A four-order sparse array design method based on sum-difference analysis

Abstract

The application relates to a four-order sparse array design method based on sum-difference analysis, which comprises the following steps: setting two sparse arrays at will, calculating the sum subarray and the difference subarray of the two sparse arrays; obtaining a four-order sparse array from the sparse array and; generating continuous virtual array elements from the four-order sparse array through four-order cumulants and four-order differences; and the method realizes the longer continuous part of the four-order difference subarray through the expansion and shift of the subarray, thereby significantly improving the number of degrees of freedom of the array.

Description

Technical Field

[0001] This invention relates to the field of array design technology, and in particular to a fourth-order sparse array design method based on sum-difference analysis. Background Technology

[0002] Array signal processing refers to assembling a sensor array consisting of several sensors arranged in a specific pattern and distributed at different locations in space. All elements in the sensor array sense signals from space and then perform specific processing on them. The most commonly used sensor array is the traditional uniform linear array (ULA), where the spacing between elements is constant and does not exceed half a wavelength to avoid spatial aliasing. For a uniform linear array (ULA) with N sensors, traditional subspace-based array signal processing methods can resolve at most N-1 signal sources. To resolve more signal sources, additional sensors need to be added, significantly increasing cost and system complexity.

[0003] The introduction of sparse arrays has provided a new direction for array design. Compared to traditional uniform arrays, sparse arrays, due to their high degrees of freedom and low mutual coupling, have been widely used in array signal processing. Current technologies typically employ sparse arrays designed based on second-order differential arrays for array signal processing. Specifically, the method involves vectorizing the covariance matrix of the array signal received by the sparse array to obtain a single-snapshot virtual received data. This virtual received data is equivalent to the array signal received by the original array's virtual differential array. Compared to a ULA with the same number of array elements, a sparse array designed based on a second-order differential array can obtain at most N... 2 With each degree of freedom, the number of signal sources that can be resolved will increase. However, for array design technology, the number of signal sources that can be resolved still does not meet the requirements, and the number of degrees of freedom needs to be improved. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a fourth-order sparse array design method based on sum-difference analysis that can significantly increase the number of degrees of freedom of the array.

[0005] The technical solution adopted in this invention is a fourth-order sparse array design method based on sum-difference analysis, which includes the following steps:

[0006] S1. Arbitrarily set two sparse arrays, and the positions of the array elements of the two sparse arrays are as follows: Calculate the sum matrix and difference matrix of the two sparse arrays, where the sum matrix is ​​represented as: The difference matrix is ​​represented as follows: Where i = 1, 2, and a and b both represent arbitrary array elements in the sparse array;

[0007] S2, Describe two sparse arrays and The continuous lengths of the sum and difference matrices are λ1 and λ2, respectively, and the continuous lengths of the difference matrices are μ1 and μ2, respectively. If μ1 ≥ λ1, then the sparse array... and A fourth-order sparse array is constructed, and the element position expression of the fourth-order sparse array is as follows: in

[0008] S3. The fourth-order sparse array obtained in step S2 generates continuous virtual array elements through fourth-order cumulants and fourth-order differences. The range of the virtual array elements is [-L, L], and the continuous degrees of freedom is 2L+1.

[0009]

[0010] The beneficial effects of this invention are as follows: The above-mentioned fourth-order sparse array design method based on sum-difference analysis realizes a longer continuous part of the fourth-order difference array by expanding and shifting the subarrays, thereby significantly increasing the number of degrees of freedom of the array; the array element positions of the proposed fourth-order sparse array have a closed expression and can be generated by many commonly used second-order sparse arrays; in addition, thanks to the expansion of the subarray spacing, the mutual coupling effect of the array designed by the proposed fourth-order sparse array design method based on sum-difference analysis is also significantly improved.

[0011] Preferably, in step S3, the specific process of generating continuous virtual array elements from the fourth-order sparse array obtained in step S2 through fourth-order cumulants and fourth-order differences includes the following steps:

[0012] S3.1. Assume D incoherent far-field narrowband targets are incident on the array. The received signal of the array element at time k is represented as x(k) = As(k) + n(k), where A represents the array manifold matrix, which consists of D steering vectors, i.e.: A = [a(θ1) a(θ2) … a(θ2) ... a(θ3) ... a(θ4) ... a(θ5) ... a(θ6) ... a(θ7) ... a(θ8) ... a(θ9 ... D )],a(θ i Let s(k) represent the steering vector of the i-th signal source, and s(k) and n(k) represent the sampled values ​​of the source vector and noise vector, respectively. The fourth-order cumulant of the received signal is represented in matrix form as follows: in, Represents the i-th signal source s i The fourth-order self-accumulator of (k), cum{·} is the cumulative operator; N represents u ×1 dimensional vector; Indicates Kronecker product u-1 times in total; [] H Indicates conjugate transpose;

