Low-redundancy array structure design method based on subarray shifting and padding strategy

Abstract

This invention discloses a low-redundancy coprime array structure design method based on subarray shifting and filling strategies, belonging to the field of array signal processing technology. The method includes: selecting coprime integers M and N to construct an extended coprime array; removing redundant array elements from the first subarray while maintaining the continuous degrees of freedom of the differential virtual array; removing the array element at the origin position from the second subarray and shifting another array element to fill part of the virtual holes; further filling the remaining holes by adding two physical array elements to obtain the final low-redundancy shifted coprime array; and finally using the spatial smoothing MUSIC algorithm for direction-of-arrival estimation. This invention, without changing the number of physical array elements, effectively expands the continuous degrees of freedom and array aperture of the differential virtual array through subarray reconstruction and shifting filling, reduces the mutual coupling effect between array elements, and significantly improves the accuracy and robustness of direction-of-arrival estimation, especially suitable for multi-source underdetermined direction-finding scenarios in complex environments.

Description

Technical Field

[0001] The present invention relates to the technical field of array signal processing, and particularly to a method for designing a low-redundancy co-prime array structure based on a sub-array shifting and filling strategy. Background Art

[0002] Array signal processing is an important branch of modern signal processing. By arranging multiple sensors in a certain geometric structure, the spatial incoming wave signals are received and processed to achieve functions such as signal detection, parameter estimation, and positioning. Among them, direction of arrival (DOA) estimation is a key technology in array signal processing and is widely used in scenarios such as radar, sonar, and wireless communication. Existing uniform linear arrays usually require the element spacing not to exceed half of the wavelength of the incident signal to avoid spatial aliasing. Therefore, when the number of array elements is limited, it is difficult to further improve the effective aperture and angular resolution ability of the array, and the number of resolvable signal sources is limited by the number of physical array elements and it is difficult to meet the multi-source or even underdetermined direction-finding requirements. To overcome the above limitations, sparse arrays expand the effective aperture through non-uniform arrangement and use differential virtual arrays to obtain higher degrees of freedom, thereby improving the multi-source resolution and DOA estimation performance. As one of the typical sparse arrays, the co-prime array has the advantage of obtaining a large degree of freedom with fewer array elements, but there are still problems in actual DOA estimation such as insufficient continuous degrees of freedom, holes or redundancies in the differential virtual array resulting in low space utilization; at the same time, non-ideal factors such as element mutual coupling and channel inconsistency will cause deviations in the array manifold, leading to spectral peak broadening, peak merging, and an increase in estimation error, affecting the estimation accuracy and robustness. Therefore, it is necessary to propose a new method for designing a co-prime sparse array structure to improve the continuous degrees of freedom and effective aperture of the differential virtual array under the condition of limited number of physical array elements, reduce the influence of redundancies and holes, and improve the stability and engineering applicability of DOA estimation. Summary of the Invention

[0003] The technical solution of the present invention to solve the above technical problems is to provide a method for designing a low-redundancy co-prime array structure based on a sub-array shifting and filling strategy, including the following steps:

[0004] S1: Select two positive integers M and N that are relatively prime to each other and satisfy M < N, and construct an extended co-prime array structure based on this;

[0005] S2: Without changing the continuous degrees of freedom of the differential virtual array generated by the extended co-prime array structure, remove the first ([ -1) physical array elements in the first sub-array;

[0006] S3: Remove the array element located at the origin in the second sub-array, and shift the physical array element located at in the second sub-array to The text also provides information on the virtual holes that can be filled by the shift operation and the locations of the newly created holes.

[0007] S4: In the subarray respectively at place and Two physical array elements are added at the point to obtain the proposed array structure; and the physical array element position expression of the array structure and the generated differential virtual array are determined.

[0008] S5: The spatial smoothing MUSIC algorithm is used to estimate the direction of arrival of the incident signal using the designed array structure.

[0009] Furthermore, in step S1, the extended coprime array structure is composed of two uniform linear arrays, and its element position expression is specifically expressed as follows:

[0010] ;

[0011] Differential virtual array generated by this array Including self-difference sets Mutual difference set ; Differential set Including and Mutual difference set include and Differential Virtual Array and the difference set Mutual difference set The specific expression is shown below;

[0012] ;

[0013] ;

[0014] .

