A broadband elevation-azimuth joint estimation method and device based on an atomic norm minimization method with low complexity

Abstract

The present application provides a broadband elevation - azimuth joint estimation method and apparatus based on a low - complexity atomic norm minimization method. The method includes: performing a fast Fourier transform on the data received by a uniform rectangular array to convert the signal into a multi - frequency form; obtaining a second - order Toeplitz matrix containing elevation angle and azimuth angle information through a reduced - scale atomic norm minimization algorithm; estimating the spatial angular frequency from the second - order Toeplitz matrix using matrix pencil matching; and converting the estimated spatial angular frequency into an elevation angle and an azimuth angle. The advantages of the present application are as follows: it can achieve DOA estimation in the case of small samples with higher efficiency; it eliminates the influence of grid mismatch; the accuracy is greatly improved; the calculation results are highly accurate; it is more in line with the actual situation and has a wider applicability.

Description

Technical Field

[0001] This application belongs to the field of array signal processing, and specifically relates to a broadband elevation-azimuth joint estimation method and device based on a low-complexity atomic norm minimization method. Background Technique

[0002] Direction of Arrival (DOA) estimation is a popular research topic in the field of array signal processing and is widely used in applications such as sonar, radar, and positioning. Early DOA estimation was mainly based on traditional beamforming techniques. Such methods are limited by the array aperture and cannot break through the Rayleigh limit. To solve this problem, DOA estimation algorithms based on subspace methods, represented by Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), emerged and are still widely used today. Such methods break through the Rayleigh limit by estimating and applying the signal subspace, greatly improving the resolution ability. However, such methods require signals to be independent, are sensitive to signal coherence, and require a large number of snapshots to obtain a reliable estimate of the signal subspace.

[0003] In recent years, the development of Compressive Sensing (CS) theory has provided a new solution for DOA estimation. By fully applying the highly sparse characteristics of signals in the DOA domain, it has become possible to estimate DOA using a small number of snapshots or even a single snapshot. By dividing the DOA domain into a dense grid, the steering vectors corresponding to each grid point can form an overcomplete dictionary, converting the DOA estimation problem into a sparse recovery problem. When the dictionary satisfies the Null Space Property (NSP) or the Restricted Isometry Property (RIP), through convex relaxation, this NP-hard l0 optimization problem can be converted into an l1 optimization problem that can be solved by programming. When the actual signal does not fall on the divided grid points, it is called grid mismatch. Grid mismatch will greatly reduce the estimation accuracy and there is also a risk of missing targets when used for positioning. For this reason, a series of off-grid algorithms have been proposed. Such algorithms improve the influence of grid mismatch by interpolating between grids, but since there are still grids, the problem has not been completely solved.

[0004] The above methods are collectively referred to as grid-based algorithms because they assume the existence of grids. Compared with grid-based algorithms, the gridless algorithms proposed in recent years do not assume the existence of grids and directly process continuous DOAs. Its representative algorithm is the Atomic Norm Minimization (ANM) method. By solving the atomic norm minimization problem, this method can obtain an accurate estimate of DOA.

[0005] Traditional ANM algorithms are designed based on one-dimensional narrowband signals. In practical applications, signals are often high-dimensional broadband. This makes it difficult to directly apply traditional one-dimensional narrowband algorithms to practice.

[0006] In recent years, a great deal of research has been conducted on broadband and high-dimensional signals respectively. For high-dimensional signals, by utilizing high-order Toeplitz matrices, they can be incorporated into the ANM framework; for broadband signals, a multi-frequency model can be used to model them in order to estimate their DOA using ANM. However, there has been little research on the ANM algorithm for broadband high-dimensional signals. Summary of the Invention

[0007] The purpose of this application is to overcome the defect that the prior art cannot estimate the elevation angle and azimuth angle for both broadband and high-dimensional signals simultaneously.

[0008] To achieve the above purpose, this application proposes a broadband elevation-azimuth joint estimation method based on low-complexity atomic norm minimization and matching method, including:

[0009] Step 1: Perform a fast Fourier transform on the data received by the uniform rectangular array to convert the signal into a multi-frequency form;

[0010] Step 2: Obtain a second-order Toeplitz matrix containing elevation angle and azimuth angle information through a reduced-scale atomic norm minimization algorithm;

[0011] Step 3: Estimate the spatial angular frequency from the second-order Toeplitz matrix using matrix pencil matching;

[0012] Step 4: Convert the estimated spatial angular frequency into elevation angle and azimuth angle.

