K-fold expansion and filling co-prime array design method

Abstract

The invention provides a k-fold expansion and filling co-prime array design method. The k-fold expansion and filling co-prime array design method comprises the following steps of 1, setting a structure of an original co-prime array; 2, constructing an expanded co-prime array, and determining the positions of holes in a non-negative part in a differential co-array; 3, constructing a k-fold expanded co-prime array, and determining the value of the sub-array spacing and the position of a non-negative part hole in a differential co-array; 4, through hole filling, constructing a k-fold expansion filling co-prime array, and calculating a physical aperture, a uniform degree of freedom, a degree of freedom, a weighting function and optimal configuration of parameters when the uniform degree of freedom is given; and step 5, filling part of array element spacing of the co-prime array by compressing k-fold expansion, constructing a k-fold expansion filling co-prime array with compressed array element spacing, and determining composition conditions of the array. According to the method, the differential common-array freedom degree is remarkably improved, the mutual coupling effect between array elements is effectively restrained, and therefore the DOA estimation precision and the resolving power are improved.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing and relates to a sparse array design technique, specifically a k-fold expansion and filling method for coprime array design. Background Technology

[0002] Direction of Arrival (DOA) estimation using spatiotemporal data acquired by arrays is a key technology in array signal processing and has been widely applied in radar, sonar, positioning, astronomy, and wireless communication. The array's geometry directly determines its angular resolution, the number of resolvable sources, and the performance of DOA estimation. Traditional DOA estimation methods are typically based on simple uniform linear arrays (ULAs). However, a ULA consisting of T elements can only identify a maximum of T-1 sources. When the number of targets is large or their spatial distribution is dense, it is necessary to increase the number of elements to expand the array aperture, leading to a significant increase in system hardware size and implementation cost. Furthermore, to avoid spatial aliasing, ULAs typically need to satisfy a half-wavelength element spacing constraint. In practical engineering, this constraint often results in excessively small distances between elements, exacerbating the mutual coupling (MC) effect and severely impacting the accuracy of DOA estimation. Therefore, under the constraints of limited platform space and the number of array elements, achieving high-resolution, multi-target direction-of-arrival (DOA) estimation has become a crucial problem that urgently needs to be solved in the field of array signal processing.

[0003] In recent years, sparse arrays have gradually become a hot research topic in the field of direction-of-arrival (DOA) estimation due to their ability to achieve larger array apertures and higher degrees of freedom (DOF) compared to uniform linear arrays (ULAs) with the same number of array elements. Furthermore, compared to ULAs, sparse arrays also have significant advantages in reducing element redundancy, suppressing mutual coupling effects, and saving hardware costs. To quantitatively describe the ability of an array to convert its physical aperture into an effective virtual aperture, the spatial efficiency (SE) metric is typically introduced. SE is defined as the ratio of the upper limit of the largest consecutive virtual array elements in a differential co-array to the array's physical aperture, and it also measures the degree of physical space utilization by the sparse array. Typical sparse arrays include nested arrays (NAs) and coprime arrays (CAs). Nested arrays (NA) consist of a densely packed uniform linear array (ULA) and a sparsely packed uniform linear array (ULA). Their difference co-array (DCA) is continuous and hole-free, achieving complete filling of all positions within their virtual array coverage area, thus achieving 100% spatial efficiency. However, due to the presence of a densely packed ULA, the mutual coupling (MC) effect between array elements is relatively severe. Coprime arrays (CA), on the other hand, consist of two sparse uniform arrays with coprime spacing. Although the presence of holes in their difference co-array (DCA) results in lower spatial efficiency and uniform degrees of freedom (uDOFs) compared to nested arrays (NA), the larger element spacing significantly suppresses the mutual coupling effect. With further research into sparse arrays, various variations and innovative designs have been developed based on nested arrays (NA) and coprime arrays (CA) to further improve the array's direction-of-arrival (DOA) estimation performance. In their paper “Padded coprime arrays for improved DOA estimation: Exploiting hole representation and filling strategies” (IEEE Trans. Signal Process, vol. 68, pp. 4597–4611, 2020.), W. Zheng et al. proposed expanding two sparse uniform arrays in a coprime array CA and introducing filling subarrays to fill holes, thereby expanding the uniform segments in the differential coprime array DCA and forming padded coprime arrays (PCAs).Compared to coprime arrays (CA), this array design not only increases the number of uniform degrees of freedom (uDOFs) but also mitigates the mutual coupling MC effect by limiting the spacing between array elements. X. Lai et al. proposed a multi-layer filled coprime array (MLFCA) in their paper "Multi-Layer Filled Coprime Arrays for DOAEstimation With Extended Hole-Free Coarray" (IEEE Transactions on Vehicular Technology, vol. 73, no. 2, pp. 2621-2632, Feb. 2024). This array design first constructs a coprime array with multiple repeating subarrays, then layers its holes, and finally fills the holes with additional subarrays. The MLFCA and its improved extended multi-layer filled coprime array (EMLFCA) have significantly more uniform degrees of freedom (uDOFs) than nested arrays. Like nested arrays (NA), multi-hole-filled coprime arrays (MLFCA) and extended multi-hole-filled coprime arrays (EMLFCA) can form hole-free differential coprime arrays (DCA) with a space efficiency of 100%. However, the mutual coupling MC effect between array elements is higher than that of coprime arrays (CA). In their paper "Bilateral Multi-Subarray Padded Coprime Array Configuration for DOA Estimation in MIMO Radar" (IEEE Transactions on Vehicular Technology, vol. 74, no. 8, pp. 12714-12726, Aug. 2025), H. Chen et al. proposed a bilateral multi-subarray coprime array (BMSCA) and introduced additional filling subarrays to form a bilateral multi-subarray padded coprime array (BMSPCA). Compared with padded coprime arrays (PCAs), this array further improves the uniform degrees of freedom (uDOFs) while maintaining a low mutual coupling MC effect. However, the design did not consider the impact of space efficiency on the application of the array in engineering. The excessively sparse array elements resulted in the array occupying too much physical space while failing to achieve the corresponding performance gains, leading to low space efficiency.Therefore, given the limited size of the array platform, this design struggles to simultaneously achieve both direction-of-arrival (DOA) estimation accuracy and engineering feasibility, thus limiting its engineering application value to some extent.

