DOA estimation method based on novel array optimization

Abstract

A DOA estimation method based on novel array optimization is provided; according to the implementation mode, after a system is initialized, an improved non-uniform array is firstly constructed, then the non-uniform array is used for receiving narrow-band far-field incoherent signals, in order to make full use of data collected by the non-uniform array, a difference sum common array is generated according to a VCAM algorithm, time delay sampling processing is conducted, then a conjugate augmentation vector is constructed, and the non-uniform array is used for receiving the narrow-band far-field incoherent signals. Secondly, constructing a pseudo data matrix, performing vectorization operation on the pseudo data matrix, constructing a covariance matrix, obtaining difference set and sum set elements containing a novel non-uniform array, performing redundancy removal and sorting, applying a matrix reconstruction thought for recovering a rank of the matrix, and performing Toeplitz matrix reconstruction, so as to obtain a covariance matrix in a conventional MUSIC algorithm; therefore, DOA estimation can be carried out by directly using the MUSIC algorithm to obtain angle information.

Description

Technical Field

[0001] The invention belongs to the field of array signal processing and relates to a DOA estimation method of a novel array arrangement. Background Art

[0002] Since the middle of the last century, research in the field of array signal processing has continued to deepen, and a variety of high-resolution methods have been proposed and widely used in modern communications and radar systems. When estimating signal parameters using array signal processing theory, the angle of arrival (DOA) estimation plays an important role. The array received data is processed to minimize interference and noise, and the processed data is used to accurately estimate the parameters of the target signal. Many scholars have spent a lot of time and energy studying the DOA estimation problem and proposed many data processing algorithms and array models with high estimation accuracy. Multi-source narrowband signal DOA estimation is one of the important applications of antenna arrays, and it is of great significance in modern warfare and intelligent communications. Due to the complexity of the actual environment, in order to improve the positioning accuracy, many fields have higher and higher requirements for DOA estimation accuracy.

[0003] Most DOA estimation methods use uniform arrays (ULAs). However, uniform linear arrays are usually limited by limited degrees of freedom and array aperture, which will have a negative impact on the number of identified signal sources and the accuracy of DOA estimation. Therefore, to detect more signal sources, more physical sensors are required, which will inevitably increase the economic cost and the difficulty of array deployment. In order to solve this problem, non-uniform arrays were considered, and the minimum redundant array and the minimum hole array were first proposed, but because there was no fixed expression, the optimal arrangement could not be effectively derived. In recent years, coprime arrays and nested arrays are two typical non-uniform arrays, which have attracted much attention due to their array geometry and the freedom of simple display expressions. The present invention proposes an improved new array, which makes a basic improvement to the arrangement of sparse arrays, shifts and improves ordinary coprime arrays, and obtains a universal model. Due to the optimization of the array, it has higher degrees of freedom than ordinary nested arrays and coprime arrays, and has higher accuracy and performance in DOA estimation. Summary of the invention

[0004] The purpose of the present invention is to propose a DOA estimation method based on a novel array optimization.

[0005] A DOA estimation method based on novel array optimization has the following steps:

[0006] Step 1: Improve and arrange the sparse non-uniform array.

[0007] In order to improve the accuracy of DOA estimation of non-uniform arrays, a sum-difference array arrangement is adopted to make the central continuous section long enough, with smaller coupling leakage and weaker coupling effect.

[0008] Take N array elements and divide them into two sub-arrays. The number of array elements in the first sub-array is M1, and the number of array elements in the second sub-array is M2. The number of array elements in the sub-arrays satisfies the following conditions

[0009]

[0010] The array position spacing is d and (2M1+1)d respectively, and the corresponding array element position set is U1={d, 2d,…, (M1-1)d}, U2={(M1+1)d, (3M1+2)d,…, ((M1+1)+(2M1+1)M2)d}, d=λ / 2, λ is the wavelength of the signal source, and the array structure of the basic nested array is as follows Figure 1 shown.

