Direction of arrival estimation method and system based on co-prime frequency sparse array motion aperture expansion model

Abstract

The invention discloses a direction of arrival estimation method and system based on a co-prime frequency sparse array motion aperture expansion model, and the method comprises the steps: enabling a preset unmanned plane cluster array to move in a preset direction at a constant speed, and receiving reflection signals of a plurality of far-field targets, sampling the reflected signals according to the two co-prime frequencies to obtain a received signal matrix; 2, according to the received signal matrix, obtaining covariance matrixes corresponding to the two co-prime frequencies respectively, vectorized expressions of the covariance matrixes, a covariance matrix between the two co-prime frequencies and vectorized expressions of the covariance matrixes, and completing construction of an aperture expansion model; and 3, constructing an array flow pattern matrix according to the aperture expansion model, and solving the array flow pattern matrix by using a group lasso method to obtain direction-of-arrival estimation. According to the method, aperture expansion is carried out by integrating time-frequency-space information at the same time, and the precision and the degree of freedom of DOA estimation by using the sparse array are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of array signal processing, and in particular to a direction of arrival estimation method and system based on a coprime frequency sparse array motion aperture expansion model. Background Art

[0002] In array signal processing, compared to traditional non-sparse uniform arrays, sparse arrays have a larger array element spacing, which can provide a larger array aperture with the same number of antennas, thereby making it possible to save costs and improve accuracy. Its specific advantages are: First, when the number of array elements is the same, sparse arrays have a larger array aperture, which improves the accuracy of direction finding and increases the maximum number of source detections; second, when the aperture size is consistent, the number of array elements required is reduced, which can reduce the construction cost of the system and the complexity of the system design; third, a wider array element spacing can be set to reduce the influence of mutual coupling errors between array elements. Therefore, considering the advantages of sparse arrays, it is of great significance to study and design sparse array formation structures with better performance and design high-precision wave direction estimation algorithms to ensure accurate parameter estimation.

[0003] With the widespread use of drones in local warfare for reconnaissance and surveillance, precision strikes, electronic jamming, camouflage, deception, and target guidance, unmanned platforms have garnered worldwide attention due to their unique tactical and strategic advantages. Combining these unique advantages with unmanned platforms can significantly enhance their role in traditional reconnaissance and positioning, possessing high research value and military application significance. According to array antenna theory, high-resolution and high-degree-of-freedom target positioning requires the support of large-aperture arrays. However, significantly increasing the system's degrees of freedom by optimizing the array structure alone is becoming increasingly difficult. Using sparse arrays with larger physical dimensions makes it difficult to balance platform design complexity and overall cost control. Therefore, to further enhance the degrees of freedom (DOF) of array structure design and achieve the ability to locate and track larger targets, synthetic aperture (SA) technology has been widely used in array signal processing in recent years. Combining sparse arrays with SA can effectively eliminate holes in the virtual array corresponding to the original sparse array through array motion, increase the length of the continuous virtual array element segment, and improve the accuracy of DOF and DOA estimation.

[0004] With limited physical element size, existing technologies mainly expand the array aperture by using additional information in time, frequency, and space.

[0005] (1) Array expansion of comprehensive airspace information

[0006] In pursuit of higher precision and greater freedom, sparse arrays based on the concept of differential mutual arrays have attracted widespread attention. Compared with traditional UAV swarm (ULA) arrays, sparse arrays can achieve a larger equivalent virtual aperture with the same number of array elements, receivers, and other front-end channels due to the use of additional spatial structure information of sparse array element distribution for differential array generation. Therefore, sparse arrays can bring about an improvement in freedom and precision. For a designed M-element sparse array, the maximum number of sparse arrays can be achieved. The essence of this improvement is to expand the aperture with the help of the spatial information obtained by the differential mutual array.

[0007] (2) Array expansion of integrated frequency domain information

[0008] This method is similar to the concept of constructing virtual elements in coprime arrays. Receiving coprime frequency reflection signals from the target via a ULA is equivalent to constructing a virtual coprime array. Based on this virtual coprime array, a differential array is then solved to further expand the virtual element aperture. By additionally utilizing the same spatial spectrum information from snapshots sampled at different frequencies, this method also improves element degrees of freedom and DOA estimation accuracy. Essentially, this method leverages frequency domain information to enhance the aperture expansion effect of a sparse array.

[0009] (3) Array expansion of integrated time domain information

[0010] As research into sparse array design deepens, increasing the array's degrees of freedom simply by optimizing the array structure becomes increasingly difficult. Furthermore, adopting sparse arrays of larger physical sizes makes it difficult to balance the design complexity and cost of the platform. Therefore, existing methods employ the concept of two-sampling motion synthesis to create virtual arrays with larger apertures and a greater number of continuous virtual elements. By utilizing additional sampling snapshots from multiple motion processes, effectively filling in the holes in the virtual array, they achieve improved element degrees of freedom and accuracy. Essentially, this approach leverages time-domain information obtained from sampling at different moments to increase the aperture expansion effect.

[0011] The aforementioned aperture expansion methods all utilize one or two types of information. For example, motion-based sparse array expansion utilizes both spatial and temporal information, while multi-frequency aperture expansion utilizes both spatial and frequency information. However, it is not possible to simultaneously integrate time, frequency, and spatial information for aperture expansion to achieve higher accuracy and greater degrees of freedom for sparse arrays. Summary of the Invention

[0012] To address the problem that existing technologies cannot simultaneously integrate time-frequency-space information to expand the aperture, thereby failing to achieve higher precision and greater degrees of freedom for sparse arrays, the present invention provides a direction-of-arrival estimation method and system based on a coprime frequency sparse array motion aperture expansion model. By designing an aperture expansion model that integrates time-frequency-space information, a closed-form expression for the distribution of synthesized virtual array elements is given. Based on the aperture expansion model, an array flow matrix is ​​constructed, and the array flow matrix is ​​solved using the group-LASSO method to obtain a direction-of-arrival (DOA) estimate. The present invention simultaneously integrates time-frequency-space information to expand the aperture, thereby improving the precision and degrees of freedom of sparse arrays.