[0004] To improve direction-of-arrival (DOA) estimation performance, the optimization design of sparse arrays typically aims to suppress mutual coupling effects and increase uniform degrees of freedom. However, in practical applications, spatial efficiency is crucial to considering the array's physical feasibility and scene adaptability. Therefore, from an engineering perspective, the design of sparse arrays must simultaneously consider the impact of mutual coupling, degrees of freedom, and spatial efficiency. However, these three factors often conflict and are difficult to optimize simultaneously. On one hand, to suppress mutual coupling effects between array elements, the element spacing needs to be increased; however, excessive element spacing reduces spatial efficiency and wastes platform space resources. On the other hand, to increase the array's uniform degrees of freedom, the number of array elements needs to be increased or supplementary subarrays need to be introduced to fill gaps. However, introducing supplementary subarrays or increasing the number of array elements within a limited space will exacerbate mutual coupling effects, ultimately leading to a decrease in DOA estimation performance. Therefore, under the condition of limited space resources, how to achieve a reasonable trade-off between mutual coupling suppression, degree of freedom enhancement and space efficiency, and design a sparse array structure that takes into account both engineering feasibility and high DOA estimation performance, has become one of the key technical problems that urgently need to be solved in the field of sparse array design. Summary of the Invention

[0005] Objective of the Invention: The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a method for designing k-fold expanded and filled coprime arrays, comprising the following steps:

[0006] Step 1: Define the structure of the original coprime array;

[0007] Step 2: Based on the original coprime array, construct an expanded coprime array and determine the location of the non-negative holes in the difference comatrix;

[0008] Step 3: Construct a k-fold expanded coprime array by copying and translating subarrays, and determine the value of the subarray spacing L and the position of the non-negative part of the holes in the differential coarray;

[0009] Step 4: Construct a k-fold expanded coprime array for hole filling, and calculate the physical aperture, uniform degree of freedom, degree of freedom, weighting function, and given uniform degree of freedom. Optimal configuration of time parameters;

[0010] Step 5: Construct a k-fold expansion filled coprime array with compressed element spacing by compressing the k-fold expansion to fill part of the coprime array element spacing, and clarify the array's construction conditions.

[0011] Step 1 includes: setting the original coprime array to contain two uniform subarrays, namely subarray one and subarray two, wherein the element spacing of subarray one is Md and the number of elements is N; the element spacing of subarray two is Nd and the number of elements is M. The unit element spacing, M and N are coprime; Subarray 1 and Subarray 2 are arranged in an overlapping manner and share the first array element. The first array element is used as the reference array element and placed at the origin of the coordinate system. The remaining array elements are arranged along the negative half-axis with a spacing of Md or Nd.

[0012] Step 2 includes: based on the original coprime array, keeping the positions of the elements of the subarray unchanged, symmetrically folding the second subarray along the origin to unfold the two overlapping subarrays, and then translating it along the positive semi-axis. This forms an expanded coprime array UCA, with a total number of array elements of the expanded coprime array UCA. Location set ,in, This means taking all integers within the range. and These are intermediate parameters; the physical aperture of the expanded coprime array UCA is... The locations of the holes in the non-negative part of the differential coarray DCA of the coprime array UCA are as follows: , , , ,in, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , This indicates rounding up to the nearest integer.

[0013] Step 3 includes: based on the expanded coprime array UCA, the subarray is copied k-1 times and translated along the positive semi-axis, so that the subarray spacing between any two adjacent subarrays is . This forms a k-fold expanded coprime array k-UCA, where k is the number of repetitions of subarray two after replication and translation; the total number of elements in the k-fold expanded coprime array k-UCA is... Array element location set Among them, the set of array element positions of subarray one , , Let be the set of positions formed by k repeating subarrays. For set The set of element positions of the i-th subarray. intermediate parameters Subarray spacing At that time, the k-fold expansion of the coprime array k-UCA maximizes the utilization of the virtual array elements;

[0014] The non-negative part of the differential comatrix DCA of the k-fold expanded coprime array k-UCA has the following hole locations: , , , ,in, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , .

[0015] Step 4 includes: making the supplementary subarray ,in, and For the set of array element positions, For intermediate parameters, Indicates rounding down; will supplement the subarray Incorporating the k-fold expanded coprime array k-UCA, thus forming the k-fold expanded filled coprime array k-UPCA; the set of element positions of the k-fold expanded filled coprime array k-UPCA. The total number of elements in the k-UPCA array filled by k-fold expansion is .

[0016] Step 4 also includes:

[0017] Step 4-a: Calculate the physical aperture, uniformity degree of freedom, and degree of freedom of the k-fold expanded and filled coprime array k-UPCA;

[0018] Step 4-b: Calculate the weighting function of the k-UPCA array with k-fold expansion and filling;

[0019] Step 4-c: Calculate the given uniform degrees of freedom At that time, the optimal configuration of parameters M, N, and k for k-UPCA of the k-re-expanded coprime array is determined.

[0020] Step 4-a includes: k-fold expansion filling of the coprime array; the aperture size of the k-UPCA physical array is... The range of uniform virtual array elements in the differential comatrix DCA of the k-UPCA (k-reinforced coprime array) is: Uniform degrees of freedom , among which, when At that time, intermediate parameters ;when At that time, intermediate parameters ;when At that time, the size of the degrees of freedom (DOF) ,when At that time, the size of the degrees of freedom (DOF) .

[0021] Step 4-b includes: weighting function for:

[0022] , , ,

[0023] Step 4-c includes: when The problem is then stated as:

[0024] ,

[0025] in Let be the set of all positive integers. This represents the total number of array elements. Given uniform degrees of freedom, `subjectto` indicates that the system is constrained by... This indicates taking the minimum value. Let M and N be the greatest common divisor, and the optimization objective be to satisfy a given uniform degree of freedom. Under the condition of making the total number of array elements Minimize, the approximate solution to the problem is , .

[0026] Step 5 includes: when the k-fold expansion fills the coprime array k-UPCA, the array satisfies the condition: there exists a factor of parameter N. , making Then, the k-fold expanded and filled coprime array k-UPCA can be further improved into a k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing, where the factor It is a positive integer, specifically in the form of a subarray. The first in The position of the first element of each subarray remains unchanged, and... The element spacing is compressed to The positions of the remaining elements in the k-fold expanded and filled coprime array k-UPCA remain unchanged, thus forming a new array design: the k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing. The set of element positions for array k-UPCACiS is... ,in, This is the set of positions of the elements in subarray one. for variants, The set of array element positions is , For the sub-array variants and The set of array element positions is , , A factor of N , To supplement the subarray; the physical aperture of the k-fold expansion filling the coprime array k-UPCACiS with compressed element spacing is: .

[0027] Compared to traditional uniform linear arrays (ULAs), sparse arrays can achieve larger apertures and higher degrees of freedom with the same number of elements. Their direction-of-arrival (DOA) estimation performance largely depends on the rational design of the array structure. In existing sparse array designs, increasing the array's degrees of freedom and suppressing mutual coupling effects between elements are the main technical approaches to improving DOA estimation performance. High space efficiency is a crucial factor in ensuring the array's engineering feasibility under space-constrained physical platform conditions. However, there is a clear interdependence between mutual coupling suppression, degree-of-freedom enhancement, and space efficiency, making it difficult for traditional array structure designs to simultaneously achieve these performance indicators within the constraints of limited physical aperture.

[0028] Based on this, the present invention proposes a k-fold expansion and filling coprime array design method. By expanding and replicating the coprime array and accurately filling the holes in the differential coarray, the method improves the uniform degree of freedom of the array while maintaining low mutual coupling effect. Furthermore, by designing a reasonable array element spacing, the spatial efficiency of the array is improved. Thus, an effective balance is achieved between mutual coupling, degree of freedom, and spatial efficiency, further improving the array's direction of arrival (DOA) estimation performance and engineering applicability.

[0029] The present invention has the following beneficial effects:

[0030] (1) Based on the original coprime array, this invention proposes a k-fold expansion and filling coprime array design method by expanding, copying, translating and filling holes of the subarray. Based on this, it proposes an improved array with compressed array element spacing k-fold expansion and filling coprime array. Compared with the traditional coprime array, the proposed array significantly improves the differential coarray degree of freedom while maintaining a reasonable distribution of array element spacing, effectively suppresses the mutual coupling effect between array elements, and thus improves the DOA estimation accuracy and resolution.

[0031] (2) This invention fully considers the impact of space efficiency on engineering applications. Higher space efficiency can ensure that the array maintains relatively good direction-of-arrival (DOA) estimation performance even when the platform space is limited. This invention achieves an effective balance between degrees of freedom, mutual coupling suppression, and space efficiency under limited platform space conditions by reasonably setting the element spacing and filling strategy, so that the array can still maintain good DOA estimation performance in practical application scenarios with limited platform space. Attached Figure Description

[0032] Figure 1 This is a flowchart of the method of the present invention.

[0033] Figure 2 This is a schematic diagram of the original coprime array.

[0034] Figure 3 This is a schematic diagram of UCA array elements and DCA (where M=4, N=9, and the position unit is d).