[0011] The degrees of freedom for this underlying non-uniform array are as follows:

[0012] DOF = 2(3M1+2)-1 = 6M1-3

[0013] The number of continuous array elements of the basic nested array is too small, the array aperture is too small, and the degree of freedom is too small to meet the accuracy requirements of array estimation. The array is disrupted for improvement. First, the first array element of the second subarray is moved to the first subarray to form M3 array elements, and then the last array element is moved to The improved subarray is constructed by filling the gaps in the traditional nested array mentioned above, expanding the length of the continuous array, increasing the number of effective array elements, and improving the DOA estimation accuracy. Figure 2 shown.

[0014] The number of improved sub-arrays meets the following conditions

[0015]

[0016] The spacing of the array positions is d, (2M3+1)d, and the corresponding array element position set is U3={d, 2d, ..., M3d}, U4={(3M3+2)d, ..., ((M3+1)+(2M3+1)(M4-1)d}. Move the last array element position to So the array position set is

[0017]

[0018] The sum and difference operations are performed on the array element position set S respectively, and the array difference set Sd For S d ={d o -d p |d o , d p ∈S, 1≤o, p≤N}, in represents the difference of the positions of the array elements located on the negative semi-axis, Represented as the difference set of the array element positions located on the positive semi-axis It can be expressed as

[0019]

[0020] The sum of the array element positions can be expressed as S s ={±(d o +d p )|d o , d p ∈S, 1≤o, p≤N}, in It is represented as the sum of the positions of the array elements located on the negative semi-axis, The sum of the positions of the array elements on the positive semi-axis is represented by It can be expressed as

[0021]

[0022] The total continuous virtual array element set S sd for

[0023]

[0024] The degrees of freedom of the improved non-uniform array are as follows:

[0025]

[0026] From the above degree of freedom formula, we can get that the degree of freedom of the non-uniform array is determined by the size of M1 and M2, so we need to consider the optimal combination of the number of arrays. For the problem of maximum degree of freedom under the constraints of this model, we can convert it into a quadratic function and solve it to find the extreme point, that is, the optimal solution, so we get

[0027]

[0028] It is rounded up. From the above formula, we can see that when N ≥ 5, the non-uniform array in this section has higher degrees of freedom and array aperture than the traditional non-uniform array, and the larger the number of array elements, the greater the degrees of freedom, and the greater the advantages compared to the traditional non-uniform array, the better the performance.

[0029] Step 2: Using the sparse non-uniform array to receive far-field narrow-band incoherent signals.

[0030] In DOA estimation, the narrowband signal can be expressed as

[0031]

[0032] where u k (t) is the amplitude function, is the phase function, ω k =2πf k is the angular frequency, and the signal delay τ k The function after is

[0033]

[0034] Because u k (t)≈u k (t-τ k )and Therefore, k (t-τ k ) can be written as

[0035]

[0036] Assume that the number of signal sources incident on the matrix is ​​K, and the incident angles are θ1, θ2, ..., θ K ,θ k ∈[-π / 2, π / 2], k=1, 2, ..., K, the first array element position is set as the reference array element, then the received data of the mth array element is

[0037]

[0038] The above formula is expressed in matrix form through statistics as

[0039] X(t)=A(θ)S(t)+N(t)

[0040] where X(t) = [x1(t), x2(t), …, x N (t)] T , A(θ)=[a(θ1), a(θ2),…, a(θ K )] is the array flow matrix, a(θ k ) is the steering vector of the kth signal, N(t) = [n1(t), n2(t), ..., n N (t)] T is a noise matrix with a mean of 0 and a variance of σ 2 .

[0041] Step 3: Use the vector conjugate augmented MUSIC algorithm to construct a conjugate augmented vector.

