Delay co-prime sparse linear array and DOA estimation method

Abstract

A DOA estimation method of a delayed co-prime sparse linear array is suitable for signal processing of DOA estimation of radar, sonar and wireless communication, and comprises the following steps: measuring a signal source by using the delayed co-prime sparse array to obtain a signal sample; constructing two uniform subarray segmentation schemes with different delays, and segmenting the signal sample to obtain data; analyzing the segmented data by using a unitary ESPRIT algorithm to obtain a winding phase; and constructing a congruence equation set for each pair of wound phases, and solving the congruence equation set by using a robust Chinese remainder theorem algorithm to obtain a DOA estimated value.

Description

Technical Field

[0001] This invention relates to the fields of signal sampling and signal processing, as well as radar, sonar, and wireless communication, and particularly to a method for estimating the Direction of Arrival (DOA) of a sparse linear array with coprime delay. Background Technology

[0002] Direction of Arrival (DOA) estimation is a crucial problem in array signal processing. Its basic principle involves using an array antenna to receive spatial signals and then processing these signals using statistical signal processing techniques to obtain DOA information. It is a key technology in fields such as radar, sonar, and wireless communication. The accuracy of DOA estimation depends on the array shape and the DOA estimation algorithm.

[0003] Traditional arrays are typically linear uniform arrays with element spacing equal to half the wavelength of the incident signal. At higher signal frequencies (shorter wavelengths), the spacing between adjacent elements in conventional arrays is very close, leading to significant inter-element coupling and signal interference, thus impacting the accuracy of DOA estimation. On the other hand, DOA estimation accuracy is generally positively correlated with array size. However, the aperture of a linear uniform array with half-wavelength spacing is severely limited by the number of elements, making it difficult to achieve large-aperture arrays with a small number of elements, and consequently, difficult to achieve high-precision DOA estimation with a limited number of elements.

[0004] To mitigate the mutual coupling effects between array elements and improve DOA estimation accuracy by expanding the array aperture with a finite number of elements, sparse linear arrays can be employed. These arrays typically consist of non-uniformly arranged elements, allowing element spacing greater than half the signal wavelength. Traditional sparse arrays require the array positions to be differentially formed into a continuous and uniform virtual array, such as minimum redundancy arrays, nested arrays, coprime arrays, and their variations. Therefore, existing sparse arrays cannot achieve arbitrary sparsity; they are inherently constrained by the minimum sparseness rule, thus preventing further reduction of the number of closely spaced elements and the reduction of coupling effects between elements. The array aperture is also constrained by the minimum sparseness rule and cannot be further increased. On the other hand, solving for the optimal sparse array that satisfies the minimum sparseness rule relies on enumeration operations, and the computational cost increases exponentially with the number of elements. This becomes computationally difficult when the number of elements is large. Therefore, it is necessary to design linear arrays that can achieve arbitrary sparse arrangement and are simple in design, to further increase the array aperture and reduce the mutual coupling effects between elements, and to develop corresponding DOA estimation methods to improve DOA estimation accuracy and computational efficiency.

[0005] The information disclosed in the background section is only for enhancing the understanding of the background of this invention, and therefore may contain information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0006] To address the shortcomings, this invention provides a method, system, medium, and device for DOA estimation of sparse linear arrays with coprime delays. It achieves a larger array aperture using a small number of array elements, reduces the influence of array element coupling, and improves DOA estimation accuracy and computational efficiency.

[0007] A method for DOA estimation of a sparse linear array with coprime delays, applicable to signal processing for direction-of-arrival estimation in radar, sonar, and wireless communication, comprising:

[0008] Step S1: Use a time-delayed coprime sparse array to measure the signal source and obtain signal samples;

[0009] Step S2: Construct two uniform subarray segmentation schemes with different delays and segment the signal samples to obtain data;

[0010] Step S3: Analyze the segmented data using the unitary ESPRIT algorithm to obtain the entangled phase;

[0011] Step S4: Construct a system of congruence equations for each pair of entangled phases, and solve them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate.