[0013] In order to achieve the above object, the technical solution of the present invention is:

[0014] The first aspect of the present invention proposes a direction of arrival estimation method based on a coprime frequency sparse array motion aperture expansion model, comprising:

[0015] Step 1: Detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies. Move the preset drone cluster array at a constant speed in a predetermined direction and receive reflected signals from multiple far-field targets. Sample the reflected signals according to the two mutually prime frequencies to obtain a corresponding received signal matrix, facilitating the construction of an aperture expansion model.

[0016] Step 2: According to the received signal matrix, the covariance matrix corresponding to the two coprime frequencies and its vectorized representation, as well as the covariance matrix between the two coprime frequencies and its vectorized representation are obtained to complete the aperture expansion model construction to facilitate subsequent direction of arrival estimation;

[0017] Step 3: Construct the array flow pattern matrix based on the aperture expansion model, and use the group lasso method to solve the array flow pattern matrix to obtain the wave direction of arrival estimation.

[0018] Furthermore, sampling the reflected signal at two mutually prime frequencies to obtain a corresponding received signal matrix specifically includes:

[0019] The reflected signal is sampled twice at frequencies f1 and f2, and phase shift compensation is performed on the second-sampled signal. The compensated signal and the first-sampled signal are combined to obtain a received signal matrix of the f1 frequency component and a received signal matrix of the f2 frequency component. The f1 frequency and the f2 frequency are coprime frequencies, which facilitates the integration of time domain and frequency domain information.

[0020] The received signal matrix of the f1 frequency component is expressed as follows:

[0021]

[0022] Where y1(t) is the received signal matrix of the f1 frequency component, f1 is the frequency, is the f1 frequency component in the first sampling signal, is the f1 frequency component in the second sampling signal, τ is the interval between two samplings, t is the time, j is the imaginary unit, A1 is the array flow pattern vector set at the f1 frequency during the first sampling, and B1 is the array flow pattern vector set at the f1 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f1, is the noise component in the second sampling signal corresponding to the frequency f1, A S,1 is the array flow vector set at the frequency f1 during sampling, and n1(t) is the noise component in the sampling signal at the frequency f1;

[0023] The received signal matrix of the f2 frequency component is expressed as follows:

[0024]

[0025] Where y2(t) is the received signal matrix of the f2 frequency component, f2 is the frequency, is the f2 frequency component in the first sampling signal, is the f2 frequency component in the second sampling signal, A2 is the array flow vector set at the f2 frequency during the first sampling, and B2 is the array flow vector set at the f2 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f2, is the noise component in the second sampling signal at the frequency f2, A S,2 is the array flow vector set at the frequency f2 during sampling, and n2(t) is the noise component in the sampling signal at the frequency f2.

[0026] Furthermore, the array position sets before and after motion synthesis corresponding to the received signal matrices of the f1 frequency component and the f2 frequency component are expressed by the following formula:

[0027] P1={nW / η,0≤n≤N-1}

[0028] P1 mov =P1∪{D / η+P1}

[0029] P2={nW,0≤n≤N-1}

[0030]

[0031] Where P1 is the array position set before movement corresponding to the received signal matrix of the frequency component f1, n is the nth UAV, W is a positive integer, N is the number of UAVs in the UAV cluster array, η is the coprime frequency ratio, P1 mov is the array position set after motion synthesis corresponding to the signal matrix of the frequency component f1, D is the motion distance of the drone array before and after the two samplings, P2 is the array position set before motion corresponding to the signal matrix of the frequency component f2, is the array position set after motion synthesis corresponding to the received signal matrix of the f2 frequency component.

[0032] Furthermore, the covariance matrix of the f1 frequency is expressed as follows:

[0033]

[0034] in, is the covariance matrix of the f1 frequency in the virtual array, E is the expected value, is the covariance matrix, is the noise power at frequency f1, I 2N is a 2N column vector of all ones, is the signal power of the kth source at the frequency f1, a s,1 (θ k ) is the steering vector of the kth source, (·) H is the conjugate transpose;

[0035] The covariance matrix of the f1 frequency signal is quantized and expressed as follows:

[0036]

[0037] in, is the vectorized representation of the covariance matrix of the f1 frequency signal, is the joint steering matrix corresponding to the frequency f1, is the joint steering matrix of the kth source at frequency f1, a1(θ k ) is the array flow vector at frequency f1, P′ 11 is the signal power vector at frequency f1.

[0038] Furthermore, the covariance matrix of the f2 frequency is expressed by the following formula:

[0039]

[0040] in, is the covariance matrix of the f2 frequencies in the virtual array, is the signal power of the kth source at frequency f2, is the noise power at frequency f2;

[0041] The covariance matrix of the f2 frequency signal is quantized and expressed as follows:

[0042]

[0043] in, is the vectorized representation of the covariance matrix of the f2 frequency signal, is the joint steering matrix corresponding to the frequency f2, is the joint steering matrix of the kth source at frequency f2, a2(θ k ) is the array flow vector at frequency f2, P′ 22 is the signal power vector at frequency f2.

