Co-prime array design method based on virtual array hole filling

Abstract

The invention belongs to the technical field of array signal processing, and particularly relates to a co-prime array design method based on virtual array hole filling, which comprises the following steps of: S1, forwards moving two sub-arrays of an extended co-prime array by a distance of Nd / 2; s2, turning over one sub-array of the shifted co-prime array to obtain an improved co-prime array; s3, generating a differential virtual array according to the improved co-prime array; s4, introducing and combining the array to carry out hole filling on the differential virtual array; s5, calculating a received signal model and a covariance matrix of the filled virtual array; and S6, DOA estimation is carried out by using an ROOT-MUSIC algorithm. The method has the advantages that better signal source resolution and lower array element mutual coupling are achieved, meanwhile, the continuous freedom degree is improved, more signal sources can be estimated at the same time, and high practical value is achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of array signal processing, in particular to a kind of based on virtual array hole filling coprime array design method. BACKGROUND

[0002] In the field of array signal processing, the direction of arrival (DOA) estimation has always been an important research direction. DOA estimation has been widely studied and applied in many fields such as wireless communication, radar, sonar, electronic countermeasures, etc. Among them, the DOA estimation technology based on uniform linear array has been the most commonly used array DOA estimation method for a long time due to its mature theoretical system and simple array structure. However, with the continuous improvement of the performance requirements of DOA estimation in related fields, the disadvantages of uniform linear array have become increasingly prominent. On the one hand, uniform linear array has a natural limitation in terms of degrees of freedom, and cannot achieve underdetermined estimation of incident signals; on the other hand, this array structure has obvious shortcomings in terms of hardware cost and implementation difficulty. Therefore, in recent years, research on coprime arrays has become increasingly widespread. Coprime arrays can significantly reduce the number of array elements while maintaining estimation performance, and the virtual array expanded in some way can increase the continuous degrees of freedom compared to the uniform linear array. However, due to the existence of holes in the virtual array itself, there are still problems such as low array element utilization and limited angle resolution. SUMMARY

[0003] (I) Technical problems to be solved

[0004] The present application patent aims to solve the above problems by shifting, flipping and other operations on the extended coprime array to obtain an improved coprime array structure. Then, by differentially expanding the improved coprime array into a virtual array, introducing a joint array into the virtual array, partial hole filling can be achieved to increase the continuous degrees of freedom. Finally, the ROOT-MUSIC algorithm is used for DOA estimation, which can effectively improve the degrees of freedom and resolution of DOA estimation while maintaining low computational complexity.

[0005] (II) Technical solutions

[0006] To achieve the above purpose, the present application specifically adopts the following technical solutions:

[0007] A coprime array design method based on virtual array hole filling, comprising the following steps:

[0008] S1: Shift the two sub-arrays of the extended coprime array by a distance of Nd / 2 in the positive direction;

[0009] S2: Flip one of the shifted coprime arrays to obtain an improved coprime array;

[0010] S3: generating a differential virtual array according to the improved coprime array;

[0011] S4: introducing and combining arrays to fill holes in the differential virtual array;

[0012] S5: calculating a filled virtual array receiving signal model and a covariance matrix;

[0013] S6: performing DOA estimation using the ROOT-MUSIC algorithm.

[0014] Further, the specific content of S1 is to move the two sub-arrays of the extended coprime array forward by Nd / 2 distance;

[0015] The element position of the sub-array 1 of the shifted extended coprime array is T1={(mN+N / 2)d / 0≤m≤2M-1};

[0016] The element position of the sub-array 2 of the shifted extended coprime array is T2={(nM+N / 2)d / 0≤n≤N-1};

[0017] The element position of the shifted extended coprime array can be expressed as:

[0018] T=T1UT2={(mN+N / 2)d / 0≤m≤2M-1}U{(nM+N / 2)d / 0≤n≤N-1}.

[0019] Further, the specific content of S2 is to flip one of the sub-arrays of the shifted coprime array to obtain an improved coprime array;

[0020] The element position of the sub-array 1 of the improved coprime array is TF1={(mN+N / 2)d / 0≤m≤2M-1};

[0021] The element position of the sub-array 2 of the improved coprime array is TF2={-(nM+N / 2)d / 0≤n≤N-1};

[0022] The element position of the improved coprime array can be expressed as:

[0023] TF=TF1UTF2={(mN+N / 2)d / 0≤m≤2M-1}U{-(nM+N / 2)d / 0≤n≤N-1}.