[0035] Figure 4 This is a schematic diagram of a k-UCA array design.

[0036] Figure 5 This is a schematic diagram of k-UCA array elements and DCA (where M=5, N=12, k=2, and the position unit is d).

[0037] Figure 6 This is a schematic diagram of k-UPCA array elements and DCA (where M=5, N=12, k=2, and the position unit is d).

[0038] Figure 7 This is a schematic diagram of the special symmetrical structure of holes and virtual array elements in the discontinuous part of DCA.

[0039] Figure 8 This is a schematic diagram of the k-UPCACiS array elements and DCA (where M=5, N=12, k=2, α=2, and the position unit is d).

[0040] Figure 9 This is a schematic diagram of the k-UPCACiS array elements and DCA (where M=5, N=12, k=2, α=3, and the position unit is d).

[0041] Figure 10 This is a schematic diagram showing the relationship between DOA estimation error and signal-to-noise ratio.

[0042] Figure 11 This is a schematic diagram showing the relationship between DOA estimation error and the number of snapshots.

[0043] Figure 12 This is a schematic diagram showing the relationship between DOA estimation error and the mutual coupling coefficient. Detailed Implementation

[0044] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.

[0045] like Figure 1 As shown, this embodiment of the invention provides a method for designing a k-fold expanded and filled coprime array, comprising the following steps:

[0046] Step 1: Define the structure of the original coprime array.

[0047] The original coprime array is assumed to contain two uniform subarrays. One subarray has an element spacing of Md and N elements; the other subarray has an element spacing of Nd and M elements. The unit element spacing, Furthermore, M and N are coprime. Subarray 1 and Subarray 2 are arranged in an overlapping configuration and share the first array element, which is used as the reference array element and placed at the origin of the coordinate system. The remaining array elements are arranged along the negative semi-axis with spacing of Md or Nd, as shown below. Figure 2 As shown.

[0048] Step 2: Based on the original coprime array, construct an expanded coprime array and determine the location of the non-negative holes in the differential comatrix.

[0049] Because the two subarrays of the original coprime array are arranged in an overlapping manner, a relatively tight element spacing is generated, resulting in a severe mutual coupling effect, and the degrees of freedom of its differential coarray are also severely restricted. Based on the original coprime array, keeping the element positions of the two subarrays unchanged, the second subarray is symmetrically folded along the origin, causing the two overlapping subarrays to unfold and translate along the positive semi-axis. This forms an Unfolded Coprime Array (UCA). One subarray of the UCA contains M elements with an element spacing of Nd. Begin by uniformly arranging the array along the positive semi-axis; another subarray contains There are n array elements with an element spacing of Md, uniformly arranged from the origin along the negative half-axis, where M and N are coprime integers and satisfy... The total number of elements in the coprime array UCA is... Its location is:

[0050] (1),

[0051] In the formula, This means taking all integers within the range. and These are intermediate parameters.

[0052] Unfolding a coprime array UCA has the following properties:

[0053] (1) The physical aperture of the coprime array UCA is .

[0054] (2) The location of the holes in the non-negative part of the differential coarray DCA of the coprime array UCA is as follows: , , and Given by the following formula

[0055] (2),

[0056] In the formula, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , This indicates rounding up to the nearest integer.

[0057] Figure 3 The figure presents the element positions of the expanded coprime array UCA and its differential comatrix DCA when M=4 and N=5. Due to the symmetry of the differential comatrix DCA, only the non-negative part of the DCA is shown in the figure. It can be seen that compared to the original coprime array, the expanded coprime array UCA reduces redundancy and thus gains more degrees of freedom (DOF) because the two uniform subarrays are expanded. Furthermore, the increased element spacing effectively suppresses the mutual coupling effect. However, the presence of holes in the differential comatrix DCA severely affects its uniform degrees of freedom (uDOFs), and the usable degrees of freedom after spatial smoothing are even less than those of a uniform array with an element spacing of half a wavelength. Therefore, to solve the above problems, the array optimization strategy is to improve the array degrees of freedom by altering the array structure of the expanded coprime array UCA through replication and shifting, and to improve the uniform degrees of freedom by identifying and filling the holes in the differential comatrix DCA.

[0058] Step 3: Construct a k-fold expanded coprime array by copying and translating the subarrays, and determine the value of the subarray spacing L and the position of the non-negative holes in the differential coarray.

[0059] Based on the unfolded coprime array UCA, the subarrays are copied k-1 times and translated along the positive semi-axis, such that the spacing between any two subarrays is . This forms a k-fold expanded coprime array, where k is the number of repetitions of subarray two after copying and translation. For example... Figure 4As shown, this array design is called a k-times unfolded coprime array (k-UCA), and its total number of elements is... The array element position set is ,gather Let the set of positions of subarray one be the set of positions. The set of positions formed by k repeating subarrays is given by the following formula:

[0060] (3),

[0061] In the formula, For set The set of positions of the i-th subarray. M and N are coprime integers and satisfy intermediate parameters . Figure 5 The diagram shows the element positions and differential coprime array (DCA) of the k-fold expanded coprime array k-UCA when M=5, N=12, and k=2.

[0062] Proposition 1: When the subarray spacing At that time, the k-fold expansion of the coprime array k-UCA maximizes the utilization of the virtual array elements.

