A method for designing a type-ii ternary primitive array based on a primitive sparse arrangement

Abstract

The application discloses a type II ternary coprime array design method based on a coprime sparse arrangement and relates to the field of array structure design. A basic coprime number pair, additional array elements and third subarray parameters are determined, two coprime positive integers M and N are selected to construct a prototype coprime array basic structure, the prototype coprime array comprises a subarray 1 with an array element number of N and a spacing of Md and a subarray 2 with an array element number of M and a spacing of Nd, and d is a unit array element spacing; one array element is added at a positive direction of the prototype coprime array, and a uniform subarray with an array element number of and a spacing of is added in reverse, so that a type II ternary coprime array is formed. The type II ternary coprime array designed by the method has the characteristics of high continuous degrees of freedom, weak mutual coupling, modularity and scalability.

Description

Technical Field

[0001] This invention relates to the field of array structure design, and more specifically to a design method for a type II ternary coprime array based on coprime sparse arrangement. Background Technology

[0002] Direction of arrival (DOA) estimation is a core task in signal processing, crucial for signal source localization and beamforming, and widely used in radar, sonar, wireless communication, sensor networks, UAVs, and autonomous driving. Sparse linear arrays have attracted much attention due to their high degrees of freedom, low coupling, and large aperture. Traditional minimum-pitch uniform linear arrays typically employ a half-wavelength equidistant configuration, exhibiting good modularity and scalability. However, because this array cannot extend its degrees of freedom through virtual elements, the number of estimable sources is limited by the number of physical elements, and it suffers from limited aperture and high coupling. Sparse arrays, through non-equidistant element settings, can achieve larger apertures and more diverse differential array sets, effectively improving estimation performance. However, existing sparse linear arrays often sacrifice the regularity of virtual elements to achieve greater degrees of freedom, resulting in shortcomings in modularity and scalability. Furthermore, traditional coprime array concepts mainly focus on binary coprime structures, failing to fully explore the potential of ternary coprime structures. Therefore, by analyzing the pairing of three sets of uniform linear arrays into coprime spacings, revealing the structure of their difference comatrix sets, and constructing integer functions to evaluate the number of elements in the element range set, this will help achieve a balance between high degrees of freedom, low coupling, large aperture, modularity, and scalability in sparse arrays, thereby better leveraging the advantages of sparse arrays. This research not only has theoretical significance but will also promote the development and application of sparse arrays in practical applications. Summary of the Invention

[0003] In view of this, the present invention provides a design method for a type II ternary coprime array based on coprime sparse arrangement.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: A design method for a type II ternary coprime array based on coprime sparse arrangement includes the following steps: Determine the basic coprime pairs, additional array elements, and parameters of the third subarray. Select two coprime positive integers M and N to construct the basic structure of the prototype coprime array. The prototype coprime array contains subarray 1 with N elements and a spacing of Md, and subarray 2 with M elements and a spacing of Nd, where d is the unit element spacing. In the positive direction of the prototype coprime array... Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarrays form a type II ternary coprime array.

[0005] Optionally, determine the matching parameters between subarray 3 and the additional array elements. Based on the hole distribution of the prototype coprime array and the hole-filling characteristics of the additional array elements, derive the matching conditions for the element spacing and number of array elements that maximize the continuous degrees of freedom through theoretical derivation. Based on the differential coarray theory, derive the analytical expressions for the dissimilar and continuous degrees of freedom of the type II ternary coprime array. Based on the positive direction of the prototype coprime array... Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarray is expanded by adding Q+1 array elements without destroying the original coprime array structure.

[0006] Optionally, performance is verified under coupled and uncoupled models. The Toeplitz-ESPRIT algorithm is used for DOA estimation, and the array performance is evaluated by the root mean square error (RMSE) and the probability of successful resolution (POR). The RMSE and POR are defined as follows: ; ; In the formula, The number of Monte Carlo experiments; Number of information sources; It is the first The true incident direction of the signal source; It is the first In this experiment The estimated DOA value; Set to an acceptable resolution. This represents the number of successful resolutions for the k-th signal.

[0007] Optionally, the difference comatrix of the prototype coprime matrix is:

[0008] In the formula, Indicates the range, For integer functions, the domain is In the formula, Indicates the domain.