[0012] In the aforementioned method for estimating the DOA of a sparse linear array with coprime delays, step S1 includes:

[0013] The array parameters L, P, and Q are determined based on the number of sensors (2M+1) and the array aperture H:

[0014] (17)

[0015] (18)

[0016] Where d is the half wavelength of the electromagnetic signal; the sparse array is composed of two uniform arrays with M+1 and M elements respectively; the product of L and d, Ld, is the spacing between the uniform subarrays in the sparse array; the product of P and d, Pd, is the forward offset distance between the two uniform subarrays in the sparse array; the product of Q and d, Qd, is the backward offset distance between the two uniform subarrays in the sparse array; The rounding up symbol,

[0017] A sparse array samples a signal source, and the sampling lasts for N sampling periods. The signal samples collected by the sparse array are represented as follows:

[0018] (19)

[0019] in This is the first signal sample collected by a uniform array of M+1 array elements in a sparse array; Let n be the second signal sample collected by a uniform array of M elements in a sparse array, where n is an index that iterates from 1 to N.

[0020] In the aforementioned method for estimating the DOA of a sparse linear array with coprime delay, P and Q are coprime positive integers and satisfy the condition that the total aperture of the array is D = M(P + Q)d.

[0021] In the aforementioned method for estimating the DOA of a delay-coprime sparse linear array,

[0022] First uniform subarray partitioning scheme:

[0023] (20)

[0024] in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the first segmentation scheme.

[0025] Second uniform subarray partitioning scheme:

[0026] (twenty one)

[0027] in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the second segmentation scheme.

[0028] In the aforementioned method for estimating the DOA of a sparse linear array with coprime delays, step S3 includes:

[0029] Step 1: Define two unitary matrices and :

[0030] (twenty two)

[0031] (twenty three)

[0032] in , and They are respectively , and The identity matrix; , and They are respectively , and The reverse identity matrix, whose anti-diagonal elements are 1; Represents a zero vector; For imaginary numbers, , This indicates the transpose operation.

[0033] Step 2: Calculate the extended covariance matrix of the two delay schemes. and :

[0034] (twenty four)

[0035] in This is the conjugate transpose operation; and Extended snapshots representing two time-delay schemes

[0036] (25)

[0037] Step 3: Calculate the real-valued covariance matrix and Then, eigenvalue decomposition is performed on it to obtain the signal subspace matrix. and :

[0038] (26)

[0039] in The real part extraction operation extracts the real part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar.

[0040] right and Eigenvalue decomposition is performed to obtain the signal subspace matrix. and :

[0041] (27)

[0042] (28)

[0043] in and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal characteristic values ​​and noise characteristic values; and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal eigenvalue matrix and noise eigenvalue matrix;

[0044] Step 4: Calculate the real-valued eigenma matrix and :

[0045] (29)

[0046] in and Let two choice matrices be represented, and their definitions are as follows: and ; The matrix formed by concatenating the zero matrix and the identity matrix: , for The zero matrix, for The identity matrix; The imaginary part extraction operation extracts the imaginary part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar. This represents the generalized inverse operation.

[0047] Step 5: Construct the combination matrix Eigenvalue decomposition is performed on the signal to obtain its eigenvalues. Then, the fuzzy phase obtained under the two delay schemes is calculated. :

[0048] (30)

[0049] in The number of information sources; k is an integer ranging from 1 to K; It is the arctangent function.

[0050] In the aforementioned method for estimating the DOA of a sparse linear array with coprime delays, step S4 includes:

[0051] For each pair of entangled phases Construct a system of congruence equations:

[0052] (31)

[0053] in and For an unknown integer, The unknown to be solved

[0054] Solving congruence equations using the robust Chinese Remainder Theorem algorithm, and calculating... To obtain an accurate DOA estimate, the value is determined. :

[0055] (32)

[0056] in It is the arcsine function.

[0057] based on For each k value, obtain the DOA estimate of K signal sources. .

[0058] In the aforementioned method for estimating the DOA of a sparse linear array with coprime delay, the unitary ESPRIT algorithm constructs a real-valued extended covariance matrix from the received data of the subarray and performs unitary transformation and eigenvalue decomposition to obtain the entangled phase estimate.

[0059] A system for performing the method includes:

[0060] The measurement module uses a time-delayed coprime sparse array to measure the signal source and obtain signal samples;

[0061] The module constructs two uniform subarray segmentation schemes with different delays and segments signal samples to obtain data.

[0062] The analysis module uses the unitary ESPRIT algorithm to analyze the segmented data and obtain the entangled phase;

[0063] The calculation module constructs a system of congruence equations for each pair of entangled phases and solves them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate.

[0064] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.

[0065] An electronic device, the electronic device comprising:

[0066] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,

[0067] The processor implements the method when executing the program.