[0063] Proof: Subarray spacing L will affect cross difference If when At that time, cross difference set If there is no overlap between the generated virtual array elements, it can be concluded that the k-fold expanded coprime array k-UCA maximizes the utilization of virtual array elements. Let represent the set of virtual array elements formed by the difference between two subarrays. We will now prove by contradiction that when... Time, difference set and Virtual array elements without overlap. When At that time, there were:

[0064] (4),

[0065] (5),

[0066] In the formula, For intermediate parameters, and , , ,therefore , , .

[0067] Assumption and If there are overlapping virtual array elements, then there exists , , Make:

[0068] (6),

[0069] Simplifying equation (6) yields:

[0070] (7),

[0071] because and ,so Therefore, equation (7) can be rewritten as:

[0072] (8),

[0073] because and Coprime, therefore, if formula (8) holds, then if and only if for If a number is a non-zero integer multiple of a, then it must have the following: This is related to This contradicts the statement. Therefore, when the subarray spacing... hour, and The k-fold expanded coprime array k-UCA, which has no overlapping virtual array elements, maximizes the utilization of virtual array elements.

[0074] when When, the location of the holes in the non-negative part of the differential comatrix DCA of the k-fold expanded coprime array k-UCA is: , , and Given by the following formula:

[0075] (9),

[0076] In the formula, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , . Figure 5 The diagram shows the element positions and differential co-array DCA of the k-fold expanded coprime array k-UCA when M=5, N=12, and k=2.

[0077] Step 4: Construct a k-fold expanded coprime array for hole filling, and calculate the physical aperture, uniform degree of freedom, degree of freedom, weighting function, and given uniform degree of freedom. The optimal configuration of time parameters M, N, and k.

[0078] Order to supplement the subarray , and For the set of array element positions, For intermediate parameters, This indicates rounding down. The supplementary subarray will be added. After incorporating the k-fold expanded coprime array k-UCA, a k-fold expanded filled coprime array is formed. The set of element positions of the k-fold expanded filled coprime array k-UPCA is as follows: The total number of elements in a k-UPCA array filled by k-fold expansion is M and N are coprime integers and satisfy . Figure 6 The diagram shows the element positions and differential co-array DCA of the k-fold expanded and filled coprime array k-UPCA when M=5, N=12, and k=2.

[0079] Step 4-a: Calculate the physical aperture, uniformity degree of freedom, and degree of freedom of the k-fold expanded and filled coprime array k-UPCA.

[0080] The k-UPCA array design with k-fold expansion and filling of coprime arrays has the following properties:

[0081] (1) Its physical array aperture size is .

[0082] (2) When At that time, the location of the first hole in the non-negative part of its differential comatrix DCA is... ;when At that time, the location of the first hole in the non-negative part of its differential comatrix DCA is... .

[0083] (3) The range of uniform virtual array elements in the differential co-array DCA is And the uniform degrees of freedom are , among which when At that time, intermediate parameters ;when At that time, intermediate parameters .

[0084] (4) When At that time, the size of the degrees of freedom (DOF) is ,when At that time, the size of the degrees of freedom (DOF) is .

[0085] Proof: The properties (1), (2) and (3) of the k-fold expanded filled coprime array k-UPCA can be directly derived from the positions of the array elements of the k-fold expanded filled coprime array k-UPCA and the special set of holes in the differential coprime array DCA, so they will not be described further.

[0086] For property (4), first calculate the subarray. Before filling, the differential comatrix DCA located in Number of virtual array elements Next, calculate the subarray. After filling, in The number of virtual array elements increased Finally, through the formula The degrees of freedom (DOF) are derived, including the intermediate parameters. .

[0087] Before the holes are filled, the differential comatrix DCA of the k-fold expanded coprime array k-UCA is located in The virtual array elements are obtained by differentially dividing two uniform arrays. Therefore, the holes and virtual array elements of this discontinuous segment have a special symmetrical structure. Figure 5 Taking discontinuous segments in differential coarrays (DCA) as an example, the special symmetric structure of these holes and virtual array elements, such as... Figure 7 As shown.

[0088] Based on the structural characteristics and the symmetry of the differential comatrix DCA as a whole, the submatrix can be obtained. Before filling, the differential comatrix DCA located in Number of virtual array elements .

[0089] when At that time, sub-array Filling will not make the interval Add virtual array elements, therefore .when At that time, only the sub-array The first array element and The difference between them will affect Therefore, let's assume , , , If it is an intermediate parameter, then The location of the hole meets the requirements. Filling the hole satisfies the equation:

[0090] (10)

[0091] Will Substituting into equation (10) and simplifying, we get ,therefore The number of elements in the solution set is Therefore, when At that time, sub-array The first array element and After The individual element differences will fill the interval. The holes in, i.e. .

[0092] Finally, according to the formula ,when At that time, the size of DOF ,when At that time, the size of DOF .

[0093] Step 4-b: Calculate the weight function of the k-UPCA for the k-fold expanded and filled coprime array.