[0009] Optionally, the total number of elements in a Type II ternary coprime array is .

[0010] Alternatively, the number of distinct degrees of freedom can be obtained through the following lemma: gather The number of distinct elements in the expression is ; The number of distinct integers contained is ; Using the above lemma, the number of distinct elements, i.e., the distinct degrees of freedom, in a type II ternary coprime matrix is: .

[0011] Optionally, the number of continuous degrees of freedom is obtained through the following lemma: when , The maximum value of the elements in the middle is ; when hour, The maximum value of the elements in the middle is ; Using the above lemma, the number of virtual elements, i.e., the continuous degrees of freedom, of a type II ternary coprime matrix is: .

[0012] As can be seen from the above technical solutions, compared with the prior art, the present invention discloses a design method for a type II ternary coprime array based on coprime sparse arrangement. The type II ternary coprime array designed by this method has the characteristics of high continuous degree of freedom, weak mutual coupling, modularity, and scalability. In the mutual coupling scenario, it has better performance than other non-extended sparse arrays. Attached Figure Description

[0013] 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 embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0014] Figure 1 The diagram shows the structure of the Type II ternary coprime array provided by this invention.

[0015] Figure 2 This is a schematic diagram of the extended type II ternary coprime array provided by the present invention; Figure 3 A schematic diagram of a type II ternary coprime array before its expansion, provided by the present invention; Figure 4 A schematic diagram of the extended Type II ternary coprime array provided by the present invention. Detailed Implementation

[0016] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] This invention discloses a design method for a type II ternary coprime array based on coprime sparse arrangement, comprising the following steps: Determine the basic coprime pairs, additional array elements, and parameters of the third subarray. Select two coprime positive integers M and N to construct the basic structure of the prototype coprime array. The prototype coprime array contains subarray 1 with N elements and a spacing of Md, and subarray 2 with M elements and a spacing of Nd, where d is the unit element spacing. In the positive direction of the prototype coprime array... Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarray is further enhanced by adding Q+1 array elements to form subarray 3, thus forming a type II ternary coprime array.

[0018] Furthermore, this embodiment also includes determining the matching parameters between subarray 3 and the additional array elements. Based on the hole distribution of the prototype coprime array and the hole-filling characteristics of the additional array elements, the matching conditions for the array element spacing and the number of array elements that maximize the continuous degrees of freedom are derived theoretically. Based on the differential coarray theory, the analytical expressions for the dissimilar and continuous degrees of freedom of the type II ternary coprime array are derived. The matching conditions are obtained by deriving the matching parameters for the array element spacing and the number of array elements in the positive direction of the prototype coprime array. Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarray is expanded by adding Q+1 array elements without destroying the original coprime array structure.

[0019] Furthermore, performance was verified under coupled and uncoupled models. The Toeplitz-ESPRIT algorithm was used for DOA estimation, and array performance was evaluated by the root mean square error (RMSE) and the probability of successful resolution (POR). The RMSE and POR are defined as follows: ; ; In the formula, The number of Monte Carlo experiments; Number of information sources; It is the first The true incident direction of the signal source; It is the first In this experiment The estimated DOA value; Set to an acceptable resolution. This represents the number of successful resolutions for the k-th signal.

[0020] In this embodiment, the positive direction of the prototype coprime array is... Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is A uniform subarray, with a total of Q+1 elements added, results in an array called a Type II ternary coprime array (TCA-II), with a total number of elements. Array structure such as Figure 1 As shown.

[0021] To analyze the differential comatrix correlation characteristics of TCA-II, the following function is constructed. ; ; ; function , , and The domains are respectively ; ; ; ; Lemma 1: The negatives of the elements in are all in middle.

[0022] Lemma 2: Set of Integers In scope Internal missing A positive integer.

[0023] Lemma 3: and There is no overlap.

[0024] Lemma 4: such that Established The number of different elements is .

[0025] Theorem 1: In Within the range, Missing There are 1 positive integers, and the missing number can be represented as . .

[0026] Based on Lemmas 5 and 6 below, Theorem 2 gives the number of distinct degrees of freedom of TCA-II.

[0027] Lemma 5: Sets The number of distinct elements in the expression is .

[0028] Lemma 6: The number of distinct integers contained is .

[0029] Theorem 2: The number of distinct elements, i.e., the distinct degrees of freedom, is .

