Acoustic vector array DOA estimation method based on quaternion matrix dimensionality reduction

Abstract

The invention aims to provide an acoustic vector array DOA (direction of arrival) estimation method based on quaternion matrix dimensionality reduction, which comprises the following steps of: constructing a CAACIS acoustic vector co-prime array at a receiving end, establishing a received signal quaternion model based on array output, estimating a quaternion covariance matrix, performing dimensionality reduction processing on the quaternion covariance matrix, and estimating the DOA of the acoustic vector array based on quaternion matrix dimensionality reduction. The method comprises the following steps: acquiring a virtual array receiving signal model of an acoustic vector co-prime array, reconstructing a Hermitian-Toeplitz matrix, performing eigenvalue decomposition on the Hermitian-Toeplitz matrix, constructing a spatial spectrum function, and acquiring a DOA estimation result through spectrum peak search. According to the method, the problem of how to realize multi-target DOA estimation with low complexity and good performance based on a quaternion frame under an acoustic vector co-prime array is solved.

Description

Technical Field

[0001] The invention relates to an acoustic vector array DOA estimation method based on quaternion matrix dimensionality reduction, and belongs to the field of sonar signal processing. Background Art

[0002] DOA estimation is an important branch of array signal processing, and plays an important role in underwater target detection based on sonar arrays. With the advancement of marine industry and marine technology, the demand for human exploration of deep seas has driven various underwater platforms to gradually develop in a miniaturized and intelligent manner, which has put forward higher requirements for the design of sonar arrays carried by the platforms. In recent years, a new array structure combining acoustic vector sensors with coprime arrays has gradually become a research hotspot.

[0003] Since the acoustic vector sensor can synchronously and concurrently obtain the sound pressure and vibration velocity information of a certain point in the underwater space, compared with the traditional sound pressure sensor array, the acoustic vector sensor array has higher array gain and spatial resolution with the same number of array elements. In order to make full use of the structural information and multi-channel signal components of the acoustic vector array, researchers proposed to apply quaternion theory to acoustic vector signal modeling, in order to achieve super-resolution DOA estimation of multiple targets under large sparse acoustic vector arrays. At present, a series of super-resolution algorithms based on the quaternion framework have been proposed, and have shown performance superior to the traditional "long vector" method.

[0004] However, when the object of the quaternion model construction involves acoustic vector coprime arrays including CAACIS, we cannot ignore the fact that a similar array structure using only a small number of physical array elements is sufficient to produce a large virtual differential common array, which will multiply the computational complexity of the above quaternion method, which is already not low. Although a few existing algorithms use some estimation methods that do not require eigenvalue decomposition in the quaternion domain to simplify the calculation process at the expense of some performance, all calculations of such methods are essentially still performed in the quaternion domain, and the more complicated spectrum peak search process cannot be avoided.

[0005] Therefore, "how to simplify the quaternion algorithm in essence and significantly reduce the computational complexity while achieving good estimation performance" has become the key to further optimize the DOA estimation algorithm based on acoustic vector coprime arrays. Summary of the invention

[0006] The object of the present invention is to provide a method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction, which has both low complexity and good estimation performance.

[0007] The object of the present invention is achieved in that the steps are as follows:

[0008] Step 1: Construct a CAACIS acoustic vector coprime array at the receiving end;

[0009] Step 2: Establish the quaternion model z of the received signal based on the output of the acoustic vector coprime array q (t):

[0010] z q (t) = [p(t) + v x (t)]+[p(t)+v y (t)]·j

[0011] =A q s(t)+n q (t),

[0012] Among them, p(t), v x (t), v y (t) are the components of the array receiving signal in the sound pressure channel, x-axis, and y-axis direction particle velocity channel respectively; A q ∈H M×K is the quaternion array manifold matrix, n q (t) is the quaternion noise vector; s(t) = [s 1 (t),s 2 (t),···,s K (t)] T is the incident signal vector, indicating that there is a group of K independent far-field narrowband signals from the horizontal direction θ = [θ 1 ,θ 2 ,···,θ k ] T ∈[-π,π] is incident on the array as a plane wave;

[0013] The received signal quaternion model z described in step 2 q (t), the position of the imaginary symbol j in the formula is closely related to the subsequent processing flow and cannot be replaced arbitrarily;

[0014] Step 3: Use z q (t) Estimate the quaternion covariance matrix R q :

[0015]

[0016] in, Represents the quaternion conjugate transpose, R s represents the signal covariance matrix, R nq represents the noise covariance matrix;

