A complementary array design method based on the concept of virtual and difference arrays

Abstract

The application discloses a complementary coprime array design method based on virtual and difference array concept. The method firstly removes part of sensors in a traditional coprime array to reduce redundancy. Then, a complementary sub-array is constructed by using the removed sensors, so that the virtual and difference array has a wide range of continuous segments, thereby improving the degree of freedom, and strictly limiting the number of sensors with small intervals, reducing the mutual coupling effect. The implementation steps are as follows: calculating the basic unit of array element spacing, determining the array element number and array element interval parameters, and determining the physical position of the array element according to the analytical expression. The application can effectively increase the degree of freedom of the virtual and difference array under the condition of the same array element number, reduce the mutual coupling effect, improve the estimation accuracy of the direction of arrival, improve the spatial target detection capability, and reduce the array operation cost.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing and relates to a sparse array optimization layout and design method suitable for direction-of-arrival estimation. Specifically, it is an array aperture extension structure based on coprime arrays, which can be used to generate virtual sum-difference arrays with high degrees of freedom and low mutual coupling. Background Technology

[0002] Direction of arrival (DOA) estimation is a crucial topic in array signal processing, playing a vital role in applications such as radar, sonar, wireless communication, and navigation. Traditional ODA methods employ uniformly arranged sensor arrays with sensor spacing less than half a wavelength. This uniform arrangement ensures the unique spatial spectrum of the signal, prevents spatial information aliasing, and allows for unique identification of spatial targets through spectral functions. However, the number of identifiable sources in a uniform linear array is severely limited by the degrees of freedom; a uniform linear array of R sensors can identify at most R-1 sources.

[0003] To address the underdetermined direction-of-arrival (DOA) estimation problem when the number of signal sources exceeds the number of array elements, a non-uniform linear array with fewer sensors, also known as a sparse array, has been proposed. Sparse arrays calculate the covariance of the received signals for each element and then vectorize the covariance matrix to obtain an equivalent virtual phased array. The virtual array elements are located at the difference positions of the actual array elements. If the virtual difference matrix can be continuously arranged, target azimuth estimation can be achieved using the virtual array. The most commonly used sparse arrays are nested arrays and coprime arrays. Nested arrays can generate hole-free virtual difference matrices, but they suffer from severe mutual coupling due to the tightly packed sensors. Coprime arrays have a sparser structure, thus mitigating mutual coupling effects, but the holes in the virtual difference matrix greatly limit the number of continuously arranged virtual sensors that can be obtained. By applying the vectorized conjugate augmented MUSIC algorithm and the vectorized non-circular MUSIC algorithm, the resulting virtual array will simultaneously contain the difference matrix and the sum matrix, and the virtual array element positions are the difference and sum of the actual array element positions. For coprime matrices, the holes in their virtual difference matrices can be filled by elements in their virtual sum matrices. Furthermore, the introduction of virtual sum matrices further expands the range of continuous virtual matrices. However, current coprime structures are designed specifically for virtual difference matrices, without considering virtual sum matrices. Therefore, there is significant redundancy between their virtual difference and sum matrices, leaving considerable room for improvement in degrees of freedom. Thus, designing a one-dimensional sparse matrix with high degrees of freedom (i.e., longer continuous segments) in its virtual sum and difference matrices, while limiting the number of closely spaced sensor pairs to reduce mutual coupling, is of great significance for practical applications. Summary of the Invention

[0004] The purpose of this invention is to provide an optimized layout and design method for one-dimensional sparse arrays for direction-of-arrival estimation. Based on a traditional coprime array, some sensors are arranged on the right side of the original array to form a supplementary subarray. With the same number of sensors, a larger virtual sum-difference array, higher array degrees of freedom, lower mutual coupling effects, and stronger space target detection capability are achieved compared to existing coprime arrays.

[0005] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution, which includes the following steps:

[0006] (1) Let d represent the basic unit of the array element spacing, and take it as half wavelength, i.e., d = λ / 2, where λ represents the wavelength of the incident signal of the array.