[0044] Furthermore, the covariance matrix between the frequencies f1 and f2 is expressed as follows:

[0045]

[0046] in, is the covariance matrix between the frequencies of f1 and f2, and are different weight coefficients, is the initial noise power of the kth source at different frequencies, is the complex noise power of the kth source at different frequencies;

[0047] The covariance matrix of the f1 and f2 frequencies is quantized as follows:

[0048]

[0049] in, is the vectorized covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2, P′ 12 is the signal power vector of frequencies f1 and f2.

[0050] Furthermore, the conjugate form of the covariance matrix of the f1 and f2 frequencies is expressed as follows:

[0051]

[0052] in, is the conjugate form of the covariance matrix of the frequencies f1 and f2, is the initial noise power of the kth source at different frequencies in conjugate form, is the complex noise power of the kth source at different frequencies in conjugate form;

[0053] The vectorized form of the conjugate form of the covariance matrix of the frequencies f1 and f2 is expressed as follows:

[0054]

[0055]

[0056] P′ 21 =(P′ 12 ) *

[0057] in, is the vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2 in conjugate form, P′ 12 is the signal power vector of frequencies f1 and f2 in conjugate form, (·) * is conjugate symmetry.

[0058] Furthermore, in step 2, the effective virtual array element set corresponding to the two mutually prime frequencies after motion synthesis can be obtained according to the aperture expansion model and expressed as follows:

[0059]

[0060] Among them, P MS P1 is the effective virtual array element set corresponding to the coprime frequency motion synthesis, self is the set of physical element positions of the subarray corresponding to the frequency f1, is the set of physical element positions of the subarray corresponding to the frequency f2, for The opposite number of is the second-order differential mutual array of P1 and P2.

[0061] Furthermore, the array flow matrix is ​​expressed by the following formula:

[0062]

[0063] P′=[(P′ 11 ) T ,(P′ 22 ) T ,(P′ 12 ) T ,(P′ 21 ) T ] T

[0064]

[0065] Among them, r is the array flow matrix, AMS is the set of joint steering matrices, P′ is the set of signal power vectors, is the noise power, and blkdiag is an operation to create a block diagonal matrix.

[0066] A second aspect of the present invention provides a direction of arrival estimation system based on a coprime frequency sparse array motion aperture expansion model, comprising:

[0067] A signal model module is used to detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies. The preset drone cluster array moves at a constant speed in a predetermined direction and receives reflected signals from multiple far-field targets. The reflected signals are sampled according to the two mutually prime frequencies to obtain a corresponding received signal matrix to facilitate the construction of an aperture expansion model.

[0068] An aperture expansion model module is used to obtain the covariance matrix corresponding to the two coprime frequencies and their vectorized representations, as well as the covariance matrix between the two coprime frequencies and their vectorized representations based on the received signal matrix, to complete the aperture expansion model construction and facilitate subsequent direction of arrival estimation;

[0069] The estimation module is used to construct the array flow pattern matrix according to the aperture expansion model and use the group lasso method to solve the array flow pattern matrix to obtain the wave direction of arrival estimation.

[0070] Beneficial effects of the present invention:

[0071] (1) This paper constructs a UAV cluster array based on a sparse array to obtain spatial information. The reflected signals are sampled at two coprime frequencies during the UAV cluster array's motion to obtain the corresponding received signal matrix, thereby acquiring both time and frequency domain information. Based on the received signal matrix, a new sparse array aperture expansion model is designed. This aperture expansion model can simultaneously integrate time-frequency-space information to achieve aperture expansion based on coprime frequency motion synthesis.

[0072] (2) The present invention provides a closed-form expression for the distribution of virtual array elements after coprime frequency motion synthesis (the array position set after coprime frequency motion synthesis), and theoretically proves that the aperture expansion effect is better than the existing aperture expansion method. Simulation experiments show that compared with the use of only one or two expansion methods, the method of the present invention achieves a significant improvement in the number of effective virtual array elements and the number of continuous virtual array elements.

[0073] (3) Based on the proposed aperture expansion model, this paper designs a DOA estimation method within the group-LASSO framework. Specifically, the array flow matrix is ​​solved using the group-LASSO method. Simulation experiments show that compared to using only one or two expansion methods, the proposed method achieves significant improvements in DOA estimation accuracy and the number of estimable sources. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 A flow chart of a direction of arrival estimation method based on a coprime frequency sparse array motion aperture expansion model provided by an embodiment of the present invention.

[0075] Figure 2 A schematic diagram of an information model provided by an embodiment of the present invention.

[0076] Figure 3 A schematic diagram of the distribution of virtual array elements before and after the movement of a drone cluster array provided by an embodiment of the present invention.

[0077] Figure 4 A schematic diagram of array element distribution before and after synthesis provided by an embodiment of the present invention.

[0078] Figure 5 A schematic diagram of spatial spectrum distribution provided by an embodiment of the present invention.

[0079] Figure 6 A schematic diagram illustrating how the root mean square error (RMS) varies with the signal-to-noise ratio (SNR) according to an embodiment of the present invention.

[0080] Figure 7 A schematic diagram showing how RMSE varies with the number of snapshots provided by an embodiment of the present invention.

[0081] Figure 8 This is an architectural diagram of a direction-of-arrival estimation system based on a coprime frequency sparse array motion aperture expansion model provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0082] To make the objectives, technical solutions, and advantages of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly described below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0083] Example 1

[0084] like Figure 1 As shown, the direction of arrival estimation method based on the coprime frequency sparse array motion aperture expansion model includes:

[0085] S101: Detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies, causing a preset drone cluster array to move at a constant speed along a predetermined direction and receive reflected signals from multiple far-field targets, and sample the reflected signals according to the two mutually prime frequencies to obtain a corresponding received signal matrix.