[0094] For array design, k-UPCA with k-fold expansion to fill coprime arrays, its weighting function is... It is given by the following formula:

[0095] (11),

[0096] Proof: For sparse arrays, This represents the number of differences of d, 2d, and 3d in the self-difference and cross-difference subsets. For the k-fold expanded filled coprime array k-UPCA, its key weight function is only related to... and It is relevant, but not relevant to other difference sets.

[0097] for ,when hour, The spacing between array elements is The number of array elements is The uniform sparse linear array, therefore the difference 3d appears in total. In this case, there are no differences d and 2d; when , There are no differences d, 2d, and 3d in it.

[0098] for ,because For the spacing between array elements ,and Inserted as a supplementary subarray In the gaps between the array elements, Each array element in the middle, except for the two on the sides Apart from the array element in question, the distance between it and the other array elements is greater than 3d. Therefore, it is only necessary to consider... The array element in the middle and the two on its sides The difference between array elements. Because It is the union of two sets. For ease of proof, The first in The difference between the individual array element and the array elements on both sides can be expressed as: and Therefore, for variants, and Having the same difference d, 2d, 3d, where This represents the modulo operation. When... At that time, the set The middle element contains only one array element, and its differences from the array elements on both sides are d and 2d, respectively. When hour, Suppose there exist two pairs of array elements whose difference is , is represented as:

[0099] (12)

[0100] In the formula, It is an intermediate parameter, and satisfies , Consider the following two cases:

[0101] 1) If and Then we have:

[0102] (13)

[0103] because and Coprime, and , Therefore, equation (13) holds only if and This means the difference It appears only once.

[0104] 2) If and Then we have:

[0105] (14)

[0106] because and Coprime, and , Therefore, for equation (14) to hold, the following conditions must be met. , However, when When it is an odd number, There is no solution at this point. When the number is even, the solution to equation (14) is and , At this point, there will be two sets of differences. This means that when hour, It contains two differences, 2d; when hour, It contains two differences, 3d.

[0107] Step 4-c: Calculate the given uniform degrees of freedom When, the optimal configuration of parameters M, N, and k for k-fold expansion filling coprime array k-UPCA is determined.

[0108] This invention provides a parameter selection scheme for selecting parameters based on a given desired uniform degree of freedom. In the case of array parameters Find the optimal configuration. When The problem is then stated as:

[0109] (15)

[0110] In the formula, T represents the total number of array elements. For a given uniform degree of freedom, This indicates taking the minimum value. Denotes the greatest common divisor of M and N. Let be the set of all positive integers, and `subject to` indicates that it is restricted to. (This is not considered for now.) The integer and coprime properties, and when When the number is even, the original problem is transformed into:

[0111] (16)

[0112] In the formula, Let be the set of all positive real numbers.

[0113] when When fixed, according to the Lagrange multiplier method, the solution is:

[0114] (17)

[0115] when When it is large, Size by Dominant approach yields approximate solutions. Finally, we check the coprimeness of integers near the optimal solutions M and N in equation (17) to obtain the optimal solution of the original problem (15).

[0116] Step 5: Construct a k-fold expansion filled coprime array with compressed element spacing by compressing the k-fold expansion to fill the element spacing of the coprime array k-UPCA, and clarify the construction conditions of the array.

[0117] Definition (space efficiency): Space efficiency SE is given by the following formula:

[0118] (18)

[0119] In the formula, Indicates space efficiency. The upper limit of the range of the largest uniform virtual array elements in the differential co-array (DCA) represents the effective virtual aperture. For array physical aperture, , Represents the set of locations of a physical array. This indicates taking the maximum value. From formula (18), it can be intuitively seen that the space efficiency... The study measured the utilization of platform space by sparse arrays and the array's ability to convert physical apertures into effective virtual apertures.

[0120] When a k-fold expansion fills a coprime array k-UPCA, the array satisfies the condition: there exists a factor of parameter N. , making , where factors If the integer is positive, the k-fold expanded and filled coprime array k-UPCA can be further improved into a k-fold expanded and filled coprime array with compressed element spacing, thereby further improving the space efficiency of the array design. The specific method is as follows: subarray The first in The position of the first element of each subarray remains unchanged, and the subarray is... The element spacing is compressed to The remaining element positions of the k-fold expanded and filled coprime array k-UPCA remain unchanged, thus forming a k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing, whose element position set is as follows. ,in, This is the set of positions of the elements in subarray one. for variants, The set of array element positions is , For the sub-array variants and The set of array element positions is , , A factor of N , To supplement the subarray, the physical aperture of the k-fold expansion filling the coprime array k-UPCACiS with compressed element spacing is: . Figure 8 and Figure 9 exist Figure 6 Based on this, compression factors are introduced respectively and .