[0042] First, the conjugate augmented correlation vector is constructed. The output x1(t) of the first array element and the output x of the mth array element are collected respectively. m (t)N p samples, we can get the vector [x1(1), x1(2), ..., x1(N p )] and the vector [x m (1+τ), x m (2+τ),…,x m (N p +τ)], then the time delay correlation function of the vector can be expressed as

[0043]

[0044] Where τ≠0. When the number of snapshots N p When ω is large enough and l ≠ω k , It can be expressed as

[0045]

[0046] in * is conjugated. can be simplified to

[0047]

[0048] Vectorization Can get

[0049] r x (τ) = Ar s (τ)

[0050] in By changing τ to -τ, we can get

[0051] [r x (-τ)] * =A * r s (τ)

[0052] In order to reduce repeated operations, subtract the first element of the array direction vector, that is, Then the above formula is converted to

[0053] [r' x (-τ)] * =(A') * r s (τ)

[0054] Combined with x (τ) and [rx (-τ)] * , we get the vectorized conjugate augmented vector

[0055]

[0056] Step 4: Construct a pseudo data matrix based on the vector conjugate augmented MUSIC algorithm, perform data processing, and then perform DOA estimation.

[0057] Assume the pseudo sampling period is T s , the number of pseudo-sampling snapshots is N s , construct the pseudo data matrix of r(τ) as

[0058] Y=[r(T s ), r(2T s ),...,r(N s T s )]=BUE

[0059] Where E represents

[0060]

[0061] Where B=[b(θ1), b(θ2),…, b(θ K )],b(θ k )=[(a′(θ k )) H , a(θ k ) T ] T ,U=diag([|u1| 2 ,…,|u K | 2 ], diag is the diagonal matrix.

[0062] The covariance matrix of r(τ) can be estimated from Y as

[0063]

[0064] in express

[0065]

[0066] Vectorizing R rr have to

[0067] z=vec(R rr )=(B * ⊙B)p

[0068] where p = [|(s R )1| 4 ,|(s R )2|4 ,...,|(s R ) Q | 4 ] T , B in the qth column * ⊙B can be expressed as

[0069]

[0070] diff1 and diff2 contain the difference elements of the array, and sum1 and sum2 contain all the sum elements of the array. v It is a vector composed of all the lags generated by the above formula. Since there are repeated redundant elements, the repeated elements are excluded, sorted in ascending order, and the central continuous part is extracted to obtain its vector matrix s c The transformation matrix Tr is defined as

[0071]

[0072] Where a=1,2,...,2l+1,b=1,2,...,(2N) 2 , σ represents the Kronecker product.

[0073] So z can be re-expressed as

[0074] z=Trz rr =TR(B * ⊙B)p

[0075] In order to obtain the full-rank virtual covariance matrix, the above formula is reconstructed into a Toeplitz matrix.

[0076]

[0077] The MUSIC algorithm is directly used to solve it and obtain its DOA estimation.

[0078] The present invention first proposes an improved method for sparse arrays, which extends the length of continuous array elements of traditional nested arrays, fills in holes, increases array aperture, expands degrees of freedom, and has higher precision. Then, according to the VCAM algorithm, the sum-difference array is obtained, which has more degrees of freedom and higher precision than ordinary difference arrays. Then, data processing is performed according to the VCAM algorithm to obtain its DOA estimation. Finally, simulation verifies the effectiveness and good precision of the new array proposed by the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0080] Figure 1 This is the array element layout diagram of the basic nested array.

[0081] Figure 2 To improve the array element layout.

[0082] Figure 3 It is a flow chart of the present invention;

[0083] Figure 4 is the directional diagram of the virtual array element;

[0084] Figure 5 is a scatter plot of the present invention;

[0085] Figure 6 is a root mean square error diagram of the present invention;

[0086] Figure 7 is a success probability diagram of the present invention; DETAILED DESCRIPTION

[0087] In order to make the above and other purposes, features and advantages of the present invention more obvious, embodiments of the present invention are specifically cited below and described in detail with reference to the accompanying drawings.

[0088] The purpose of the present invention is to propose a DOA estimation method based on a novel array optimization.

[0089] In order to achieve the above object, the present invention adopts the following technical solutions:

[0090] Step 1: Improve and arrange the sparse non-uniform array.

[0091] In order to improve the accuracy of DOA estimation of non-uniform arrays, a sum-difference array arrangement is adopted to make the central continuous section long enough, with smaller coupling leakage and weaker coupling effect.

