Orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method, medium and equipment

Abstract

The invention provides an orthogonal propagation operator direction-of-arrival estimation method based on a Newton interpolation method, a medium and equipment. Firstly, an array antenna of a uniform linear array structure is used for receiving a signal, and a received signal is obtained at a receiving end; then, calculating to obtain a received signal covariance matrix by using the received signal; then, constructing a Toeplitz matrix by using the received signal covariance matrix; secondly, using off-diagonal elements in the first row and the first column of the Toeplitz matrix, estimating diagonal elements by using a Newton interpolation method, and replacing the original diagonal elements; and finally, DOA estimation is carried out by using an orthogonal propagation operator method. Compared with cubic spline interpolation and linear prediction interpolation methods, the method provided by the invention not only maintains the original DOA estimation performance, but also has lower algorithm complexity.

Description

Technical Field

[0001] The present invention relates to direction of arrival (DOA) estimation, belonging to the technical fields of radar positioning, navigation, and wireless communication, and particularly relates to a DOA estimation method, medium, and device based on Newton interpolation method for orthogonal propagator. Background Art

[0002] DOA estimation is an important issue in the field of array signal processing and has been widely studied for decades. Among them, subspace-based algorithms and high-resolution algorithms, such as multiple signal classification (MUSIC) and estimating signal parameter via rotational invariance techniques (ESPRIT), have attracted much attention due to their asymptotically infinite resolution and unbiased estimation performance.

[0003] However, the above technologies require eigenvalue decomposition or singular value decomposition of the received signal covariance matrix, which requires a large amount of computing power and cost for applications with large amounts of data and high real-time requirements. For example, a multiple-input multiple-output system with a large number of antennas, and there may be certain limitations due to the high computational complexity of these methods and they may not be directly applicable in some field scenarios.

[0004] To solve this problem, the concept of "propagator" is introduced in array signal processing. Based on this concept, many techniques have been proposed, such as the propagator method (PM) and the orthogonal propagator method (OPM). These methods only use linear operations to find the propagator without the need for eigenvalue decomposition or singular value decomposition. The propagator has been proven to be effective in DOA estimation. However, the propagator-based methods do not consider background noise, and their performance will be affected in low signal-to-noise ratio scenarios. To eliminate the influence of noise, when the noise is additive white Gaussian noise in time and space, the noise only affects the main diagonal elements of the received signal covariance matrix. Therefore, the influence of noise can be reduced by restoring the noise-free diagonal elements of the received signal covariance matrix. When the received signal covariance matrix is a Toeplitz matrix, the diagonal elements can be estimated by interpolation using the off-diagonal elements of the first column and the first row of the matrix, and only one interpolation is required.

[0005] Cubic spline interpolation and linear prediction have been proven to be effective in removing diagonal element noise and improving the estimation performance. However, cubic spline interpolation requires second and third derivatives and solving multiple equations; the linear prediction method has a large matrix dimension when the number of receiving arrays is large. Therefore, both have a high computational complexity. Thus, a new technical solution is needed to solve the above problems. Summary of the Invention

[0006] To solve the limitation of high computational complexity of cubic spline interpolation and linear prediction methods in large-scale arrays, the present invention provides an orthogonal propagation operator direction-of-arrival estimation method, medium, and device based on Newton interpolation method, by constructing a Toeplitz received signal covariance matrix, using Newton interpolation method to recover diagonal elements, and finally using the orthogonal propagation operator for DOA estimation.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] In a first aspect, the present invention provides an orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method, including the following steps:

[0009] S1: Use the array antennas of a uniform linear array to receive signals and obtain the received signals at the receiving end;

[0010] S2: Calculate the received signal covariance matrix according to the received signals;

[0011] S3: Reconstruct the received signal covariance matrix into a Toeplitz matrix;

[0012] S4: Use the norms of the off-diagonal elements of the first row and the first column of the Toeplitz matrix for Newton interpolation method to estimate the diagonal elements;

[0013] S5: Use the estimated diagonal elements to replace the diagonal elements of the Toeplitz matrix to construct a new covariance matrix;

[0014] S6: Use the orthogonal propagation operator method to perform DOA estimation on the new covariance matrix.

[0015] Optionally, in step S1, the uniform linear array is composed of a uniform linear array with N array elements and an element spacing of λ / 2, where λ is the carrier wavelength.

