Two-dimensional DOA estimation method based on Capon dimensionality reduction for mutually prime arrays

Abstract

The present invention discloses a dimensionality-reduced Capon algorithm for two-dimensional DOA estimation of multiple signals using a coprime array. The algorithm first calculates the covariance matrix of the received signal, then reduces the two-dimensional spectral function to one dimension. Leveraging the linear relationship between true and fuzzy DOA estimates, a one-dimensional partial spectrum search is performed on a local region to obtain a fuzzy DOA estimate that correlates with the theoretical DOA, significantly reducing computational complexity. Finally, based on the properties of the coprime array, a true DOA estimate is obtained. Compared to the traditional coprime array two-dimensional Capon algorithm, this method achieves nearly identical DOA estimation performance while avoiding the significant computational complexity associated with a two-dimensional spectral peak search.

Description

Technical Field

[0001] The present invention relates to the field of direction of arrival estimation, and in particular to a two-dimensional DOA estimation method of a coprime array reduced-dimensional Capon. Background Art

[0002] Estimating the direction of arrival (DOA) of multiple narrowband signal sources is a fundamental issue in array signal processing and is of significant importance in various engineering applications, including wireless communications, medical imaging, radar, sonar, and navigation. Coprime arrays have attracted significant attention due to their ability to reduce mutual coupling, increase degrees of freedom, and improve resolution. Various DOA estimation methods for coprime arrays have been proposed. To eliminate phase ambiguity, a searchless DOA algorithm based on quasi-projection can significantly reduce computational complexity. Alternatively, a spectral peak search method can be used to derive the DOA from the results of two subarrays and the properties of coprime arrays.

[0003] When estimating the direction of arrival of a two-dimensional signal source, a coprime symmetric sparse cross array can be used for DOA estimation, and an effective algorithm is proposed by constructing a high-order matrix. Alternatively, the traditional two-dimensional multi-signal classification algorithm (MUSIC algorithm) can be applied to coprime planar arrays. This full-spectrum search method has excellent DOA estimation performance, but the computational complexity is high. While partial spectrum search methods reduce complexity to a certain extent, they still require a two-dimensional spectrum peak search, resulting in a high computational load and high cost in practical engineering. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to address the problem of excessive computational complexity of two-dimensional spectral peak search involved in the background technology. A method of dimensionality reduction processing of spectral functions is adopted to avoid the huge computational complexity brought by two-dimensional spectral peak search, reduce the time and space cost of the algorithm, and at the same time maintain its good estimation performance.

[0005] The present invention adopts the following technical solution to solve the above technical problems: a two-dimensional DOA estimation method of a coprime array with reduced-dimensional Capon:

[0006] Step 1: Divide the coprime array into two uniform arrays. Next, consider these two uniform arrays separately and calculate the covariance matrix of the received signal. Perform dimension reduction on the spectral function and construct Q(v).

[0007] Step 2, in v∈[0,2 / M j ]A one-dimensional local search is performed in the area to find is the largest K peaks in the (1,1)th element, and the corresponding v k (k=1,2,…,K) estimated value.

[0008] Step 3, based on the estimated v k (k=1,2,…,K), find K vectors Then use the least squares method to get u k (k=1,2,…,K) estimated value.

[0009] Step 4: Calculate all feasible (u, v) pairs based on the phase difference between the true direction of arrival and the ambiguous direction of arrival, and then calculate the difference function d c And select K minimum values, and then average them to get the final estimate

[0010] Step 5: Calculate the elevation angle and azimuth angle of the signal source to obtain a two-dimensional DOA estimate.

[0011] Preferably, the specific steps of constructing the covariance matrix and Q(v) in step 1 are as follows:

[0012] Step 1.1, divide the coprime array into two uniform arrays. Next, consider these two uniform arrays separately, and the covariance matrix of the received signal is Can be achieved through Make an estimate, where L is the number of snapshots.