[0024] Further, the specific content of S3 is to generate a differential virtual array according to the improved coprime array;

[0025] The element position expression of the differential virtual array is:

[0026] TF CDA= ± {(Mn + Nm + N)d / 0≤n≤N-1, 0≤m≤2M-1} U ± {M(n1 + n2)d / 0≤n1, n2≤N-1} U ± {N(m1 - m2)d | 0≤m1, m2≤2M-1} ;

[0027] Further, the specific content of S4 is to introduce and combine array to fill the hole of differential virtual array.

[0028] And the expression of combined array element position is:

[0029] TF SCA = ± {(Mn + Nm + N)d / 0≤n≤N-1, 0≤m≤2M-1} U ± {M(n1 + n2)d / 0≤n1, n2≤N-1} U ± {N(m1 - m2)d | 0≤m1, m2≤2M-1} ;

[0030] Calculate the continuous range of the filled virtual array elements, which is continuous in the range of {-(2MN + M + N - 1), (2MN + M + N - 1)}.

[0031] Further, the specific content of S5 is to calculate the filled virtual array received signal model and the covariance matrix.

[0032] The calculation method is: assuming that there are K incident signals of different directions θ incident to the improved coprime array, the virtual array received data expression is:

[0033] X(t) = A(θ)S(t) + n(t)

[0034] In the formula, S(t) = [S1(t), S2(t), …, S K (t)] T , …, S K K(t)] represents the incident signal vector, n(t) is an additive Gaussian white noise vector, A(θ) = [a(θ1), a(θ2), …, a(θ K )] is an array manifold matrix, and the direction vector a(θ j2 ) = [e π(2MN+M+N-1)d j2π(2MN+N-1)d ,..., e -j2π(2MN+N-1)d -j2π(2MN+M+N-1)d ,..., e T ];

[0035] The covariance matrix can be calculated from the received data expression, and the covariance matrix expression is:

[0036]

[0037] Further, the specific content of S6 is to use the ROOT-MUSIC algorithm for DOA estimation.​

[0038] Eigenvalue decomposition is performed on R obtained in S5 to obtain signal subspace U S and noise subspace U N Since each column vector in A(θ) is orthogonal to the noise subspace, a spatial spectrum function can be constructed:

[0039]

[0040] By replacing the denominator part of the spatial spectrum, we can obtain:

[0041]

[0042] In the formula, p(z) is a 2(2MN+M+N-1) order polynomial, which contains 2(2MN+M+N-1) roots that are mirror images of each other relative to the unit circle, and the K roots closest to the unit circle are taken to determine the signal direction of arrival estimate:

[0043]

[0044] (Three) beneficial effects

[0045] Compared with the prior art, the present application provides a kind of design method of relatively prime array based on virtual array hole filling, with the following beneficial effects:

[0046] (1) the present application applies joint array to virtual array hole filling, constructs continuous uniform linear array, increases continuous degree of freedom, can estimate more signal sources simultaneously;

[0047] (2) the present application improves array aperture by subarray flip array transformation means, improves DOA estimation angle resolution ability, is more practical;

[0048] (3) the present application adopts ROOT-MUSIC algorithm to estimate DOA, thereby greatly reducing the calculation complexity; DETAILED DESCRIPTION

[0049] Figure 1 The present application provides an implementation flowchart of the design method of relatively prime array based on virtual array hole filling;

[0050] Figure 2 The structure diagram of the improved relatively prime array designed by the present application;

[0051] Figure 3 DOA estimation result of the method described in the present application;

[0052] Figure 4 ​A comparison chart of the improved coprime array designed by the present application and other coprime arrays in terms of MUSIC spectrum;

[0053] Figure 5 A chart of test results of the angular resolution of the method of the present application;

[0054] Figure 6 A simulation chart of the relationship between the root mean square error of DOA estimation and the signal-to-noise ratio of the improved coprime array designed by the present application and other coprime arrays;

[0055] Figure 7 A simulation chart of the relationship between the root mean square error of DOA estimation and the number of snapshots of the improved coprime array designed by the present application and other coprime arrays; DETAILED DESCRIPTION

[0056] The present application can be implemented in many different forms, and the embodiments described below are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present application. Therefore, the following detailed description of the embodiments of the present application provided in the accompanying drawings is not intended to limit the scope of the claimed patent, but only to represent selected embodiments of the present application. Based on the embodiments in the present application, other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present application.