[0068] Compared with existing technologies, this invention has the following advantages: This invention utilizes a sparse arrangement of finite array elements to achieve a large-aperture array, reducing the mutual coupling effect of array elements and improving the accuracy and computational efficiency of DOA estimation. The invented sparse array design is simple, and the DOA estimation method is efficient and stable. Attached Figure Description

[0069] Various other advantages and benefits of the present invention will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0070] In the attached diagram:

[0071] Figure 1 This is a flowchart of the time-delayed coprime sparse linear array and DOA estimation method proposed in this invention;

[0072] Figure 2 The schematic diagram shows the principle of the proposed time-delay coprime sparse array.

[0073] Figure 3 A schematic diagram illustrating the construction of two uniform sub-segmentation schemes with different delays;

[0074] Figure 4 A schematic diagram comparing the DOA estimation errors of the designed time-delay coprime sparse array with those of existing arrays;

[0075] Figure 5 A schematic diagram comparing the DOA estimation time of the designed delay coprime sparse array with that of the existing array.

[0076] The present invention will be further explained below with reference to the accompanying drawings and embodiments. Detailed Implementation

[0077] Specific embodiments of the invention will now be described in more detail with reference to the accompanying drawings. While specific embodiments of the invention are shown in the drawings, it should be understood that the invention may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this invention will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

[0078] It should be noted that certain terms are used in the specification and claims to refer to specific components. Those skilled in the art will understand that different terms may be used to refer to the same component. This specification and claims do not distinguish components based on differences in terminology, but rather on differences in function. The terms "comprising" or "including" used throughout the specification and claims are open-ended and should be interpreted as "comprising but not limited to." The following descriptions are preferred embodiments for carrying out the invention; however, these descriptions are for the purpose of understanding the general principles of the specification and are not intended to limit the scope of the invention. The scope of protection of this invention is determined by the appended claims.

[0079] To facilitate understanding of the embodiments of the present invention, further explanations and descriptions will be provided below with reference to the accompanying drawings and specific embodiments. The accompanying drawings do not constitute a limitation on the embodiments of the present invention.

[0080] like Figures 1 to 5 As shown, a method for DOA estimation of a sparse linear array with coprime delays is presented. This method is applicable to signal processing for direction-of-arrival estimation in radar, sonar, and wireless communication. The method includes the following steps:

[0081] Step S1: Use a time-delayed coprime sparse array to measure the signal source and obtain signal samples;

[0082] Step S2: Construct two uniform subarray segmentation schemes with different delays and segment the signal samples to obtain data; construct two uniform subarray segmentation schemes with different delays by decimating array elements every other element. Accordingly, segment the measured signal samples according to the different uniform subarrays from which they originate.

[0083] Step S3: Analyze the segmented data using the unitary ESPRIT algorithm to obtain the wrapped phase; ESPRIT is a classic algorithm in the field of array signal processing, and Unitary-ESPRIT refers to performing a unitary transform before ESPRIT. Therefore, Unitary-ESPRIT can also be called unitary ESPRIT.

[0084] Step S4: Construct a system of congruence equations for each pair of entangled phases and solve them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate. The robust Chinese Remainder Theorem algorithm is a branch of algorithms developed based on the Chinese Remainder Theorem, which solves the problem of poor robustness to remainder errors in traditional algorithms. It can be considered a standard term in the field of signal processing.

[0085] In a preferred embodiment of the DOA estimation method for a delay-coprime sparse linear array, step S1 includes:

[0086] The array parameters L, P, and Q are determined based on the number of sensors (2M+1) and the array aperture H:

[0087] (1)

[0088] (2)

[0089] Where d is the half wavelength of the electromagnetic signal; the sparse array is composed of two uniform arrays with M+1 and M elements respectively; the product of L and d, Ld, is the spacing between the uniform subarrays in the sparse array; the product of P and d, Pd, is the forward offset distance between the two uniform subarrays in the sparse array; the product of Q and d, Qd, is the backward offset distance between the two uniform subarrays in the sparse array; The rounding up symbol,

[0090] L represents the integer distance of the interval between the uniform subarrays in the designed sparse linear array, divided by d; P represents the integer distance of the forward offset distance between two uniform subarrays in the designed sparse linear array, divided by d; Q represents the integer distance of the forward offset distance between two uniform subarrays in the designed sparse linear array, divided by d; the values ​​of L, P, and Q are all non-negative integers, and satisfy L = P + Q. Therefore, L × d is the interval between the uniform subarrays in the sparse array; P × d is the forward offset distance between two uniform subarrays in the sparse array; Q × d is the backward offset distance between two uniform subarrays in the sparse array.