[0013] As an improvement of the above method, Step 2 includes:

[0014] For the noise-free case, construct the following atomic norm minimization problem:

[0015]

[0016] where, represents the received noise-free signal; Y represents the received signal containing noise;

[0017]

[0018] where, c i represents the amplitude of the i-th signal; r represents the number of far-field broadband plane wave signals; ‖·‖ is the l2 norm, N f represents the number of sub-bands into which a signal is divided; ⊙ represents the Khatri-Rao product;

[0019]

[0020] a1(f,ω 1i) = [1, exp(j2πfω 1i ),..., exp(j2πfω 1i ·(N1 - 1)); a2(f, ω 2i ) = [1, exp(j2πfω 2i ),..., exp(j2πfω 2i ·(N2 - 1)); ω i denotes the spatial angular frequency vector of the i-th signal, ω 1i denotes the horizontal component of ω i ; ω 2i denotes the vertical component of ω i ; N1 represents the number of array elements in the horizontal direction; N2 represents the number of array elements in the vertical direction; f0 represents the sampling frequency / N f ; c represents the wave propagation speed, d represents the element spacing; θ i denotes the elevation angle of the i-th signal; φ i denotes the azimuth angle of the i-th signal; T represents the vector transpose;

[0021] The atomic norm is defined as:

[0022]

[0023] where t represents the scaling constant; inf{·} represents the infimum; conv(·) represents the convex hull; the atom set is defined as:

[0024]

[0025] The second-order Toeplitz matrix Toep 2D (μ) is obtained by solving the following equation:

[0026]

[0027] where W represents the optimization intermediate variable; Tr(·) represents taking the trace of the matrix; denotes the conjugate transpose;

[0028] denotes the Kronecker product;

[0029]

[0030] is the optimization intermediate variable;

[0031] N represents noise;

[0032] The mapping operator is defined as:

[0033]

[0034] where, is the adjoint of; is the adjoint of;

[0035]

[0036] Define

[0037]

[0038] where, ω1 represents the horizontal component of the spatial angular frequency; ω2 represents the vertical component of the spatial angular frequency.

[0039] As an improvement to the above method, step 2 includes:

[0040] For the case with noise, construct the following atomic norm minimization problem:

[0041]

[0042] where, represents the received noise-free signal; Y represents the received signal containing noise; β represents the noise power; ‖·‖2 represents the 2-norm of the matrix;

[0043]

[0044] where, c i represents the amplitude of the i-th signal; r represents the number of far-field broadband plane wave signals; ‖·‖ is the l2 norm, N f represents the number of sub-bands into which a signal is divided; ⊙ represents the Khatri-Rao product;

[0045]

[0046] a1(f, ω 1i ) = [1, exp(j2πfω 1i ),..., exp(j2πfω 1i ·(N1 - 1)); a2(f, ω 2i ) = [1, exp(j2πfω 2i ),..., exp(j2πfω2i ·(N2 - 1)); ω i represents the spatial angular frequency vector of the i-th signal, ω 1i represents ω i the horizontal component; ω 2i represents ω i the vertical component; N1 represents the number of array elements in the horizontal direction; N2 represents the number of array elements in the vertical direction; f0 represents the sampling frequency / N f ; c represents the wave propagation speed, d represents the element spacing; θ i represents the elevation angle of the i-th signal; φ i represents the azimuth angle of the i-th signal; T represents the vector transpose;

[0047] Atomic norm is defined as:

[0048]

[0049] where t represents the scaling constant; inf{·} represents the infimum; conv(·) represents the convex hull; the atom set is defined as:

[0050]

[0051] The second-order Toeplitz matrix Toep 2D (μ) is obtained by solving the following equation:

[0052]

[0053] where ∈ is the noise power; W represents the optimization intermediate variable; Tr(·) represents taking the trace of the matrix; represents the conjugate transpose;

[0054] represents the Kronecker product;

[0055]

[0056] is the optimization intermediate variable;

[0057] N represents the noise;

[0058] Mapping operator is defined as:

[0059]

[0060] wherein, is the adjoint of ; is the adjoint of ;

[0061]

[0062] Define

[0063]

[0064] where ω1 represents the horizontal component of the spatial angular frequency; ω2 represents the vertical component of the spatial angular frequency.