[0057] EMBODIMENT

[0058] As shown in the figure, an embodiment of the present application proposes a coprime array design method based on virtual array hole filling, comprising the following steps: Figures 1-7

[0059] S1: moving two sub-arrays of the extended coprime array forward by a distance of Nd / 2;

[0060] The element position of the sub-array 1 of the shifted extended coprime array: T1={(mN+N / 2)d / 0≤m≤2M-1};

[0061] The element position of the sub-array 2 of the shifted extended coprime array: T2={(nM+N / 2)d / 0≤n≤N-1};

[0062] The element position of the shifted extended coprime array can be expressed as:

[0063] T=T1UT2={(mN+N / 2)d / 0≤m≤2M-1}U{(nM+N / 2)d / 0≤n≤N-1}.

[0064] ​S2: Flip one of the shifted subarrays to obtain an improved coprime array;

[0065] Element positions of the subarray 1 of the improved coprime array: TF1 = {(mN + N / 2)d / 0≤m≤2M-1};

[0066] Element positions of the subarray 2 of the improved coprime array: TF2 = {-(nM + N / 2)d / 0≤n≤N-1};

[0067] All element positions of the improved coprime array can be expressed as:

[0068] TF = TF1 U TF2 = {(mN + N / 2)d / 0≤m≤2N-1} U {-(nM + N / 2)d / 0≤n≤N-1}.

[0069] S3: Generate a differential virtual array according to the improved coprime array;

[0070] The expression of the element positions of the differential virtual array is:

[0071] TF DCA = ±{(Mn + Nm + N)d / 0≤n≤N-1,0≤m≤2M-1} U ±{M(n1-n2)d / 0≤n1,n2≤N-1}∪±{N(m1-m2)d|0≤m1,m2≤2M-1};

[0072] S4: Introduce and combine arrays to fill holes in the differential virtual array;

[0073] The expression of the element positions of the sum-union array is:

[0074] TF SCA = ±{(Mn - Nm)d / 0≤n≤N-1,0≤m≤2M-1} U ±{M(n1+n2)d / 0≤n1,n2≤N-1}∪±{-N(m1+m2)d|0≤m1,m2≤2M-1};

[0075] Calculate the continuous range of the filled virtual array elements, which is continuous in the range {-(2MN + M + N - 1), (2MN + M + N - 1)}.

[0076] S5: Calculate the received signal model and covariance matrix of the filled virtual array;

[0077] The calculation method is as follows: assuming that K incident signals of different directions θ are incident to the improved coprime array, the expression of the virtual array received data is:

[0078] X(t) = A(θ)S(t) + n(t)

[0079] In the formula, S(t)=[S1(t), S2(t), …, S K (t)] T represents an incident signal vector, n(t) is an additive Gaussian white noise vector, A(theta)= [a(theta1), a(theta2), …, a(theta K )] is an array manifold matrix, and the direction vector

[0080] a(theta K )=[e i2π(2MN+M+N-1)d ,e i2π(2MN+N-1)d ,...,e -j2π(2MN+N-1)d ,e j2π(2MN+M+N-1)d ] T ;

[0081] The covariance matrix can be calculated from the received data expression, and the covariance matrix expression is as follows:

[0082]

[0083] S6: using the root-MUSIC algorithm to estimate the DOA.

[0084] Eigenvalue decomposition is performed on R obtained in S5 to obtain a signal subspace U S and a noise subspace U N Since each column vector in A(theta) is orthogonal to the noise subspace, a spatial spectrum function can be constructed:

[0085]

[0086] The spatial spectrum denominator part is replaced by , and the following can be obtained:

[0087]

[0088] In the formula, p(z) is a 2(2MN+M+N-1) order polynomial, contains 2(2MN+M+N-1) roots that are mirror images of each other relative to the unit circle, and the K roots closest to the unit circle are taken to determine the signal direction of arrival estimation value:

[0089]

[0090] The implementation flowchart of the method for designing a coprime array based on virtual array hole filling provided by the application is as follows: Figure 1As shown in the figure, the two sub-arrays of the extended coprime array are moved by N / 2d distance; one of the sub-arrays of the shifted coprime array is flipped to obtain an improved coprime array; a differential virtual array is generated according to the improved coprime array; the differential virtual array is hole-filled by introducing and combining an array; a virtual array signal model and a covariance matrix after filling are calculated; finally, the DOA is estimated by using the ROOT-MUSIC algorithm.

