A design method of type I ternary primitive array based on primitive sparse arrangement

Abstract

This invention discloses a design method for a Type I ternary coprime array based on coprime sparse arrangement, relating to the field of array structure design. Two coprime positive integers M and N are selected to construct the basic structure of a prototype coprime array. A subarray with n elements and n spacing is added in reverse to the prototype coprime array to form a Type I ternary coprime array. Based on the distribution law of holes in the prototype coprime array, the lower bound expression of the optimal parameters of the subarray can be obtained, ensuring that the holes in the prototype coprime array are completely filled. Analytical expressions for the distinct and continuous degrees of freedom of the Type I ternary coprime array are derived. The method is scalable, allowing for the expansion of array degrees of freedom without destroying the prototype coprime array structure by adding array elements in reverse. The Type I ternary coprime array designed using this method exhibits weak mutual coupling, scalability, and high distinct degrees of freedom.

Description

Technical Field

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

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A design method for a type I ternary coprime array based on coprime sparse arrangement includes the following steps:

[0006] Choose two coprime positive integers M and N to construct the basic structure of a prototype coprime matrix. The prototype coprime matrix contains submatrix 1 with N elements and a spacing of Md, and submatrix 2 with M elements and a spacing of Nd, where d is the unit element spacing. Then, add submatrix 2 with Q elements and a spacing of Md to the prototype coprime matrix in reverse order. Subarray 3 forms a type I ternary coprime matrix;

[0007] By determining the optimal lower bound of parameters for subarray 3, and based on the distribution law of holes in the prototype coprime array, a lower bound of the number of array elements that completely fills the holes is obtained. ,when At that time, the difference set between subarray 3 and subarrays 1 and 2 covers the hole region of the prototype coprime array;

[0008] Based on the theory of differential coprime, analytical expressions for the distinct and continuous degrees of freedom of a type I ternary coprime matrix are derived. The distinct degrees of freedom are determined by the number of distinct virtual elements in the mutual difference set between subarrays, and the continuous degrees of freedom are determined by the number of continuous segment virtual elements in the mutual difference set between subarrays.

[0009] Type I ternary coprime arrays are scalable, do not destroy the original coprime array structure, and expand the array degrees of freedom by increasing the number of array elements in reverse.

[0010] Optionally, the difference comatrix of the prototype coprime matrix is:

[0011]

[0012] In the formula, Indicates the range, For integer functions, the domain is In the formula, Indicates the domain.

[0013] Optionally, the element positions of a type I ternary coprime array are represented as follows:

[0014] ;

[0015] by and Let these represent the cross differences between subarray 3 and subarrays 1 and 2 in TCA-I:

[0016] ;

[0017] ;

[0018] The domains are respectively:

[0019] ;

[0020] .

[0021] Optionally, determine the optimal lower bound of parameters for subarray 3, and based on the distribution law of holes in the prototype coprime array, obtain the lower bound of the number of array elements that completely fill the holes. ,when When the cross difference set of subarray 3 with subarrays 1 and 2 covers the hole region of the prototype coprime matrix, the theorem states that the hole in the prototype coprime matrix can always be filled. Values ,at this time Including integer range and All integers in the intersection of .

[0022] Optionally, the calculation of distinct degrees of freedom is based on the following lemma derivation:

[0023] The number of distinct elements in the expression is ;

[0024] The number of distinct elements in the expression is ;

[0025] Combining the above lemmas, we get The number of distinct elements in the expression is .

[0026] Optionally, the calculation of continuous degrees of freedom is based on the following lemma derivation:

[0027] Anything between All integers within are in the set middle;

[0028] when Not less than hour, contain within One integer;

[0029] Combining the above lemma, we obtain when Not less than hour, The maximum value is .

[0030] Optionally, the Type I ternary coprime array is scalable by increasing the spacing by Q in the reverse direction. The array elements enable array expansion; the expansion does not destroy the original coprime array structure, but only expands the array elements based on the original array; the Type I ternary coprime array can achieve an increase in dissimilar degrees of freedom, compared with the original coprime array, the dissimilar degrees of freedom are increased by . .