[0065] As an improvement of the above method, step 4 includes:

[0066] The elevation angle θ i and the azimuth angle φ i in the i-th signal are respectively obtained by the following formulas:

[0067]

[0068] This application also provides a wideband elevation-azimuth joint estimation device based on a low-complexity atomic norm minimization and matching method, which is implemented based on the above method. The device includes:

[0069] A Fourier transform module, configured to perform a fast Fourier transform on the data received by the uniform rectangular array and convert the signal into a multi-frequency form;

[0070] A second-order Toeplitz matrix calculation module, configured to obtain a second-order Toeplitz matrix containing elevation angle and azimuth angle information through a reduced-scale atomic norm minimization algorithm;

[0071] A spatial angular frequency estimation module, configured to estimate the spatial angular frequency from the second-order Toeplitz matrix using matrix pencil matching;

[0072] An elevation angle and azimuth angle conversion module, configured to convert the estimated spatial angular frequency into an elevation angle and an azimuth angle.

[0073] This application also provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, the method described in any one of the above is implemented.

[0074] This application also provides a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, the processor is caused to execute the method described in any one of the above.

[0075] Compared with the prior art, the advantages of the present application are as follows:

[0076] 1. Compared with the traditional two-dimensional DOA estimation method based on the subspace method, the method proposed by the present invention can achieve DOA estimation in the case of small samples with higher efficiency;

[0077] 2. Compared with the traditional gridding algorithm, the method proposed by the present invention realizes gridless estimation and eliminates the influence of grid mismatch;

[0078] 3. Compared with the traditional narrowband two-dimensional DOA estimation algorithm based on ANM, the method proposed by the present invention realizes broadband DOA estimation, makes full use of the acquired information, and significantly improves the accuracy;

[0079] 4. Compared with the existing broadband one-dimensional DOA estimation algorithm based on ANM, the method proposed by the present invention extends the dimension to two dimensions, which is more in line with the actual situation and has a wider applicability;

[0080] 5. Compared with the existing 2DMFANM method and other traditional methods, the calculation results are highly accurate. Description of the Drawings

[0081] Figure 1 It shows a schematic diagram of a uniform rectangular array (URA) receiving broadband far-field plane wave signals;

[0082] Figure 2 It shows a flow chart of a broadband elevation angle - azimuth angle joint estimation method based on the atomic norm minimization method;

[0083] Figure 3(a) shows one of the comparison of the estimation performances of the 2DMFANM method and the traditional method for the elevation angle and azimuth angle;

[0084] Figure 3(b) shows another comparison of the estimation performances of the 2DMFANM method and the traditional method for the elevation angle and azimuth angle;

[0085] Figure 3(c) shows the third comparison of the estimation performances of the 2DMFANM method and the traditional method for the elevation angle and azimuth angle;

[0086] Figure 3(d) shows one of the comparison of the estimation performances of the 2DMFANM method and the traditional method for the spatial angular frequency;

[0087] Figure 3(e) shows another comparison of the estimation performances of the 2DMFANM method and the traditional method for the spatial angular frequency;

[0088] Figure 3(f) shows the third comparison of the estimation performances of the 2DMFANM method and the traditional method for the spatial angular frequency;

[0089] Fig. 4(a) shows the comparison of the curves of the root mean square error of the spatial angular frequency estimated by the 2DMFANM method and the traditional method versus SNR;

[0090] Fig. 4(b) shows the comparison of the curves of the root mean square error of the elevation angle and azimuth angle estimated by the 2DMFANM method and the traditional method versus SNR;

[0091] Fig. 4(c) shows the comparison of the curves of the higher estimation times of the 2DMFANM method and the traditional method versus SNR;

[0092] Fig. 5(a) shows the estimation results of the elevation angle and azimuth angle by the 2DMFANM method;

[0093] Fig. 5(b) shows the estimation results of the elevation angle and azimuth angle by the 2DMFANM_SizeRedu method;

[0094] Fig. 5(c) shows the estimation results of the spatial angular frequency by the 2DMFANM method;

[0095] Fig. 5(d) shows the estimation results of the spatial angular frequency by the 2DMFANM_SizeRedu method. Detailed implementation manners

[0096] The technical solutions of the present application will be described in detail below with reference to the accompanying drawings.

[0097] The present invention proposes a broadband elevation-azimuth joint estimation method (2DMFANM_SizeRedu) and device based on a low-complexity atomic norm minimization method. This method is optimized based on a broadband elevation-azimuth joint estimation method (2DMFANM) based on the atomic norm minimization method. The 2DMFANM method uses a multi-frequency model to model broadband signals and proposes a corresponding atomic set. The elevation angle and azimuth angle are obtained by constructing an atomic norm minimization problem. In order to solve the atomic norm minimization problem, a two-dimensional mapping operator is proposed, which enables the construction of a multivariate trigonometric polynomial constraint, so that the bounded lemma can be applied to convert the ANM problem into an SDP problem. To solve the problem of excessive computational complexity of the 2DMFANM method, the present invention further proposes a fast algorithm 2DMFANM_SizeRedu to accelerate the calculation by removing redundant information.