[0091] A sparse array samples a signal source, and the sampling lasts for N sampling periods. The signal samples collected by the sparse array are represented as follows:

[0092] (3)

[0093] in This is the first signal sample collected by a uniform array of M+1 array elements in a sparse array; Let n be the second signal sample collected by a uniform array of M elements in a sparse array, where n is an index that iterates from 1 to N.

[0094] In a preferred embodiment of the method for estimating the DOA of a delay-coprime sparse linear array, P and Q are coprime positive integers and satisfy the total array aperture D = M(P + Q)d.

[0095] In a preferred embodiment of the method for estimating the DOA of a delay-coprime sparse linear array,

[0096] First uniform subarray partitioning scheme:

[0097] (4)

[0098] in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the first segmentation scheme.

[0099] Second uniform subarray partitioning scheme:

[0100] (5)

[0101] in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the second segmentation scheme.

[0102] In a preferred embodiment of the DOA estimation method for a delay-coprime sparse linear array, step S3 includes:

[0103] Step 1: Define two unitary matrices and :

[0104] (6)

[0105] (7)

[0106] in , and They are respectively , and The identity matrix; , and They are respectively , and The reverse identity matrix, whose anti-diagonal elements are 1; Represents a zero vector; For imaginary numbers, , This indicates the transpose operation.

[0107] Step 2: Calculate the extended covariance matrix of the two delay schemes. and :

[0108] (8)

[0109] in This is the conjugate transpose operation; and Extended snapshots representing two time-delay schemes

[0110] (9)

[0111] Step 3: Calculate the real-valued covariance matrix and Then, eigenvalue decomposition is performed on it to obtain the signal subspace matrix. and :

[0112] (10)

[0113] in The real part extraction operation extracts the real part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar.

[0114] right and Eigenvalue decomposition is performed to obtain the signal subspace matrix. and :

[0115] (11)

[0116] (12)

[0117] in and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal characteristic values ​​and noise characteristic values; and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal eigenvalue matrix and noise eigenvalue matrix;

[0118] Step 4: Calculate the real-valued eigenma matrix and :

[0119] (13)

[0120] in and Let two choice matrices be represented, and their definitions are as follows: and ; The matrix formed by concatenating the zero matrix and the identity matrix: , for The zero matrix, for The identity matrix; The imaginary part extraction operation extracts the imaginary part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar. This represents the generalized inverse operation.

[0121] Step 5: Construct the combination matrix Eigenvalue decomposition is performed on the signal to obtain its eigenvalues. Then, the fuzzy phase obtained under the two delay schemes is calculated. :

[0122] (14)

[0123] in The number of information sources; k is an integer ranging from 1 to K; It is the arctangent function.

[0124] In a preferred embodiment of the DOA estimation method for a delay-coprime sparse linear array, step S4 includes:

[0125] For each pair of entangled phases Construct a system of congruence equations:

[0126] (15)

[0127] in and For an unknown integer, The unknown to be solved

[0128] Solving congruence equations using the robust Chinese Remainder Theorem algorithm, and calculating... To obtain an accurate DOA estimate, the value is determined. :

[0129] (16)

[0130] in It is the arcsine function.

[0131] based on For each k value, obtain the DOA estimate of K signal sources. .

[0132] In a preferred embodiment of the DOA estimation method for a delay-coprime sparse linear array, the unitary ESPRIT algorithm constructs a real-valued extended covariance matrix from the received data of the subarray and performs unitary transformation and eigenvalue decomposition to obtain an estimate of the entangled phase.

[0133] A system for performing the method includes:

[0134] The measurement module uses a time-delayed coprime sparse array to measure the signal source and obtain signal samples;

[0135] The module constructs two uniform subarray segmentation schemes with different delays and segments signal samples to obtain data.

[0136] The analysis module uses the unitary ESPRIT algorithm to analyze the segmented data and obtain the entangled phase;

[0137] The calculation module constructs a system of congruence equations for each pair of entangled phases and solves them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate.

[0138] A computer storage medium including computer instructions that, when run on a computer, cause the computer to perform the method.

[0139] An electronic device, the electronic device comprising:

[0140] Memory, processor, and computer programs stored in memory and executable on the processor, wherein,

[0141] The processor implements the method when executing the program.

[0142] In one embodiment, the delayed coprime sparse linear array and DOA estimation method include the following steps:

[0143] (1) Design a time-delay coprime sparse array and use it to measure the signal source;

[0144] In this exemplary example, it is assumed that the given number of array elements is 19, that is Array aperture half wavelength of electromagnetic signal The key parameters L, P, and Q of the array are respectively...