[0015] Further, step S2 specifically includes: for the extended coprime array, when M is even and M is greater than or equal to 2, or when M is odd and M is greater than or equal to 3, before removing the first subarray ( -1) redundant array elements, at which point the differential virtual array... The expression is:

[0016] ;

[0017] In the formula, , , ;

[0018] At this point, the specific hole distribution in the differential virtual array can be represented as:

[0019] .

[0020] Further, step S3 specifically includes: removing the array element located at point 0 in the second subarray, and placing the array element located at point 0 in the second subarray... The physical array element at that location is shifted to The position of the physical array element after the step is expressed as:

[0021] ;

[0022] Differential virtual array generated from physical array The specific expression is:

[0023] ;

[0024] In the formula:

[0025] ;

[0026] The expression for the virtual hole position that can be filled after the shift operation is:

[0027] ;

[0028] The expression for the additional hole position due to the shift operation is:

[0029] .

[0030] Furthermore, in step S4, the physical element position expression of the proposed array structure RSACA is:

[0031] ;

[0032] In the formula:

[0033] ;

[0034] Differential virtual array generated by the array structure Specifically, it is expressed as follows:

[0035] ;

[0036] in:

[0037] Furthermore, the constructed RSACA array has the following properties:

[0038] Property 1: The degrees of freedom of a RSACA array are represented as:

[0039]

[0040] Property 2: The continuous degrees of freedom of the RSACA array are Its continuous range is or ;

[0041] Property 3: The array aperture of the RSACA array reaches... ;

[0042] Property 4: When d equals 1, 2, 3, the weight function expression of the RSACA array is:

[0043] .

[0044] Furthermore, the process of using the spatial smoothing MUSIC algorithm for direction-of-arrival estimation includes:

[0045] Calculate the sample covariance matrix based on the received signal. ;

[0046] Eigenvalue decomposition of the sample covariance matrix yields the noise subspace. ;

[0047] Constructing the spatial spectrum function The angle of the incident signal is calculated by finding the peak value of the spatial spectrum.

[0048] Compared with existing technologies, this invention provides a low-redundancy coprime array structure design method based on subarray shifting and filling strategies, which has the following advantages:

[0049] 1. This invention aims to improve array performance by optimizing the layout between subarrays. Addressing the insufficient mutual coupling suppression capability of traditional coprime arrays, redundant physical elements in coprime arrays are removed without altering the array's continuous degrees of freedom, and an expression for the element positions in the proposed array structure is provided. This method significantly expands the continuous degrees of freedom of the virtual array without increasing the number of physical elements, reduces the coupling effect between elements, and thus improves the performance of DOA estimation.

[0050] 2. Based on the removal of redundant array elements, this invention further introduces optimization strategies such as subarray shifting and subarray filling to obtain an improved coprime array structure. Then, it is extended into a virtual array through difference joint operation, in which the continuous degrees of freedom and array aperture of the array are improved. Finally, the spatial smoothing MUSIC algorithm is used for angle estimation, and the estimation performance is evaluated in various scenarios. This invention is not only applicable to ideal conditions, but also has good applicability and robustness to complex non-ideal environments. Attached Figure Description

[0051] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0052] Figure 1 This is a schematic diagram of the physical array structure proposed in this invention;

[0053] Figure 2 This is a schematic diagram of the non-negative part of the virtual array generated by the difference joint method in this invention;

[0054] Figure 3 This figure shows the simulation results of the root mean square error of the present invention versus several typical improved array structures as a function of snapshot number under an uncoupled model.

[0055] Figure 4 The figure shows the simulation results of the root mean square error of the present invention versus the signal-to-noise ratio of several typical improved array structures under the uncoupled model.

[0056] Figure 5 This figure shows the simulation results of the root mean square error of the present invention versus several typical improved array structures as a function of the number of snapshots under the coupled model.

[0057] Figure 6 This figure shows the simulation results of the root mean square error of the present invention versus the signal-to-noise ratio of several typical improved array structures under the coupled model.

[0058] Figure 7 This is a power spectrum comparison image of the array structure of the present invention and several typical improved array structures. Detailed Implementation

[0059] This invention proposes a low-redundancy coprime array structure design method based on subarray shifting and filling strategies, aiming to improve array performance by optimizing the layout between subarrays.