[0098] Principle description of the broadband elevation-azimuth joint estimation method based on the low-complexity atomic norm minimization method:

[0099] The signal processed by the method is a broadband far-field plane wave signal received by a uniform rectangular array (URA), and the function is to obtain the elevation angle and azimuth angle estimates of the signal source. The schematic diagram is as Figure 1 shown.

[0100] In the first step of the method, the data received by the uniform rectangular array is first subjected to a fast Fourier transform (FFT) to convert the signal into a multi-frequency form. In the second step, a second-order Toeplitz matrix containing elevation angle and azimuth angle information is obtained through a reduced-scale atomic norm minimization algorithm. In the third step, matrix pencil matching (MaPP) is used to estimate the spatial angular frequency from the second-order Toeplitz matrix. In the fourth step, the spatial angular frequency estimated in the third step is converted into elevation angle and azimuth angle. The algorithm flow is as Figure 2 shown.

[0101] Example 1

[0102] The broadband elevation-azimuth joint estimation method based on the atomic norm minimization method specifically includes:

[0103] Consider that r far-field broadband plane waves illuminate a uniform rectangular array (URA) of N1×N2. The array has N1 and N2 array elements in the x-axis and y-axis directions respectively. The elevation angle and azimuth angle of the i-th wave are θ i and φ i . Performing an N f -bit fast Fourier transform (FFT) on the broadband source can divide the signal into N f sub-bands, and the center frequency of each sub-band is f·f0, where f ∈ {1, 2,..., N f}, and f0 is the sampling frequency / N f . The i-th signal s i can be expressed as

[0104]

[0105] where ‖·‖ is the l2 norm. Here c i represents the amplitude of the i-th signal, contains the power of N f frequency bands. Let J be the set containing the array element indices, defined as J = {0, 1,..., N1 - 1} × {0, 1,..., N2 - 1}. The signal received by the k-th array element in the f-th sub-band can be expressed as:

[0106]

[0107] where k ∈ J, c represents the wave propagation speed, and d represents the array element spacing.

[0108] Defining the spatial angular frequency vector of the i-th signal we can obtain

[0109]

[0110] It should be noted that the angular information including the pitch angle and the azimuth angle can be estimated through ω i obtained. Combining the signals of N f subbands, the signal received by the k-th array element can be organized as:

[0111]

[0112] where ⊙ represents the Khatri-Rao product.

[0113] Combining the signals of the f-th subband of all array elements together, the following matrix representation can be obtained:

[0114]

[0115] It can be further organized as:

[0116]

[0117] where represents the steering vector,

[0118] Define the matrix A(f, ω i ) as: A(f, ω i ) = a2(f, ω 2i ) · a1(f, ω 1i ) T .

[0119] Vectorizing (5) can obtain the following representation for the f-th subband:

[0120]

[0121] where represents the Kronecker product.

[0122] Combining (4) and (6), the complete signal model can be obtained as follows:

[0123]

[0124] where

[0125] The present invention obtains the DOA information to be extracted through . In the case of noiseless, the method directly processes while in the actual application scenario with noise, the received signal can be modeled as

[0126]

[0127] where N represents noise.

[0128] Based on (7), define the following atomic set

[0129]

[0130] To extract the DOA information from the observed signals, construct the following atomic norm minimization problem:

[0131] For the noise-free case:

[0132]

[0133] For the noisy signals:

[0134]

[0135] Here the atomic norm is defined as:

[0136]

[0137] where \(t\) represents the dilation constant; \(\inf\{\cdot\}\) represents the infimum; \(\text{conv}(\cdot)\) represents the convex hull; \(\beta\) represents the noise power.

[0138] Since the atomic norm cannot be directly solved, the optimization problem needs to be converted into a semi-definite programming (SDP) problem that can be solved by programming. For simplicity, only the SDP conversion process for the noise-free case (10) will be given in detail below, and the noisy case can be obtained similarly.

