DOA estimation method based on hybrid analog-digital array architecture design

Abstract

The application provides a DOA estimation method based on a hybrid analog-digital array architecture design, which comprises the following steps: a hybrid analog-digital array is arranged at a receiving end architecture; the hybrid analog-digital array adopts a low-complexity HAD array architecture; the hybrid analog-digital array architecture is used to receive an incident signal and model; an initial value of an analog phase shifter network is designed; a difference virtual array of the low-complexity HAD array architecture is calculated; an initial angle estimation value is obtained from the difference virtual array receiving signal by using a classical direction finding algorithm; after the initial angle estimation value is obtained, the phase shifter network weight is redesigned, and then a difference virtual array output signal is obtained; and a DOA estimation result is obtained from the classical direction finding algorithm; the method can effectively avoid the problems of phase ambiguity and beam scanning, can reduce the cost and complexity, and can effectively approach the estimation performance of a full-digital array architecture.

Description

Technical Field

[0001] This invention relates to a DOA estimation method based on a hybrid analog-digital array architecture design, belonging to the field of signal processing technology. Background Technology

[0002] Direction-of-Arrival (DOA) estimation is an important branch of array signal processing. It refers to the process of using an array antenna to receive spatial signals and then processing the received signals through statistical signal processing techniques and various optimization methods to recover the direction of arrival information of the incident signal. It has wide applications in radar, sonar, voice and wireless communication.

[0003] Very large-scale arrays (VLS) are an evolution of large-scale arrays, deploying an extremely large number of antennas, sometimes reaching thousands, which brings ultra-high angular resolution and accuracy to DOA estimation. However, due to the enormous number of antennas, a matching number of radio frequency (RF) links is required, including expensive components such as analog-to-digital converters (ADCs), which greatly increases hardware costs and power consumption.

[0004] To address the issues of high hardware cost and power consumption, Hybrid Analog and Digital (HAD) array architectures with finite RF chains are widely adopted. In sub-connected HAD architectures, the number of phase shifters equals the number of antennas, and multiple phase shifters are connected to a single RF chain, making integration and expansion easy. However, HAD architectures introduce phase ambiguity, requiring multiple time slots to eliminate it. As the number of sub-array antennas increases, the DOA measurement delay increases linearly, necessitating the design of low-complexity HAD architectures for DOA estimation. Furthermore, due to the altered HAD architecture, the array output differs from that of traditional HAD arrays, requiring the design of DOA estimation methods based on hybrid analog-digital array architectures.

[0005] The above-mentioned issues are problems that should be considered and resolved in the process of DOA estimation based on hybrid analog-digital array architecture design. Summary of the Invention

[0006] The purpose of this invention is to provide a DOA estimation method based on a hybrid analog-digital array architecture to solve the problems of phase ambiguity and the need to reduce complexity in existing HAD architectures.

[0007] The technical solution of this invention is:

[0008] A DOA estimation method based on a hybrid analog-digital array architecture includes the following steps:

[0009] S1. The receiver is equipped with a hybrid analog-digital array architecture, which adopts a low-complexity HAD array architecture.

[0010] S2. Utilize the architecture's hybrid analog-digital array to receive and model the incident signal;

[0011] S3. Design the initial values ​​for the analog phase shifter network;

[0012] S4. Calculate the received signal of the differential virtual array in the low-complexity HAD array architecture.

[0013] S5. Based on the received signal of the differential virtual array The initial angle estimate is obtained using the classical direction-finding algorithm.

[0014] S6. Using the obtained initial angle estimate After redesigning the phase shifter network weights, the differential virtual array output signal is obtained. The DOA estimation results are obtained by the classical direction finding algorithm.

[0015] Furthermore, in step S1, the hybrid analog-digital array adopts a low-complexity HAD array architecture, specifically,

[0016] S11. The hybrid analog-digital array adopts a low-complexity HAD array architecture. The hybrid analog-digital array contains N array elements, which are divided into K subarrays.

[0017] S12, the first subarray of the K subarrays contains M1 array elements, each array element is connected to an RF chain, and the next K-1 subarrays each contain M2 array elements, where M2 = M1 + 1;

[0018] S13. Consider the output of each RF chain as the output of a virtual array element. The overall virtual array structure is a two-level nested array, and the virtual array element position set is S = {q1, q2, ..., q}. i ,...,q I}={0,1,...,M1-1,M1,2M2-1,...,(K-1)M2-1}, q i This is the position index of the i-th virtual array element.