[0092] Take N array elements and divide them into two sub-arrays. The number of array elements in the first sub-array is M1, and the number of array elements in the second sub-array is M2. The number of array elements in the sub-arrays satisfies the following conditions

[0093]

[0094] The array position spacing is d and (2M1+1)d respectively, and the corresponding array element position set is U1={d, 2d,…, (M1-1)d}, U2={(M1+1)d, (3M1+2)d,…, ((M1+1)+(2M1+1)M2)d}, d=λ / 2, λ is the wavelength of the signal source, and the array structure of the basic nested array is as follows Figure 1 shown.

[0095] The degrees of freedom for this underlying non-uniform array are as follows:

[0096] DOF = 2(3M1+2)-1 = 6M1-3

[0097] The number of continuous array elements of the basic nested array is too small, the array aperture is too small, and the degree of freedom is too small to meet the accuracy requirements of array estimation. The array is disrupted for improvement. First, the first array element of the second subarray is moved to the first subarray to form M3 array elements, and then the last array element is moved to The improved subarray is constructed by filling the gaps in the traditional nested array mentioned above, expanding the length of the continuous array, increasing the number of effective array elements, and improving the DOA estimation accuracy. Figure 2 shown.

[0098] The number of improved sub-arrays meets the following conditions

[0099]

[0100] The spacing of the array positions is d, (2M3+1)d, and the corresponding array element position set is U3={d, 2d, ..., M3d}, U4={(3M3+2)d, ..., ((M3+1)+(2M3+1)(M4-1)d}. Move the last array element position to So the array position set is

[0101]

[0102] The sum and difference operations are performed on the array element position set S respectively, and the array difference set S d For S d ={d o -d p |d o , d p ∈S, 1≤o, p≤N}, in represents the difference of the array element positions located on the negative semi-axis, Represented as the difference set of the array element positions located on the positive semi-axis It can be expressed as

[0103]

[0104] The sum of the array element positions can be expressed as S s ={±(d o +d p )|d o , d p ∈S, 1≤o, p≤N}, in It is represented as the sum of the positions of the array elements located on the negative semi-axis, The sum of the positions of the array elements on the positive semi-axis is represented by It can be expressed as

[0105]

[0106] The total continuous virtual array element set S sd for

[0107]

[0108] The degrees of freedom of the improved non-uniform array are as follows:

[0109]

[0110] From the above degree of freedom formula, we can get that the degree of freedom of the non-uniform array is determined by the size of M1 and M2, so we need to consider the optimal combination of the number of arrays. For the problem of maximum degree of freedom under the constraints of this model, we can convert it into a quadratic function and solve it to find the extreme point, that is, the optimal solution, so we get

[0111]

[0112] It is rounded up. From the above formula, we can see that when N ≥ 5, the non-uniform array in this section has higher degrees of freedom and array aperture than the traditional non-uniform array, and the larger the number of array elements, the greater the degrees of freedom, and the greater the advantages compared to the traditional non-uniform array, the better the performance.

[0113] Step 2: Using the sparse non-uniform array to receive far-field narrow-band incoherent signals.

[0114] In DOA estimation, the narrowband signal can be expressed as

[0115]

[0116] where u k (t) is the amplitude function, is the phase function, ω k =2πf k is the angular frequency, and the signal delay τk The function after is

[0117]

[0118] Because u k (t)≈u k (t-τ k )and Therefore, k (t-τ k ) can be written as

[0119]

[0120] Assume that the number of signal sources incident on the matrix is ​​K, and the incident angles are θ1, θ2, ..., θ K ,θ k ∈[-π / 2, π / 2], k=1, 2, ..., K, the first array element position is set as the reference array element, then the received data of the mth array element is

[0121]

[0122] The above formula is expressed in matrix form through statistics as

[0123] X(t)=A(θ)S(t)+N(t)

[0124] where X(t) = [x1(t), x2(t), …, x N (t)] T , A(θ)=[a(θ1), a(θ2),…, a(θ K )] is the array flow matrix, a(θ k ) is the steering vector of the kth signal, N(t) = [n1(t), n2(t), ..., n N (t)] T is a noise matrix with a mean of 0 and a variance of σ 2 .

[0125] Step 3: Use the vector conjugate augmented MUSIC algorithm to construct a conjugate augmented vector.