[0016] Optionally, in step S2, the received signal covariance matrix is expressed as:

[0017] R = E{x(t)x H (t)}

[0018] = AE{s(t)s H (t)}AH +E{n(t)n H (t)};

[0019] =ASA H +σ 2 I

[0020] wherein, R represents the received signal covariance matrix, E{·} represents taking the mathematical expectation, x(t) is the received signal of the array antenna, A is the direction matrix of the array, s(t) is the source signal, n(t) is the received noise signal, σ 2 is the noise variance, and I represents the identity matrix.

[0021] Optionally, in step S3, the calculation of reconstructing the received signal covariance matrix into a Toeplitz matrix is as follows:

[0022] The Toeplitz matrix R T is constructed by calculating the first row of the matrix:

[0023]

[0024] wherein, r t (m) represents the (m + 1)-th element of the first row of R T , and R(n, m + n) represents the element in the n-th row and the (m + n)-th column of the covariance matrix R;

[0025] The constructed Toeplitz matrix R T is expressed as:

[0026]

[0027] wherein, * represents conjugate.

[0028] Optionally, in step S4, the calculation of estimating the diagonal elements is as follows:

[0029]

[0030] wherein, N(0) represents the noise-free diagonal element, and f[·] represents the difference quotient.

[0031] Optionally, in step S5, the new covariance matrix is expressed as:

[0032]

[0033] wherein, R nf represents the new covariance matrix.

[0034] In a second aspect, the present invention provides a computer-readable storage medium storing a computer program, which causes a computer to execute the direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method as described in the first aspect.

[0035] In a third aspect, the present invention provides an electronic 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 direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method as described in the first aspect is implemented.

[0036] The beneficial effects of the present invention are as follows: The present invention uses an array antenna with a uniform linear array structure to receive signals, and from the perspectives of Toeplitz reconstruction and reducing computational complexity, proposes a direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method, retaining the main advantages of the prior art in estimating diagonal elements using off-diagonal elements and improving in terms of computational complexity. Compared with the prior art, the present invention maintains the original DOA estimation accuracy after being processed by the Newton interpolation method and has lower computational complexity. Description of the Drawings

[0037] Figure 1 is a flowchart of the direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method in this embodiment.

[0038] Figure 2 is a schematic diagram of an array antenna with a uniform linear array structure receiving signals in this embodiment.

[0039] Figure 3 is a comparison diagram of RMSE of the method in this embodiment with cubic spline interpolation and linear prediction algorithms under different signal-to-noise ratios.

[0040] Figure 4 is a comparison diagram of the algorithm running time of the method in this embodiment with cubic spline interpolation and linear prediction algorithms under different signal-to-noise ratios.

[0041] Figure 5 is a comparison diagram of RMSE of the method in this embodiment with cubic spline interpolation and linear prediction algorithms under different numbers of snapshots.

[0042] Figure 6 is a comparison diagram of the algorithm running time of the method in this embodiment with cubic spline interpolation and linear prediction algorithms under different numbers of snapshots. Detailed Embodiments

[0043] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application.

[0044] In one embodiment, the present invention proposes a method for estimating the direction of arrival of an orthogonal propagation operator based on Newton interpolation method, and its design principle is as follows:

[0045] 1. Use an array antenna with a uniform linear array structure to receive signals, and propose a method for estimating the direction of arrival of an orthogonal propagation operator based on Newton interpolation method from the perspective of Toeplitz reconstruction and reducing computational complexity, so as to solve the problem of high computational complexity of interpolation techniques represented by cubic spline interpolation and linear prediction techniques.

[0046] 2. When using cubic spline interpolation to estimate diagonal elements, multiple equations and unknowns need to be set, so there is a high computational complexity; linear prediction does not require setting equations, but matrix inversion operations are required. When the dimension of the received signal is large, the complexity at this time will also increase. The use of Newton interpolation method improves this process, and the coefficients of the polynomial are obtained through simple division, thus greatly reducing the computational complexity. After the preprocessing operation, the OPM algorithm is selected to estimate the DOA of the received signal.

[0047] The flow of the method for estimating the direction of arrival of an orthogonal propagation operator based on Newton interpolation method is as Figure 1 shown, and mainly includes the following aspects.