[0019] Further, in step S2, the incident signal is received and modeled using a hybrid analog-digital array architecture, specifically,

[0020] S21. Assuming the far-field narrowband signal is incident at an azimuth angle θ, the array-received signal can be expressed as follows:

[0021]

[0022] in, Let s(t) be the signal source, t be the baseband signal, j be the imaginary unit, and f be the signal source. c Let n(t) be the carrier frequency, n(t) be additive white Gaussian noise, and a(θ) be the array manifold vector, expressed as a(θ) = [1, e^(-1 / 2)]. jπsin(θ) ,...,e jπ(N-1)sin(θ) ] T Where e is the natural constant, N is the number of array elements, and (·) T Indicates transpose;

[0023] S22. Phase calibration is performed after passing through an analog phase shifter network. The calibrated received signal is represented as follows:

[0024]

[0025] in,(·) H Let V denote the transpose conjugate, and V be the analog beamforming matrix, denoted as: The weight matrix of the first subarray , Let v represent an M1×M1 dimensional identity matrix. k Let w(t) be the simulated beamforming vector of the k-th subarray, where k ≥ 2, and w(t) is noise, expressed as w(t) = V H n(t);

[0026] S23, Calibrated Received Signal After passing through a parallel RF chain, it is down-converted into a baseband signal y(t)=V H a(θ)s(t)+w(t), the baseband signal is converted into an output signal y(n)=V by an analog-to-digital converter (ADC). H a(θ)s(n)+w(n), where y(n) is the output signal, s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots.

[0027] Further, in step S3, the initial values ​​of the simulated phase shifter network are designed, specifically, when k≥2, the simulated beamforming vector v of the kth subarray is... k : [R1](:,1) represents the first column of matrix R1. Where E[·] represents the expectation, and x1(n) is the array received signal of the first M2 array elements.

[0028] Furthermore, in step S4, the received signal of the differential virtual array of the low-complexity HAD array architecture is calculated. Specifically,

[0029] S41, The covariance matrix of the array output signal is expressed as:

[0030]

[0031] in, Indicates the incident signal power. This represents noise power, and D is the gain coefficient matrix. Let represent the array manifold vector of a second-order nested array, where e is the natural constant, j represents the imaginary unit, N is the number of array elements, and (·) T This indicates transpose, q1...qI are indices of the (1-I)th virtual array element, and B = V. H V, where V is the simulated beamforming matrix, (·) H Indicates transpose and conjugate;

[0032] S42, the covariance matrix R of the array output signal yy Perform vectorization to obtain a vector:

[0033]

[0034] Where vec(·) represents the vectorization operation, This represents the Kronecker product operation, where b = vec(B);

[0035] S43. The gain matrix of the array under ideal conditions is obtained as follows:

[0036]

[0037] Among them, I M1 Let M1 represent the M1×M1 dimensional identity matrix, M2 be the number of elements in each subarray of the (2-K)th subarray, and I K-1 Represents a (K-1)×(K-1) dimensional identity matrix;

[0038] S44, Let the vector The differential virtual array is obtained by subtracting the physical elements of the two-level nested array. The vector z′ is then deredundantized and rearranged according to the order of its elements to obtain the received signal from the differential virtual array. .

[0039] Further, in step 43, the gain matrix under ideal conditions of the array is obtained, specifically,

[0040] In the ideal case where the phase of the simulated phase shifter is precisely aligned with the incident angle, i.e., the ideal simulated beamforming vector... When, where k≥2, the gain of the (2-K)th subarray in, Let be the steering vector of the k-th subarray, where |·| represents the absolute value operation. Therefore, the gain matrix of the array under ideal conditions is obtained as follows:

[0041]

[0042] in, Let M1 represent the M1×M1 dimensional identity matrix, M2 be the number of elements in each subarray of the (2-K)th subarray, and I K-1 It represents a (K-1)×(K-1) dimensional identity matrix.

[0043] Furthermore, in steps S5 and S6, the classical direction-finding algorithm employs a spatial smoothing method and a sparse reconstruction compressed sensing DOA estimation method.