[0017] Step 4: Quaternion covariance matrix R q Perform dimensionality reduction; reorganize the matrix and expand it into a complex adjoint matrix Will Reorganize the blocks to get the reduced-dimensional array covariance matrix R iq ;

[0018] Step 5: Obtain the virtual array receiving signal model of the acoustic vector coprime array; vectorize R iq , delete the redundant rows in the obtained vector and arrange them in ascending order according to the index of the virtual array element position to obtain the equivalent virtual array receiving signal

[0019] Step 6: Utilization Reconstructing the Hermitian-Toeplitz matrix

[0020] Step 7: Perform eigenvalue decomposition on the reconstructed Hermitian-Toeplitz matrix:

[0021]

[0022] Among them, U s and U n Represent the signal subspace and the noise subspace respectively, and their corresponding eigenvalue matrices are Λ s and Λ n ; and construct the spatial spectrum function:

[0023]

[0024] Among them, α(θ) is the steering vector corresponding to the virtual array;

[0025] Step 8: Obtain the DOA estimation result through spectrum peak search; traverse θ∈[-π,π] to find the peak of the spatial spectrum, and the angle corresponding to each peak is the final DOA estimation result.

[0026] Further, the construction process of the CAACIS mutual prime acoustic vector array described in step 1 can be described as follows: using a plurality of two-dimensional acoustic vector sensors including a sound pressure channel and two particle velocity channels facing the x-axis and the y-axis respectively to construct an array model; arranging a plurality of the above-mentioned acoustic vector sensors linearly along the y-axis direction to form two columns of uniform linear sub-arrays respectively; the two columns of sub-arrays are respectively composed of M 1 and M 2 The sensor array elements are composed of M 2 d and (M 1 / ρ)d; where d is the half wavelength of the signal and ρ is the compression factor, satisfy and M 2 Mutually prime; the compression factor ρ is a value [2,M 1] is an integer, the selection of which affects the number of additional virtual array elements generated; finally, the No. 1 array elements of the two columns of sub-arrays are aligned and superimposed into a new sparse linear acoustic vector array, namely CAACIS; the CAACIS array contains M = M 1 +M 2 -1 physical array element, if m is used 1 、m 2 Respectively represent the serial numbers of certain array elements in the two columns of sub-arrays, then the set of their physical array element positions p M for

[0027]

[0028] The quaternion model of the received signal described in step 2: q (t) = [p(t) + v x (t)]+[p(t)+v y (t)]·j, the position of the imaginary number symbol j in the formula is closely related to the subsequent processing flow and cannot be replaced arbitrarily.

[0029] The dimensionality reduction process described in step 4 is as follows: first, A q Expand to A q =A 1 +A 2 j, and reorganize the quaternion covariance matrix R q In the following form

[0030]

[0031] Among them, A 1 =A(θ p +Θ x ), A 2 =A(θ p +Θ y ), A 1 ,A 2 ∈C M×K , A represents the original array manifold matrix; Represents the noise power; I M and 0 M Represent the M-dimensional identity matrix and the M-dimensional zero matrix respectively; Θ p , Θ x , Θ y ∈R K×K are all diagonal matrices, satisfying

[0032]

[0033] The sorted R q Expressed as And expand it into complex adjoint matrix form

[0034]

[0035] Complex adjoint matrix Matrix block

[0036]

[0037] Reorganize the block sub-matrices to obtain the reduced-dimensional covariance matrix

[0038]

[0039] In the formula, R iq ∈C M×M It contains all the sound pressure, velocity components and complete array structure information, and the dimension is only the original matrix R q half.

[0040] The Hermitian-Toeplitz matrix described in step 6 It can be obtained by:

[0041]

[0042] in For vector The former M vc The elements are arranged in reverse order, For vector The back M vc elements, M vc is the maximum number of consecutive virtual elements that can be obtained through the array.

[0043] The construction process of the spatial spectrum function described in step 7 is specifically as follows: constructing the MUSIC spatial spectrum function according to the eigenvalue decomposition result is as follows:

[0044]

[0045] Among them, α(θ) is the corresponding steering vector of the virtual array, satisfying

[0046]

[0047] in, λ is the signal wavelength.

[0048] Compared with the prior art, the advantages of the present invention are:

[0049] (1) The present invention utilizes the intrinsic connection between the complex-quaternion mapping and the quaternion covariance matrix structure to halve the dimension of the quaternion covariance matrix without losing the received signal information. This process significantly reduces the computational burden of the DOA estimation algorithm and is particularly effective when processing large virtual differential co-arrays generated by CAACIS arrays.