[0031] Optionally, performance verification is also included: under coupled and uncoupled models, the OMP algorithm is used for DOA estimation, and array performance is evaluated using the root mean square error (RMSE) and the probability of successful resolution (POR). The RMSE and POR are defined as follows:

[0032] ;

[0033] ;

[0034] 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.

[0035] As can be seen from the above technical solutions, compared with the prior art, the present invention discloses a design method for a type I ternary coprime array based on coprime sparse arrangement. The type I ternary coprime array designed by this method has the characteristics of weak mutual coupling, scalability, and high degree of freedom, and has better performance than other non-extended sparse arrays in the mutual coupling scenario. Attached Figure Description

[0036] 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.

[0037] Figure 1 The diagram shows the structure of a type I ternary coprime array provided by this invention.

[0038] Figure 2 A schematic diagram illustrating the formation of continuous degrees of freedom in a type I ternary coprime matrix provided by the present invention;

[0039] Figure 3 This is a schematic diagram of the expansion of the Type I ternary coprime array provided by the present invention;

[0040] Figure 4 A schematic diagram of a type I ternary coprime array before its expansion, provided by the present invention;

[0041] Figure 5 A schematic diagram of the extended Type I ternary coprime array provided by the present invention. Detailed Implementation

[0042] 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.

[0043] This invention discloses a design method for a type I ternary coprime array based on coprime sparse arrangement, comprising the following steps:

[0044] Choose two coprime positive integers M and N to construct the basic structure of a prototype coprime matrix. The prototype coprime matrix contains submatrix 1 with N elements and a spacing of Md, and submatrix 2 with M elements and a spacing of Nd, where d is the unit element spacing. Then, add submatrix 2 with Q elements and a spacing of Md to the prototype coprime matrix in reverse order. Subarray 3 forms a type I ternary coprime matrix;

[0045] By determining the optimal lower bound of parameters for subarray 3, and based on the distribution law of holes in the prototype coprime array, a lower bound of the number of array elements that completely fills the holes is obtained. ,when At that time, the difference set between subarray 3 and subarrays 1 and 2 covers the hole region of the prototype coprime array;

[0046] Based on the theory of differential coprime, analytical expressions for the distinct and continuous degrees of freedom of a type I ternary coprime matrix are derived. The distinct degrees of freedom are determined by the number of distinct virtual elements in the mutual difference set between subarrays, and the continuous degrees of freedom are determined by the number of continuous segment virtual elements in the mutual difference set between subarrays.

[0047] Type I ternary coprime arrays are scalable, do not destroy the original coprime array structure, and expand the array degrees of freedom by increasing the number of array elements in reverse.

[0048] Specifically, the prototype coprime matrix utilizes the coprime property between two subarrays to generate rich degrees of freedom and reduce array coupling. Therefore, it's possible to further improve the performance of sparse linear arrays by generating multiple coprime relationships based on the prototype coprime matrix. According to the positional pattern of holes in the differential comatrix of the prototype coprime matrix, integer multiples of... Add an integer multiple or It can fill the holes in a prototype coprime matrix. It is also known that… and Coprime, then Must be with , Coprime positive integers, therefore , , If three positive integers are pairwise coprime, then a sparse linear array constructed using the triple coprime relationship can generate a richer set of degrees of freedom.

[0049] Based on the prototype coprime matrix, add array elements with the number Q and spacing of q in reverse. Subarray 3 is used to construct a new array structure called Type I ternary coprime array (TCA-I), with a total number of array elements. Array structure such as Figure 1 As shown.

[0050] The element positions of TCA-I can be represented as:

[0051] ;

[0052] by and Let these represent the cross difference sets between subarray 3 and subarrays 1 and 2 in TCA-I, respectively.

[0053] ;

[0054] ;

[0055] The domains are respectively:

[0056] ;

[0057] ;

[0058] Theorem 1 below gives the conditions for filling holes in a prototype coprime matrix. The lower realm, once Beyond this lower bound, the holes in the prototype coprime array cannot be filled.

