Metasurface-assisted multi-channel super-resolution DOA estimation method based on off-grid sparse Bayesian learning

Abstract

The invention discloses a metasurface-assisted multi-channel super-resolution DOA (direction of arrival) estimation method based on off-grid sparse Bayesian learning. The implementation mode of the method is as follows: giving an incoming wave signal, and establishing a signal model for the incoming wave signal to reach a metasurface array element; according to the transmission amplitude and the transmission phase of the transmission type metasurface, establishing a signal model of an incoming wave signal transmitting out of the metasurface; establishing a signal model of a receiving end of the receiving antenna according to the position relationship between the receiving antenna and the metasurface; grid division is carried out on the airspace, and a sparse flow pattern array is constructed; according to a Bayesian learning algorithm, carrying out hyper-parameter initialization by utilizing the collected signal sample value, and carrying out iteration on hyper-parameters according to the idea of an expectation maximization algorithm; and finally, constructing a power spectrum function, and visualizing a calculation result to obtain a direction of arrival estimation result. Compared with an existing DOA estimation method, the method has better estimation performance and higher resolution under the condition that the number of receiving channels is the same. According to the invention, high-precision target estimation and super-resolution capability are guaranteed, the number of receiving channels is reduced, the complexity of a receiving system is reduced, and the system cost is saved.

Description

Technical Field

[0001] The present invention belongs to the field of signal processing, and specifically relates to a multi-channel super-resolution DOA estimation method based on a transmission metasurface. Background Art

[0002] Direction of arrival (DOA) estimation, a key topic in array signal processing, has attracted widespread attention. Its primary focus is determining the angular direction of a signal arriving at a receiving array. Related technologies have found widespread application in wireless communication systems, underwater acoustic communications, radar sensing, source localization, integrated sensing, and communications. DOA estimation algorithms have a long history: the classic beamforming method, a beam scanning method based on Fourier analysis, was developed in the early 1960s and is the earliest research approach. However, this method has certain limitations: its resolution is limited by the Rayleigh criterion. Therefore, to improve resolution, it is necessary to increase both the aperture and the number of metasurface elements in the array. However, the high economic cost associated with this enhancement has limited its practical application. In 1986, ROSchMit proposed the Multiple Signal Classification (MUSIC) algorithm, which pioneered the study of subspace-based array signal processing algorithms. This breakthrough exceeded the Rayleigh limit and marked the beginning of super-resolution algorithms. The MUSIC algorithm has remained popular over the years, and many improved algorithms have been proposed, furthering the development of DOA estimation.

[0003] However, improving accuracy and performance under varying DOA estimation conditions remains a long-standing and challenging open problem. In recent years, with the development of compressed sensing, DOA estimation methods based on sparse signal processing have attracted widespread attention. The core idea of ​​this approach is to transform DOA estimation into a sparse signal reconstruction problem. Some researchers have used sparse Bayesian learning to solve DOA estimation for arbitrary linear arrays. Because these grid-based models assume that the signal's DOAs lie precisely on a predefined grid, the actual angles of arrival may not fall precisely on the grid points, leading to grid mismatch and reducing estimation accuracy. Therefore, an off-grid model based on polynomial expansions has been proposed. When combined with a sparse Bayesian algorithm, relatively impressive results can be achieved. However, these approaches rely on traditional antenna arrays with multiple radio frequency channels to receive signals, which is very complex in hardware implementation. Summary of the Invention

[0004] In view of the above technical background, the present invention proposes a metasurface-assisted multi-channel super-resolution DOA estimation method based on off-grid sparse Bayesian learning, which reduces the number of receiving channels while ensuring a certain estimation performance.