[0048] I. Data model

[0049] Suppose there are K narrowband far-field signals incident on a uniform linear array with N array elements. The distance between adjacent two array elements is d and the wavelength of the input signal is λ, and d / λ = 0.5 is satisfied between the two, Figure 2 which is the schematic diagram of the uniform linear array used in this embodiment. The received signal of the nth array element can be expressed as:

[0050]

[0051] where, x k (t) represents the source signal from the direction θ k at time t = 1, 2,...., L; L is the number of snapshots; n n (t) represents an additive Gaussian white noise vector with a mean of 0 and a variance of σ 2 . Expressing the array received signal in the form of matrix algebra, we can get:

[0052] x(t) = As(t) + n(t)

[0053] where, r(t) = [r0(t), r1(t), …, r N-1 (t)] T represents the received signal vector, A = [a(θ1), a(θ2),..., a(θ K)] is the direction matrix of the array, is the direction vector from the θ k direction, and the source signal s(t) = [s1(t), s2(t), …, s K (t)] T is a narrowband far-field signal, and n(t) = [n0(t), n1(t),..., n N-1 (t)] T is the received noise signal.

[0054] Assuming that the signal and the noise are independent of each other, the received signal covariance matrix R can be expressed as:

[0055] R = E{x(t)x H (t)}

[0056] = AE{s(t)s H (t)}A H + E{n(t)n H (t)}

[0057] = ASA H + σ 2 I

[0058] where E{.} represents taking the mathematical expectation, and S represents the source signal covariance matrix. Due to the existence of noise and the finite number of snapshots, the received signal covariance matrix R is not a Toeplitz matrix, so it needs to be reconstructed. The Toeplitz matrix R T can be constructed by calculating the first row of the matrix:

[0059]

[0060] where r t (m) represents the (m + 1)-th element of the first row of R T The Toeplitz matrix R T can be expressed as:

[0061]

[0062] where * represents the conjugate symbol.

[0063] II. Newton interpolation method

[0064] The received signal covariance matrix can also be expressed as:

[0065]

[0066] where m, n ∈ [1, …, N], N is the number of arrays, is the power of the k-th signal. It can be seen from the above formula that noise only exists in the diagonal elements of the covariance matrix. However, with a finite number of snapshots, the off-diagonal elements contain residuals from the noise. Therefore, the diagonal elements can be estimated using the off-diagonal elements. Since the received signal covariance matrix is a Toeplitz matrix, only one diagonal element needs to be estimated. The Newton interpolation method can be used to recover the noise-free diagonal elements.

[0067] Define the Newton interpolation polynomial as:

[0068] N n (x) = a0 + a1(x - x0) + a2(x - x0)(x - x1) + … + a n (x - x0)…(x - x n-1 )

[0069] where a k (k = 0, 1, 2, …, n) are undetermined coefficients.

[0070] The ratio of the difference between the independent variable and the dependent variable is called the divided difference. Define the average rate of change of the function y = f(x) over the interval [x i , x i+1 :

[0071]

[0072] is called the first-order divided difference of f(x) with respect to x i , x i+1 and is denoted as f[x i , x i+1 . It can be derived that the m-th divided difference is:

[0073]

[0074] By transformation, we get:

[0075] f(x) = f(x0) + (x - x0)f[x, x0]

[0076] f[x, x0] = f[x0, x1] + (x - x1)f[x, x0, x1]

[0077] f[x, x0, …, x n-1 = f[x0, x1, …, x n + (x - x n )f[x, x0, …, x n

[0078] Substituting successively, we get the Newton interpolation polynomial:

[0079] ​f(x) = f(x0) + (x - x0)f[x0, x1] + … + (x - x0)(x - x1)…(x - x n-1 )f[x0, x1, …, x n

[0080] The undetermined coefficient a k = f[x0, x1, …, x k (k = 0, 1, …, n).

[0081] In this embodiment, the Newton interpolation method is applied to the method for estimating the noise - free diagonal elements, and the specific steps are as follows:

[0082] Step 1: Construct each divided - difference f[·] according to the first - row elements r T (m) and the first - column elements r t (m) of the Toeplitz matrix R t * .