[0126] First, the conjugate augmented correlation vector is constructed. The output x1(t) of the first array element and the output x of the mth array element are collected respectively. m (t)N p samples, we can get the vector [x1(1), x1(2), ..., x1(N p )] and the vector [x m (1+τ), x m (2+τ),…,x m (N p+τ)], then the time delay correlation function of the vector can be expressed as

[0127]

[0128] Where τ≠0. When the number of snapshots N p When ω is large enough and l ≠ω k , It can be expressed as

[0129]

[0130] in * is conjugated. can be simplified to

[0131]

[0132] Vectorization Can get

[0133] r x (τ) = Ar s (τ)

[0134] in By changing τ to -τ, we can get

[0135] [r x (-τ)] * =A * r s (τ)

[0136] In order to reduce repeated operations, subtract the first element of the array direction vector, that is, Then the above formula is converted to

[0137] [r' x (-τ)] * =(A') * r s (τ)

[0138] Combined with x (τ) and [r x (-τ)] * , we get the vectorized conjugate augmented vector

[0139]

[0140] Step 4: Construct a pseudo data matrix based on the vector conjugate augmented MUSIC algorithm, perform data processing, and then perform DOA estimation.

[0141] Assume the pseudo sampling period is T s , the number of pseudo-sampling snapshots is Ns , construct the pseudo data matrix of r(τ) as

[0142] Y=[r(T s ),r(2T s ), ..., r(N s T s )]=BUE

[0143] Where E represents

[0144]

[0145] Where B=[b(θ1), b(θ2),…, b(θ K )],b(θ k )=[(a'(θ k )) H , a(θ k ) T ] T ,U=diag([|u1| 2 , ..., |u K | 2 ], diag is the diagonal matrix.

[0146] The covariance matrix of r(τ) can be estimated from Y as

[0147]

[0148] in express

[0149]

[0150] Vectorizing R rr have to

[0151] z=vec(R rr )=(B * ⊙B)p

[0152] where p = [|(s R )1| 4 ,|(s R )2| 4 , ..., |(s R ) Q | 4 ] T , B in the qth column * ⊙B can be expressed as

[0153]

[0154] diff1 and diff2 contain the difference elements of the array, and sum1 and sum2 contain all the sum elements of the array. v It is a vector composed of all the lags generated by the above formula. Since there are repeated redundant elements, the repeated elements are excluded, sorted in ascending order, and the central continuous part is extracted to obtain its vector matrix s c The transformation matrix Tr is defined as

[0155]

[0156] Where a=1,2,...,2l+1,b=1,2,...,(2N) 2 , σ represents the Kronecker product.

[0157] So z can be re-expressed as

[0158] z=Trz rr =TR(B * ⊙B)p

[0159] In order to obtain the full-rank virtual covariance matrix, the above formula is reconstructed into a Toeplitz matrix.

[0160]

[0161] The MUSIC algorithm is directly used to solve it and obtain its DOA estimation.

[0162] The present invention first proposes an improved method for sparse arrays, which extends the length of continuous array elements of traditional nested arrays, fills in holes, increases array aperture, expands degrees of freedom, and has higher precision. Then, according to the VCAM algorithm, the sum-difference array is obtained, which has more degrees of freedom and higher precision than ordinary difference arrays. Then, data processing is performed according to the VCAM algorithm to obtain its DOA estimation. Finally, simulation verifies the effectiveness and good precision of the new array proposed by the present invention.

[0163] Simulation 1: In this simulation experiment, a sparse non-uniform array with 10 elements is selected. The array position set is [1, 2, 3, 4, 5, 17, 28, 39, 45, 56]*d, the distance is d=0.5λ, and the incident angle is 0°. The directional pattern received by the array under the VCAM algorithm is simulated to verify that the algorithm can effectively estimate DOA and prove its effectiveness and accuracy. The directional pattern is shown in Figure 4 shown.