[0086] S102: Obtain the covariance matrices and vectorized representations corresponding to the two coprime frequencies and the covariance matrix between the two coprime frequencies according to the received signal matrix, and complete the construction of the aperture expansion model.

[0087] S103: Constructing an array flow pattern matrix according to the aperture expansion model, and solving the array flow pattern matrix using the group lasso method to obtain a direction of arrival estimate.

[0088] The present invention samples the reflected signal at coprime frequencies to obtain the corresponding received signal matrix. It then integrates time-frequency-space information to design an aperture expansion model. Based on the aperture expansion model, it constructs an array flow pattern matrix, which is then solved to obtain a direction of arrival estimate. This invention also integrates time-frequency-space information to perform aperture expansion, thereby improving the accuracy and degrees of freedom of sparse arrays.

[0089] Example 2

[0090] Based on the above embodiment, the present invention proposes an information model corresponding to step S101, which specifically includes:

[0091] like Figure 2 As shown in Figure 1, N drones each carry one antenna element and corresponding receiving and synchronization equipment to form a UAV array. The spacing between the UAV array elements is d0, and the target is detected using a single-frequency detection signal with two mutually prime frequencies f1 and f2 (f2>f1). The mutually prime frequency relationship is defined as follows:

[0092] f2=ηf1

[0093] Where η is the ratio of the mutually prime frequencies.

[0094] Correspondingly, the relationship between the frequency and the wavelength can be described as:

[0095] ηλ2=λ1

[0096] Among them, λ1 is the wavelength of frequency f1, and λ2 is the frequency f2.

[0097] For the convenience of expression, the reference interval d of the array elements in the UAV array is defined as u =λ2 / 2, the physical array element spacing is d0=Wd u , where W is a positive integer.

[0098] The UAV array moves along the axis at a constant speed v. During the movement, it detects the reflected signals of K far-field targets with incident angles {θ1,θ2,...,θ KSince the UAV array's motion distance is essentially negligible compared to the target's distance from the receiving array, the angle at which the reflected signal enters the receiving array can be assumed to remain unchanged. The UAV array completes two samplings during its motion, with each sampling taking Q snapshots. The interval between the two samplings is τ, and the distance between the two samplings is D = vτ, where D is the distance between the two samplings.

[0099] Assume that the lth sampling f i The reflected signal of frequency can be expressed as Then the f1 frequency component in the first sampling signal during the motion process can be expressed as:

[0100]

[0101] in, is the f1 frequency component in the first sampling signal, ρ 1,k (t) is the complex reflection coefficient of the kth target at frequency f1, which can be considered to be independently distributed under different frequencies and receiving array elements, and the channel state can be considered to be unchanged in the short time of each sampling; is the noise component in the first sampling signal corresponding to the frequency f1, which can be regarded as a Gaussian distribution. is the Doppler shift generated by the array motion at the frequency f1, a1(θ k ) is the array flow pattern vector at frequency f1.

[0102]

[0103] Writing the above array response equation in matrix form yields:

[0104]

[0105] A1=[a1(θ1),a1(θ2),…,a1(θ K )]

[0106]

[0107] Among them, A1 is the array flow vector set at the frequency f1 at the first sampling, is the set of intermediate values, is the middle value, is the noise component in the first sampling signal corresponding to the frequency f1.

[0108] Similarly, the f1 frequency component of the second sampling signal can be expressed as:

[0109]

[0110] Let b1(θk )=exp(-j2πf1Dd u sinθ k / c)a1(θ k )

[0111] B1=[b1(θ1),b1(θ2),…,b1(θ K )]

[0112]

[0113] The above formula can be rewritten as:

[0114]

[0115] in, is the f1 frequency component in the second sampling signal, τ is the interval between two samplings, t is the time, j is the imaginary unit, B1 is the array flow vector set at the f1 frequency during the second sampling, is the set of intermediate values ​​of the second sampling, is the middle value of the second sampling, is the noise component in the second sampling signal corresponding to the frequency f1, and b1(θ1) is the array flow vector at the frequency f1 during the second sampling.

[0116] It can be noted that due to the narrowband nature of the received signal, the time delay can be compensated by phase shift, that is:

[0117]

[0118] Therefore, the second sampling signal of the f1 frequency component is phase-shift compensated and then combined with the first sampling signal to obtain the received signal matrix of the f1 frequency component, which can be expressed as follows:

[0119]

[0120] A s,1 =[a s,1 (θ1),a s,1 (θ2),…,a s,1 (θ K )]

[0121] a s (θ k )=[a1 T (θ k ),b1 T (θ k )] T

[0122] Where y1(t) is the received signal matrix of the f1 frequency component, f1 is the frequency, is the f1 frequency component in the first sampling signal, is the f1 frequency component in the second sampling signal, τ is the interval between two samplings, t is the time, j is the imaginary unit, A1 is the array flow pattern vector set at the f1 frequency during the first sampling, and B1 is the array flow pattern vector set at the f1 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f1, is the noise component in the second sampling signal corresponding to the frequency f1, A S,1 is the array flow vector set at the frequency f1 during sampling, and n1(t) is the noise component in the sampling signal at the frequency f1.