[0091] The improved coprime array structure provided by the application is shown in the figure Figure 2 The array element position expression is:

[0092] TF=TF1UTF2={(nN+N / 2)d / 0≤m≤2M-1}U{-(nM+N / 2)d / 0≤n≤N-1}.

[0093] The DOA estimation result graph of the method is shown in the figure Figure 3 The simulation parameters are set as follows: the number of sources K=12, the incident signal angle values are (-45, -33, -22, -16, -7, 9, 14, 24, 35, 60, 75, 87), the number of snapshots is set to 100, the signal-to-noise ratio is set to 5dB, and the improved coprime array is set as shown in the figure Figure 2 (M=3, N=5, d is 1 / 2 of the incident signal wavelength).

[0094] As shown in the figure Figure 3 It can be seen that the DOA estimation method of the application can accurately identify 12 incident angle angles, and the angle error is small, and has good DOA estimation performance.

[0095] The spatial spectrum comparison results of the improved coprime array structure and the extended coprime array (ACA), the shifted coprime array (SCA), and the thin coprime array (TCA) when the MUSIC algorithm is used are shown in the figure Figure 4

[0096] As shown in the figure Figure 4 It can be seen that the improved coprime array can accurately distinguish 12 incident angle angles, and the MUSIC spatial spectrum peak is more obvious, and the extended coprime array, the shifted coprime array and the thin coprime array cannot accurately distinguish the signal source. Compared with the above three coprime arrays, the improved coprime array can distinguish more signal sources and can realize underdetermined estimation.

[0097] The angle resolution capability test result graph of the method is shown in the figure Figure 5 ​The simulation parameter is set as follows: the number of sources K=12, the incident signal angle values are (-45, -42, -39, -16, -14, -12, 14, 16, 18, 23, 25, 27), the number of snapshots is set as 100, the signal-to-noise ratio is set as 5dB, and the improved co-prime array is set as shown in Figure 2 As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal).

[0098] As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal). Figure 5 It can be seen that the method can accurately distinguish 12 incident signal angles with an angle interval of 2°, and has a relatively ideal angle resolution.

[0099] The relationship curve between the root mean square error and the signal-to-noise ratio when the improved co-prime array structure and the extended co-prime array (ACA), the shift co-prime array (SCA), and the thin co-prime array (TCA) apply the ROOT-MUSIC algorithm to DOA estimation is as shown in Figure 6 The simulation parameter is set as follows: the number of sources K=3, the incident signal angle values are (-45, 34, 60), the number of snapshots is set as 100, the signal-to-noise ratio is set as -10dB to 10dB with a step of 5dB, and the improved co-prime array is set as shown in Figure 2 As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal). 500 independent Monte Carlo experiments are performed, and MATLAB simulation software is used for simulation analysis.

[0100] As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal). Figure 6 It can be seen that the root mean square error of the improved co-prime array and the other three co-prime array structures decreases with the increase of the signal-to-noise ratio, wherein the root mean square error of the improved co-prime array structure is obviously smaller than that of the other three array structures, and has a better DOA estimation performance.

[0101] The relationship curve between the root mean square error and the number of snapshots when the improved co-prime array structure and the extended co-prime array (ACA), the shift co-prime array (SCA), and the thin co-prime array (TCA) apply the ROOT-MUSIC algorithm to DOA estimation is as shown in Figure 7 The simulation parameter is set as follows: the number of sources K=3, the incident signal angle values are (-45, 34, 60), the number of snapshots is set as 100 to 500 with a step of 100, and the signal-to-noise ratio is set as -10dB, and the improved co-prime array is set as shown in Figure 2 As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal). 500 independent Monte Carlo experiments are performed, and MATLAB simulation software is used for simulation analysis.

[0102] As shown in the figure (wherein M=3, N=5, and d is 1 / 2 of the wavelength of the incident signal). Figure 7It can be seen that the root mean square errors of the improved coprime array and the other three coprime array structures decrease with the increase of the number of snapshots, and the root mean square error of the improved coprime array structure is obviously smaller than that of the other three array structures, and the DOA estimation performance is better.

[0103] Finally, it should be noted that the specific embodiments described herein are merely examples of the present application. Various modifications, supplements, or substitutions of similar means can be made to the described specific embodiments by those skilled in the art without departing from the present application or exceeding the scope defined by the appended claims.