[0145] (17)

[0146] (18)

[0147] A schematic diagram of the designed array is shown below. Figure 1 As shown, from left to right, it is composed of two uniform arrays with an interval of (P+Q)d, consisting of the 1st, 3rd, 5th, ..., 19th sensors and the 2nd, 4th, 6th, ..., 18th sensors. The forward offset distance between the two uniform arrays is Pd, and the backward offset distance is Qd, where (P+Q)d, Pd, and Qd represent the products of P+Q, P, Q, and d, respectively.

[0148] The array was used to measure signal sources. In this example, there were two signal sources with DOA angles of 20° and 65°, respectively. The signal-to-noise ratio (SNR) was set to 5dB, and the measurement period lasted for 100 sampling cycles. The obtained data were as follows: and .

[0149] (2) Construct two uniform array partitioning schemes with different delays;

[0150] In this exemplary example, the partitioning scheme of the first uniform array is as follows: the first M array elements of the first uniform array are selected to form the second uniform array, which are then used as two sub-arrays, such as... Figure 3 As shown in (a).

[0151] (4)

[0152] In this exemplary example, the partitioning scheme for the second uniform array is as follows: the last M elements of the second uniform array and the first uniform array are selected to form two sub-arrays, respectively, as follows: Figure 3 As shown in (b).

[0153] (5)

[0154] (3) The segmented data is analyzed using the unitary ESPRIT algorithm to obtain the entangled phase;

[0155] In this example, M=9, and the first step is to define the unitary matrix. and

[0156] (19)

[0157] (20)

[0158] in

[0159] (twenty one)

[0160] (twenty two)

[0161] Step 2: Calculate the extended covariance matrix of the two delay schemes. and :

[0162] (8)

[0163] in

[0164] (9)

[0165] Step 3: Calculate the real-valued covariance matrix and Then, eigenvalue decomposition is performed on it to obtain the signal subspace matrix. and :

[0166] (10)

[0167] right and Eigenvalue decomposition is performed to obtain the signal subspace matrix. and :

[0168] (11)

[0169] (12)

[0170] Step 4: Calculate the real-valued eigenma matrix and :

[0171] (13)

[0172] in and Let two choice matrices be represented, and their definitions are as follows: and , The matrix formed by concatenating the zero matrix and the identity matrix: , for The zero matrix, for The identity matrix, in this example, M=9, therefore

[0173] (twenty three)

[0174] Step 5: Construct the combination matrix Eigenvalue decomposition is performed on the signal to obtain its eigenvalues. Then, the fuzzy phase obtained under the two delay schemes is calculated. :

[0175] In this example,

[0176] (twenty four)

[0177] (25)

[0178] (26)

[0179] (27)

[0180] (28)

[0181] (29)

[0182] (30)

[0183] (4) Construct a system of congruence equations for each pair of entangled phases, and solve them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate. In this example, for and The constructed systems of congruence equations are as follows:

[0184] (31)

[0185] (31)

[0186] Solving the system of congruence equations using the robust Chinese remainder theorem yields the following results. and The value is

[0187] (32)

[0188] Further according to (34)

[0189] The DOA estimate is obtained by solving the problem.

[0190] (25)

[0191] Compared to the true value of 65 and 20 The absolute errors are only 0.0216° and -0.0122°, and the relative errors are only 0.033% and 0.056%. This preliminarily proves its effectiveness in DOA identification.

[0192] To further illustrate the advantages of the proposed method in terms of DOA estimation accuracy and computational efficiency compared to conventional uniform arrays, the proposed delayed coprime array is compared with a conventional uniform array with the same number of elements (19 elements). The conventional uniform array is an equidistant linear array with half-wavelength intervals. 200 tests were conducted with a signal-to-noise ratio (SNR) of 5 dB. The estimation results for the delayed coprime array and the conventional uniform array are as follows: Figure 3As shown in (a) and (b) in the figure, a magnified view of the DOA estimation details is provided for easier observation. It can be seen that the DOA estimation values ​​for the delayed coprime array are more concentrated and closer to the true values, indicating higher DOA estimation accuracy. For further quantitative analysis, the root mean square error (RMSE) of 200 estimation results for both arrays was calculated, and the results are shown in the figure. Figure 4 As shown, under the same signal-to-noise ratio, the delayed coprime array of the present invention has a lower root mean square error than the conventional uniform array, indicating that it has higher DOA estimation accuracy.