[0059] Theorem 1: Any hole in a prototype coprime matrix can be filled. Values ,at this time Including integer range and All integers in the intersection of .

[0060] Proof: Let For integer intervals number within There are two scenarios.

[0061] Scenario 1: , It can be written as ,set up The following inequality can be established:

[0062] ;

[0063] again It cannot be an integer, and has the following characteristics:

[0064] ;

[0065] From the above formula, we can obtain that Exceeding This contradicts the previous assumption, therefore there is .

[0066] like , It will also be greater than This also contradicts the previous assumption, therefore .

[0067] when At that time, between Integers within It must be The elements in, when hour, Even with more elements, the conclusion remains valid.

[0068] Scenario 2: Similarly, it can be proven that between Integers within It must be The elements in.

[0069] In addition, when Reduce to At that time, it can no longer be guaranteed Cover all the holes in the prototype coprime array. For example, At this time, 22 is a hole in the prototype coprime array. It is 1, but 22 is not. It is not in the middle either. The middle. Therefore, the lower bound is called the middle. , represented as .

[0070] After understanding the design of TCA-I, we analyze its distinct degrees of freedom. The self-difference set of TCA-I is a subset of the cross-difference set; therefore, the position of the difference comatrix is ​​determined solely by the cross-difference set. The specific expression for the difference comatrix is:

[0071] ;

[0072] Based on Lemma 1 and Lemma 2 below, Theorem 2 gives the number of virtual array elements, i.e., distinct degrees of freedom, of the difference comatrix of TCA-I.

[0073] Lemma 1: The number of distinct elements in the expression is .

[0074] Lemma 2: The number of distinct elements in the expression is .

[0075] Theorem 2: Includes 1 distinct integer.

[0076] Proof: By imitating the proof idea of ​​Lemma 2, it is not difficult to obtain:

[0077] ;

[0078] Combining Lemma 1 and Lemma 2, we can obtain The number of distinct elements contained in the sample is:

[0079] ;

[0080] Given the total number of array elements In order to achieve the maximum degrees of freedom, an optimization problem is constructed.

[0081] ;

[0082] The Lagrange function in the above equation can be expressed as:

[0083] ;

[0084] In the formula, It is a Lagrange multiplier.

[0085] Taking the derivatives of the Lagrange function separately, we can obtain

[0086] ;

[0087] ;

[0088] ;

[0089] ;

[0090] By solving the above equation, we can obtain:

[0091] ;

[0092] because The maximum number of distinct degrees of freedom can be divided into several cases. The maximum number of distinct degrees of freedom under a given number of array elements is shown in Table 1.

[0093] Table 1. Values ​​of the maximum continuous degrees of freedom in TCA-I

[0094]

[0095] According to the function , , Definition, The maximum value is less than And because and Coprime, Not included Therefore, regardless of TCA-I How to choose the value, in the position of its difference matrix There are holes everywhere. At the same time, when When it is greater than another lower bound, the maximum continuous difference comatrix Contains between All integers within the range, this lower bound is The following provides detailed proofs for Lemmas 3 and 4 and Theorem 3.

[0096] Lemma 3: Any between All integers within are in the set middle.

[0097] Lemma 4: When Not less than hour, contain within A number of integers.

[0098] Theorem 3: When Not less than hour, The maximum value is .

[0099] Proof: For , its in All missing integers within the range are in the union. In the middle, therefore contain All integers. By Lemma 8, when Not less than hour, Any integer contained in in, so contain All integers, and Not included Therefore The maximum value is .

[0100] by For example, Figure 2 The process of continuous degrees of freedom formation in TCA-I is shown, in which and Fillable middle These holes eventually exist at position 42, at which point the maximum value of the largest continuous segment of the TCA-I differential comatrix is ​​41, and the single-sided continuous degree of freedom is 42.