[0060] The design method of a low-redundancy co-prime array structure based on sub-array shift and filling strategy proposed by the present invention will be described in the following specific embodiments:

[0061] In the technical solution of this embodiment, a design method of a low-redundancy co-prime array structure based on sub-array shift and filling strategy includes the following steps:

[0062] S1: Select two positive integers M and N that are relatively prime to each other and satisfy M < N, and construct an extended co-prime array structure accordingly;

[0063] S2: Without changing the continuous degrees of freedom of the difference virtual array generated by the extended co-prime array structure, remove the first ([ -1) physical array elements in the first sub-array;

[0064] S3: Remove the array element located at the origin in the second sub-array, and shift the physical array element located at to , and at the same time give the virtual holes that can be filled by this shift operation and the newly added hole positions;

[0065] S4: Add two physical array elements in the sub-array at and respectively to obtain the proposed array structure; and determine the physical array element position expression of this array structure and the difference virtual array generated by it;

[0066] S5: Use the spatial smoothing MUSIC algorithm to estimate the direction of arrival of the incident signal by using the designed array structure.

[0067] Furthermore, in the step S1, the extended co-prime array structure is composed of two uniform linear arrays, and its array element position expression is specifically expressed as:

[0068] ;

[0069] The difference virtual array generated by this array includes the auto-difference set , the cross-difference set ; the auto-difference set includes and , the cross-difference set includes and ; the difference virtual array , and the specific expressions of the auto-difference set and the cross-difference set are as follows;

[0070] ;

[0071] ;

[0072] .

[0073] Further, step S2 specifically includes: for the extended coprime array, when M is even and M is greater than or equal to 2, or when M is odd and M is greater than or equal to 3, before removing the first subarray ( -1) redundant array elements, at which point the differential virtual array... The expression is:

[0074] ;

[0075] In the formula, , , ;

[0076] At this point, the specific hole distribution in the differential virtual array can be represented as:

[0077] .

[0078] Further, step S3 specifically includes: removing the array element located at point 0 in the second subarray, and placing the array element located at point 0 in the second subarray... The physical array element at that location is shifted to The position of the physical array element after the step is expressed as:

[0079] ;

[0080] Differential virtual array generated from physical array The specific expression is:

[0081] ;

[0082] In the formula:

[0083] ;

[0084] The expression for the virtual hole position that can be filled after the shift operation is:

[0085] ;

[0086] The expression for the additional hole position due to the shift operation is:

[0087] .

[0088] Furthermore, in step S4, the physical element position expression of the proposed array structure RSACA is:

[0089] ;

[0090] In the formula:

[0091] ;

[0092] Differential virtual array generated by the array structure Specifically, it is expressed as follows:

[0093] ;

[0094] in:

[0095] Furthermore, the constructed RSACA array has the following properties:

[0096] Property 1: The degrees of freedom of a RSACA array are represented as:

[0097]

[0098] Property 2: The continuous degrees of freedom of the RSACA array are Its continuous range is or ;

[0099] Property 3: The array aperture of the RSACA array reaches... ;

[0100] Property 4: When d equals 1, 2, 3, the weight function expression of the RSACA array is:

[0101] .

[0102] Furthermore, the process of using the spatial smoothing MUSIC algorithm for direction-of-arrival estimation includes:

[0103] Calculate the sample covariance matrix based on the received signal. ;

[0104] Eigenvalue decomposition of the sample covariance matrix yields the noise subspace. Constructing the spatial spectral function The angle of the incident signal is calculated by finding the peak value of the spatial spectrum.

[0105] Specifically, the SS-MUSIC algorithm is adopted, and spatial spectrum and RMSE are used as evaluation indicators. The DOA estimation performance of the proposed array structure and typical array structures are compared and analyzed under various simulation conditions to verify the effectiveness of the proposed array structure.

[0106] There are K far-field uncorrelated narrowband signal sources. When incident on an array of P elements, the unit element spacing is defined as d, which can be expressed by the expression... The calculation yielded, where Let λ be the wavelength of the incident signal. Therefore, the expression for the received signal of the array can be obtained as shown in the following equation.

[0107]

[0108] in For an array manifold matrix, it can be specifically represented as , It is a direction vector. Let be a signal vector, and its expression is: , This indicates that the mean is 0 and the variance is . Gaussian noise vector.

[0109] The covariance matrix can be calculated from the above expression for the received signal. The expression for is shown in the following formula.