[0044] Furthermore, in step S6, the obtained initial angle estimate is used... After redesigning the phase shifter network weights, the differential virtual array output signal is obtained. DOA estimation results obtained from classical direction finding algorithm Specifically,

[0045] S61. Using the initial angle estimate The phase shifter network weights have been redesigned, and the output signal is now r(n) = V. H a(θ)s(n)+w(n), where s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots, where a(θ) is the array manifold vector, and V is the analog beamforming matrix, expressed as in, Denotes the M1×M1 dimensional identity matrix, and the simulated beamforming vector of the k-th subarray. Where k≥2;

[0046] S62. For the output signal r(n), calculate the received signal of the differential virtual array.

[0047] S63, Based on the differential virtual array output signal DOA estimation results were obtained using the classical direction-finding algorithm.

[0048] Further, in step S62, for the output signal r(n), the received signal of the differential virtual array is calculated. Specifically,

[0049] S621, the covariance matrix of the output signal r(n) is:

[0050]

[0051] in, Indicates the incident signal power. D1 represents the noise power, and D1 is the gain matrix. Denotes the array manifold vector of a second-order nested array, where e is the natural constant, j is the imaginary unit, and q i Let B be the index of the i-th virtual element, and B = V. H V, where V is the simulated beamforming matrix, (·) H Indicates transpose and conjugate;

[0052] S622, Regarding the covariance matrix R rr Perform vectorization operations.

[0053]

[0054] Where vec(·) represents the vectorization operation, This represents the Kronecker product operation, where p is the signal power and b = vec(B).

[0055] S623, Let the vector The differential virtual array is obtained by subtracting the physical elements of the two-level nested array. The vector r′ is then deredundantized and rearranged according to the order of its elements to obtain the received signal from the differential virtual array.

[0056] The beneficial effects of this invention are:

[0057] I. This DOA estimation method based on a hybrid analog-digital array architecture design, by designing a novel HAD array architecture, fully utilizes the characteristics of nested arrays, effectively avoids phase ambiguity and beam scanning problems, reduces cost and complexity, and ensures DOA estimation accuracy to a certain extent.

[0058] II. Compared with the prior art, this invention effectively reduces hardware costs and complexity by combining the HAD array architecture and the concept of nested arrays.

[0059] Third, this DOA estimation method based on a hybrid analog-digital array architecture can effectively approximate the estimation performance of an all-digital array architecture by redesigning the phase shifter network weight vector, and can be used for DOA estimation scenarios of ultra-large-scale arrays. Attached Figure Description

[0060] Figure 1 This is a flowchart illustrating the DOA estimation method based on a hybrid analog-digital array architecture design according to an embodiment of the present invention.

[0061] Figure 2 This is a schematic diagram illustrating the HAD array architecture in the embodiment.

[0062] Figure 3 This is a schematic diagram comparing the performance of the DOA estimation method based on the hybrid analog-digital array architecture design with existing array architecture methods. Detailed Implementation

[0063] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0064] Example

[0065] A DOA estimation method based on a hybrid analog-digital array architecture includes the following steps:

[0066] S1. The receiver is equipped with a hybrid analog-digital array architecture, which adopts a low-complexity HAD array architecture.

[0067] S11. The hybrid analog-digital array contains N array elements and is divided into K subarrays;

[0068] S12, the first subarray of the K subarrays contains M1 array elements, each array element is connected to an RF chain, and the next K-1 subarrays each contain M2 array elements, where M2 = M1 + 1;

[0069] S13. Consider the output of each RF chain as the output of a virtual array element. The overall virtual array structure is a two-level nested array, and the virtual array element position set is S = {q1, q2, ..., q}. i ,...,q I}={0,1,...,M1-1,M1,2M2-1,...,(K-1)M2-1}, q i This is the position index of the i-th virtual array element.