[0030] Based on Lemmas 7 and 8 below, Theorem 3 gives the number of continuous degrees of freedom of TCA-II.

[0031] Lemma 7: When , The maximum value of the elements in the middle is .

[0032] Lemma 8: When hour, The maximum value of the elements in the middle is .

[0033] Theorem 3:

[0034] Proof: By Lemmas 15 and 16, the largest continuous segment in the difference comatrix of TCA-II is... The number of virtual array elements, i.e., the continuous degrees of freedom, is 1. .

[0035] Given the total number of array elements, construct a Lagrange optimization problem to achieve the maximum continuous degrees of freedom. The values ​​of the maximum continuous degrees of freedom are shown in Table 1.

[0036] Table 1. Values ​​of the maximum continuous degrees of freedom for TCA-II

[0037] TCA-II possesses scalability, and in the positive direction of the prototype coprime matrix... Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The array can be expanded by adding Q+1 array elements to a uniform subarray. The structure is as follows: Figure 2 As shown, the expansion does not destroy the original configuration; it simply expands the array elements on the basis of the original array to achieve an increase in continuous degrees of freedom.

[0038] Taking a total of 12 array elements as an example, the array element parameters are set to... ,right When normalized, the array structure is as follows: Figure 3 As shown, the red markers indicate the physical element positions, and the blue markers indicate the virtual element positions of the differential array. Adding three more elements with a spacing of 7 in the reverse direction results in the array shown below. Figure 4As shown, red represents the original 12 array elements, green represents the added array elements, forming an expanded array, and black represents the positions of virtual array elements in the differential array of the expanded array. Figure 3 and Figure 4 It can be observed that after the TCA-II extension, the number of continuous degrees of freedom increases significantly, by 42.

[0039] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to in the method section.

[0040] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A design method for a type II ternary coprime array based on coprime sparse arrangement, characterized in that, Includes the following steps: Determine the basic coprime pairs, additional array elements, and parameters of the third subarray. Select two coprime positive integers M and N to construct the basic structure of the prototype coprime array. The prototype coprime array contains subarray 1 with N array elements and Md spacing and subarray 2 with M array elements and Nd spacing, where d is the unit array element spacing. In the positive direction of the prototype coprime matrix Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarrays form a type II ternary coprime array; The matching parameters between subarray 3 and additional array elements are determined. Based on the hole distribution of the prototype coprime array and the hole-filling characteristics of the additional array elements, the matching conditions between the array element spacing and the number of array elements that maximize the continuous degrees of freedom are derived theoretically. Based on the theory of differential coprime, the analytical expressions for the distinct and continuous degrees of freedom of a type II ternary coprime matrix are derived. By using the positive direction based on the prototype coprime matrix Add one array element at the point, then add array elements in the reverse direction to make the total number of array elements. Spacing is The uniform subarray is expanded by adding Q+1 array elements, and the expansion does not destroy the original coprime array structure. The number of distinct degrees of freedom is obtained through the following lemma: gather The number of distinct elements in the expression is ; The number of distinct integers contained is ; Using the above lemma, the number of distinct elements, i.e., the distinct degrees of freedom, in a type II ternary coprime matrix is: ; The number of continuous degrees of freedom is obtained through the following lemma: when , The maximum value of the elements in the middle is ; when hour, The maximum value of the elements in the middle is ; Using the above lemma, the number of virtual elements, i.e., the continuous degrees of freedom, of a type II ternary coprime matrix is: .

2. The design method for a type II ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that, Performance was verified under coupled and uncoupled models. The Toeplitz-ESPRIT algorithm was used for DOA estimation, and array performance was evaluated by root mean square error (RMSE) and probability of successful resolution (POR). The RMSE and POR are defined as follows: ； ； In the formula, The number of Monte Carlo experiments; Number of information sources; It is the first The true incident direction of the signal source; It is the first In this experiment The estimated DOA value; Set to an acceptable resolution. This represents the number of successful resolutions for the k-th signal.

3. The design method for a type II ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that, The difference comatrix of the prototype coprime matrix is: In the formula, Indicates the range, For integer functions, the domain is In the formula, Indicates the domain.

4. The design method for a type II ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that, The total number of elements in a Type II ternary coprime array is .

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