[0101] Furthermore, in this embodiment, TCA-I has scalable characteristics, allowing for the addition of Q spacing units in the reverse direction. The array elements can be used to expand the array, with the structure as follows: Figure 3 As shown. The expansion does not change the original configuration; it simply adds array elements to the original array, allowing for an increase in distinct degrees of freedom. Taking a total of 10 array elements as an example, the element parameters are set to... ,Will When normalized, the array structure is as follows: Figure 4 As shown, the red markers indicate the physical array element positions, and the blue markers indicate the virtual array element positions of the differential array. Adding three more array elements with a spacing of 7 units in the reverse direction results in the array shown below. Figure 5 As shown, red represents the original 10 array elements, green represents the added array elements, forming the expanded array, and black represents the virtual array elements in the differential array of the expanded array. Figure 4 and Figure 5 It can be observed that after extending TCA-I, the continuous degrees of freedom remain unchanged, while the dissimilar degrees of freedom increase by 36.

[0102] In this embodiment, for performance verification: under coupled and uncoupled models, the OMP 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:

[0103] ;

[0104] ;

[0105] 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.

[0106] 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 it corresponds to the method disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0107] 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 I ternary coprime array based on coprime sparse arrangement, characterized in that, Includes the following steps: Two co-prime positive integers M and N are selected to construct a prototype co-prime matrix base structure, the prototype co-prime matrix comprises a sub-matrix 1 with N array elements and a spacing of Md, and a sub-matrix 2 with M array elements and a spacing of Nd, d is a unit array element spacing; a sub-matrix 3 with Q array elements and a spacing of is reversely added on the basis of the prototype co-prime matrix to form a type I ternary co-prime matrix; The optimal parameter lower bound of the subarray 3 is determined, and based on the hole distribution law of the prototype coprime array, a lower bound of the number of array elements for completely filling the holes is obtained When The difference set of the subarray 3 and the subarray 1 and the subarray 2 covers the hole region of the prototype coprime array. Based on the theory of differential coprime, analytical expressions for the distinct and continuous degrees of freedom of a type I ternary coprime matrix are derived. The distinct degrees of freedom are determined by the number of distinct virtual elements in the mutual difference set between subarrays, and the continuous degrees of freedom are determined by the number of continuous segment virtual elements in the mutual difference set between subarrays. Type I ternary coprime array is scalable and does not destroy the original coprime array structure. The array degrees of freedom can be expanded by increasing the number of array elements in reverse. The element positions of a type I ternary coprime array are represented as follows: ； with and denote the symmetric difference between the subarrays 3 and 1, 2 in TCA-I, respectively: ； ； The domains are respectively: ； 。 2. The method of claim 1, wherein the method is a method of designing a Type-I ternary array based on a sparse permutation of coprime arrays, characterized in that, The difference comatrix of the prototype coprime matrix is: wherein denotes the value range, is an integer function whose domain is , denotes the domain.

3. The design method for a type I ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that, By determining the optimal lower bound of parameters for subarray 3, and based on the distribution law of holes in the prototype coprime array, a lower bound of the number of array elements that completely fills the holes is obtained. ,when When the cross difference set of subarray 3 with subarrays 1 and 2 covers the hole region of the prototype coprime matrix, the theorem states that the hole in the prototype coprime matrix can always be filled. Values ,at this time Including integer range and All integers in the intersection of .

4. The method of claim 1, wherein, The calculation of distinct degrees of freedom is based on the following lemma derivation: The number of different elements in the middle is ; The number of different elements is ; Combining the above lemma, we have The number of different elements in the middle is .

5. The method of claim 1, wherein, The calculation of continuous degrees of freedom is based on the following lemma derivation: Anything between All integers within are in the set middle; when Not less than hour, contain within One integer; Combining the above lemma, we obtain when Not less than hour, The maximum value is .

6. The design method for a type I ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that, Type I ternary coprime arrays are scalable, and can be increased by adding Q spacing elements in reverse. The array elements enable array expansion; and the expansion does not change the original configuration, but only expands the array elements based on the original array; the Type I ternary coprime array can achieve an increase in dissimilar degrees of freedom, compared with the prototype coprime array, the dissimilar degrees of freedom are increased by .

7. The design method for a type I ternary coprime array based on coprime sparse arrangement according to claim 1, characterized in that... Under both coupled and uncoupled models, the OMP algorithm is used for DOA estimation. 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.

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