[0083] Step 2: Substitute each divided - difference into the Newton interpolation polynomial:

[0084]

[0085] Step 3: When x = 0, the noise - free diagonal element N(0) can be obtained:

[0086]

[0087] Step 4: Replace the diagonal element R(0) of R T with N(0) to construct a new covariance matrix R nf :

[0088]

[0089] After the noise - free diagonal elements are estimated, the OPM algorithm can be used for DOA estimation. OPM divides the matrix A into two sub - matrices:

[0090]

[0091] where A1 and A2 are (K×K) and ((N - K)×K) - dimensional matrices respectively. Under the conditions that A is full - rank and A1 is non - singular, the propagation operator P can be defined as:

[0092] P H A1 = A2

[0093] or

[0094] [P H , - I N-K A = Q H ​A = 0

[0095] where I N-K is an ((N - K)×(N - K)) - dimensional identity matrix. The above equation indicates that the direction vector A and Q are column - orthogonal. Therefore, the DOA can be estimated by searching for the peak positions of the OPM pseudo - spectrum:

[0096]

[0097] The propagation operator P needs to be obtained from the covariance matrix R nf First, the covariance matrix R nf is partitioned as follows:

[0098] R nf = [G, H]

[0099] where G and H are (N×K) and (N×(N - K)) - dimensional matrices respectively. The relationship between the propagation operator P and these two matrices is:

[0100] H = GP

[0101] Therefore, the propagation operator P can be directly obtained by the least - squares solution:

[0102] P = G + H

[0103] where + represents the Moore - Penrose inverse operator.

[0104] III. Experimental Analysis

[0105] To compare and verify the effectiveness of the above - mentioned method and correctly analyze the performance of the algorithm, multiple simulation experiments were carried out on the algorithm and the experimental simulation results were analyzed. The specific process is as follows:

[0106] (1) Experimental performance evaluation metrics

[0107] The performance evaluation metric for DOA estimation uses the root mean square error (RMSE), and its definition is as follows:

[0108]

[0109] where is the angle estimation value during the q - th Monte Carlo simulation experiment of this method, Q is the number of Monte Carlo experiments, and K is the total number of signal sources.

[0110] The performance evaluation metric for algorithm complexity estimation uses the algorithm running time CPU - time.

[0111] (2) Experimental simulation effect diagram

[0112] Figure 3 When two far-field signal sources (θ1 = -6°, θ2 = 7°) are incident on a uniform linear array, the number of snapshots is set to 500, the signal-to-noise ratio varies from -5 dB to 20 dB, and 500 Monte Carlo simulations are run. This figure shows the comparison of the RMSE performance of the method of this embodiment with the cubic spline interpolation splne and the linear prediction LP algorithm at different signal-to-noise ratios. Among them, the number of array elements N of the uniform linear array is 21. It can be seen from the figure that the root mean square error of all methods decreases as the signal-to-noise ratio increases. The method of using Newton interpolation to estimate the diagonal elements proposed in this embodiment has approximate estimation performance with the other two algorithms, and has a slight advantage at high signal-to-noise ratios.

[0113] Figure 4 When two far-field signal sources (θ1 = -6°, θ2 = 7°) are incident on a uniform linear array, the number of snapshots is set to 500, the signal-to-noise ratio varies from -5 dB to 20 dB, and 500 Monte Carlo simulations are run. With other conditions unchanged, this figure shows the comparison of the algorithm operation time of the method of this embodiment with the cubic spline interpolation splne and the linear prediction LP algorithm at different signal-to-noise ratios. It can be seen from the figure that the cubic spline interpolation has the longest operation time at any signal-to-noise ratio, followed by the linear prediction algorithm, but the Newton interpolation method proposed in this embodiment has the lowest operation time.

[0114] Figure 5 When two far-field signal sources (θ1 = -6°, θ2 = 7°) are incident on a uniform linear array, the signal-to-noise ratio is set to 10 dB, the number of snapshots varies from 200 to 800, and 500 Monte Carlo simulations are run. With other conditions unchanged, this figure shows the comparison of the RMSE performance of the method of this embodiment with the cubic spline interpolation splne and the linear prediction LP algorithm at different numbers of snapshots. It can be seen from the figure that the root mean square error of all methods decreases as the number of snapshots increases. The method of using Newton interpolation to estimate the diagonal elements proposed in this embodiment has approximate estimation performance with the other two algorithms and has a slight advantage.

[0115] Figure 6 When two far-field signal sources (θ1 = -6°, θ2 = 7°) are incident on a uniform linear array, the signal-to-noise ratio is set to 10 dB, the number of snapshots varies from 200 to 800, and 500 Monte Carlo simulations are run. With other conditions unchanged, this figure shows the comparison of the algorithm operation time of the method of this embodiment with the cubic spline interpolation splne and the linear prediction LP algorithm at different numbers of snapshots. It can be seen from the figure that the cubic spline interpolation has the longest operation time at any number of snapshots, followed by the linear prediction algorithm, but the Newton interpolation method proposed in this embodiment has the lowest operation time.