[0013] S3.2, convert the fourth-order cumulant matrix C 4,x (u) Vectorization yields the vectorized fourth-order cumulant data: z = vec{C 4,x (u)}=V(θ)p, taking p as the equivalent signal vector, that is: Treat V(θ) as the equivalent array manifold matrix, i.e.: V(θ)=[v(θ1) v(θ2) … v(θ) D )],in,

[0014] S3.3, Define the fourth-order difference matrix as: in, Represents a second-order difference matrix. This represents the set of physical array element positions, and continuous virtual array elements are generated using the fourth-order difference array. Attached Figure Description

[0015] Figure 1 This is a flowchart illustrating a fourth-order sparse array design method based on sum and difference analysis according to the present invention.

[0016] Figure 2 The sparse array based on the fourth-order difference matrix design in this invention is in A schematic diagram of the array element positions at that time;

[0017] Figure 3 This diagram illustrates the comparison of the root mean square error (RMSE) versus signal-to-noise ratio (SNR) curves when performing DOA estimation on a fourth-order sparse array designed using the method of this invention with three other commonly used fourth-order sparse arrays. Detailed Implementation

[0018] The invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can implement it based on the description. The scope of protection of the invention is not limited to these specific embodiments.

[0019] This invention relates to a fourth-order sparse array design method based on sum-difference analysis, the method comprising the following steps:

[0020] S1. Determine the unit length between array elements, where the unit length between array elements is denoted by d. λ represents the wavelength of the incident signal incident on the array, and the positions of the array elements are all normalized by a unit length d; two sparse arrays are arbitrarily set, and the positions of the array elements of the two sparse arrays are as follows: Calculate the sum matrix and difference matrix of the two sparse arrays, where the sum matrix is ​​represented as: The difference matrix is ​​represented as follows: Where i = 1, 2, and a and b both represent arbitrary array elements in the sparse array;

[0021] S2, Describe two sparse arrays and The continuous lengths of the sum and difference matrices are λ1 and λ2, respectively, and the continuous lengths of the difference matrices are μ1 and μ2, respectively. If μ1 ≥ λ1, then the sparse array... and The constructed fourth-order sparse array has the following element position expression: in

[0022] S3. The fourth-order sparse array obtained in step S2 generates continuous virtual array elements through fourth-order cumulants and fourth-order differences. The range of the virtual array elements is [-L, L], and the continuous degrees of freedom is 2L+1.

[0023]

[0024] The aforementioned fourth-order sparse array design method based on sum-difference analysis achieves a longer continuous portion of the fourth-order difference array by expanding and shifting subarrays, thus significantly increasing the number of degrees of freedom. The element positions of the proposed fourth-order sparse array have a closed-form expression and can be generated from many commonly used second-order sparse arrays. Furthermore, thanks to the expansion of subarray spacing, the mutual coupling effect of the array designed by the proposed sum-difference analysis-based fourth-order sparse array design method is also significantly improved. The sparse array designed based on fourth-order cumulants and fourth-order difference arrays can further increase the number of degrees of freedom, achieving up to N... 4 There are several degrees of freedom. As the number of array elements increases, the gap in degrees of freedom will widen further.

[0025] Preferably, in step S3, the specific process of generating continuous virtual array elements from the fourth-order sparse array obtained in step S2 through fourth-order cumulants and fourth-order differences includes the following steps:

[0026] S3.1. Assume D incoherent far-field narrowband targets are incident on the array. The received signal of the array element at time k is represented as x(k) = As(k) + n(k), where A represents the array manifold matrix, which consists of D steering vectors, i.e.: A = [a(θ1) a(θ2) … a(θ2) ... a(θ3) ... a(θ4) ... a(θ5) ... a(θ6) ... a(θ7) ... a(θ8) ... a(θ9 ... D )],a(θ i) represents the steering vector of the i-th signal source, determined by the physical array element position and angle of arrival. s(k) and n(k) represent the sampled values ​​of the source vector and noise vector, respectively. The fourth-order cumulant of the received signal is represented in matrix form as follows: in, Represents the i-th signal source s i The fourth-order self-accumulator of (k), cum{·} is the cumulative operator; N represents u ×1 dimensional vector; Indicates Kronecker product u-1 times in total; [] H Indicates conjugate transpose;

