows: The normalized DFT matrix is defined as Where, T = 2M b +1 indicates the element range [-M b ,M b ], the total number of elements of the continuous fourth-order virtual array within the matrix F with coordinates (p, q) is The kth non-Gaussian source corresponds to the continuous range [-M b ,M b The direction vector of ] is expressed as a(θ k )∈C T×1 , multiply the F matrix on the left, and rewrite the direction vector to get The qth element and 1≤q≤T is represented by When Tsinθ k / 2=q k When is an integer, There is only one qth=q k elements are not 0, and the rest are 0; Source angle θ k Initial estimate of pass The positions of the largest K elements in are determined. Assuming that the vectorized received signal after DFT transformation is y ini =Fz1 The horizontal coordinates corresponding to the K largest peaks of the spatial spectrum are marked as The initial estimate of DOA is 6. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 5 is characterized in that: The phase rotation matrix Φ(η) is constructed as follows: The phase rotation matrix is defined as Wherein, η∈(-π / T,π / T) is the phase offset, and T is the total number of virtual array elements in the virtual array.
7. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 6 is characterized in that: Step 6 searches for the phase offset in the range (-π / T, π / T) to obtain the optimal phase offset η. k , get the angle estimate after phase rotation DFT precise estimation The details are as follows: Direction vector The phase shift is superimposed The qth element is represented by Rewrite Among them, 1≤q≤T; when Tsinθ k When / 2 is an integer, the phase shift η is 0; when Tsinθ k When / 2 is not an integer, the value of η is searched in (-π / T,π / T) so that Tsinθ k / 2+Tη k / (2π) is an integer; Phase shift η k Obtained by searching in (-π / T,π / T) in, is the matrix F The computational complexity of this part is O(GKT), and G is the number of quantized grids in the region [-π / T,π / T]; The precise estimate after phase rotation is 。 8. The non-Gaussian signal DOA estimation method based on three-level nested array vectorized DFT-Taylor algorithm according to claim 1 is characterized in that: Step 7: Error compensation based on Taylor expansion, combined with the angle estimate of DFT precision estimation and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k , as follows: The direction vector a(θ k )exist Expand, yes in, Ignoring the second-order derivative and above, and ignoring the influence of noise, we have Among them, the direction vector Error matrix Δ = diag(δ1,δ2,...,δ K ), w = Δp; Perform matrix transformation on the formula to get By performing variant derivation on the formula, we can obtain Δ=w · / p The final DOA estimate is 9. A non-Gaussian signal DOA estimation system based on a three-level nested array vectorized DFT-Taylor algorithm, characterized in that: The system is used to implement the non-Gaussian signal DOA estimation method based on the three-level nested array vectorized DFT-Taylor algorithm as described in any one of claims 1 to 8, and the system includes a DOA estimation model construction module, a sparse array model construction module, a vector z calculation module, a DFT initial estimation value calculation module, a phase rotation matrix construction module, a DFT precise estimation angle estimation value calculation module, and a final angle estimation value calculation module, wherein: DOA estimation model building module, used to establish a DOA estimation model for non-Gaussian signals; The sparse array model building module is used to construct a sparse array model with three-layer nesting relationship, which is a three-level nested array; Vector z calculation module, used to calculate the fourth-order cumulant matrix C of the received signal x(t) 4,x , and for C 4,x Perform vectorization, sorting and remove redundancy to obtain vector z; The DFT initial estimate calculation module is used to intercept the vector z and obtain continuous virtual array element range [-M b ,M b ] corresponding to the part z1, construct the matrix F, and obtain the initial estimate of DFT And k = 1, 2, ..., K; A phase rotation matrix building module, used to construct a phase rotation matrix Φ(η); The angle estimation value calculation module of DFT precise estimation searches for the optimal phase offset η in the range (-π / T, π / T) k , get the angle estimate after phase rotation DFT precise estimation The final angle estimation calculation module performs error compensation based on Taylor expansion and combines the angle estimation value accurately estimated by DFT and error compensation δ k , calculate the final angle estimate θ of the DFT-Taylor algorithm k .
10. A mobile terminal comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the program, the non-Gaussian signal DOA estimation method based on the three-level nested array vectorized DFT-Taylor algorithm as described in any one of claims 1 to 8 is implemented.