[0110]

[0111] in, This represents the power of the k-th source. This represents the noise covariance matrix. In practical applications, since the theoretical covariance matrix is ​​difficult to calculate, the sample covariance matrix is ​​generally used. To approximate it.

[0112] Eigendecomposition of the sample covariance matrix yields the noise subspace and the noise subspace, as shown below:

[0113]

[0114] In the formula, For signal subspace; Let be the noise subspace; based on the orthogonality between the noise eigenvectors and the signal vectors, the expression for the array space spectral function can be obtained:

[0115]

[0116] By finding the peak value of the aforementioned spatial spectral function, the angle of arrival can be determined. The estimate.

[0117] like Figure 1 As shown, the array consists of The array consists of 4 physical array elements, where 4 < M < N. The array comprises four subarrays, all located on the positive semi-axis, with subarray 1 containing... Subarray 1 contains N-2 elements, subarray 2 contains (N-2) elements, subarray 3 contains 2 elements, and subarray 4 contains 1 physical element.

[0118] like Figure 2 As shown, to more intuitively understand the proposed array structure, M=5 and N=6 are chosen for further explanation. When M=5 and N=6, , , , At this point, the physical element positions of the RSACA array and the element positions of the differential virtual array are shown in the figure.

[0119] like Figure 7 As shown, without changing the number of physical sensors in the array structure described above, the number of incident signal sources is increased from 15 to 35, and they are evenly distributed in... Within the interval, all other simulation settings remain the same as described above. We then compare the DOA estimation performance of different array structures under mutually coupled conditions under underdetermined conditions.

[0120] In the spatial spectrum diagram, the red dashed line represents the actual DOA, while the blue curve represents the estimated spatial spectrum. It can be observed that, under mutual coupling conditions, only the proposed RSACA array and SSACA array can correctly distinguish and locate all incident signal sources. This indicates that the RSACA array has better robustness for underdetermined DOA estimation in the presence of mutual coupling. Figure 3 and Figure 4 The figures show the relationship curves between the root mean square error (RMSE) of DOA estimation and the number of snapshots and signal-to-noise ratio (SNR). All simulation experiments used 200 Monte Carlo runs, and the root mean square error (RMSE) is defined as follows:

[0121]

[0122] In the formula, K is the number of signal sources, and Q is the number of Monte Carlo experiments. This represents the true angle of the k-th incident signal source. This represents the estimated angle of the k-th incident signal in the q-th Monte Carlo experiment. A smaller root mean square error indicates better DOA estimation performance.

[0123] Simulation results are as follows Figure 3 As shown, under an uncoupled model, the root mean square error of the proposed array structure, along with kCPA, TCA, NSCA, SCA, and SSACA arrays, is investigated as a function of the number of snapshots. The selected array... Fifteen incident signal sources were uniformly distributed within an angular range, with a signal-to-noise ratio (SNR) of 10 dB. The number of snapshots varied from 200 to 2000, with a step size of 200. The number of Monte Carlo experiments was set to 200. The RMSE of various array structures monotonically decreased with increasing snapshot number. When the number of snapshots exceeded 1600, the RMSE of each algorithm tended to stabilize. The RMSE of the RSACA array was consistently lower than that of the other structures, indicating that the proposed array structure's DOA estimation performance was significantly better than that of the other array structures.

[0124] Simulation results are as follows Figure 4 As shown, under an uncoupled model, the root mean square error of the above six array structures is investigated as a function of signal-to-noise ratio (SNR). The selected arrays are... Fifteen incident signal sources were uniformly distributed within an angular range, with 500 snapshots and a signal-to-noise ratio (SNR) varying from 0 to 30 dB in a step size of 5. The number of Monte Carlo experiments was set to 200. This figure compares the root mean square error (RMSE) performance of various array structures under different SNRs. As the SNR increases from 0 dB to 30 dB, the RMSE of all array structures monotonically decreases and tends to stabilize at high SNRs. Throughout the entire SNR range, the RSACA array structure consistently exhibits the lowest RMSE, demonstrating a more significant performance advantage over other array structures, especially in low SNR scenarios.