Claims

1. A coprime array design method based on virtual array hole filling, characterized in that: The following steps are involved: S1: Move the two sub-arrays of the extended coprime array forward by a distance of Nd / 2; S2: Flip one of the sub-arrays of the shifted coprime array to obtain an improved coprime array; S3: Generate a differential virtual array based on the improved coprime array; S4: introducing and combining arrays to fill holes in differential virtual arrays; S5: Calculate the received signal model and covariance matrix of the filled virtual array; S6: Use the ROOT-MUSIC algorithm to estimate DOA.

2. The method for designing a coprime array based on virtual array hole filling according to claim 1, characterized in that: The specific content of S1 is: moving the two sub-arrays of the extended coprime array forward by a distance of Nd / 2; The element position of submatrix 1 of the shifted extended coprime matrix is: T1 = {(mN+N / 2)d / 0≤m≤2M-1}; The element position of submatrix 2 of the shifted extended coprime matrix is: T2 = {(nM+N / 2)d / 0≤n≤N-1}; The positions of all elements in the shifted extended coprime array can be expressed as: T=T1UT2={(mN+N / 2)d / 0≤m≤2M-1}U{(nM+N / 2)d / 0≤n≤N-1}.

3. The method for designing a coprime array based on virtual array hole filling according to claim 2, characterized in that: The specific content of S2 is: flipping one of the sub-arrays of the shifted coprime array to obtain an improved coprime array; Improve the element position of the coprime array subarray 1: TF1 = {(mN+N / 2)d / 0≤m≤2M-1}; Improve the element position of the coprime array subarray 2: TF2 = {-(nM+N / 2)d / 0≤n≤N-1}; The positions of all elements of the improved coprime array can be expressed as: TF=TF1UTF2={(mN+N / 2)d / 0≤m≤2M-1}U{-(nM+N / 2)d / 0≤n≤N-1}.

4. The method for designing a coprime array based on virtual array hole filling according to claim 3, characterized in that: The specific content of S3 is: generating a differential virtual array according to the improved coprime array; The expression of the element position of the differential virtual array is: TF DCA =±{(Mn+Nm+N)d / 0≤n≤N-1,0≤m≤2M-1}U±{M(n1-n2)d / 0≤n1,n2≤N-1}∪±{N(m1-m2)d|0≤m1,m2≤2M-1}。 5. The method for designing a coprime array based on virtual array hole filling according to claim 4, characterized in that: The specific content of S4 is: introducing and combining arrays to fill holes in the differential virtual array; The expression of the combined array element position is: TF SCA =±{(Mn-Nm)d / 0≤n≤N-1,0≤m≤2M-1}U±{M(n1+n2)d / 0≤n1,n2≤N-1}∪±{-N(m1+m2)d|0≤m1,m2≤2M-1}; The continuous range of the virtual array elements after filling is calculated. The virtual array is continuous within the range of {-(2MN+M+N-1), (2MN+M+N-1)}.

6. The method for designing a coprime array based on virtual array hole filling according to claim 5, characterized in that: The specific content of S5 is: calculating the virtual array receiving signal model and covariance matrix after filling; The calculation method is: Assuming that there are K incident signals θ from different directions incident on the improved coprime array, the expression of the virtual array receiving data is: X(t)=A(θ)S(t)+n(t) Where, S(t)=[S1(t),S2(t),…,S K (t)] T represents the incident signal vector, n(t) is the additive white Gaussian noise vector, A(θ)=[a(θ1),a(θ2),...,a(θ K )] is the array manifold matrix, the direction vector a(θ K )=[and j2π(2MN+M+N-1)d ,And j2π(2MN+N-1)d ,...,And -j2π(2MN+N-1)d ,And -j2π(2MN+M+N-1)d ] T ; The covariance matrix can be calculated from the received data expression. The covariance matrix expression is:

7. The method for designing a coprime array based on virtual array hole filling according to claim 6, characterized in that: The specific content of S6 is: using the ROOT-MUSIC algorithm to estimate DOA; Perform eigenvalue decomposition on R obtained in S5 to obtain the signal subspace U S And the noise subspace U N , since each column vector in A(θ) is orthogonal to the noise subspace, the spatial spectrum function can be constructed: use Replacing the denominator of the spatial spectrum, we can get: Where p(z) is a 2(2MN+M+N-1) degree polynomial, which contains 2(2MN+M+N-1) roots that are mirror images of each other relative to the unit circle. The K roots closest to the unit circle are taken. To determine the direction of arrival estimate of the signal:

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