[0007] (2) Based on the total number of array elements R given by the array, select two coprime integers M and N as array parameters, satisfying R = 2M + N - 1, where N > M ≥ 3;

[0008] (3) Construct a conventional coprime matrix consisting of two uniform linear submatrices. The first submatrix contains N sensors with an element spacing of Md, denoted as . The second subarray has 2M sensors with an inter-element spacing of Nd, denoted as... The two subarrays share the sensor located at position 0;

[0009] (4) Remove from the second subarray Several unimportant sensors, located in The remaining sensors of the second subarray are denoted as

[0010] (5) Construct a supplementary subarray using the unimportant sensors removed in step (4), denoted as .

[0011] (6) The resulting complementary coprime matrix is ​​defined as Use its array elements to generate virtual sum-difference comatrix;

[0012] First, virtual difference matrix The location can be calculated get;

[0013] Secondly, virtual and array The location can be calculated get;

[0014] Finally, the virtual sum-difference matrix is ​​composed of the virtual difference matrix. and virtual array The union of the sets is denoted as .

[0015] (7) Find the largest continuous segment of the virtual sum difference matrix [-L u d,L u d], then the array degrees of freedom L can be obtained. u Using this virtual continuous uniform linear array, various wave direction estimation algorithms based on subspace decomposition can be executed to accurately estimate the wave direction of arrival of spatial signals.

[0016] Furthermore, the degrees of freedom of the aforementioned supplementary coprime matrix are:

[0017] Furthermore, given a total number of array elements R = 2M + N - 1, when M and N respectively take... At that time, the virtual array has degrees of freedom L u Maximum, maximum degrees of freedom is However, the optimal values ​​of M and N are generally not coprime integers. For most sensor numbers R, a coprime integer pair around the optimal value can be chosen as the values ​​of M and N.

[0018] Compared with the prior art, the present invention has the following advantages:

[0019] (1) This invention utilizes the traditional coprime matrix... The unimportant sensors are repositioned to the right of the original array to form a supplementary subarray, which reduces the redundancy between the virtual difference array and the sum array and expands the continuous range of the virtual sum and difference array; the present invention can increase the array aperture, increase the number of degrees of freedom and effectively reduce the mutual coupling effect between antennas.

[0020] (2) Under the same array element condition, the supplementary coprime array structure proposed in this invention can obtain a larger virtual continuous aperture and array degree of freedom through virtual sum and difference array, which effectively improves the one-dimensional direction of arrival estimation accuracy, increases spatial resolution, increases the number of space target detections, and reduces array operating costs.

[0021] (3) The array arrangement position with the maximum degree of freedom can be obtained based on the total number of array elements R, which is convenient for theoretical design and practical engineering application. Attached Figure Description

[0022] Figure 1 This is an example diagram of a supplementary coprime matrix proposed in this invention, where M = 3 and N = 4. In this case, the submatrix... and The selected array element positions can be represented as:

[0023]

[0024] Figure 2This is a schematic diagram of the virtual difference matrix, virtual sum matrix, and virtual sum-difference matrix of the supplementary coprime matrix proposed in this invention, where M=3 and N=4.

[0025] Figure 3 This is a comparison diagram of the spatial target orientation estimation results of the structure designed in this invention with five other sparse structures.

[0026] Figure 4(a) is a comparison of the mean square error of the direction of arrival estimation of the structure designed in this invention with that of five other sparse structures as a function of signal-to-noise ratio.

[0027] Figure 4(b) is a comparison of the mean square error of the direction of arrival estimation of the structure designed in this invention with that of five other sparse structures as a function of snapshot quantity.

[0028] Figure 4(c) is a comparison of the mean square error of the direction of arrival estimation of the structure designed in this invention with that of five other sparse structures as a function of mutual coupling strength. Detailed Implementation

[0029] The invention will be further described below with reference to the accompanying drawings and embodiments.