[0050] (2) At the same time, the rational selection of strategies in the dimensionality reduction process enables the present invention to maintain the additional orthogonal constraints of the quaternion space while reducing the matrix dimensionality, and to demonstrate superior spatial resolution and estimation accuracy over similar super-resolution methods under the same conditions.

[0051] (3) In addition, to address the rank deficiency problem of the equivalent virtual array covariance matrix, the present invention draws on the idea of ​​using single snapshot data to achieve coherent target estimation, reconstructs a Hermitian-Toeplitz matrix, and then realizes multi-target robust DOA estimation under the acoustic vector coprime array without sacrificing the array aperture. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 CAACIS acoustic vector coprime array system: M 1 and M 2 are the number of array elements of subarray 1 and subarray 2 respectively, d is the array element spacing, and ρ is the compression factor;

[0053] Figure 2 It is a DOA estimation flow chart of the present invention;

[0054] Figure 3 The spatial spectrum of each method (in the figure, ICQ-MUSIC is the method proposed by the present invention, CLV-MUSIC and ACQ-MUSIC are existing methods, the same below): SNR = 0 dB, number of snapshots L = 300, 9 independent targets;

[0055] Figure 4 The target resolution probability of each method with the angle interval as a variable: SNR = 0 dB, number of snapshots L = 300, 1000 Monte Carlo experiments;

[0056] Figure 5 is the computational complexity of each method with the number of physical array elements as the variable: number of snapshots L = 500, number of spectrum peak search points N = 2000. DETAILED DESCRIPTION

[0057] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0058] Combination Figure 1-5 , the steps of the present invention are as follows:

[0059] (1) Figure 1 As shown, M=M is ​​used at the receiving end. 1 +M 2 -1 two-dimensional acoustic vector sensor to construct CAACIS acoustic vector coprime array;

[0060] (2) Establish the quaternion model of the received signal based on the output of the acoustic vector coprime array q (t):

[0061] z q (t) = [p(t) + v x (t)]+[p(t)+v y (t)]·j

[0062] =A q s(t)+n q (t), where p(t), v x (t), v y (t) are all i-complex vectors, representing the components of the array receiving signal in the sound pressure channel, x-axis, and y-axis directions of the particle velocity channel; j is an imaginary number, i and j are orthogonal to each other; A q ∈H M×K is the quaternion array manifold matrix, n q (t) is the quaternion noise vector; s(t) = [s 1 (t),s 2 (t),···,s K (t)] T is the incident signal vector, indicating that there is a group of K independent far-field narrowband signals from the horizontal direction θ = [θ 1 ,θ 2 ,···,θ k ] T ∈[-π,π] is incident on the array as a plane wave;

[0063] (3) Using z q (t) Estimate the quaternion covariance matrix R q :

[0064]

[0065] in, Represents the quaternion conjugate transpose, R s represents the signal covariance matrix, R nq represents the noise covariance matrix;

[0066] (4) Quaternion covariance matrix R q Perform dimensionality reduction processing; first, A q Expand to A q =A 1 +A 2 j, and reorganize the quaternion covariance matrix R q In the following form

[0067]

[0068] Among them, IM represents the M-dimensional identity matrix, 0 M represents an M-dimensional zero matrix; represents the noise power; and A 1 =A(θ p +Θ x ), A 2 =A(θ p +Θ y ), A 1 ,A 2 ∈C M×K , A represents the original array manifold matrix; Θ p , Θ x , Θ y ∈R K×K are all diagonal matrices, satisfying

[0069]

[0070] The sorted R q Expressed as And expand it into complex adjoint matrix form

[0071]

[0072] Complex adjoint matrix Matrix block

[0073]

[0074] Reorganize the block sub-matrices to obtain the reduced-dimensional covariance matrix

[0075]

[0076] In the formula, R iq ∈C M×M It contains all the sound pressure, velocity components and complete array structure information, and the dimension is only the original matrix R q half of;

[0077] (5) Obtain the virtual array receiving signal model of the acoustic vector coprime array; first, vectorize R iq :

[0078]

[0079] in, represents the Kronecker product, ⊙ represents the Khatri-Rao product; h = vec{I M}, vec{·} represents the vectorization operation; g is the single-snap equivalent received signal vector; vector r is extracted according to the index information of the virtual array element position iq, delete the redundant rows, and rearrange them in ascending order according to the virtual array element numbers to obtain the equivalent virtual array receiving signal

[0080]