[0070] S2. Utilize the architecture's hybrid analog-digital array to receive and model the incident signal;

[0071] S21. Assuming the far-field narrowband signal is incident at an azimuth angle θ, the array-received signal can be expressed as follows:

[0072]

[0073] in, Let s(t) be the signal source, t be the baseband signal, j be the imaginary unit, and f be the signal source. c Let n(t) be the carrier frequency, n(t) be additive white Gaussian noise, and a(θ) be the array manifold vector, expressed as a(θ) = [1, e^(-1 / 2)]. jπsin(θ) ,...,e jπ(N-1)sin(θ) ]T Where e is the natural constant, N is the number of array elements, and (·) T Indicates transpose;

[0074] S22. Phase calibration is performed after passing through an analog phase shifter network. The calibrated received signal is represented as follows:

[0075]

[0076] in,(·) H Let V denote the transpose conjugate, and V be the analog beamforming matrix, denoted as: The weight matrix of the first subarray Let v represent an M1×M1 dimensional identity matrix. k Let w(t) be the simulated beamforming vector of the k-th subarray, where k ≥ 2, and w(t) is noise, expressed as w(t) = V H n(t);

[0077] S23, Calibrated Received Signal After passing through a parallel RF chain, it is down-converted into a baseband signal y(t)=V H a(θ)s(t)+w(t), the baseband signal is converted into an output signal y(n)=V by an analog-to-digital converter (ADC). H a(θ)s(n)+w(n), where y(n) is the output signal, s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots.

[0078] S3. Design the initial values ​​for the analog phase shifter network;

[0079] In step S3, the initial values ​​of the simulated phase shifter network are designed. Specifically, when k≥2, the simulated beamforming vector v of the kth subarray is... k : [R1](:,1) represents the first column of matrix R1. Where E[·] represents the expectation, and x1(n) is the array received signal of the first M2 array elements.

[0080] S4. Calculate the received signal of the differential virtual array in the low-complexity HAD array architecture. ;

[0081] S41, The covariance matrix of the array output signal is expressed as:

[0082]

[0083] in, Indicates the incident signal power. This represents noise power, and D is the gain coefficient matrix. Let represent the array manifold vector of a second-order nested array, where e is the natural constant, j represents the imaginary unit, N is the number of array elements, and (·) T Indicates transpose, q1...q I B = V, which is the index of the (1-I)th virtual element. H V, where V is the simulated beamforming matrix, (·) H Indicates transpose and conjugate;

[0084] S42, the covariance matrix R of the array output signal yy Perform vectorization to obtain a vector:

[0085]

[0086] Where vec(·) represents the vectorization operation, This represents the Kronecker product operation, where b = vec(B);

[0087] S43. The gain matrix of the array under ideal conditions is obtained as follows:

[0088]

[0089] in, Let M1 represent the M1×M1 dimensional identity matrix, M2 be the number of elements in each subarray of the (2-K)th subarray, and I K-1 Represents a (K-1)×(K-1) dimensional identity matrix;

[0090] In step 43, the gain matrix under ideal conditions of the array is obtained, specifically,

[0091] In the ideal case where the phase of the simulated phase shifter is precisely aligned with the incident angle, i.e., the ideal simulated beamforming vector... When, where k≥2, the gain of the (2-K)th subarray in, Let be the steering vector of the k-th subarray, where |·| represents the absolute value operation. Therefore, the gain matrix of the array under ideal conditions is obtained as follows:

[0092]

[0093] in, Let M1 represent the M1×M1 dimensional identity matrix, M2 be the number of elements in each subarray of the (2-K)th subarray, and I K-1 It represents a (K-1)×(K-1) dimensional identity matrix.

[0094] S44, Let the vector The differential virtual array is obtained by subtracting the physical elements of the two-level nested array. The vector z′ is then deredundantized and rearranged according to the order of its elements to obtain the received signal from the differential virtual array.

[0095] S5. Based on the received signal of the differential virtual array The initial angle estimate is obtained using the classical direction-finding algorithm.

[0096] S6. Using the obtained initial angle estimate After redesigning the phase shifter network weights, the differential virtual array output signal is obtained. DOA estimation results obtained from classical direction finding algorithm

[0097] S61. Using the initial angle estimate The phase shifter network weights have been redesigned, and the output signal is now r(n) = V. H a(θ)s(n)+w(n), where s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots, where a(θ) is the array manifold vector, and V is the analog beamforming matrix, expressed as in, Denotes the M1×M1 dimensional identity matrix, and the simulated beamforming vector of the k-th subarray. Where k≥2;

[0098] S62. For the output signal r(n), calculate the received signal of the differential virtual array.