[0164] Simulation 2: In this simulation experiment, a sparse non-uniform array with 10 elements was selected. The array position set is [1, 2, 3, 4, 5, 17, 28, 39, 45, 56]*d, the distance is d=0.5λ, the array configuration is consistent with the above experiment 1, the incident angles are 30°, 40°, 50°, snr=10dB, 50 Monte Carlo experiments are performed, and the scatter plot is as follows: Figure 5 shown.

[0165] Simulation 3: This simulation experiment selected a sparse non-uniform array with 10 elements. The array position set is [1, 2, 3, 4, 5, 17, 28, 39, 45, 56]*d, the distance is d=0.5λ, the array configuration is consistent with the above experiment 1, the incident angles are 30°, 40°, 50°, snr=2dB-18dB, and 100 Monte Carlo experiments are performed. The root mean square error and success probability graph are calculated. Figure 6 is the root mean square error diagram, Figure 7 The success probability graph.

[0166] from Figure 6 It can be seen that the root mean square error of the new optimized array proposed in the present invention gradually decreases with the increase of the signal-to-noise ratio. The closer the signal is to the real signal, the more accurate the estimated angle is, which is in line with the simulation logic of the algorithm. Figure 7 This is the success probability diagram of DOA estimation. It can be seen that the success probability increases with the increase of signal-to-noise ratio and gradually tends to 1. The closer the signal is to the real signal, the more accurate the estimated angle is, which is in line with the simulation logic of the algorithm. Combining these two figures can verify the effectiveness and accuracy of the algorithm proposed in this section.

[0167] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modifications to the same effective embodiments using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, same change and modification made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still falls within the scope of the technical solution of the present invention.