[0139] The dual problem of (10) is

[0140]

[0141] where represents the dual norm of the atomic norm in (12), and is defined as

[0142]

[0143] where \(\sup\) represents the supremum; using the definition in (7) and the definition we can obtain:

[0144]

[0145] Let we can obtain:

[0146]

[0147] To represent (16) in the form of a multivariate trigonometric polynomial to use the bounded lemma, introduce the following mapping operator. Let:

[0148]

[0149] According to 's definition, there is:

[0150]

[0151] In order to construct the mapping operator, the following definitions are made:

[0152]

[0153] where

[0154] Let:

[0155]

[0156] and

[0157]

[0158] where

[0159] According to (18)-(21), it can be seen that can be regarded as 's partial observation. Therefore, the mapping operator is defined as follows:

[0160]

[0161] Let:

[0162]

[0163] From (7), (22) and (23), it can be seen that Let:

[0164]

[0165] where is 's adjoint. Substituting (24) into the definition of Ψ(Q,ω) gives:

[0166]

[0167] Therefore, (16) can be transformed into:

[0168]

[0169] It is deduced that:

[0170]

[0171] Note that the right side of (27) is a multivariate trigonometric polynomial inequality. According to the bounded lemma and appropriate sum of squares relaxation (SOS), this inequality can be written in the following SDP form:

[0172]

[0173] where is a half-space, is a Toeplitz matrix, with 1 on its diagonal and 0 elsewhere, i = 1, 2, is the N f ×N f identity matrix.

[0174] Therefore, (13) can be converted into the following SDP problem:

[0175]

[0176] where denotes the conjugate transpose.

[0177] The SDP form of the original problem (10) can be obtained by calculating the dual problem of (29), and the result is as follows:

[0178]

[0179] where W represents the optimization intermediate variable; Toep 2D (μ) is a second-order multi-layer Toeplitz matrix,

[0180] The spatial angular frequency vector ω = [ω1, ω2] can be obtained from Toep 2D (μ) by the MaPP method. Finally, the pitch angle and azimuth angle can be obtained respectively by the following formulas:

[0181]

[0182] It is worth mentioning that current general SDP solvers, such as SDPT3, can give the solution of the dual problem simultaneously after solving the original problem. In actual operation, since the solution of the ANM dual problem (29) is faster, it is recommended to directly solve (29), and Toep 2D (μ) in (30) is the dual variable of P0 in (29), which can be directly given by the solver. Here, the SDP form of the dual problem in the presence of noise is given as follows:

[0183]

[0184] where β is the noise power; ‖·‖2 represents the 2-norm of the matrix.

[0185] The process of the 2DMFANM method is shown in Table 1.

[0186] Table 1 2DMFANM method

[0187]

[0188] The improvements of the 2DMFANM_SizeRedu method over the 2DMFANM method are as follows:

[0189] The main complexity of the 2DMFANM algorithm comes from H in (29), which has a large sparsity and contains redundant information. By removing this redundant information, the algorithm speed can be significantly improved. Specifically, define

[0190]

[0191] Let

[0192]

[0193] , where Let where

[0194]

[0195] From this, we can obtain

[0196]

[0197] From (37), a corresponding new mapping operator can be constructed such that where, is the adjoint of;

[0198] And define There is:

[0199]

[0200] Through analysis similar to (26)-(28), we can obtain:

[0201]

[0202] From (39), it can be seen that P r and H ris much smaller than P0 and H0, so the complexity of the problem can be greatly reduced. It is worth mentioning that since the right end of (39) is more restrictive than the left end, it will affect the algorithm performance. By analyzing the dual form of (39), the ANM algorithm with reduced size can be obtained:

[0203]

[0204] Among them, is the optimized intermediate variable;

[0205] When there is noise, the algorithm is modified as follows:

[0206]

[0207] where ∈ is the noise power.

[0208] The flow of the 2DMFANM_SizeRedu algorithm is shown in Table 2.

[0209] Table 2 2DMFANM_SizeRedu algorithm

[0210]

[0211] To illustrate the effectiveness of the method of the present invention, the following simulation experiments were conducted. Consider a 5×5 URA, and the azimuth and elevation angles of the signal are set to (-60°, -45°, -30°) and (30°, 40°, 45°) respectively. Set the wave speed to v = 1500 m / s. Set the fundamental frequency f0 = 1000 Hz in the multi-frequency signal model. The element spacing is set to half of the maximum wavelength, that is Assume that the signals defined as (1) are independent and follow a complex Gaussian distribution Assume that the noise is also Gaussian white noise. The solution of the optimization problem in the simulation is implemented using CVX. The following results are obtained through 100 Monte Carlo experiments.

[0212] In this experiment, we mainly compared the 2DMFANM method with the sparse Bayesian (SBL) algorithm and the traditional ANM-based narrowband algorithm to demonstrate its effectiveness. Set N in the 2DMFANM method f = 5. When there is no noise, the estimation results are as Figures 3(a)-3(f) shown. Figures 3(d), 3(e) and 3(f) compare the estimation performance of the 2DMFANM method and the SBL method and the traditional method for the spatial angular frequency, while Figures 3(a), 3(b) and 3(c) compare the estimation performance for the elevation angle and azimuth angle. From Figures 3(a)-3(f)It can be seen that by utilizing the information of multiple frequencies, the estimation results of the 2DMFANM method are concentrated near the true value, while the estimation results of the traditional method are relatively dispersed, demonstrating the superiority of the 2DMFANM method.