[0030] Taking an incident signal frequency of 20 GHz and a total number of array elements R = 9 as an example, the supplementary coprime array of the present invention (such as...) Figure 1 The construction (as shown) will be described in detail.

[0031] (1) Calculate the wavelength of the incident signal λ = c / f = 3.0 × 10⁻⁶ 8 / 20×10 9 =1.5cm; Let d represent the basic unit of array element spacing, and take it as half wavelength, that is, d = λ / 2 = 7.5mm;

[0032] (2) Based on the given total number of array elements R = 9, select a set of M and N that satisfy R = 2M + N - 1, where N > M ≥ 3; in this example, M = 3 and N = 4 were selected.

[0033] (3) Construct a conventional coprime matrix based on the selected M and N; in this example, the conventional coprime matrix consists of two uniform linear submatrices. and Composition; the two subarrays share a sensor located at 0;

[0034] (4) Subarray A less critical sensor located at {8, 16} × 7.5 mm is placed to the right of the original array, forming a supplementary subarray. The remaining sensors are denoted as

[0035] (5) The positions of the elements of the proposed supplementary coprime matrix are as follows: Figure 1 As shown, it can be represented as in

[0036]

[0037] The actual array element placement positions are

[0038] {0, 22.5, 30, 45, 67.5, 90, 150, 232.5, 315} mm;

[0039] (6) Calculate the virtual difference matrix of the complementary coprime matrix (e.g., Figure 2 (a) shown), virtual and array (as shown) Figure 2 (b) shown), virtual sum-difference matrix (as shown in) Figure 2 (as shown in (c)); since all three types of virtual matrices are symmetric about the zero point. Figure 2 Only the positive half of the virtual array is shown in the drawing;

[0040] First, by taking the differences between each pair of array element positions, a set of differences is obtained, which constitutes a virtual difference matrix.

[0041] Then, by summing the positions of the array elements pairwise, a series of sums are obtained. The set containing these sums and their opposites constitutes the virtual sum array.

[0042] Finally, by and The union of the sets constitutes a virtual sum-difference matrix, denoted as . It is continuous in the range [-40, 40] × 7.5 mm; therefore, the degree of freedom of the complementary coprime matrix in this example is L. u =40.

[0043] The effects of the present invention will be further described below with reference to simulation examples.

[0044] Simulation Example: With the total number of array elements R = 17, the following six array structures are compared: Prototype Coprime Array (PCA) with M = 7, N = 11; Conventional Coprime Array (CCA) with M = 5, N = 8; Diff-sum Nested Array (DsNA); Transformed Nested Array-1 (TNA-1) with N1 = 9, N2 = 8; Transformed Nested Array-2 (TNA-2) with N1 = 9, N2 = 8; and Supplementary Coprime Array (SCA) of this invention with M = 5, N = 8. Based on the analytical expressions of these six array structures, their normalized array element positions (omitting the spacing element d) can be obtained as follows:

[0045]

[0046] Assume 31 uncorrelated narrowband sources are incident on the aforementioned sparse array, with incident angles uniformly distributed between -60° and 60°. The signal-to-noise ratio of the incident signals is 0 dB, and a total of 800 snapshots are collected. The coupling coefficient amplitude is set to 0.3. The estimation results of the direction of arrival for the six sparse arrays are as follows: Figure 3 As shown, in the presence of strong mutual coupling, the supplementary coprime matrix SCA proposed in this invention can accurately identify all 31 sources; DsNA, TNA-1, and TNA-2 lose a small number of sources and some spurious peaks appear in the spatial spectrum because these three types of nested structures suffer from severe mutual coupling; spurious peaks also exist in the spatial spectrum of CCA and PCA, and the direction-of-arrival estimation accuracy is also degraded to some extent because they can only provide a limited number of degrees of freedom.