[0193] Furthermore, to verify the computational efficiency advantage of the proposed method, the DOA estimation time for delayed coprime arrays and conventional uniform arrays was calculated by varying the number of array elements. The test environment consisted of an i5-10400 CPU, 32GB of RAM, and MATLAB 2020a. The results are as follows: Figure 5 As shown, it can be seen that, with the same number of array elements, the invented delayed coprime array and its DOA method require less computation time, demonstrating its advantage in computational efficiency.

[0194] This invention proposes a sparse linear array and an efficient DOA estimation method. It utilizes a sparse arrangement of finite array elements to achieve large-aperture arrays, reducing element coupling effects and improving DOA estimation accuracy and computational efficiency. The invented sparse array design is simple, and the DOA estimation method is efficient and stable.

[0195]

Application Examples

[0196] In this exemplary example, it is assumed that the given number of array elements is 19, that is Array aperture half wavelength of electromagnetic signal The key parameters L, P, and Q of the array are respectively...

[0197] (17)

[0198] (18)

[0199] A schematic diagram of the designed array is shown below. Figure 1 As shown, from left to right, it is composed of two uniform arrays with an interval of (P+Q)d, consisting of the 1st, 3rd, 5th, ..., 19th sensors and the 2nd, 4th, 6th, ..., 18th sensors. The forward offset distance between the two uniform arrays is Pd, and the backward offset distance is Qd, where (P+Q)d, Pd, and Qd represent the products of P+Q, P, Q, and d, respectively.

[0200] The array was used to measure signal sources. In this example, there were two signal sources with DOA angles of 20° and 65°, respectively. The signal-to-noise ratio (SNR) was set to 5dB, and the measurement period lasted for 100 sampling cycles. The obtained data were as follows: and .

[0201] Construct two uniform array partitioning schemes with different delays. The partitioning scheme for the first uniform array is as follows: select the first M array elements of the first uniform array and the second uniform array as two subarrays, respectively. Figure 3 As shown in (a).

[0202] (4)

[0203] The second uniform array partitioning scheme is as follows: select the last M elements from the second uniform array and the first uniform array to form two sub-arrays, such as... Figure 3 As shown in (b).

[0204] (5)

[0205] The unitary ESPRIT algorithm is used to analyze the segmented data to obtain the entangled phase;

[0206] In this example, M=9, and the first step is to define the unitary matrix. and

[0207] (19)

[0208] (20)

[0209] in

[0210] (twenty one)

[0211] (twenty two)

[0212] Step 2: Calculate the extended covariance matrix of the two delay schemes. and :

[0213] (8)

[0214] in

[0215] (9)

[0216] Step 3: Calculate the real-valued covariance matrix and Then, eigenvalue decomposition is performed on it to obtain the signal subspace matrix. and :

[0217] (10)

[0218] right and Eigenvalue decomposition is performed to obtain the signal subspace matrix. and :

[0219] (11)

[0220] (12)

[0221] Step 4: Calculate the real-valued eigenma matrix and :

[0222] (13)

[0223] in and Let two choice matrices be represented, and their definitions are as follows: and , The matrix formed by concatenating the zero matrix and the identity matrix: , for The zero matrix, for The identity matrix, in this example, M=9, therefore

[0224] (twenty three)

[0225] Step 5: Construct the combination matrix Eigenvalue decomposition is performed on the signal to obtain its eigenvalues. Then, the fuzzy phase obtained under the two delay schemes is calculated. :

[0226] In this example,

[0227] (twenty four)

[0228] (25)

[0229] (26)

[0230] (27)

[0231] (28)

[0232] (29)

[0233] (30)

[0234] Finally, a system of congruence equations is constructed for each pair of entangled phases, and solved using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate. In this example, for... and The constructed systems of congruence equations are as follows:

[0235] (31)

[0236] (32)

[0237] Solving the system of congruence equations using the robust Chinese remainder theorem yields the following results. and The value is

[0238] (76)

[0239] Further according to (33)

[0240] The DOA estimate is obtained by solving the problem.

[0241] (34)

[0242] Compared to the true value of 65 and 20 The absolute errors are only 0.0216° and -0.0122°, and the relative errors are only 0.033% and 0.056%. This preliminarily proves its effectiveness in DOA identification.