[0123] Similarly, the received signal matrix after two motion sampling and phase shift compensation of the f2 frequency component signal can be expressed as

[0124]

[0125] Where y2(t) is the received signal matrix of the f2 frequency component, f2 is the frequency, is the f2 frequency component in the first sampling signal, is the f2 frequency component in the second sampling signal, A2 is the array flow vector set at the f2 frequency during the first sampling, and B2 is the array flow vector set at the f2 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f2, is the noise component in the second sampling signal at the frequency f2, A S,2 is the array flow vector set at the frequency f2 during sampling, and n2(t) is the noise component in the sampling signal at the frequency f2.

[0126] Example 3

[0127] Based on the above embodiments, the present invention proposes an aperture expansion model, which specifically includes:

[0128] like Figure 3 As shown, the received signal matrix corresponding to the f1 frequency component, the array position set before and after motion synthesis at the f1 frequency (in d u As units) can be expressed as:

[0129] P1={nW / η,0≤n≤N-1}

[0130] P1 mov =P1∪{D / η+P1}

[0131] Where P1 is the array position set before movement corresponding to the received signal matrix of the frequency component f1, n is the nth UAV, W is a positive integer, N is the number of UAVs in the UAV cluster array, η is the coprime frequency ratio, P1 mov is the array position set after motion synthesis corresponding to the received signal matrix of the f1 frequency component.

[0132] Similarly, the array position sets before and after motion synthesis at frequency f2 can be expressed as:

[0133] P2={nW,0≤n≤N-1}

[0134]

[0135] Among them, P2 is the array position set before movement corresponding to the received signal matrix of the f2 frequency component, is the array position set after motion synthesis corresponding to the received signal matrix of the f2 frequency component.

[0136] The covariance matrix (self-lag) of the f1 frequency is expressed as follows:

[0137]

[0138] in, is the covariance matrix of the f1 frequency in the virtual array, E is the expected value, is the covariance matrix, is the noise power at frequency f1, I 2N is a 2N column vector of all ones, is the signal power of the kth source at the frequency f1, a s,1 (θ k ) is the steering vector of the kth source, (·) H is the conjugate transpose.

[0139] Similarly, the self-lag of frequency f2 can be expressed as:

[0140]

[0141] in, is the covariance matrix of the f2 frequencies in the virtual array, is the signal power of the kth source at frequency f2, is the noise power at frequency f2.

[0142] The covariance matrix (cross-lag) of the frequencies f1 and f2 can be expressed as follows:

[0143]

[0144] in, is the covariance matrix between the frequencies of f1 and f2, and are different weight coefficients, is the initial noise power of the kth source at different frequencies, is the complex noise power of the kth source at different frequencies.

[0145] Similarly, the conjugate form of the covariance matrix can be obtained It is specifically expressed as follows:

[0146]

[0147] in, is the conjugate form of the covariance matrix of the frequencies f1 and f2, is the initial noise power of the kth source at different frequencies in conjugate form, is the complex noise power of the kth source at different frequencies in conjugate form.

[0148] The covariance matrix of the f1 frequency signal is vectorized as follows:

[0149]

[0150] in, is the vectorized representation of the covariance matrix of the f1 frequency signal, is the joint steering matrix corresponding to the frequency f1, is the joint steering matrix of the kth source at frequency f1, a1(θ k ) is the array flow vector at the frequency f1, and P1′1 is the signal power vector at the frequency f1.

[0151] The vectorized representation of the covariance matrix of the f1 frequency signal can be equivalently represented as a set of effective virtual array elements on the virtual array domain from the perspective of the array element. The effective virtual array element set on the virtual array domain corresponding to the vectorized representation of the covariance matrix of the f1 frequency signal can be represented as:

[0152] P 11 =P1 self ∪{P1 self +D / η}∪{P1 self -D / η}

[0153] Among them P1 self is the second-order differential mutual array of P1.

[0154] The covariance matrix of the f2 frequency signal is vectorized as follows:

[0155]

[0156] in, is the vectorized representation of the covariance matrix of the f2 frequency signal, is the joint steering matrix corresponding to the frequency f2, is the joint steering matrix of the kth source at frequency f2, a2(θ k ) is the array flow vector at frequency f2, P′ 22 is the signal power vector at frequency f2.

[0157] From the perspective of array elements, the covariance matrix vectorization of the f2 frequency signal can be equivalently expressed as a set of effective virtual array elements on the virtual array domain. The effective virtual array element set on the virtual array domain corresponding to the covariance matrix vectorization of the f2 frequency signal can be expressed as:

[0158]

[0159] in is the second-order differential mutual array of P2.

[0160] The covariance matrix (cross-lag) between different frequencies can be vectorized as follows:

[0161]

[0162] in, is the vectorized covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2, P′ 12 is the signal power vector of frequencies f1 and f2.

[0163] From the perspective of array elements, the vectorization of the covariance matrix between different frequencies can be equivalently expressed as a set of effective virtual array elements on the virtual array domain. The corresponding set of effective virtual array elements on the virtual array domain of the vectorization of the covariance matrix between different frequencies can be expressed as:

[0164]

[0165] in is the second-order differential mutual array of P1 and P2.

[0166] The vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2 is given by the following formula:

[0167]

[0168] P′ 21 =(P′ 12 ) *

[0169] in, is the vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2 in conjugate form, P′ 12 is the signal power vector of frequencies f1 and f2 in conjugate form, (·) * is conjugate symmetry.

[0170] From the perspective of array elements, the vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2 can be equivalently expressed as a set of effective virtual array elements on the virtual array domain. The effective virtual array element set on the virtual array domain corresponding to the vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2 can be expressed as:

[0171]

[0172] in is the second-order differential mutual array of P1 and P2, and They are opposite numbers.