[0081] in, represents the position of the elements of the rearranged continuous virtual array, and λ represents the half wavelength of the signal; M vc is the maximum number of continuous virtual array elements that can be obtained through the array;

[0082] (6)Use Reconstructing the Hermitian-Toeplitz matrix

[0083]

[0084] in For vector The former M vc The elements are arranged in reverse order, For vector The back M vc elements, M vc =M 1 M 2 -M 1 (M 2 -1) / ρ,M 1 and M 2 are the number of sub-array elements respectively, ρ is the compression factor;

[0085] (7) Perform eigenvalue decomposition on the reconstructed Hermitian-Toeplitz matrix:

[0086]

[0087] Among them, U s and U n Represent the signal subspace and the noise subspace respectively, and their corresponding eigenvalue matrices are Λ s and Λ n ; and construct the spatial spectrum function:

[0088]

[0089] Among them, α(θ) is the corresponding steering vector of the virtual array, satisfying

[0090]

[0091] in,

[0092] (8) Obtain the DOA estimation result through spectrum peak search; traverse θ∈[-π,π] to find the peak of the spatial spectrum, and the angle corresponding to each peak is the final DOA estimation result.

[0093] Figure 2 The process of DOA estimation is described. Since the present invention performs dimensionality reduction processing on the quaternion covariance matrix based on the complex-quaternion mapping characteristics, the matrix dimension is reduced to half of the original without losing the received signal information, and the rank of the virtual covariance matrix is ​​restored by using the Hermitian-Toeplitz matrix reconstruction technology, it is possible to achieve multi-target DOA estimation with low complexity and good estimation performance under the acoustic vector coprime array such as CAACIS.

[0094] It should be pointed out that the method of "using the complex-quaternion mapping characteristics to reduce the dimension of the quaternion covariance matrix" proposed in the present invention has a certain universality and can be considered as a part of the data modeling process. Therefore, this method is applicable to various types of acoustic vector linear arrays (the present invention takes the CAACIS acoustic vector coprime array as an example), and can also be combined with a variety of DOA estimation methods (this article takes MUSIC as an example) to expand the application scenarios of the method or improve the performance. Without departing from the technical principles of the present invention, technicians in this field can also make several variations of the method (for example, applying the method to other similar common acoustic vector arrays, or combining the method with other existing algorithms), and these variations should also be within the scope of protection of the present invention.

[0095] Application examples of the present invention:

[0096] Figure 3 , Figure 4 , Figure 5 The performance and computational complexity of the proposed method (ICQ-MUSIC) and the existing methods (CLV-MUSIC, ACQ-MUSIC) are compared. In the above simulations, it is assumed that: signals from different directions arrive at the array in the form of plane waves at the same time and remain independent of each other; the sensitivity of each receiving array element is the same and does not interfere with each other, and has an equal signal-to-noise ratio for any signal; the sound pressure and particle velocity components inside the array element are independent of each other. Unless otherwise specified, all methods are compared under exactly the same conditions.

[0097] Figure 3 The spatial spectra of 9 independent targets estimated by each method under the conditions of SNR = 0 dB and snapshot number L = 300 are shown. Although the three methods can accurately estimate all preset target azimuths, the ICQ-MUSIC method proposed in the present invention has the sharpest spectrum peak.

[0098] Figure 4 The target resolution probability of each method based on 1000 Monte Carlo experiments is shown, where SNR = 0 dB and the number of snapshots L = 300. At different angle intervals, the ICQ-MUSIC method has the highest target resolution probability.

[0099] Figure 5 The computational complexity of each method with the number of physical array elements as a variable is shown under the conditions of the number of snapshots L = 500 and the number of spectrum peak search points N = 2000. It can be found that the ICQ-MUSIC method proposed in the present invention has a very significant advantage in terms of computational complexity.

[0100] The above simulation results show that compared with the existing similar methods, the method proposed in the present invention has significantly reduced computational complexity while also achieving certain advantages in estimation accuracy and spatial resolution, thereby achieving efficient multi-target DOA estimation under the acoustic vector array.