[0243] To further illustrate the advantages of the proposed method in terms of DOA estimation accuracy and computational efficiency compared to conventional uniform arrays, the proposed delayed coprime array is compared with a conventional uniform array with the same number of elements (19 elements). The conventional uniform array is an equidistant linear array with half-wavelength intervals. 200 tests were conducted with a signal-to-noise ratio (SNR) of 5 dB. The estimation results for the delayed coprime array and the conventional uniform array are as follows: Figure 3 As shown in (a) and (b) in the figure, a magnified view of the DOA estimation details is provided for easier observation. It can be seen that the DOA estimation values ​​for the delayed coprime array are more concentrated and closer to the true values, indicating higher DOA estimation accuracy. For further quantitative analysis, the root mean square error (RMSE) of 200 estimation results for both arrays was calculated, and the results are shown in the figure. Figure 4 As shown, under the same signal-to-noise ratio, the delayed coprime array of the present invention has a lower root mean square error than the conventional uniform array, indicating that it has higher DOA estimation accuracy.

[0244] Furthermore, to verify the computational efficiency advantage of the proposed method, the DOA estimation time for delayed coprime arrays and conventional uniform arrays was calculated by varying the number of array elements. The test environment consisted of an i5-10400 CPU, 32GB of RAM, and MATLAB 2020a. The results are as follows: Figure 5 As shown, it can be seen that, with the same number of array elements, the invented delayed coprime array and its DOA method require less computation time, demonstrating its advantage in computational efficiency.

[0245] Furthermore, the delayed coprime array of the present invention consists of two sets of staggered uniform subarrays with a spatial interval of (P+Q)d (d being half a wavelength), and there is a relative offset between the two subarrays, Pd and Qd, where P and Q are coprime positive integers. This structure does not need to satisfy the "continuous virtual array" or "minimum redundancy difference set" conditions relied upon by traditional sparse arrays (such as nested arrays and coprime arrays), thus enabling more flexible sparse layout, significantly reducing physically adjacent array element pairs, and fundamentally weakening the mutual coupling effect; at the same time, with the same number of array elements, the total array aperture expands to nearly (N-1)(P+Q)d / 2, which is much larger than that of traditional half-wavelength uniform arrays, thereby greatly improving the angular resolution.

[0246] Secondly, this invention designs two subarray partitioning schemes with different spatial delays: constructing subarray pairs with delays of Pd and Qd respectively. This dual-delay mechanism enables the same incident signal to generate phase entanglement with different periods in the two subsystems, providing an independent information source for subsequent phase unwinding.

[0247] Furthermore, the unitary ESPRIT algorithm is used to process the two sets of subarray data separately. Its unitary transformation converts the complex-valued covariance matrix into a real-valued form, which not only reduces computational complexity but also improves noise robustness, thereby extracting the two sets of entangled spatial phases with high accuracy. Since the subarray spacing is greater than half a wavelength, the phase estimation result is the entanglement result of the true phase, but its entanglement period is determined by P and Q, respectively.

[0248] Most importantly, this invention transforms the Direction of Arrival (DOA) estimation problem into a noise-tolerant congruence equation solving problem, and introduces the Robust Chinese Remainder Theorem (RCRT) algorithm to jointly solve the entangled phase. Benefiting from the coprime nature of P and Q, theoretically, the unambiguous phase can be uniquely recovered within the modulus PQ range; furthermore, the Robust Chinese Remainder Theorem algorithm allows for a certain degree of error in phase estimation (caused by noise), still achieving high-probability correct deambiguity resolution, avoiding the stringent requirement of precise remainders in traditional Chinese Remainder Theorem algorithms. The entire process requires no spectral search, no sparse dictionary, and no high-dimensional matrix factorization; high-precision DOA estimation can be obtained solely through closed-form algebraic operations.

[0249] Although embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, and not restrictive. Those skilled in the art can make many other forms based on the guidance of this specification and without departing from the scope of protection of the claims of the present invention, and all of these are within the scope of protection of the present invention.

Claims

1. A method for estimating the DOA of a sparse linear array with coprime delay, characterized in that, The method is applicable to signal processing for direction-of-arrival estimation in radar, sonar, and wireless communication, and includes the following steps: Step S1: Use a time-delayed coprime sparse array to measure the signal source and obtain signal samples; Step S2: Construct two uniform subarray segmentation schemes with different delays and segment the signal samples to obtain data; Step S3: Analyze the segmented data using the unitary ESPRIT algorithm to obtain the entangled phase; Step S4: Construct a system of congruence equations for each pair of entangled phases, and solve them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate.