[0005] The technical solution for achieving the purpose of the present invention is:

[0006] A multi-channel super-resolution DOA estimation method based on off-grid sparse Bayesian learning and metasurface assistance is proposed. The specific steps are as follows:

[0007] Step 1: Establish a signal model for the incoming signal reaching the metasurface array element;

[0008] Step 2: Based on the transmission amplitude and transmission phase of the transmissive metasurface, a signal model of the incoming signal transmitted through the metasurface is established;

[0009] Step 3: Based on the positional relationship between the receiving antenna and the metasurface, a signal model of the receiving end of the receiving antenna is established;

[0010] Step 4: Grid the airspace and construct a sparse flow pattern array;

[0011] Step 5: Based on the Bayesian learning algorithm, the collected signal sample values ​​are used to initialize the hyperparameters, and the hyperparameters are iterated according to the expected maximum algorithm to obtain the grid error coefficient;

[0012] Step 6: Construct the power spectrum function and visualize the calculation results based on the grid points obtained in step 5. Then, obtain the angle of arrival estimation result based on the peak value of the spectrum function.

[0013] Compared with the prior art, the present invention has the following notable features:

[0014] (1) The multi-channel super-resolution DOA estimation method assisted by the sparse Bayesian learning proposed in this invention improves the performance of the existing DOA estimation model;

[0015] (2) The model proposed in this paper is highly robust and can adapt to harsh environments such as low snapshot count and high noise interference;

[0016] (3) The present invention has a simple structure and is easy to implement.

[0017] The objectives and other advantages of the present invention may be realized and obtained through the structures particularly pointed out in the written description, claims, and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a flow chart of the multi-channel super-resolution DOA estimation method based on sparse Bayesian learning assisted by the super-surface proposed in the present invention.

[0019] Figure 2 This is a model diagram of the multi-channel super-resolution DOA estimation method based on sparse Bayesian learning assisted by the supersurface proposed in the present invention.

[0020] Figure 3This is a single simulation diagram of the multi-channel super-resolution DOA estimation method assisted by the sparse Bayesian learning-based metasurface proposed in the present invention compared with other methods.

[0021] Figure 4 This is a performance comparison chart under different numbers of snapshots when off-grid operation is performed in an example of the present invention.

[0022] Figure 5 This is a performance comparison chart under different signal-to-noise ratios when off-grid operation is performed in an example of the present invention.

[0023] Figure 6 This is a comparison chart of the root mean square error of the algorithm in the example of the present invention and the existing DOA estimation method as the number of snapshots changes.

[0024] Figure 7 This is a comparison chart of the root mean square error of the algorithm in the example of the present invention and the existing DOA estimation method as the signal-to-noise ratio changes.

[0025] Figure 8 This is a comparison chart of the root mean square error of the algorithm in the example of the present invention and the existing DOA estimation method as a function of the angular interval. DETAILED DESCRIPTION

[0026] It is easy to understand that, based on the technical solution of the present invention, a person of ordinary skill in the art can imagine various embodiments of the present invention without changing the essential spirit of the present invention. Therefore, the following specific embodiments and drawings are merely illustrative of the technical solution of the present invention and should not be regarded as the whole of the present invention or as a limitation or limitation of the technical solution of the present invention. On the contrary, the purpose of providing these embodiments is to enable those skilled in the art to understand the present invention more thoroughly. The preferred embodiments of the present invention are described in detail below in conjunction with the accompanying drawings, wherein the accompanying drawings constitute a part of this application and are used together with the embodiments of the present invention to illustrate the innovative concept of the present invention.

[0027] A multi-channel super-resolution DOA estimation method based on off-grid sparse Bayesian learning and metasurface assistance, comprising:

[0028] Step 1: Given an incoming wave signal, establish a signal model of the incoming wave signal reaching the metasurface array element;

[0029] Assume that there exists a K ] are projected onto the N-unit metasurface. The total number of receiving antennas is M. Therefore, the signal received by the n-th metasurface unit is as follows:

[0030]

[0031] Where β = 2π / λ is the wave number, β is the wavelength, D is the spacing between metasurfaces, and s k(t) is the kth incident signal, θ k is the incident angle of the kth signal, n n (t) is the Gaussian white noise generated by the nth metasurface unit.

[0032] Define the flow array A, signal vector s(t), and noise vector n'(t) as:

[0033]

[0034] Then the vector form of the signal received by the metasurface array is expressed as:

[0035] x(t)=Αs(t)+n′(t)

[0036] That is, the signal model of the incoming wave signal reaching the metasurface array element

[0037] Step 2: Based on the transmission amplitude and transmission phase of the transmissive metasurface, a signal model of the incoming signal transmitted through the metasurface is established; wherein the signal of the incoming signal transmitted through the nth metasurface unit is specifically:

[0038]

[0039] Among them, ξ n is the amplitude displacement of the nth metasurface unit, φ n is the phase shift of the nth metasurface unit.