[0213] To demonstrate the performance of the 2DMFANM method in the presence of noise, the signal-to-noise ratio (SNR) was set from -2 dB to 20 dB for simulation. An estimation was considered successful when the root mean square error (RMSE) of the spatial angular frequency estimation was less than 0.2. Fig. 4(c) shows the number of successful estimations of the three methods. When calculating the RMSE of the estimation, only the successful estimation results were considered. Figs. 4(a) and 4(b) show the curves of the estimated RMSE varying with the SNR. As can be seen from Fig. 4(c), the number of successful estimations of all methods increases with the increase of the SNR. Compared with the narrowband algorithm (ANM), the 2DMFANM method has a higher success rate at low SNR, indicating that the performance can be improved by using the information of multiple frequencies. As can be seen from Fig. 4(a), the performance of the 2DMFANM method in estimating the spatial angular frequency is not significantly better than that of the SBL method at low SNR. However, as can be seen from Fig. 4(b), the 2DMFANM method is significantly superior to the SBL method in estimating the elevation angle and azimuth angle. This is mainly because the correspondence between the spatial angular frequency and the elevation angle and azimuth angle is non-linear. Since the SBL method is grid-based, the estimation results all fall on the fixed grid. While the proposed gridless method enables the estimation results to continuously aggregate near the true value. Therefore, when the spatial angular frequency is converted into the elevation angle and azimuth angle, the 2DMFANM method will achieve better estimation results. The advantage of the gridless method can also be reflected by the RMSE in the noise-free case, where the RMSE of the 2DMFANM method is much lower than that of the SBL.

[0214] To verify the performance of the proposed fast algorithm 2DMFANM_SizeRedu, considering the noise-free case, N f was set from 1 to 5, and simulations were carried out on the original algorithm 2DMFANM and the fast algorithm 2DMFANM_SizeRedu. Figures 5(a)-5(d) Shows the estimation results of each algorithm when N f = 5.

[0215] An estimation was considered successful when the RMSE of the spatial angular frequency estimation was less than 0.05. The number of successful estimations is summarized in Table 3. It can be seen from it that the success rates of both the 2DMFANM method and the fast algorithm 2DMFANM_SizeRedu increase significantly with the increase of N f .

[0216] Table 3 Number of successful estimations of the fast algorithm

[0217]

[0218] The operation times of the two algorithms are summarized in Table 4. It can be seen that the operation time of the original 2DMFANM algorithm increases significantly with N f increasing. 2DMFANM_SizeRedu has the ability to reduce the operation time.

[0219] Table 4 Operation Time of the Fast Algorithm Unit: Seconds

[0220]

[0221] The RMSEs of the pitch angle - azimuth angle estimation and the spatial angular frequency estimation by the two algorithms are summarized in Table 5 and Table 6 respectively. It can be seen from this that with the increase of N f the accuracy of the 2DMFANM estimation increases significantly. The estimation accuracy of the fast algorithm 2DMFANM_SizeRedu increases with the increase of N f as well. The performance of the fast algorithm can be summarized as follows: When there are sufficient operation resources in the actual application scenario, 2DMFANM_SizeRedu can significantly improve the estimation accuracy.

[0222] Table 5 RMS of the Fast Algorithm for Estimating Pitch Angle and Azimuth Angle

[0223]

[0224] Table 6 RMSE of the Fast Algorithm for Estimating Spatial Angular Frequency

[0225]

[0226] Example 2

[0227] This application also provides a wideband pitch angle - azimuth angle joint estimation device based on a low - complexity atomic norm minimization and matching method, which is implemented based on the above - mentioned method. The device includes:

[0228] A Fourier transform module, which is used to perform a fast Fourier transform on the data received by the uniform rectangular array to convert the signal into a multi - frequency form;

[0229] A module for calculating the second - order Toeplitz matrix, which is used to obtain a second - order Toeplitz matrix containing pitch angle and azimuth angle information through a reduced - scale atomic norm minimization algorithm;

[0230] A module for estimating the spatial angular frequency, which is used to estimate the spatial angular frequency from the second - order Toeplitz matrix using matrix pencil matching;

[0231] A module for converting the pitch angle and azimuth angle, which is used to convert the estimated spatial angular frequency into the pitch angle and azimuth angle.