[0116] In summary, the direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method proposed in this embodiment provides a simple method for estimating the interpolation of diagonal elements. From aspects such as signal-to-noise ratio and number of snapshots, the RMSE estimation performance and algorithm operation time of the proposed technique are evaluated through simulations of multiple signals. Compared with existing interpolation methods, this method maintains the existing DOA estimation performance while reducing the computational burden and having a lower algorithm complexity.

[0117] In another embodiment, the present invention proposes a computer-readable storage medium storing a computer program that causes a computer to execute the direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method in the foregoing embodiment.

[0118] In another embodiment, the present invention proposes an electronic 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 direction-of-arrival estimation method of the orthogonal propagation operator based on the Newton interpolation method in the foregoing embodiment is implemented.

[0119] In the embodiments disclosed in the present application, the computer storage medium may be a tangible medium that may contain or store a program for use by or in connection with an instruction execution system, apparatus, or device. The computer storage medium may include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of the computer storage medium would include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination of the foregoing.

[0120] Those of ordinary skill in the art can realize that the units and algorithm steps of each example described in combination with the embodiments disclosed in the present application can be implemented by electronic hardware or a combination of computer software and electronic hardware. Whether these functions are executed in a hardware or software manner depends on the specific application and design constraints of the technical solution. A professional technician can use different methods for each specific application to implement the described functions, but such implementation should not be considered to exceed the scope of the present application.

[0121] The above are only the preferred embodiments of the present invention. The protection scope of the present invention is not limited to the above embodiments. Any technical solutions falling within the concept of the present invention belong to the protection scope of the present invention. It should be noted that for those of ordinary skill in the art, several improvements and refinements made without departing from the principle of the present invention should be regarded as within the protection scope of the present invention.

Claims

1. An orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method, characterized in that, It includes the following steps: S1: Use an array antenna with a uniform linear array to receive signals, and obtain the received signals at the receiving end; S2: Calculate the received signal covariance matrix based on the received signals; S3: Reconstruct the received signal covariance matrix into a Toeplitz matrix; S4: Use the norms of the off-diagonal elements in the first row and the first column of the Toeplitz matrix for Newton interpolation to estimate the diagonal elements; S5: Use the estimated diagonal elements to replace the diagonal elements of the Toeplitz matrix to construct a new covariance matrix; S6: Use the orthogonal propagation operator method to perform DOA estimation on the new covariance matrix.

2. The orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method according to claim 1, characterized in that: In step S1, the uniform linear array consists of a uniform linear array with N array elements and an element spacing of λ / 2, where λ is the carrier wavelength.

3. The orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method according to claim 1, wherein: In step S2, the received signal covariance matrix is expressed as: Wherein, R represents the received signal covariance matrix, E{·} represents taking the mathematical expectation, x(t) is the signal received by the array antenna, A is the direction matrix of the array, s(t) is the source signal, n(,) is the received noise signal, and σ 2 is the noise variance, and I represents the identity matrix.

4. The orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method according to claim 2, characterized in that: In step S3, the calculation of reconstructing the received signal covariance matrix into a Toeplitz matrix is as follows: Toeplitz matrix R T Constructed by calculating the first row of the matrix: where r t (m) represents the T element in the (m + 1)-th position of the first row, and R(n, m + n) represents the element in the n-th row and (m + n)-th column of the covariance matrix R; Constructed Toeplitz matrix R T is expressed as: In the formula, * represents conjugate.

5. The method for estimating the direction of arrival of an orthogonal propagation operator based on Newton interpolation method according to claim 4, characterized in that: In step S4, the calculation of estimating the diagonal elements is as follows: In the formula, N(0) represents the noise-free diagonal elements, and f[·] represents the divided difference.

6. The orthogonal propagation operator direction-of-arrival estimation method based on Newton interpolation method according to claim 5, characterized in that: In step S5, the new covariance matrix is expressed as: In the formula, R nf represents the new covariance matrix.

7. A computer-readable storage medium storing a computer program, characterized in that, The computer program causes the computer to execute the orthogonal propagation operator direction of arrival estimation method based on Newton interpolation according to any one of claims 1-6.

8. An electronic device, characterized in that, It includes: A memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, it implements the orthogonal propagation operator direction of arrival estimation method based on Newton interpolation according to any one of claims 1-6.

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