[0099] S621, the covariance matrix of the output signal r(n) is:

[0100]

[0101] in, Indicates the incident signal power. D1 represents the noise power, and D1 is the gain matrix. Denotes the array manifold vector of a second-order nested array, where e is the natural constant, j is the imaginary unit, and q i Let B be the index of the i-th virtual element, and B = V. H V, where V is the simulated beamforming matrix, (·) H Indicates transpose and conjugate;

[0102] S622, Regarding the covariance matrix R rr Perform vectorization operations.

[0103]

[0104] Where vec(·) represents the vectorization operation, This represents the Kronecker product operation, where p is the signal power and b = vec(B).

[0105] S623, Let the vector The differential virtual array is obtained by subtracting the physical elements of the two-level nested array. The vector r′ is then deredundantized and rearranged according to the order of its elements to obtain the received signal from the differential virtual array.

[0106] S63, Based on the differential virtual array output signal DOA estimation results were obtained using the classical direction-finding algorithm.

[0107] In steps S5 and S6, classic direction-finding algorithms employ spatial smoothing methods and sparse reconstruction compressed sensing DOA estimation methods. Taking spatial smoothing as an example, the Toeplitz matrix reconstruction method is used to achieve spatial smoothing, and angle estimates are obtained through multiple signal classification algorithms (MUSIC algorithm) or root-MUSIC algorithm.

[0108] This DOA estimation method based on a hybrid analog-digital array architecture design, by designing a novel HAD array architecture, fully utilizes the characteristics of nested arrays, effectively avoids phase ambiguity and beam scanning problems, reduces cost and complexity, and ensures DOA estimation accuracy to a certain extent.

[0109] Compared with existing technologies, this invention effectively reduces hardware costs and complexity by combining the HAD array architecture with the concept of nested arrays.

[0110] This DOA estimation method based on a hybrid analog-digital array architecture can effectively approximate the estimation performance of an all-digital array architecture by redesigning the phase shifter network weight vector, and can be used for DOA estimation scenarios of ultra-large-scale arrays.

[0111] The effects of the present invention will be further described below with reference to simulation examples.

[0112] Simulation example: A far-field narrowband signal is incident at a direction of 30°. The array contains 120 array elements, divided into 11 subarrays. The first subarray has 10 array elements and 400 snapshots.

[0113] The DOA estimation method based on the hybrid analog-digital array architecture of this embodiment is compared with the estimation performance of other existing array architecture methods as follows: Figure 3 As shown. By Figure 3It can be seen that the proposed method can approximate the estimation performance of the all-digital array architecture well, and its estimation performance is better than that of the traditional HAD array architecture when the number of RF chains is the same. The above results demonstrate the effectiveness of the method proposed in this invention.

[0114] In summary, the proposed method combines the nested array concept with the HAD array structure, effectively reducing hardware costs and complexity. At the same time, the proposed method can also approximate the performance of a fully digital array architecture to a certain extent.