[0040] Step 3: Based on the positional relationship between the receiving antenna and the metasurface, a signal model of the receiving end of the receiving antenna is established;

[0041] The signal received by the mth receiving antenna is expressed as follows:

[0042]

[0043] Among them, y n (t) represents the signal transmitted through the nth metasurface array, d n,m represents the distance between the nth metasurface and the mth receiving antenna, and w(t) represents the noise caused by the modulation of the metasurface unit.

[0044] To further simplify the model, the signal received by the mth antenna can be obtained in the following form:

[0045]

[0046] Where, ω m represents the phase shift caused by the wave path when the signal is transmitted from the metasurface to the receiving antenna; Ξ represents the amplitude and phase characteristics of the metasurface, and x(t) represents the signal received by the metasurface array. Specifically, they are:

[0047]

[0048] x(t)=[x1(t),x2(t),...,x N (t)] T

[0049] Let r(t)=[r1(t),r2(t),...,r M (t)] T , r m (t) is the signal received by the mth antenna. Therefore, the signal vector received by M antennas can be expressed as:

[0050]

[0051] Where n(t) = ΩΞn'(t) + w(t) represents the total noise generated by the MSA system during the entire process, and r(t) is the signal at the receiving end of the receiving antenna.

[0052] Step 4: Divide the airspace into grids and construct a sparse flow pattern array:

[0053] Direction of Arrival Range Follow the Grid Divide the grid into two parts, where the number of grids is G>>M>K, and the grid spacing is

[0054] remember This is a multivariate vector function whose value depends on θ k , D and other factors. If there is a true direction Where k∈{1,...,K}, Taylor approximation expansion can be performed:

[0055]

[0056] in Indicates the closest angle to the true The grid points, g k ∈{1,...,G},

[0057] Remember the grid matrix in Therefore, for the steering vector, Taylor approximation expansion can be performed:

[0058]

[0059] After accounting for the amplitude control and phase shift effects on the signal reaching the metasurface, the grid model becomes:

[0060]

[0061] So the received signal can be rewritten as:

[0062] r(t)=Ψ(β)s(t)+n(t)

[0063] where s(t)=[s1(t),...,s G (t)] T ,n(t)=[n1(t),...,n M (t)] T Only when the angles corresponding to the grid points are expanded is there an estimate and Otherwise g (t) = β g = 0. It is worth noting that Represents the grid closest to the true incident angle, from which g can be inferred k Indicates the corresponding index.

[0064] Assuming the number of snapshots is T, the above analytical expression can be written in the form of an observation matrix:

[0065] R=Ψ(β)S+N

[0066] where it is defined

[0067]

[0068] It is worth noting that since G>>K, the matrix S is sparse.

[0069] Step 5: Based on the Bayesian learning algorithm, the collected signal sample values ​​are used to initialize the hyperparameters, and the hyperparameters are iterated according to the idea of ​​the expected maximum algorithm to obtain the grid error coefficient:

[0070] Assume that the noise N follows a Gaussian distribution:

[0071]

[0072] where α0=σ -1 Represents the noise accuracy, σ represents the variance in the formula. Assume that α0 is unknown and it obeys the Γ distribution, that is:

[0073] p(α0; c, d) = Γ(α0; c, d)

[0074] Where the hyperparameters c,d→0.

[0075] Based on the operational properties of Gaussian distribution, the probability density function of the received signal can be obtained as:

[0076]

[0077] Assume that the signals S between snapshots are independent of each other and adopt a two-stage hierarchical prior p(S|ρ)=∫p(S|α)p(α;ρ)dα. And ρ is a hyperparameter representing the power of S, where Υ=diag(α) and:

[0078]

[0079]

[0080] According to the Bayesian formula, the posterior probability can be expressed as:

[0081] p(S,α0,α,β|R)=p(S|R,α0,α,β)p(α0,α,β|R)