Claims

1. A method for estimating DOA of acoustic vector array based on quaternion matrix dimensionality reduction, characterized in that: The following steps are included: Step 1: Construct a CAACIS acoustic vector coprime array at the receiving end; Step 2: Establish the quaternion model z of the received signal based on the output of the acoustic vector coprime array q (t): z q (t)=[p(t)+v x (t)]+[p(t)+v y (t)]·j =A q s(t)+n q (t), Among them, p(t), v x (t), v y (t) are the components of the array receiving signal in the sound pressure channel, x-axis, and y-axis direction particle velocity channel respectively; A q ∈H M×K is the quaternion array manifold matrix, n q (t) is the quaternion noise vector; s(t) = [s1(t), s2(t), …, s K (t)] T is the incident signal vector, indicating that there is a group of K independent far-field narrowband signals from the horizontal direction θ=[θ1,θ2,…,θ k ] T ∈[-π,π] is incident on the array as a plane wave; Step 3: Use z q (t) Estimate the quaternion covariance matrix R q : in,(·) Δ Represents the quaternion conjugate transpose, R s represents the signal covariance matrix, R nq represents the noise covariance matrix; Step 4: Quaternion covariance matrix R q Perform dimensionality reduction; reorganize the matrix and expand it into a complex adjoint matrix Will Reorganize the blocks to get the reduced-dimensional array covariance matrix R iq ; Step 5: Obtain the virtual array receiving signal model of the acoustic vector coprime array; vectorize R iq , delete the redundant rows in the obtained vector and arrange them in ascending order according to the index of the virtual array element position to obtain the equivalent virtual array receiving signal Step 6: Utilization Reconstructing the Hermitian-Toeplitz matrix Step 7: Perform eigenvalue decomposition on the reconstructed Hermitian-Toeplitz matrix: Among them, U s and U n Represent the signal subspace and the noise subspace respectively, and their corresponding eigenvalue matrices are Λ s and Λ n ; and construct the spatial spectrum function: Among them, α(θ) is the steering vector corresponding to the virtual array; Step 8: Obtain the DOA estimation result through spectrum peak search; traverse θ∈[-π,π] to find the peak of the spatial spectrum, and the angle corresponding to each peak is the final DOA estimation result.

2. The method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction according to claim 1, characterized in that: The specific method of step one is: An array model is constructed by using a number of two-dimensional acoustic vector sensors including a sound pressure channel and two particle velocity channels facing the x-axis and the y-axis respectively; Arrange multiple acoustic vector sensors linearly along the y-axis direction to form two uniform linear sub-arrays; the two sub-arrays are composed of M1 and M2 sensor array elements, respectively, and the array element spacing is M2d and (M1 / ρ)d, respectively; where d is the half wavelength of the signal, ρ is the compression factor, and satisfy and M2 are mutually prime; the compression factor ρ is an integer with a value of [2, M1], and its selection affects the number of additional virtual array elements generated; Finally, the No. 1 array elements of the two columns of sub-arrays are aligned and superimposed into a new sparse linear acoustic vector array, namely CAACIS; the CAACIS array contains M = M1 + M2-1 physical array elements. If m1 and m2 are used to represent the sequence numbers of an array element in the two columns of sub-arrays, then the set of its physical array element positions p M for:

3. The method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction according to claim 1, characterized in that: The received signal quaternion model z described in step 2 q (t), the position of the imaginary symbol j in the formula is closely related to the subsequent processing flow and cannot be replaced arbitrarily.

4. The method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction according to claim 1, characterized in that: The dimensionality reduction process described in step 4 is as follows: first, A q Expand to A q =A1+A2j, and reorganize the quaternion covariance matrix R q In the following form: Where A1=A(θ p +Θ x ), A2=A(Θ p +Θ y ), A1, A2∈C M×K , A represents the original array manifold matrix; Represents the noise power; I M and 0 M Represent the M-dimensional identity matrix and the M-dimensional zero matrix respectively; Θ p , Θ x , Θ y ∈R K×K Are all diagonal matrices, satisfying: The sorted R q Expressed as And expand it into complex adjoint matrix form: Complex adjoint matrix Divide the matrix into blocks: Reorganize the block sub-matrices to obtain the reduced-dimensional covariance matrix: In the formula, R iq ∈C M×M It contains all the sound pressure, velocity components and complete array structure information, and the dimension is only the original matrix R q half.

5. The method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction according to claim 1, characterized in that: The Hermitian-Toeplitz matrix described in step 6 Obtained by: in For vector The former M vc The elements are arranged in reverse order, For vector The back M vc elements, M vc is the maximum number of consecutive virtual elements that can be obtained through the array.

6. The method for estimating DOA of an acoustic vector array based on quaternion matrix dimensionality reduction according to claim 1, characterized in that: The construction process of the spatial spectrum function described in step 7 is specifically as follows: constructing the MUSIC spatial spectrum function according to the eigenvalue decomposition result is as follows: Among them, α(θ) is the steering vector corresponding to the virtual array, satisfying: in, λ is the signal wavelength.

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