[0173] because and It does not have a symmetrical structure, so the set of mutual difference components cannot be simply expressed. In order to verify the effect of the aperture expansion model of the present invention, the effective virtual array element set corresponding to the coprime frequency motion synthesis of the aperture expansion model is given for verification. The effective virtual array element set corresponding to the coprime frequency motion synthesis can be finally expressed as:

[0174]

[0175] Among them, P MS P1 is the effective virtual array element set corresponding to the coprime frequency motion synthesis, self is the set of physical element positions of the subarray corresponding to the frequency f1, is the set of physical element positions of the subarray corresponding to the frequency f2.

[0176] As an implementation method, taking W=20, η=2 / 5, D=5 as an example, the aperture synthesis effect of the aperture expansion model integrating time-frequency-space information is demonstrated, such as Figure 4 (a) shows the physical array and motion distribution before synthesis, as shown in Figure 4 (b) shows the distribution of all virtual array elements after synthesis.

[0177] It can be seen that using a sparse array of drones with 5 elements, the number of continuous virtual elements after one coprime frequency motion synthesis reaches 129, and the number of effective virtual elements reaches 149, which is a huge improvement compared to single array motion synthesis or coprime frequency synthesis, laying the foundation for more accurate DOA estimation.

[0178] Example 4

[0179] Based on the above embodiment, the present invention proposes a method for solving the array flow matrix, which is as follows:

[0180] The array flow matrix is ​​expressed as follows:

[0181]

[0182] P′=[(P′ 11 ) T ,(P′ 22 ) T ,(P′ 12 ) T ,(P′ 21 ) T ] T

[0183]

[0184] Among them, r is the array flow matrix, A MS is the set of joint steering matrices, P′ is the set of signal power vectors, is the noise power, blkdiag is to create a block diagonal matrix operation, 0 2N Represents a 2N×2N matrix of all zeros.

[0185] The four column vectors that make up the array flow matrix The angular distributions of the corresponding K target sources are consistent, so the DOA estimation problem after coprime frequency motion synthesis can be viewed as a group sparse problem of locating non-zero entries in the angular search space. The present invention uses the group-LASSO method to solve it. The sparse representation of the array flow matrix is ​​as follows:

[0186]

[0187] in, is the steering moment matrix, G is the number of angle grid searches, is the concatenation matrix, is the noise concatenation matrix, is the signal space distribution vector under grid search, 4N 2 A column vector of all ones, 2×8N 2 The all-zero matrix, Expand vector for grid search.

[0188] The first 4G items represent the source angle distribution of 2 samples at 2 frequencies, and the source angle distribution contained in different samples is the same. Therefore, the angle estimation result can be transformed into the following optimization problem:

[0189]

[0190] in, is the direction of arrival estimation result, where each column represents the grid search under a certain frequency and a certain sampling, such as ξ(P 0 )∈C 4G×1 , ξ(·) is the l2 norm operator that returns each row of the input matrix, and ε is the upper bound of the error.

[0191] The optimization problem can be equivalently expressed as a convex optimization problem, which is expressed as follows:

[0192]

[0193] Among them, λ is the regularization parameter.

[0194] The convex optimization problem is computationally simplified, and the final spatial spectrum estimation result can be expressed as P 0 =ξ(P 0 ).

[0195] Solve the spatial spectrum estimation results and get P 0 , P 0 This is the direction of arrival estimation result.

[0196] Example 5

[0197] Based on the above embodiments, the present invention proposes a comparative experiment of the aperture expansion model, which specifically includes:

[0198] To demonstrate the superiority of the present invention, the DOF (degrees of freedom) performance of the proposed aperture expansion model (i.e., an aperture synthesis method integrating time-frequency-space information), the prior art sparse uniform linear array direction of arrival estimation based on coprime frequency continuous wave signals (i.e., a sparse array coprime frequency synthesis method integrating frequency-space information), and the prior art sparse array motion-based direction of arrival estimation (i.e., a moving sparse array aperture synthesis method integrating time-space information) were evaluated, assuming the same number of array elements. The number of physical array elements for all three methods was set to 5. For the proposed method, the coprime frequencies W = 20, η = 2 / 5, and D = 5 were set. For the sparse uniform linear array direction of arrival estimation based on coprime frequency continuous wave signals, the coprime frequency ratio η = 2 / 5 and W = 5 were set. For the sparse uniform linear array direction of arrival estimation using sparse array motion, the f2 frequency component was selected, W = 3, and D = 1. This results in the SULA (sparse uniform linear array) formation with the best single-motion aperture synthesis. In the experiments, the group-LASSO method was used for DOA estimation, and the regularization parameter was set to 0.25.

[0199] The number of targets is fixed to 20, evenly distributed in the [-60, 60] angle range. The search interval is set to 0.1. The received signal-to-noise ratio of all signals is set to 10dB. The sampling snapshot of each movement is set to 500. The experimental results can be observed as follows Figure 5 As shown, Figure 5 (a) is the spatial spectrum distribution of the direction of arrival estimation of a sparse uniform linear array based on a coprime frequency continuous wave signal, where the number of sources (targets) is K = 20. Figure 5 (b) is the spatial spectrum distribution of the direction of arrival estimated using sparse array motion, where K = 20, Figure 5 (c) is the spatial spectrum distribution of the present invention, where K = 20, Figure 5 (d) is the spatial spectrum distribution of the present invention, where the number of sources (targets) K=30.