[0082] To solve the above probability formula, we need to consider these two items separately. First, for the first item, it is the posterior distribution of S, and its form satisfies the Gaussian distribution, that is:

[0083]

[0084] μ(t)=α0ΣΨ H r(t)

[0085] Σ=(α0Ψ H Ψ+Υ -1 ) -1

[0086] To calculate μ(t) and Σ, we need to determine the values ​​of the hyperparameters α0, α, and β. This is achieved by maximizing the conditional probability p(α0,α,β|R). Since the joint probability can be expressed as p(R,α0,α,β)=p(α0,α,β|R)p(R), and p(R) is independent of the hyperparameters, the problem is transformed into maximizing the joint probability p(R,α0,α,β). Considering S as a hidden variable, this joint probability is proportional to p(S,R,α0,α,β), and its specific expression can be derived from the total probability formula:

[0087] p(S,R,α0,α,β)=p(R|S,α0,β)p(S|α)p(α)p(β)p(α0)

[0088] Then the maximum likelihood estimation method is used to process the above formula.

[0089] The update expression of the hyperparameter is:

[0090]

[0091] in:

[0092]

[0093] Furthermore, estimating β is equivalent to maximizing p(R|S,α0,β)p(β). To do this, we need to make an additional assumption, namely At this time, the maximum likelihood estimate of β is further equivalent to maximizing E{lnp(R|S,α0,β)}, so the problem can be transformed into:

[0094]

[0095] in:

[0096]

[0097] This is a convex function, and the point where the partial derivative is zero is the point where the minimum value is located:

[0098] β new =Q -1 v

[0099] Step 6: Finally, construct the power spectrum function and visualize the calculation results to obtain the angle of arrival estimation result:

[0100] According to the above steps, only the relevant parameters of the signal are obtained. However, if the angle of the incoming signal needs to be solved specifically, the power spectrum function needs to be constructed to visualize the calculation results:

[0101] Based on the previous steps, α, α0 and β are iteratively updated through sparse Bayesian learning until convergence, which means that the grid points The domain is updated as follows: However, in order to further identify the true value of the incident angle, an indicator is needed to measure each grid point. Therefore, inspired by classic algorithms (such as MUSIC), a relative power spectrum is constructed to find the peak:

[0102]

[0103] If the grid index of the highest K peaks Where k∈[1,...,K], the estimated result of the Kth incident signal is

[0104] Example 1

[0105] The present invention is verified by computer simulation method, and all steps and conclusions are verified to be correct on MATLAB.

[0106] Initialization system parameters include: carrier frequency of 10 GHz, initial signal-to-noise ratio of 15 dB, noise caused by the metasurface n nThe signal-to-noise ratio of (t) is set to 100 dB, the number of metasurface elements is 32, the element spacing is 1 / 2 wavelength, the number of receiving antennas is 8, the receiving antenna spacing is 1 / 2 wavelength, the incident angles are 0° and 20°, and the number of snapshots is 50. The metasurface parameter ρ is 0.8, φ is a random phase between [0°, 180°], and the random phase generated in this example is

[0107] [0,π,π,0,0,π,π,0,π,0,π,0,π,0,π,0,0,π,0,0,0,π,0,π,0,0,π,0,π,π,0,0]

[0108] Assume that there exists a K ] is incident on the metasurface of N units. The total number of receiving antennas is M. Therefore, the signal received by the nth metasurface unit is as follows:

[0109]

[0110] Where β = 2π / λ is the wave number, β is the wavelength, D is the spacing between metasurfaces, and s k (t) is the kth incident signal, θ k is the incident angle of the kth signal, n n (t) is the Gaussian white noise generated by the nth metasurface unit, and its mean is set to 0.

[0111] Define the flow array A, signal vector s(t), and noise vector n'(t) as:

[0112]

[0113] Then the vector form of the signal received by the metasurface array is expressed as:

[0114] x(t)=Αs(t)+n′(t)

[0115] Next, a signal model is established based on the amplitude and phase shift of the transmissive metasurface. The signal transmitted through the nth metasurface unit can be expressed as:

[0116]

[0117] Among them, ξ n is the amplitude displacement of the nth metasurface unit, φ n is the phase shift of the nth metasurface unit.