[0125] Simulation results are as follows Figure 5 As shown, under the coupled model, the root mean square error of the above six array structures is investigated as a function of the number of snapshots. The selected arrays are... Fifteen incident signal sources were uniformly distributed within an angular range, with a signal-to-noise ratio (SNR) of 10 dB. The number of snapshots varied from 200 to 2000 with a step size of 200. The number of Monte Carlo experiments was set to 200. As the number of snapshots increased from 200 to 2200, the RMSE of all array structures continuously decreased and eventually stabilized. Throughout the range, the RMSE of the proposed array structure was consistently lower than that of other array structures, and even with a smaller number of snapshots, it was still significantly superior to the other arrays, indicating that the proposed array structure has better robustness to mutual coupling effects.

[0126] Simulation results are as follows Figure 6 As shown, under the coupled model, the root mean square error of the above six array structures is investigated as a function of signal-to-noise ratio (SNR). The following is a selection of... Fifteen incident signal sources were uniformly distributed within an angular range, with 500 snapshots and a signal-to-noise ratio (SNR) varying from 0 to 30 dB in a step size of 5. The number of Monte Carlo experiments was set to 200. As the SNR increased from 0 dB to 30 dB, the RMSE values ​​of all array structures showed a stepwise decrease and eventually stabilized. Compared to other array structures, the RMSE of the RSACA array was significantly lower throughout the entire SNR variation range, indicating that this array has better mutual coupling suppression capability.

[0127] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A low-redundancy coprime array structure design method based on subarray shifting and filling strategies, characterized in that, It includes the following steps: S1: Select two relatively prime positive integers M and N, and satisfy M < N, and construct an extended co-prime array structure based on this; S2: Without changing the continuous degrees of freedom of the differential virtual array generated by the extended coprime array structure, remove the first (...) of the first subarray. -1) physical array elements; S3: Remove the array element located at the origin in the second subarray, and then remove the array element located at the origin in the second subarray. The physical array element at that location is shifted to The text also provides information on the virtual holes that can be filled by the shift operation and the locations of the newly created holes. S4: In the subarray respectively at place and Two physical array elements are added at the point to obtain the proposed array structure; And determine the physical array element position expression of this array structure and the generated difference virtual array; S5: Adopt the spatial smoothing MUSIC algorithm and use the designed array structure to estimate the direction of arrival of the incident signal.

2. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 1, characterized in that, In the step S1, the extended co-prime array structure is composed of two uniform linear arrays, and its array element position expression is specifically expressed as: ； Differential virtual array generated by this array Including differential sets Mutual difference set ; Differential set Including and Mutual difference set include and Differential Virtual Array and the difference set Mutual difference set The specific expression is shown below; ； ； 。 3. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 2, characterized in that, Step S2 specifically includes: For the extended coprime array, when M is even and M is greater than or equal to 2, or when M is odd and M is greater than or equal to 3, before removing the first subarray ( -1) redundant array elements, at which point the differential virtual array... The expression is: ； In the formula, , , ; At this time, the specific hole distribution in the difference virtual array can be expressed as: 。 4. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 3, characterized in that, Step S3 specifically includes: removing the array element located at point 0 in the second subarray, and placing the array element located at point 0 in the second subarray... The physical array element at that location is shifted to The position of the physical array element after the step is expressed as: ； Differential virtual array generated from physical array The specific expression is: ； In the formula: ； The virtual hole position expression that can be filled after the shift operation is: ； The hole position expression newly added due to the shift operation is: 。 5. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 1, characterized in that, In the step S4, the physical array element position expression of the proposed array structure RSACA is: ； In the formula: ； Differential virtual array generated by the array structure Specifically, it is expressed as follows: ； Among them: 。 6. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 5, characterized in that, The constructed RSACA array has the following properties: Property 1: The degrees of freedom of a RSACA array are expressed as: ; Property 2: The continuous degrees of freedom of the RSACA array are Its continuous range is or ; Property 3: The array aperture of the RSACA array reaches ; Property 4: When d is equal to 1, 2, 3, the weight function expression of the RSACA array is: 。 7. The low-redundancy coprime array structure design method based on subarray shifting and filling strategy according to claim 1, characterized in that, The process of estimating the direction of arrival by using the spatial smoothing MUSIC algorithm includes: Calculate the sample covariance matrix based on the received signal. ; Eigenvalue decomposition of the sample covariance matrix yields the noise subspace. Constructing the spatial spectral function The angle of the incident signal is calculated by finding the peak value of the spatial spectrum.

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