[0200] from Figure 5 As can be seen from (a), (b), and (c), the synthetic array corresponding to the aperture expansion model of the present invention can clearly distinguish all targets, while the number of non-negative virtual array elements generated by the two existing technologies is 19 and 14 respectively, so both cannot accurately distinguish all targets. To further verify the degree of freedom of the proposed array, the number of targets is increased to 30, as shown in the figure. Figure 5 As shown in (d), all targets can still be clearly distinguished in the present invention.

[0201] Next, we analyze the variation of the root mean square error (RMS) of the present invention with the signal-to-noise ratio (SNR), and compare it with the two methods proposed in the prior art. The parameters of each synthetic array and the signal-to-noise ratio of the signal source are consistent with the above experiment. The number of targets is set to 6 and evenly distributed in the interval [-60°, 60°]. The number of Monte Carlo runs is 200. The SNR varies from -5 to 15 with a step size of 2, and the SNR of all target signals is consistent. The curve of the root mean square error with the SNR is shown in Figure 2. Figure 6 As shown in FIG. 1 , it can be seen that the present invention has the best RMSE (root mean square error) performance.

[0202] Finally, the variation of the root mean square error with the number of snapshots was simulated. The signal-to-noise ratio was set to 10dB, and the number of snapshots varied from 100 to 1000, with a step size of 50 in the initial stage and 100 in the subsequent stages. The parameters of the synthetic array, the signal source parameters, and the number of Monte Carlo simulations were kept consistent with the above experiments. Figure 7 It can be seen that the aperture expansion model constructed by the present invention has the best RMSE performance compared with the two comparison methods.

[0203] Example 6

[0204] Based on the above embodiments, Figure 8 As shown, the present invention proposes a direction of arrival estimation system based on a coprime frequency sparse array motion aperture expansion model, comprising:

[0205] The signal model module is used to detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies, so that a preset drone cluster array moves at a constant speed in a predetermined direction and receives reflected signals from multiple far-field targets. The reflected signals are sampled according to the two mutually prime frequencies to obtain a corresponding received signal matrix.

[0206] The aperture expansion model module is used to obtain the covariance matrix corresponding to the two coprime frequencies and its vectorized representation and the covariance matrix between the two coprime frequencies and its vectorized representation according to the received signal matrix, so as to complete the aperture expansion model construction.

[0207] The estimation module is used to construct the array flow pattern matrix according to the aperture expansion model and use the group lasso method to solve the array flow pattern matrix to obtain the wave direction of arrival estimation.

[0208] It should be noted that the wave direction estimation system based on the coprime frequency sparse array motion aperture expansion model provided in the embodiment of the present invention is to realize the above-mentioned wave direction estimation method based on the coprime frequency sparse array motion aperture expansion model. Its specific functions can be referred to the above-mentioned method embodiments and will not be repeated here.

[0209] In summary, the present invention constructs a drone cluster array based on a sparse array to obtain spatial information, and samples the reflected signal according to two coprime frequencies when the drone cluster array moves to obtain the corresponding received signal matrix, thereby obtaining time domain and frequency domain information. A new sparse array aperture expansion model is designed based on the received signal matrix, which can simultaneously integrate time-frequency-space information and complete aperture expansion based on coprime frequency motion synthesis. The present invention provides a closed-form expression of the distribution of virtual array elements after coprime frequency motion synthesis (the array position set after coprime frequency motion synthesis), and theoretically proves that the aperture expansion effect is better than the existing aperture expansion method. Simulation experiments show that compared with the use of one or two expansion methods alone, the method of the present invention has achieved significant improvement in the number of effective virtual array elements and the number of continuous virtual array elements. Based on the proposed aperture expansion model, the present invention designs a DOA estimation method based on the group-LASSO framework, that is, the array flow matrix is ​​solved by the group-LASSO (group lasso) method. Simulation experiments show that compared with the use of only one or two extension methods, the method of the present invention has achieved significant improvements in DOA estimation accuracy and the number of estimable sources.

[0210] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A method for estimating the direction of arrival based on a coprime frequency sparse array motion aperture expansion model, characterized in that: include: Step 1: Detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies. Move the preset drone cluster array at a constant speed along a predetermined direction and receive reflected signals from multiple far-field targets. Sample the reflected signals according to the two mutually prime frequencies to obtain a corresponding received signal matrix. Step 2: According to the received signal matrix, the covariance matrix corresponding to the two coprime frequencies and its vectorized representation and the covariance matrix between the two coprime frequencies and its vectorized representation are obtained to complete the aperture expansion model construction; Step 3: Construct the array flow pattern matrix based on the aperture expansion model, and use the group lasso method to solve the array flow pattern matrix to obtain the wave direction of arrival estimation.

2. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 1, wherein: The sampling of the reflected signal according to the two mutually prime frequencies to obtain the corresponding received signal matrix specifically includes: The reflected signal is sampled twice according to the frequency f1 and the frequency f2, and the phase shift of the second sampled signal is compensated. The compensated signal and the first sampled signal are combined to obtain the received signal matrix of the frequency component f1 and the received signal matrix of the frequency component f2; wherein the frequency f1 and the frequency f2 are coprime frequencies; The received signal matrix of the f1 frequency component is expressed as follows: Where y1(t) is the received signal matrix of the f1 frequency component, f1 is the frequency, is the f1 frequency component in the first sampling signal, is the f1 frequency component in the second sampling signal, τ is the interval between two samplings, t is the time, j is the imaginary unit, A1 is the array flow pattern vector set at the f1 frequency during the first sampling, and B1 is the array flow pattern vector set at the f1 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f1, is the noise component in the second sampling signal corresponding to the frequency f1, A S,1 is the array flow vector set at the frequency f1 during sampling, is the noise component in the sampled signal at the frequency f1; The received signal matrix of the f2 frequency component is expressed as follows: Where y2(t) is the received signal matrix of the f2 frequency component, f2 is the frequency, is the f2 frequency component in the first sampling signal, is the f2 frequency component in the second sampling signal, A2 is the array flow vector set at the f2 frequency during the first sampling, and B2 is the array flow vector set at the f2 frequency during the second sampling. is the set of intermediate values, is the noise component in the first sampling signal corresponding to the frequency f2, is the noise component in the second sampling signal at the frequency f2, A S,2 is the array flow vector set at the frequency f2 during sampling, and n2(t) is the noise component in the sampling signal at the frequency f2.

3. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 1, wherein: The array position sets before and after motion synthesis corresponding to the received signal matrices of the f1 frequency component and the f2 frequency component are expressed by the following formula: P1={nW / η,0≤n≤N-1} P2={nW,0≤n≤N-1} Where P1 is the array position set before movement corresponding to the received signal matrix of the frequency component f1, n is the nth UAV, W is a positive integer, N is the number of UAVs in the UAV cluster array, η is the coprime frequency ratio, is the array position set after motion synthesis corresponding to the signal matrix of the frequency component f1, D is the motion distance of the drone array before and after the two samplings, P2 is the array position set before motion corresponding to the signal matrix of the frequency component f2, is the array position set after motion synthesis corresponding to the received signal matrix of the f2 frequency component.

4. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 2, wherein: The covariance matrix of the f1 frequency is expressed as follows: in, is the covariance matrix of the f1 frequency in the virtual array, E is the expected value, is the covariance matrix, is the noise power at frequency f1, I 2N is a 2N column vector of all ones, is the signal power of the kth source at the frequency f1, a s,1 (θ k ) is the steering vector of the kth source, (·) H is the conjugate transpose; The covariance matrix of the f1 frequency signal is quantized and expressed as follows: in, is the vectorized representation of the covariance matrix of the f1 frequency signal, is the joint steering matrix corresponding to the frequency f1, is the joint steering matrix of the kth source at frequency f1, a1(θ k ) is the array flow vector at frequency f1, P′ 11 is the signal power vector at frequency f1.

5. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 4, characterized in that: The covariance matrix of the f2 frequency is expressed as follows: in, is the covariance matrix of the f2 frequencies in the virtual array, is the signal power of the kth source at frequency f2, is the noise power at frequency f2; The covariance matrix of the f2 frequency signal is quantized and expressed as follows: in, is the vectorized representation of the covariance matrix of the f2 frequency signal, is the joint steering matrix corresponding to the frequency f2, is the joint steering matrix of the kth source at frequency f2, a2(θ k ) is the array flow vector at frequency f2, P′ 22 is the signal power vector at frequency f2.

6. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 4, wherein: The covariance matrix between the f1 and f2 frequencies is expressed as follows: in, is the covariance matrix between the frequencies of f1 and f2, and are different weight coefficients, is the initial noise power of the kth source at different frequencies, is the complex noise power of the kth source at different frequencies; The covariance matrix of the f1 and f2 frequencies is quantized as follows: in, is the vectorized covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2, P′ 12 is the signal power vector of frequencies f1 and f2.

7. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 6, wherein: The conjugate form of the covariance matrix of the f1 and f2 frequencies is expressed as follows: in, is the conjugate form of the covariance matrix of the frequencies f1 and f2, is the initial noise power of the kth source at different frequencies in conjugate form, is the complex noise power of the kth source at different frequencies in conjugate form; The vectorized form of the conjugate form of the covariance matrix of the frequencies f1 and f2 is expressed as follows: in, is the vectorization of the conjugate form of the covariance matrix of the frequencies f1 and f2, is the joint steering matrix of the kth source at frequencies f1 and f2 in conjugate form, P′ 12 is the signal power vector of frequencies f1 and f2 in conjugate form, (·) * is conjugate symmetry.

8. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 3, wherein: In the step 2, the effective virtual array element set corresponding to the two mutually prime frequencies after motion synthesis can be obtained according to the aperture expansion model and expressed as follows: Among them, P MS is the effective virtual array element set corresponding to the coprime frequency motion synthesis, is the set of physical element positions of the subarray corresponding to the frequency f1, is the set of physical element positions of the subarray corresponding to the frequency f2, for The opposite number of is the second-order differential mutual array of P1 and P2.

9. The method for estimating direction of arrival based on a coprime frequency sparse array motion aperture expansion model according to claim 7, wherein: The array flow matrix is ​​expressed as follows: Among them, r is the array flow matrix, A MS is the set of joint steering matrices, P′ is the set of signal power vectors, is the noise power, and blkdiag is an operation to create a block diagonal matrix.

10. A direction of arrival estimation system based on a coprime frequency sparse array motion aperture expansion model, characterized in that: include: A signal model module is used to detect multiple far-field targets using a single-frequency detection signal with two mutually prime frequencies, causing a preset drone cluster array to move at a constant speed in a predetermined direction and receive reflected signals from multiple far-field targets, and sampling the reflected signals according to the two mutually prime frequencies to obtain a corresponding received signal matrix; An aperture expansion model module is used to obtain the covariance matrix corresponding to the two coprime frequencies and their vectorized representations, as well as the covariance matrix between the two coprime frequencies and their vectorized representations according to the received signal matrix, to complete the aperture expansion model construction; The estimation module is used to construct the array flow pattern matrix according to the aperture expansion model and use the group lasso method to solve the array flow pattern matrix to obtain the wave direction of arrival estimation.

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