[0118] The signal is then sent to the receiving antenna through the metasurface element, so that the signal received by the mth receiving antenna can be expressed as follows:

[0119]

[0120] Among them, y n (t) represents the signal transmitted through the nth metasurface array, d n,m represents the distance between the nth metasurface and the Mth receiving antenna, and w(t) represents the noise caused by the modulation of the metasurface unit. To further simplify the model, the signal received by the Mth antenna can be obtained as follows:

[0121]

[0122] Where, ω m represents the phase shift caused by the wave path when the signal is transmitted from the metasurface to the receiving antenna; Ξ represents the amplitude and phase characteristics of the metasurface, and x(t) represents the signal received by the metasurface array. Its specific form is:

[0123]

[0124] Let r(t)=[r1(t),r2(t),...,r M (t)] T , r m (t) is the signal received by the mth antenna. Therefore, the signal vector received by M antennas can be expressed as:

[0125]

[0126] Where n(t)=ΩΞn'(t)+w(t) represents the total noise generated by the MSA system during the entire process.

[0127] Then, the airspace needs to be gridded to construct a sparse flow array: the range of the wave direction of arrival Follow the Grid Divide the grid into G>>M>K. Generally speaking, the grid interval is Divide the grid evenly. This is a multivariate vector function whose value depends on θ k , D and other factors. If there is a true direction Where k∈{1,...,K}, Taylor approximation expansion can be performed:

[0128]

[0129] in Indicates the closest angle to the true The grid points, g k ∈{1,...,G},

[0130] Remember the grid matrix in Therefore, for the steering vector, Taylor approximation expansion can be performed:

[0131]

[0132] After accounting for the amplitude control and phase shift effects on the signal reaching the metasurface, the grid model becomes:

[0133]

[0134] So the received signal can be rewritten as:

[0135] r(t)=Ψ(β)s(t)+n(t)

[0136] where s(t)=[s1(t),...,s G (t)] T ,n(t)=[n1(t),...,n M (t)] T Only when the angles corresponding to the grid points are expanded is there an estimate and Otherwise g (t) = β g = 0. It is worth noting that Represents the grid closest to the true incident angle, from which g can be inferred k Indicates the corresponding index.

[0137] Assuming the number of snapshots is T, the above analytical expression can be written in the form of an observation matrix:

[0138] R=Ψ(β)S+N

[0139] where it is defined

[0140]

[0141] It is worth noting that since G>>K, the matrix S is sparse.

[0142] After constructing this model, we start to solve it. First, we need to make some assumptions. Assume that the noise N follows a Gaussian distribution:

[0143]

[0144] where α0=σ -1 Represents the noise accuracy, σ represents the variance in the formula. Assume that α0 is unknown and it obeys the Γ distribution, that is:

[0145] p(α0; c, d) = Γ(α0; c, d)

[0146] The hyperparameters c, d→0 are both set to 0.0001 in this example.

[0147] Based on the operational properties of Gaussian distribution, the probability density function of the received signal can be obtained as:

[0148]

[0149] Assume that the signals S between snapshots are independent of each other and adopt a two-stage hierarchical prior p(S|ρ)=∫p(S|α)p(α;ρ)dα, which is given by

[27] . And ρ is a hyperparameter representing the power of S, where Υ=diag(α) and:

[0150]

[0151] According to the Bayesian formula, the posterior probability can be expressed as:

[0152] p(S,α0,α,β|R)=p(S|R,α0,α,β)p(α0,α,β|R)

[0153] To solve the above probability formula, we need to consider these two items separately. First, for the first item, it is the posterior distribution of S, and its form satisfies the Gaussian distribution, that is:

[0154]

[0155] μ(t)=α0ΣΨ H r(t)

[0156] Σ=(α0Ψ H Ψ+γ -1 ) -1

[0157] In order to calculate μ(t) and Σ, it is necessary to determine the values ​​of the hyperparameters α0, α, and β. This is achieved by maximizing the conditional probability p(α0,α,β|R), which corresponds to the second part of the formula obtained above. Since the joint probability can be expressed as p(R,α0,α,β)=p(α0,α,β|R)p(R), and p(R) is independent of the hyperparameters, the problem is transformed into maximizing the joint probability p(R,α0,α,β). Considering S as a hidden variable, this joint probability is proportional to p(S,R,α0,α,β), and its specific expression can be derived from the total probability formula:

[0158] p(S,R,α0,α,β)=p(R|S,α0,β)p(S|α)p(α)p(β)p(α0)

[0159] Then use the maximum likelihood estimation method to process the above formula. The update expression of the hyperparameter is:

[0160]

[0161] in:

[0162]

[0163] Furthermore, estimating β is equivalent to maximizing p(R|S,α0,β)p(β). To do this, we need to make an additional assumption, namely At this time, the maximum likelihood estimate of β is further equivalent to maximizing E{ln p(R|S,α0,β)}, so the problem can be transformed into:

[0164]

[0165] in:

[0166]

[0167]

[0168] To this end, the problem is transformed into finding the maximum value of a function related to β. Although the specific expression of C in the formula is quite complicated, it can be ignored when iteratively updating β. Therefore, the iterative expression for β is:

[0169]

[0170] This is a convex function, and the point where the partial derivative is zero is the point where the minimum value is located:

[0171] β new =Q -1 v

[0172] According to the above steps, only the relevant parameters of the signal are obtained. However, if the angle of the incoming signal needs to be solved specifically, the power spectrum function needs to be constructed to visualize the calculation results:

[0173] Based on the previous steps, α, α0 and β are iteratively updated through sparse Bayesian learning until convergence, which means that the grid points The domain is updated as follows: However, in order to further identify the true value of the incident angle, an indicator is needed to measure each grid point. Therefore, inspired by classic algorithms (such as MUSIC), a relative power spectrum is constructed to find the peak:

[0174]

[0175] If the grid index of the highest K peaks Where k∈[1,...,K], the estimated result of the Kth incident signal is

[0176] First, the legend for the simulation figures is as follows: "OGSBL-MSA" represents the metasurface-assisted system using the proposed off-grid sparse Bayesian learning method and equipped with eight receive antennas. "32-MUSIC" represents the system using the MUSIC algorithm and equipped with 32 receive antennas but without a metasurface. The naming convention for the other systems is similar. To demonstrate the superiority of the metasurface, all metasurface-assisted systems are equipped with eight receive antennas.

[0177] Simulation 1: Set the true angle value to 0° and 20°. Figure 3 As can be seen, for the "OGSBL-MSA" case, the power spectrum is very clear, sufficient for visualization. Furthermore, the power spectrum peaks used for angle estimation are very distinct, with no other interfering peaks affecting the judgment, aside from the peaks corresponding to the two true values. Therefore, the proposed model can indeed achieve accurate angle of arrival estimation at a low cost.

[0178] Simulation 2: To verify the superiority of the first-order expansion (off-grid operation) after meshing in this paper, this simulation compares the performance with and without the first-order expansion term. The true incident angle is set to [0.14°, 20.05°], and the number of Monte Carlo trials is 200.

[0179] like Figure 4 、 5 As shown in the figure, when the estimated angle does not fall exactly on the grid point, the off-grid operation will produce a smaller root mean square error under the same signal-to-noise ratio and number of snapshots, showing a clear advantage.

[0180] Simulation 3: 200 Monte Carlo experiments were performed, with the snapshot numbers set to [1, 5, 10, 20, 30, 50, 70, 90, 120], and other parameters were the same as the initial parameters given in the example. The off-grid sparse Bayesian learning algorithm and the MUSIC algorithm proposed in this invention were used to simulate and analyze the 32-element traditional array, the 8-element traditional array, and the 32-element 8-channel random phase metasurface. The root mean square error results are as follows: Figure 6 .

[0181] Simulation 4: 200 Monte Carlo experiments were performed, with the signal-to-noise ratio values ​​being [0dB, 5dB, 10dB, 15dB, 20dB, 25dB] respectively. Other parameters were the same as the initial parameters given in the example. The off-grid sparse Bayesian learning algorithm and the MUSIC algorithm proposed in this invention were used to simulate and analyze the 32-element traditional array, the 8-element traditional array, and the 32-element 8-channel random phase metasurface. The root mean square error results were as follows: Figure 7 .