[0047] Subsequently, 500 Monte Carlo experiments were performed, and the accuracy of the direction-of-arrival (DOA) estimation was evaluated based on the root mean square error (RMSE). Here, it is assumed that 16 uncorrelated narrowband signals are incident on the six sparse structures described above, with incident angles uniformly distributed between -60° and 60°. The RMSE of the DOA estimation, corresponding to different signal-to-noise ratios (SNR), is shown in Figure 4(a), where the number of snapshots is 800 and the coupling coefficient amplitude is set to 0.2. Figure 4(b) shows the RMSE results as a function of the number of snapshots, where the SNR is set to 0 dB and the coupling coefficient amplitude is set to 0.2. As can be seen from the figures, the array proposed in this invention achieves better estimation performance than other sparse arrays, and there is still a significant performance gap between them when the SNR or the number of snapshots is large. This is because the proposed array structure can significantly expand the range of virtual and difference arrays while limiting the number of sensor pairs with small spacing. Figure 4(c) illustrates the RMSE results with different coupling coefficient magnitudes, where 0 dB SNR and 800 snapshots were used. It can be seen that DsNA, TNA-1, and TNA-2 provide more accurate direction-of-arrival estimation in the weak coupling case. When the coupling coefficient magnitude is greater than 0.1, the proposed array structure outperforms other sparse arrays.

[0048] In summary, the array structure proposed in this invention is suitable for direction-of-arrival (DOA) estimation based on virtual sum-difference arrays. It has more virtual array elements compared to existing coprime arrays and is less affected by mutual coupling than nested structures. When significant mutual coupling exists, this structure exhibits superior performance in array DOA estimation and space target detection.

Claims

1. A method for designing a complementary coprime matrix based on the concept of a virtual sum-difference matrix, characterized in that, Includes the following steps: (1) Let d represent the basic unit of the array element spacing, and take it as half wavelength, i.e., d = λ / 2, where λ represents the wavelength of the incident signal of the array. (2) Based on the total number of array elements R given by the array, select two coprime integers M and N as array parameters, satisfying R = 2M + N - 1, where N > M ≥ 3; (3) Construct a conventional coprime matrix consisting of two uniform linear submatrices. The first submatrix contains N sensors with an element spacing of Md, denoted as . The second subarray has 2M sensors with an inter-element spacing of Nd, denoted as... The two subarrays share the sensor located at position 0; (4) Remove from the second subarray Several unimportant sensors, located in The remaining sensors of the second subarray are denoted as (5) Construct a supplementary subarray using the unimportant sensors removed in step (4), denoted as . (6) The resulting complementary coprime matrix is ​​defined as Use its array elements to generate virtual sum-difference comatrix; First, virtual difference matrix The location can be calculated get; Secondly, virtual and array The location can be calculated get; Finally, the virtual sum-difference comatrix is ​​composed of the virtual difference matrix. and virtual array The union of the sets is denoted as . (7) Find the largest continuous segment of the virtual sum difference matrix [-L u d,L u d], then the array degrees of freedom L can be obtained. u Using this virtual continuous uniform linear array, various wave direction estimation algorithms based on subspace decomposition can be executed to accurately estimate the wave direction of arrival of spatial signals.

2. The method for designing a complementary coprime matrix based on the concept of a virtual sum-difference matrix according to claim 1, characterized in that, Supplementing the coprime matrix involves using the traditional coprime matrix... The sensors are rearranged to the right of the original array to form a supplementary subarray.

3. The method for designing a complementary coprime matrix based on the concept of a virtual sum-difference matrix according to claim 1, characterized in that, The array has the following degrees of freedom:

4. The method for designing a complementary coprime matrix based on the concept of a virtual sum-difference matrix according to claim 1, characterized in that, Given a total number of array elements R = 2M + N - 1, when M and N respectively take... At that time, the virtual array has degrees of freedom L u Maximum, maximum degrees of freedom is However, the optimal values ​​of M and N are generally not coprime integers. For most sensor numbers R, a coprime integer pair around the optimal value can be chosen as the values ​​of M and N.

Images