[0232] Example 3

[0233] The present application may also provide a computer device, including: at least one processor, a memory, at least one network interface, and a user interface. Each component in the device is coupled together through a bus system. It can be understood that the bus system is used to realize the connection and communication between these components. In addition to the data bus, the bus system also includes a power bus, a control bus, and a status signal bus.

[0234] Among them, the user interface may include a display, a keyboard, or a pointing device. For example, a mouse, a trackball, a touchpad, or a touch screen, etc.

[0235] It can be understood that the memory in the disclosed embodiments of the present application may be a volatile memory or a non-volatile memory, or may include both volatile and non-volatile memories. Among them, the non-volatile memory may be a read-only memory (ROM), a programmable ROM (PROM), an erasable programmable ROM (EPROM), an electrically erasable programmable ROM (EEPROM), or a flash memory. The volatile memory may be a random access memory (RAM), which is used as an external cache. By way of example but not limitation, many forms of RAM are available, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate synchronous DRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), synchlink DRAM (SLDRAM), and direct rambus RAM (DRRAM). The memory described herein is intended to include but not be limited to these and any other suitable types of memory.

[0236] In some embodiments, the memory stores the following elements, executable modules, or data structures, or subsets thereof, or extended sets thereof: an operating system and application programs.

[0237] Among them, the operating system includes various system programs, such as the framework layer, the core library layer, the driver layer, etc., which are used to implement various basic services and handle hardware-based tasks. The application programs include various application programs, such as Media Player, Browser, etc., which are used to implement various application services. The program for implementing the method of the embodiments of the present disclosure may be included in the application programs.

[0238] In the above-mentioned embodiments, the program or instruction stored in the memory may also be called. Specifically, it may be the program or instruction stored in the application program. The processor is used for:

[0239] Execute the steps of the above method.

[0240] The above method can be applied to or implemented by the processor. The processor may be an integrated circuit chip with signal processing capabilities. During the implementation process, each step of the above method can be completed by the integrated logic circuit in the hardware of the processor or the instruction in the form of software. The above-mentioned processor may be a general-purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components. It can implement or execute the various methods, steps, and logic block diagrams disclosed above. The general-purpose processor may be a microprocessor or the processor may also be any conventional processor, etc. Combining the steps of the above-disclosed method can be directly embodied as being completed by the hardware decoding processor, or completed by a combination of the hardware and software modules in the decoding processor. The software module may be located in a mature storage medium in the art, such as a random access memory, a flash memory, a read-only memory, a programmable read-only memory, or an electrically erasable programmable memory, a register, etc. This storage medium is located in the memory, and the processor reads the information in the memory and combines its hardware to complete the steps of the above method.

[0241] It can be understood that the embodiments described in this application can be implemented using hardware, software, firmware, middleware, microcode, or a combination thereof. For hardware implementation, the processing unit can be implemented in one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), general-purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in this application, or a combination thereof.

[0242] For software implementation, the techniques of this application can be implemented by executing the functional modules of this application (such as procedures, functions, etc.). The software code can be stored in a memory and executed by a processor. The memory can be implemented inside or outside the processor.

[0243] Embodiment 4

[0244] This application can also provide a non-volatile storage medium for storing a computer program. When the computer program is executed by a processor, the various steps in the above method embodiments can be implemented.

[0245] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this application and not to limit them. Although this application has been described in detail with reference to the embodiments, those of ordinary skill in the art should understand that any modification or equivalent replacement of the technical solutions of this application does not depart from the spirit and scope of the technical solutions of this application, and they should all be covered within the scope of the claims of this application.