[0182] As can be seen, the root mean square error (RMS) of each experimental group gradually decreases with increasing snapshot counts and signal-to-noise ratio values. The use of the metasurface improves the performance of both the MUSIC and OGSBL algorithms, as can be seen in the OGSBL-MSA and MUSIC-MSA models. Although the performance of the metasurface-assisted system is inferior to that of the 32-MUSIC and 32-OGSBL models, it significantly improves compared to the case with the same number of antennas, demonstrating the feasibility of improving estimation quality while reducing cost.

[0183] Simulation 4: 200 Monte Carlo experiments were performed with the angle interval ranging from 1° to 10°. Other parameters were the same as the initial parameters given in the example. Two different algorithms were used to simulate and analyze the array with and without the assistance of the metasurface. The root mean square error results were as follows: Figure 8 .

[0184] from Figure 8 As can be seen in the experimental results with a 1° step size, the root mean square error (RMS) of the OGSBL-MSA system remains significantly lower than that of traditional array systems such as the 8-OGSBL and 8-MUSIC systems. Although the RMS error of the OGSBL-MSA system shows a slight rebound trend, this does not affect its superiority, as its RMS error remains lower than that of the other control groups.

Claims

1. A multi-channel super-resolution DOA estimation method based on off-grid sparse Bayesian learning assisted by a metasurface, characterized by: The specific steps are: Step 1: Establish a signal model for the incoming signal reaching the metasurface array element; Step 2: Based on the transmission amplitude and transmission phase of the transmissive metasurface, a signal model of the incoming signal transmitted through the metasurface is established; Step 3: Based on the positional relationship between the receiving antenna and the metasurface, a signal model of the receiving end of the receiving antenna is established; Step 4: Grid the airspace and construct a sparse flow pattern array; Step 5: Based on the Bayesian learning algorithm, the collected signal sample values ​​are used to initialize the hyperparameters, and the hyperparameters are iterated according to the expected maximum algorithm to obtain the grid error coefficient; Step 6: Construct the power spectrum function and visualize the calculation results based on the grid points obtained in step 5. Then, obtain the angle of arrival estimation result based on the peak value of the spectrum function.

2. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 1, characterized in that: The signal model of the incoming signal reaching the metasurface array element is specifically: x(t)=Αs(t)+n′(t) Where A is the flow array, s(t) is the signal vector, and n'(t) is the noise vector. The flow array is specifically: Where β is the wavelength, D is the distance between metasurfaces, N is the number of metasurface units, and θ k is the incident angle of the kth signal, k = 1, 2, ..., K, K is the number of signal directions.

3. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 1, characterized in that: The signal model of the incoming signal transmitted through the metasurface is specifically: Among them, ξ n is the amplitude displacement of the nth metasurface unit, φ n is the phase shift of the nth metasurface unit, x n (t) is the received signal of the nth metasurface unit.

4. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 1, characterized in that: The signal model at the receiving end of the receiving antenna is specifically: Where n(t) = ΩΞn'(t) + w(t) represents the total noise generated by the MSA system during the entire process, r(t) is the signal at the receiving end of the receiving antenna, and d n,m represents the distance between the nth metasurface and the mth receiving antenna, n = 1, 2, ..., N; m = 1, 2, ..., M; N is the number of metasurface units, M is the total number of receiving antennas, and β is the wavelength.

5. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 4, characterized in that: The specific method of gridding the airspace and constructing a sparse flow pattern array is as follows: Direction of Arrival Range Follow the Grid Divide the grid into G>>M>K, where M is the total number of receiving antennas, K is the number of signal directions, and the grid spacing is remember If there is a true direction Where k∈{1,...,K}, perform Taylor approximation expansion: in Indicates the closest angle to the true The grid points, g k ∈{1,...,G}, represents the derivative vector; Remember the grid matrix in is the error coefficient, and for the steering vector, Taylor approximation expansion is performed: The mesh model becomes: The received signal is rewritten as: r(t)=Ψ(β)s(t)+n(t) where s(t)=[s1(t),...,s G (t)] T ,n(t)=[n1(t),...,n M (t)] T Only when the angles corresponding to the grid points are expanded is there an estimate and Otherwise g (t) = β g =0, where represents the index position of the kth signal in the sparse signal vector s(t); Assume that the number of snapshots is T, then the receiving matrix is ​​written as the observation matrix of multiple snapshots as follows: R=Ψ(β)S+N where it is defined They represent the receiving matrix, signal source matrix, and additive noise matrix respectively.

6. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 5, characterized in that: According to the Bayesian learning algorithm, the collected signal sample values ​​are used to initialize the hyperparameters, and the hyperparameters are iterated according to the expected maximum algorithm. The specific method to obtain the grid error coefficient is as follows: Assume that the noise N obeys Gaussian distribution: where α0=σ -1 represents the noise accuracy, σ represents the standard deviation of the noise distribution, and its size reflects the noise power. α0 obeys the Γ distribution, that is: p(α0; c, d) = Γ(α0; c, d) Among them, c and d are the parameters of the probability distribution function, which are used to describe the mean and variance of the probability distribution function; Based on the operational properties of Gaussian distribution, the probability density function of the received signal is obtained as: Assume that the signals S between snapshots are independent of each other and adopt a two-stage hierarchical prior p(S|ρ)=∫p(S|(αp(α;ρ)dα), where the parameter ρ represents the power of S, Υ=diag(α) represents the diagonalization of the vector, I represents an identity matrix, and the specific expression of the probability involved in the expression of p(S|ρ) is as follows: According to the Bayesian formula, the specific expression of the posterior probability is: p*S,α0,α,β|R)=p9S|R,α0,α,β)p(α0,α,β|R) For the previous term of the posterior probability p(S|R,α0,α,β), it is the posterior distribution of S and formally satisfies the Gaussian distribution, that is: μ(t)=α0ΣΨ H r(t) Σ=(α0Ψ H P+Y -1 ) -1 Where (μ(t), Σ) represent the mean and variance of the Gaussian distribution, respectively. According to the expression of p(S,α0,α,β|R), the problem is solved by maximizing the value of the conditional probability p(α0,α,β|R) hyperparameter. Since the joint probability is expressed as: p(R,α0,α,β)=p(α0,α,β|R)p(R), and the probability density distribution function p(R) is not affected by parameters such as α0, the problem is transformed into maximizing the joint probability p(R,α0,α,β). Considering S as a hidden variable, p(R|S,α0,β) is proportional to p(S,R,α0,α,β), and its specific expression is derived from the following total probability formula: p(S,R,α0,α,β)=p(R|S,α0,β)p(S|α)p(α)p(β)p(α0) Then use the maximum likelihood estimation method to process the above formula: Logarithmize p(S,R,α0,α,β); By treating α0 and α as independent variables, taking the derivative of ln(p(S,R,α0,α,β)) and taking the zero point of its first-order derivative, the update expression is: Among them S g represents the g-th row of the signal matrix, It means taking the expectation of the square of its two norm, and the expressions of the variables involved are as follows: where ∑ gg represents the g-th diagonal element of Σ; In addition, the estimation of parameter β is transformed into maximizing p(R|S,α0,β)p(β), assuming Obey uniform distribution; Logarithmize p(R|S,α0,β)p(β); since β obeys a uniform distribution and does not affect the derivation process, the maximum likelihood estimation of β is further equivalent to maximizing E{lnp(R|S,α0,β)}, taking the zero point of its first-order derivative function, and obtaining: in: This is a convex function, and the point where the partial derivative is zero is the point where the minimum value is located: β new =Q -1 v Substitute the hyperparameters α0, α, and β into μ(t) and Σ; Using the obtained values ​​of μ(t) and Σ, update the values ​​of the hyperparameters according to the calculation formulas of the hyperparameters α0, α, and β; continue to iterate and update until the convergence requirements are met.

7. The method for multi-channel super-resolution DOA estimation based on off-grid sparse Bayesian learning and assisted by a metasurface according to claim 6, characterized in that: The specific method of step 6 is: Construct a relative power spectrum to find peaks: in, Represents an estimate of U, if the grid index of the highest K peak is Where k∈[1,...,K], the estimated result of the kth incident signal is Traverse the search peak and obtain the estimated values ​​of K angles.

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