Claims

1. A DOA estimation method based on novel array optimization is as follows: Step 1: Improve and arrange the sparse non-uniform array: In order to improve the accuracy of DOA estimation of non-uniform array, a sum-difference array arrangement is adopted to make the central continuous section long enough, with smaller coupling leakage and weaker coupling effect. Take N array elements and divide them into two sub-arrays. The number of array elements in the first sub-array is M1, and the number of array elements in the second sub-array is M2. The number of array elements in the sub-arrays satisfies the following conditions is rounded down; the array position spacing is d and (2M1+1)d respectively, and the corresponding array element position set is U1={d, 2d,…, (M1-1)d}, U2={(M1+1)d, (3M1+2)d,…, ((M1+1)+(2M1+1)M2)d}, d=λ / 2, λ is the signal source wavelength, and the array structure of the basic nested array is shown in Figure 1; The degrees of freedom for this underlying non-uniform array are as follows: DOF = 2(3M1+2)-1 = 6M1-3 The number of continuous array elements of the basic nested array is too small, the array aperture is too small, and the degree of freedom is too small to meet the accuracy requirements of array estimation. The array is disrupted for improvement. First, the first array element of the second subarray is moved to the first subarray to form M3 array elements, and then the last array element is moved to To form an improved subarray, the gaps in the traditional nested array mentioned above are filled, and the length of the continuous part of the array is expanded, so that the number of effective array elements is increased, and the DOA estimation accuracy is increased; the improved array structure arrangement diagram is shown in FIG2; The number of improved sub-arrays meets the following conditions The spacing of the array positions is d, (2M3+1)d, and the corresponding array element position set is U3={d, 2d, ..., M3d}, U4={(3M3+2)d, ..., ((M3+1)+(2M3+1)(M4-1)d}. Move the last array element position to So the array position set is The sum and difference operations are performed on the array element position set S respectively, and the array difference set S d For S d ={d o -d p |d o , d p ∈S, 1≤o, p≤N}, in represents the difference of the array element positions located on the negative semi-axis, Represented as the difference set of the array element positions located on the positive semi-axis It can be expressed as The sum of the array element positions can be expressed as S s ={±(d o +d p )|d o , d p ∈S, 1≤o, p≤N}, in It is represented as the sum of the positions of the array elements located on the negative semi-axis, The sum of the positions of the array elements on the positive semi-axis is represented by It can be expressed as The total continuous virtual array element set S sd for The degrees of freedom of the improved non-uniform array are as follows: From the above degree of freedom formula, it can be obtained that the degree of freedom of the non-uniform array is determined by the size of M1 and M2, so it is necessary to consider the optimal combination of the number of arrays; for the problem of maximum degree of freedom under the constraints of this model, it can be converted into a quadratic function and solved to find the extreme point, that is, the optimal solution, so we get is rounded up; from the above formula, it can be seen that when N ≥ 5, the non-uniform array in this section has higher degrees of freedom and array aperture than the traditional non-uniform array, and the larger the number of array elements, the greater the degrees of freedom, and the greater the advantage over the traditional non-uniform array, the better the performance; Step 2: using the sparse non-uniform array to receive far-field narrow-band incoherent signals; In DOA estimation, the narrowband signal can be expressed as where u k (t) is the amplitude function, is the phase function, ω k =2πf k is the angular frequency, and the signal delay τ k The function after is Because u k (t)≈u k (t-τ k )and Therefore, k (t-τ k ) can be written as Assume that the number of signal sources incident on the matrix is ​​K, and the incident angles are θ1, θ2, …, θ K ,θ k ∈[-π / 2, π / 2], k=1, 2, ..., K, the first array element position is set as the reference array element, then the received data of the mth array element is The above formula is expressed in matrix form through statistics as X(t)=A(θ)S(t)+N(t) where X(t) = [x1(t), x2(t), …, x N (t)] T , A(θ)=[a(θ1), a(θ2),…, a(θ K )] is the array flow matrix, a(θ k ) is the steering vector of the kth signal, N(t) = [n1(t), n2(t), ..., n N (t)] T is a noise matrix with a mean of 0 and a variance of σ 2 ; Step 3: Use the vector conjugate augmented MUSIC algorithm to construct a conjugate augmented vector; First, construct the conjugate augmented correlation vector; collect the output x1(t) of the first array element and the output x of the mth array element respectively. m (t)N p samples, we can get the vector [x1(1), x1(2), ..., x1(N p )] and the vector [x m (1+τ), x m (2+τ),…,x m (N p +τ)], then the time delay correlation function of the vector can be expressed as Where τ≠0; when the number of snapshots N p When ω is large enough and l ≠ω k , It can be expressed as in * is conjugation; combination relationship can be simplified to Vectorization Can get r x (τ)=Ar s (t) in By changing τ to -τ, we can get [r x (-t)] * =A * r s (t) In order to reduce repeated operations, subtract the first element of the array direction vector, that is, Then the above formula is converted to [r′ x (-t)] * =(A′) * r s (t) Combined with x (τ) and [r x (-τ)] * , we get the vectorized conjugate augmented vector Step 4: Construct a pseudo data matrix based on the vector conjugate augmented MUSIC algorithm, perform data processing, and then perform DOA estimation: Assume the pseudo sampling period is T s , the number of pseudo-sampling snapshots is N s , construct the pseudo data matrix of r(τ) as Y=[r(T s ),r(2T s ),...,r(N s T s )]=BUE Where E represents Where B=[b(θ1), b(θ2),…, b(θ K )],b(θ k )=[(a′(θ k )) H , a(θ k ) T ] T ,U=diag([|u1| 2 ,…,|u K | 2 ], diag is the diagonal matrix; The covariance matrix of r(τ) can be estimated from Y as in express Vectorizing R rr have to z=more(R rr )=(B * ⊙B)p where p = [|(s R )1| 4 ,|(s R )2| 4 , ..., |(s R ) Q | 4 ] T , B in the qth column * ⊙B can be expressed as diff1 and diff2 contain the difference elements of the array, sum1 and sum2 contain all the sum elements of the array; let s v It is a vector composed of all the lags generated by the above formula. Since there are repeated redundant elements, the repeated elements are excluded, sorted in ascending order, and the central continuous part is extracted to obtain its vector matrix s c ; The transformation matrix Tr is defined as Where a=1,2,...,2l+1,b=1,2,...,(2N) 2 ,σ represents the Kronecker product; So z can be re-expressed as from=Trz rr =TR(B * ⊙B)p In order to obtain the full-rank virtual covariance matrix, the above formula is reconstructed by Toeplitz matrix; The MUSIC algorithm is directly used to solve it and obtain its DOA estimation.

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