Claims

1. A broadband elevation-azimuth joint estimation method based on low-complexity atomic norm minimization and matching method, comprising: Step 1: Perform a fast Fourier transform on the data received by the uniform rectangular array to convert the signal into a multi-frequency form; Step 2: Obtain a second-order Toeplitz matrix containing elevation angle and azimuth angle information through a reduced-scale atomic norm minimization algorithm; Step 3: Use matrix pencil matching to estimate the spatial angular frequency from the second-order Toeplitz matrix; Step 4: Convert the estimated spatial angular frequency into an elevation angle and an azimuth angle; The said Step 2 includes: For the noise-free case, construct the following atomic norm minimization problem: Among them, represents the received signal without noise; Y represents the received signal containing noise; Among them, c i represents the amplitude of the i-th signal; r represents the number of far-field wideband plane wave signals; ‖·‖ is the l2 norm, N f represents the number of sub-bands into which a signal is divided; ⊙ represents the Khatri-Rao product; a1(f, ω 1i ) = [1, exp(j2πfω 1i ),..., exp(j2πfω 1i ·(N1 - 1)); a2(f, ω 2i ) = [1, exp(j2πfω 2i ),..., exp(j2πfω 2i ·(N2 - 1)); ω i represents the spatial angular frequency vector of the i-th signal, ω 1i represents the horizontal component of ω i ; ω 2i represents the vertical component of ω i ; N1 represents the number of array elements in the horizontal direction; N2 represents the number of array elements in the vertical direction; f0 represents the sampling frequency / N f ; c represents the wave propagation speed, d represents the element spacing; θ i represents the elevation angle of the i-th signal; φ i represents the azimuth angle of the i-th signal; T represents the vector transpose; Atomic norm is defined as: where \(t\) represents the scaling constant; \(\inf\{\cdot\}\) represents the infimum; \(\conv(\cdot)\) represents the convex hull; the atom set is defined as: The second-order Toeplitz matrix Toep is obtained by solving the following equation 2D (μ): Among them, W represents the optimized intermediate variable; Tr(·) represents taking the trace of a matrix; represents the conjugate transpose; denotes the Kronecker product; Denote optimized intermediate variables; N represents noise; Mapping operator is defined as: Among them, is the adjoint of ; is the adjoint of ; Definition where, ω1 represents the horizontal component of the spatial angular frequency; ω2 represents the vertical component of the spatial angular frequency; For the noisy case, construct the following atomic norm minimization problem: Among them, represents the received noiseless signal; Y represents the received signal containing noise; β represents the noise power; ‖·‖2 represents the 2-norm of the matrix; where c i represents the amplitude of the i-th signal; r represents the number of far-field wideband plane wave signals; ‖·‖ is the l2 norm, and N f represents the number of sub-bands into which a signal is divided; ⊙ represents the Khatri-Rao product; a1(f, ω 1i ) = [1, exp(j2πfω 1i ),..., exp(j2πfω 1i ·(N1 - 1)); a2(f, ω 2i ) = [1, exp(j2πfω 2i ),..., exp(j2πfω 2i ·(N2 - 1)); ω i represents the spatial angular frequency vector of the i-th signal, ω 1i represents the component of ω i in the horizontal direction; ω 2i represents the component of ω i in the vertical direction; N1 represents the number of array elements in the horizontal direction; N2 represents the number of array elements in the vertical direction; f0 represents the sampling frequency / N f ; c represents the wave propagation speed, d represents the element spacing; θ i represents the elevation angle of the i-th signal; φ i represents the azimuth angle of the i-th signal; T represents the vector transpose; Atomic norm is defined as: where \(t\) represents the dilation constant; \(\inf\{\cdot\}\) represents the infimum; \(\conv(\cdot)\) represents the convex hull; the atom set is defined as: The second-order Toeplitz matrix Toep is obtained by solving the following equation 2D (μ): Among them, ∈ is the noise power; W represents the optimized intermediate variable; Tr(·) represents taking the trace of a matrix; represents the conjugate transpose; denotes the Kronecker product; Indicates an optimized intermediate variable; N represents noise; Mapping operator is defined as: Among them, is the adjoint of ; is the adjoint of ; Definition where, ω1 represents the horizontal component of the spatial angular frequency; ω2 represents the vertical component of the spatial angular frequency.

2. The broadband elevation-azimuth joint estimation method based on the low-complexity atomic norm minimization and matching method according to claim 1, characterized in that The said Step 4 includes: The elevation angle θ in the i-th signal i and the azimuth angle φ i are respectively obtained by the following formulas:

3. A broadband elevation-azimuth joint estimation device based on a low-complexity atomic norm minimization and matching method, implemented based on any one of the methods described in claims 1-2, characterized in that, The said device includes: A Fourier transform module, configured to perform a fast Fourier transform on the data received by the uniform rectangular array to convert the signal into a multi-frequency form; A module for calculating the second-order Toeplitz matrix, configured to obtain a second-order Toeplitz matrix containing elevation angle and azimuth angle information through a reduced-scale atomic norm minimization algorithm; A module for estimating the spatial angular frequency, configured to estimate the spatial angular frequency from the second-order Toeplitz matrix using matrix pencil matching; and A module for converting the elevation angle and the azimuth angle, configured to convert the estimated spatial angular frequency into an elevation angle and an azimuth angle.

4. A computer device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that, When the said processor executes the said computer program, it implements the method described in any one of claims 1 to 2.

5. A computer-readable storage medium, characterized in that, The said computer-readable storage medium stores a computer program, and when the said computer program is executed by the processor, it causes the processor to execute the method described in any one of claims 1 to 2.

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