[0121] The technical effects of this invention can be illustrated by the following simulation experiments. To demonstrate the advantages of the k-UPCA (k-layer expanded filled coprime array) and its improved design, the k-UPCACiS (k-layer expanded filled coprime array with compressed element spacing), Table 1 presents a parameter comparison of different sparse arrays at T=24. These include the BMSPCA (bilateral multi-subarray filled coprime array) proposed in the paper "Bilateral Multi-Subarray Padded Coprime Array Configuration for DOA Estimation in MIMO Radar" (IEEE Transactions on Vehicular Technology, vol. 74, no. 8, pp. 12714-12726, Aug. 2025), the ePCA (extended filled coprime array) proposed in the paper "Padded coprimearrays for improved DOA estimation: Exploiting hole representation and filling strategies" (IEEE Trans. Signal Process, vol. 68, pp. 4597–4611, 2020), and the Multi-Layer Filled Coprime Arrays for DOA Estimation With Extended Hole-Free. The extended multilayer void-filled coprime array EMLFCA proposed in "Coarray" (IEEE Transactions on Vehicular Technology, vol. 73, no. 2, pp. 2621-2632, Feb. 2024.) shows in Table 1 that the proposed array design achieves an effective balance between mutual coupling, degrees of freedom, and space efficiency. The k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing further improves the space efficiency based on the k-fold expanded and filled coprime array k-UPCA.

[0122] Table 1

[0123]

[0124] Furthermore, simulation experiments were conducted using the MUSIC algorithm to compare the direction-of-arrival (DOA) estimation accuracy of the proposed array with that of the two-sided multi-subarray filled coprime array BMSPCA, the extended filled coprime array ePCA, and the extended multi-layered perforated filled coprime array EMLFCA. The root mean square error (RMSE) of DOA estimation was defined as:

[0125] (19)

[0126] In the formula, the number of Monte Carlo trials is 1000. Indicates the first The Monte Carlo trial The number of information sources. Indicates the first One source, Indicates the first The direction of arrival of each information source, For the first In this experiment The estimated value of the direction of arrival (DOA).

[0127] Simulation Experiment 1: The variation of Direction of Arrival (DOA) estimation error with signal-to-noise ratio;

[0128] This experiment was set with 1000 snapshots, a total of 24 different sparse array elements (T=24), and 25 signal sources (J=25). All signal sources were uniformly distributed across... Coupling coefficient within the angular range The signal-to-noise ratio varies from -10dB to 20dB. Figure 10 The relationship between DOA estimation error and signal-to-noise ratio for different sparse arrays is presented. Figure 10 As can be seen, the array design proposed in this invention achieves a reasonable trade-off between mutual coupling suppression, increased degrees of freedom, and spatial efficiency. The direction-of-arrival (DOA) estimation results of the k-UPCA array with compressed element spacing and its improved design, k-UPCACiS, have smaller errors.

[0129] Simulation Experiment 2: The variation of Direction of Arrival (DOA) estimation error with the number of snapshots;

[0130] In this experiment, the signal-to-noise ratio was set to 0dB, the total number of different sparse array elements was T=24, the number of information sources was J=25, and all information sources were uniformly distributed in... Coupling coefficient within the angular range The number of snapshots varies from 100 to 3000. Figure 11The relationship between DOA estimation error and snapshot number for different sparse arrays is presented. Figure 11 As can be seen, compared with other sparse array designs, the k-UPCA array with k-fold expansion and filling coprime and its improved design, the k-UPCACiS array with compressed element spacing, have smaller errors in the direction of arrival (DOA) estimation results.

[0131] Simulation Experiment 3: The variation of Direction of Arrival (DOA) estimation error with mutual coupling coefficient;

[0132] This experiment was set with a signal-to-noise ratio of 10dB, a snapshot count of 1500, a total of 24 different sparse array elements (T=24), and 25 signal sources (J=25). All signal sources were uniformly distributed across... Coupling coefficient within the angular range The range of variation is 0 to 0.6. Figure 12 The relationship between DOA estimation error and cross-coupling coefficient for different sparse arrays is presented. Figure 12 It can be seen from this that when When the values ​​are small, the accuracy of the direction-of-arrival (DOA) estimation mainly depends on the uniform degrees of freedom (uDOFs), which makes the extended multilayer hole-filled coprime array EMLFCA outperform other arrays. However, as... As the number of elements increases, the array estimation accuracy is increasingly affected by the mutual coupling MC effect. The advantages of the k-fold expanded filled coprime array k-UPCA and the k-fold expanded filled coprime array k-UPCACiS with compressed element spacing become more obvious, while the performance of the extended multilayer hole-filled coprime array EMLFCA deteriorates rapidly.

[0133] Simulation experiments show that the k-fold expanded filled coprime array k-UPCA proposed in this invention and its improved design k-fold expanded filled coprime array k-UPCACiS with compressed element spacing have high degrees of freedom and effectively suppress mutual coupling effects. A reasonable trade-off is achieved between mutual coupling suppression, degree of freedom improvement and space efficiency, thus obtaining higher direction of arrival (DOA) estimation performance.

[0134] This invention provides a method for designing k-fold expanded and filled coprime arrays. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.

Claims

1. A method for designing a k-fold expanded and filled coprime array, characterized in that, Includes the following steps: Step 1: Define the structure of the original coprime array; Step 2: Based on the original coprime array, construct an expanded coprime array and determine the location of the non-negative holes in the difference comatrix; Step 3: Construct a k-fold expanded coprime array by copying and translating subarrays, and determine the value of the subarray spacing L and the position of the non-negative part of the holes in the differential coarray; Step 4: Construct a k-fold expanded coprime array for hole filling, and calculate the physical aperture, uniform degree of freedom, degree of freedom, weighting function, and given uniform degree of freedom. Optimal configuration of time parameters; Step 5: Construct a k-fold expansion filled coprime array with compressed element spacing by compressing the k-fold expansion to fill part of the coprime array element spacing, and clarify the formation conditions of the array.