[0115] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A DOA estimation method based on a hybrid analog-digital array architecture design, characterized in that: Includes the following steps, S1. The receiver is equipped with a hybrid analog-digital array architecture, which adopts a low-complexity HAD array architecture. In step S1, the hybrid analog-digital array adopts a low-complexity HAD array architecture, specifically, S11. The hybrid analog-digital array contains N array elements and is divided into K subarrays; S12, the first subarray of the K subarrays contains Each of the K-1 array elements is connected to an RF chain. Each of the subsequent K-1 subarrays contains... Each array element, among which... ; S13. Consider the output of each RF chain as the output of a virtual array element. The overall virtual array structure is a two-level nested array, and the virtual array element position set is... , This is the position index of the i-th virtual array element; S2. Utilize the architecture's hybrid analog-digital array to receive and model the incident signal; S21. Assuming the far-field narrowband signal is incident at an azimuth angle θ, the array-received signal can be expressed as follows: ， in, Let s(t) be the signal source, t be the baseband signal, t be the time, and j be the imaginary unit. Let n(t) be the carrier frequency, n(t) be additive white Gaussian noise, and a(θ) be the array manifold vector, denoted as... Where e is the natural constant and N is the number of array elements. Indicates transpose; S22. Phase calibration is performed after passing through an analog phase shifter network. The calibrated received signal is represented as follows: ， in, Let V denote the transpose conjugate, and V be the analog beamforming matrix, denoted as: The weight matrix of the first subarray , express 3D identity matrix Let w(t) be the simulated beamforming vector of the k-th subarray, where k ≥ 2, and w(t) is noise, denoted as... ; S23, Calibrated Received Signal After passing through a parallel RF chain, it is down-converted into a baseband signal. The baseband signal is converted into an output signal by an analog-to-digital converter (ADC). Where y(n) is the output signal, s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots; S3. Design the initial values ​​for the analog phase shifter network; specifically, At that time, the simulated beamforming vector of the k-th subarray : , Representation matrix Column 1 ,in, Indicates the expectation. For the front The array of array elements receives signals; S4. Calculate the received signal of the differential virtual array in the low-complexity HAD array architecture. ; S41, The covariance matrix of the array output signal is expressed as: ， in, Indicates the incident signal power. Indicates noise power. It is the gain coefficient matrix. Let e ​​be the array manifold vector of a second-order nested array, where e is the natural constant, j represents the imaginary unit, and N is the number of array elements. Indicates transpose. This is the index of the (1-I)th virtual array element. Where V is the simulated beamforming matrix, Indicates transpose and conjugate; S42, Covariance matrix of the array output signal Perform vectorization to obtain a vector: ， in, Indicates vectorization operation, This represents the Kronecker product operation. ; S43. The gain matrix of the array under ideal conditions is obtained as follows: ， in, express 3D identity matrix Let be the number of array elements in each of the (2-K)th subarrays. express 3D identity matrix; S44, Let the vector The difference between the physical elements of the two-level nested array is obtained by subtracting from its difference virtual array. Then, according to the order of the elements in the difference virtual array, the vectors are processed... Redundancy removal and rearrangement are performed to obtain the received signal of the differential virtual array. ; S5. Based on the received signal of the differential virtual array The initial angle estimate is obtained through the classical direction-finding algorithm. ; S6. Using the obtained initial angle estimate After redesigning the phase shifter network weights, the differential virtual array output signal is obtained. The DOA estimation results are obtained by the classical direction finding algorithm. ; S61. Using the initial angle estimate The phase shifter network weights have been redesigned, and the output signal is now... Where s(n) is the signal, w(n) is additive white Gaussian noise, and n represents the number of snapshots. It is an array manifold vector. It is a simulated beamforming matrix, represented as ,in, , express An identity matrix of dimension 1, the simulated beamforming vector of the k-th subarray. ,in, ; S62, Regarding the output signal The received signal of the differential virtual array is calculated. ; S63, Based on the differential virtual array output signal The DOA estimation results were obtained using the classical direction-finding algorithm. .

2. The DOA estimation method based on a hybrid analog-digital array architecture as described in claim 1, characterized in that: In step 43, the gain matrix under ideal conditions of the array is obtained, specifically, In the ideal case where the phase of the simulated phase shifter is precisely aligned with the incident angle, i.e., the ideal simulated beamforming vector... At that time, among them Gain of the 2nd to Kth subarray ,in, Let be the guiding vector of the k-th subarray, where This represents the absolute value operation; therefore, the gain matrix of the array in the ideal case is... ， in, express 3D identity matrix Let be the number of array elements in each of the (2-K)th subarrays. express 3D identity matrix.

3. The DOA estimation method based on a hybrid analog-digital array architecture as described in claim 1, characterized in that: In steps S5 and S6, the classic direction finding algorithm employs a spatial smoothing method and a sparse reconstruction compressed sensing DOA estimation method.

4. The DOA estimation method based on a hybrid analog-digital array architecture as described in claim 1, characterized in that: In step S62, for the output signal The received signal of the differential virtual array is calculated. Specifically, S621, Output Signal The covariance matrix is ， in, Indicates the incident signal power. Indicates noise power. It is the gain matrix. Let represent the array manifold vector of a second-order nested array, where e is the natural constant and j is the imaginary unit. Let be the index of the i-th virtual array element. ,in, It is a simulated beamforming matrix. Indicates transpose and conjugate; S622, Regarding the covariance matrix Perform vectorization operations. ， in, Indicates vectorization operation, This represents the Kronecker product operation. For signal power, ; S623, Let the vector The difference between the physical elements of the two-level nested array is obtained by subtracting from its difference virtual array. Then, according to the order of the elements in the difference virtual array, the vectors are processed... Redundancy removal and rearrangement are performed to obtain the received signal of the differential virtual array. .

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