2. The DOA estimation method for a sparse linear array with coprime delay according to claim 1, characterized in that, Preferably, step S1 includes: The array parameters L, P, and Q are determined based on the number of sensors (2M+1) and the array aperture H: (1) (2) Where d is the half wavelength of the electromagnetic signal; the sparse array is composed of two uniform arrays with M+1 and M elements respectively; the product of L and d, Ld, is the spacing between the uniform subarrays in the sparse array; the product of P and d, Pd, is the forward offset distance between the two uniform subarrays in the sparse array; the product of Q and d, Qd, is the backward offset distance between the two uniform subarrays in the sparse array; The rounding up symbol, A sparse array samples a signal source, and the sampling lasts for N sampling periods. The signal samples collected by the sparse array are represented as follows: (3) in This is the first signal sample collected by a uniform array of M+1 array elements in a sparse array; Let n be the second signal sample collected by a uniform array of M elements in a sparse array, where n is an index that iterates from 1 to N.

3. The method for estimating the DOA of a sparse linear array with coprime delay according to claim 2, characterized in that, P and Q are coprime positive integers and satisfy the total array aperture D=M(P+Q)d.

4. The method for estimating the DOA of a sparse linear array with coprime delay according to claim 2, characterized in that, First uniform subarray partitioning scheme: (4) in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the first segmentation scheme. Second uniform subarray partitioning scheme: (5) in and The snapshot signal is composed of sample vectors measured from the first and second subarrays in the second segmentation scheme.

5. The method for estimating the DOA of a sparse linear array with coprime delay according to claim 1, characterized in that, Step S3 includes: Step 1: Define two unitary matrices and : (6) (7) in , and They are respectively , and The identity matrix; , and They are respectively , and The reverse identity matrix, whose anti-diagonal elements are 1; Represents a zero vector; For imaginary numbers, , This indicates the transpose operation. Step 2: Calculate the extended covariance matrix of the two delay schemes. and : (8) in This is the conjugate transpose operation; and Extended snapshots representing two time-delay schemes (9) Step 3: Calculate the real-valued covariance matrix and Then, eigenvalue decomposition is performed on it to obtain the signal subspace matrix. and : (10) in The real part extraction operation extracts the real part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar. right and Eigenvalue decomposition is performed to obtain the signal subspace matrix. and : (11) (12) in and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal characteristic values ​​and noise characteristic values; and They represent The noise subspace matrix of the signal subspace matrix; and They represent The signal eigenvalue matrix and noise eigenvalue matrix; Step 4: Calculate the real-valued eigenma matrix and : (13) in and Let two choice matrices be represented, and their definitions are as follows: and ; The matrix formed by concatenating the zero matrix and the identity matrix: , for The zero matrix, for The identity matrix; The imaginary part extraction operation extracts the imaginary part from a complex matrix / vector / scalar to form a new real-valued matrix / vector / scalar. This represents the generalized inverse operation. Step 5: Construct the combination matrix Eigenvalue decomposition is performed on the signal to obtain its eigenvalues. Then, the fuzzy phase obtained under the two delay schemes is calculated. : (14) in The number of information sources; k is an integer ranging from 1 to K; It is the arctangent function.

6. The method for estimating the DOA of a sparse linear array with coprime delay according to claim 5, characterized in that, Step S4 includes: For each pair of entangled phases Construct a system of congruence equations: (15) in and For an unknown integer, The unknown to be solved Solving congruence equations using the robust Chinese Remainder Theorem algorithm, and calculating... To obtain an accurate DOA estimate, the value is determined. : (16) in It is the arcsine function. based on For each k value, obtain the DOA estimate of K signal sources. .

7. The method for estimating the DOA of a sparse linear array with coprime delay according to claim 1, characterized in that, The unitary ESPRIT algorithm constructs a real-valued extended covariance matrix from the subarray received data and performs unitary transformation and eigenvalue decomposition to obtain the entangled phase estimate.

8. A system for performing the method as described in any one of claims 1-7, characterized in that, It includes: The measurement module uses a time-delayed coprime sparse array to measure the signal source and obtain signal samples; The module constructs two uniform subarray segmentation schemes with different delays and segments signal samples to obtain data. The analysis module uses the unitary ESPRIT algorithm to analyze the segmented data and obtain the entangled phase; The calculation module constructs a system of congruence equations for each pair of entangled phases and solves them using the robust Chinese Remainder Theorem algorithm to obtain the DOA estimate.

9. A computer storage medium, characterized in that, The storage medium includes computer instructions that, when executed on a computer, cause the computer to perform the method as described in any one of claims 1-7.

10. An electronic device, characterized in that, The electronic device includes: Memory, processor, and computer programs stored in memory and executable on the processor, wherein, When the processor executes the program, it implements the method as described in any one of claims 1-7.

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