2. The method according to claim 1, characterized in that, Step 1 includes: setting the original coprime array to contain two uniform subarrays, namely subarray one and subarray two, wherein the element spacing of subarray one is Md and the number of elements is N; the element spacing of subarray two is Nd and the number of elements is M. The unit element spacing, M and N are coprime; Subarray 1 and Subarray 2 are arranged in an overlapping manner and share the first array element. The first array element is used as the reference array element and placed at the origin of the coordinate system. The remaining array elements are arranged along the negative half-axis with a spacing of Md or Nd.

3. The method according to claim 2, characterized in that, Step 2 includes: based on the original coprime array, keeping the positions of the elements of the subarray unchanged, symmetrically folding the second subarray along the origin to unfold the two overlapping subarrays, and then translating it along the positive semi-axis. This forms an expanded coprime array UCA, with a total number of array elements of the expanded coprime array UCA. Location set ,in, This means taking all integers within the range. and These are intermediate parameters; the physical aperture of the expanded coprime array UCA is... The locations of the holes in the non-negative part of the differential coarray DCA of the coprime array UCA are as follows: , , , ,in, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , This indicates rounding up to the nearest integer.

4. The method according to claim 3, characterized in that, Step 3 includes: based on the expanded coprime array UCA, the subarray is copied k-1 times and translated along the positive semi-axis, so that the subarray spacing between any two adjacent subarrays is . This forms a k-fold expanded coprime array k-UCA, where k is the number of repetitions of subarray two after replication and translation; the total number of elements in the k-fold expanded coprime array k-UCA is... Array element location set Among them, the set of array element positions of subarray one , , Let be the set of positions formed by k repeating subarrays. For set The set of element positions of the i-th subarray. intermediate parameters Subarray spacing At that time, the k-fold expansion of the coprime array k-UCA maximizes the utilization of the virtual array elements; The non-negative part of the differential comatrix DCA of the k-fold expanded coprime array k-UCA has the following hole locations: , , , ,in, , and For the set of hole locations, intermediate parameters It is a positive integer and satisfies , , .

5. The method according to claim 4, characterized in that, Step 4 includes: making the supplementary subarray ,in, and For the set of array element positions, For intermediate parameters, Indicates rounding down; will supplement the subarray Incorporating the k-fold expanded coprime array k-UCA, thus forming the k-fold expanded filled coprime array k-UPCA; the set of element positions of the k-fold expanded filled coprime array k-UPCA. The total number of elements in the k-UPCA array filled by k-fold expansion is .

6. The method according to claim 5, characterized in that, Step 4 also includes: Step 4-a: Calculate the physical aperture, uniformity degree of freedom, and degree of freedom of the k-fold expanded and filled coprime array k-UPCA; Step 4-b: Calculate the weighting function of the k-UPCA array with k-fold expansion and filling; Step 4-c: Calculate the given uniform degrees of freedom At that time, the optimal configuration of parameters M, N, and k for k-UPCA of the k-re-expanded coprime array is determined.

7. The method according to claim 6, characterized in that, Step 4-a includes: k-fold expansion filling of the coprime array; the aperture size of the k-UPCA physical array is... The range of uniform virtual array elements in the differential comatrix DCA of the k-UPCA (k-reinforced coprime array) is: Uniform degrees of freedom , among which, when At that time, intermediate parameters ;when At that time, intermediate parameters ;when At that time, the size of the degrees of freedom (DOF) ,when At that time, the size of the degrees of freedom (DOF) .

8. The method according to claim 7, characterized in that, Step 4-b includes: weighting function for: ， ， 。 9. The method according to claim 8, characterized in that, Step 4-c includes: when The problem is then stated as: ， in Let be the set of all positive integers. This represents the total number of array elements. Given uniform degrees of freedom, "subject to" indicates that the system is constrained by... This indicates taking the minimum value. Let M and N be the greatest common divisor, and the optimization objective be to satisfy a given uniform degree of freedom. Under the condition of making the total number of array elements Minimize, the approximate solution to the problem is , .

10. The method according to claim 9, characterized in that, Step 5 includes: when the k-fold expansion fills the coprime array k-UPCA, the array satisfies the condition: there exists a factor of parameter N. , making Then, the k-fold expanded and filled coprime array k-UPCA can be further improved into a k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing, where the factor It is a positive integer, specifically in the form of a subarray. The first in The position of the first element of each subarray remains unchanged, and... The element spacing is compressed to The positions of the remaining elements in the k-fold expanded and filled coprime array k-UPCA remain unchanged, thus forming a new array design: the k-fold expanded and filled coprime array k-UPCACiS with compressed element spacing. The set of element positions for array k-UPCACiS is... ,in, This is the set of positions of the elements in subarray one. for variants, The set of array element positions is , For the sub-array variants and The set of array element positions is , , A factor of N , To supplement the subarray; the physical aperture of the k-fold expansion filling the coprime array k-UPCACiS with compressed element spacing is: .

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