[0027] S3.2, convert the fourth-order cumulant matrix C 4,x (u) Vectorization (independent of u) yields the vectorized fourth-order cumulant data: z = vec{C 4,x (u)}=V(θ)p, taking p as the equivalent signal vector, that is: Treat V(θ) as the equivalent array manifold matrix, i.e.: V(θ)=[v(θ1) v(θ2) … v(θ) D )],in,

[0028] S3.3, Define the fourth-order difference matrix as: in, Represents a second-order difference matrix. Let V(θ) represent the set of physical array element positions. Analyzing the expression of V(θ), we can see that V(θ) is still composed of D steering vectors, and the i-th steering vector is obtained by performing the Kronecker product on the i-th steering vector of the original physical array, i.e. Mathematical calculations show that v(θ) i If V(θ) corresponds exactly to the steering vector of the fourth-order difference array, then V(θ) corresponds to the array manifold matrix of the fourth-order difference array; therefore, the vectorized fourth-order cumulant z = V(θ)p can be regarded as a single snapshot data received by the virtual array (fourth-order difference array); by using the longer continuous segments in the fourth-order difference array for DOA estimation, a much higher degree of freedom than the physical array can be achieved, and more signal sources can be analyzed.

[0029] The following simulation experiments illustrate the superiority of the fourth-order sparse array design method based on sum-difference analysis proposed in this invention.

[0030] In the simulation experiments, we compared the proposed fourth-order sparse array (SD-FODC) based on sum-difference analysis with three commonly used fourth-order sparse arrays. The three commonly used fourth-order sparse arrays are a four-layer nested array (FL-NA), a simplified and improved four-layer nested array (SE-FL-NA), and a fourth-order fractal array (FO-Fractal). We set the number of physical array elements to N=9, the number of incoherent sources to K=16, the incident angle to be uniformly distributed between -60 degrees and 60 degrees, the number of snapshots to 12500, and the signal-to-noise ratio (SNR) in 5 dB increments. We tested the estimation performance of each fourth-order sparse array from -5 dB to 20 dB. For each SNR, we performed 1000 Monte Carlo simulations and calculated the final root mean square error (RMSE). The DOA estimation algorithm uniformly adopted the smoothed spatial MUSIC algorithm, and the results are shown below. Figure 3 As shown. From Figure 3 As can be seen, under all signal-to-noise ratio conditions, the SD-FODC array proposed in this invention has the lowest root mean square error and the best performance.

Claims

1. A fourth-order sparse array design method based on sum-difference analysis, characterized in that: The method includes the following steps: S1. Determine the unit length between array elements, wherein the unit length between array elements is... express, , This represents the wavelength of the incident signal incident on the array, and the positions of the array elements are all divided by a unit length. Normalization; two sparse arrays are arbitrarily defined, and the element positions of the two sparse arrays are as follows: , Calculate the sum matrix and difference matrix of the two sparse arrays, where the sum matrix is ​​represented as: The difference matrix is ​​represented as: ,in, , a and b Both represent any array element in a sparse array; S2, Describe two sparse arrays and The continuous lengths of the sum and subarrays are respectively and The continuous lengths of the difference matrix are respectively and ,like Then it is composed of sparse arrays and A fourth-order sparse array is constructed, and the element position expression of the fourth-order sparse array is as follows: ,in , ; S3. The fourth-order sparse array obtained in step S2 generates continuous virtual array elements through fourth-order cumulants and fourth-order differences. The range of the virtual array elements is... The continuous degrees of freedom are ,in, .

2. The fourth-order sparse array design method based on sum-difference analysis according to claim 1, characterized in that: In step S3, the specific process of generating continuous virtual array elements from the fourth-order sparse array obtained in step S2 through fourth-order cumulants and fourth-order differences includes the following steps: S3.1, Settings D An incoherent far-field narrowband target is incident on the array, k The received signal of the time array element is represented as ,in Represents an array manifold matrix, by D It consists of several guide vectors, namely: , Indicates the first i The steering vector of each signal source, and Let represent the sampled values ​​of the source vector and noise vector, respectively. The fourth-order cumulant of the received signal is represented in matrix form as follows: ,in, Indicates the first i One signal source The fourth-order self-accumulator, , For cumulative quantity operators; express dimensional vector; Indicates Kronecker product Common use u -1 time; Indicates conjugate transpose; S3.2, convert the fourth-order cumulant matrix Vectorization yields the vectorized fourth-order cumulant data: ,Will As an equivalent signal vector, that is: ,Will As an equivalent array manifold matrix, that is: ,in, ; S3.3, Define the fourth-order difference matrix as: ,in, Represents a second-order difference matrix. This represents the set of physical array element positions, and continuous virtual array elements are generated using the fourth-order difference array.

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