[0013] Step 1.2, the spatial spectrum function of the two-dimensional Capon is:

[0014] Where the direction vector The function V i (u,v) is defined as It can also be expressed as In the formula

[0015] Preferably, in step 2, the one-dimensional peak search is performed to obtain v k The specific steps for (k=1,2,…,K) are as follows:

[0016] Step 2.1, It is a quadratic optimization problem. Consider using eliminate The trivial solution of This optimization problem can be reformulated as follows:

[0017] Construct the following cost function In the formula, λ is a constant. xi (u) Derivative

[0018]

[0019] according to It can be concluded that a xi(u)=μQ -1 (v)e1, where μ is a constant. Combined with a xi (u)=μQ -1 (v)e1 gives

[0020] so

[0021] In addition, we can also get v k Estimation of (k=1,2,…,K)

[0022]

[0023] Step 2.2, assume that there is only one signal source incident on M i ×M i The two-dimensional direction of arrival of the mutually prime plane sub-array is (θ p ,φ p ), assuming (θ a ,φ a ) corresponds to (θ p ,φ p ), it is known that the exponential function with the natural constant as the base is a periodic function with a period of 2π, which means that the phase difference between the true direction of arrival and the ambiguous direction of arrival is

[0024] 2πd i (u p -u a ) / λ=2k ui π and 2πd i (v p -v a ) / λ=2k vi π, where u p = sinθ p cosφ p ,u a = sinθ a cosφ a , v p = sinθ p sinφ p , v a = sinθ a sinφ a , is an integer, d i =M j λ / 2,(i,j∈{1,2},i≠j). because u a ∈[-1,1],v a∈[0,1], these two ranges are not only applicable to each range individually, but should also be considered together, that is, they should also satisfy Since the element spacing of traditional uniform arrays is less than half a wavelength, k is used in DOA estimation. ui and k vi can only take 0, which means there is no fuzzy DOA value. i =M j λ / 2,(i,j∈{1,2},i≠j), k ui and k vi M j and Except for those infeasible k ui and k vi In addition to The value of 2πd i (u p -u a ) / λ=2k ui π and 2πd i (v p -v a ) / λ=2k vi π exists, but there is only one pair of k ui and k vi The values ​​are consistent with the theoretical DOA.

[0025] For simplicity, it can also be expressed as and where k ui ∈(-M j ,M j ), k vi ∈(-M j / 2,M j / 2) and i,j∈{1,2},i≠j.

[0026] It can be seen that there is a linear relationship between each true angle parameter and the corresponding blur angle parameter in the transform domain. For the i-th sub-matrix, the difference is An integer multiple of , so it can be Instead of searching in the total sector v∈[0,1], we obtain K peaks on any sector of v∈[0,1].

[0027] Without losing generality, for the i-th sub-matrix, we choose v∈[0,2 / M j ]In the region Perform a one-dimensional local search and find K peaks, where is the maximum value of the (1, 1)th element. The largest K peaks Corresponding to vk = sinθ k sinφ k (k=1,2,…,K).

[0028] Preferably, the least squares method in step 3 is used to solve u k The specific steps for (k=1,2,…,K) are as follows:

[0029] according to K vectors can be obtained

[0030] g k =-angle(a xi (u k ))=u k q=[0,2πd i u K / λ,…,2πd i (M i -1)u K / λ] T Where q=[0,2πd i / λ,…,2πd i (M i -1) / λ] T .

[0031] Now use the least squares method to estimate u k According to the least squares criterion where ||·|| F represents the Frobenius norm, c k0 is the parameter error estimate.

[0032] Preferably, the step 4 solves The specific steps for the final estimate are as follows:

[0033] First, all other feasible fuzzy (u,v) estimates are calculated using linear features.

[0034] Although phase ambiguity may occur when the array element spacing is greater than half a wavelength, the true direction of arrival estimation can be obtained by utilizing the characteristics of the coprime array. and It can be concluded that and Since M1 and M2 are relatively prime integers, only k u1 =k u2 =0 and k v1 =k v2= 0, the above equation is valid. This means that the two sub-arrays can only obtain the real DOA estimation value related to the theoretical DOA at the same time. However, in the presence of noise, the DOA estimation results of the two sub-arrays cannot be completely consistent, so the closest one is qualified.

[0035] Define the difference function d c

[0036]

[0037] Where (u 1m , v 1m ) and (u 2n , v 2n ) represent the mth and nth fuzzy DOA estimated using two sub-matrices. By choosing d c By obtaining the K smallest correlation (u, v) pairs, the true DOA estimation value can be obtained.

[0038] as well as in and are the true DOA estimates of the two sub-arrays respectively.

[0039] Preferably, the specific steps of solving the elevation angle and azimuth angle of the source direction in step 5 are as follows:

[0040] The elevation angle and azimuth angle of the source direction are

[0041]

[0042]

[0043] Where k = 1, 2,…, K.

[0044] Beneficial effects: Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0045] 1. The algorithm exploits the linear relationship between the true and fuzzy DOA estimates and only requires a one-dimensional partial search in the local area, significantly reducing the computational cost.

[0046] 2. The estimation performance of this algorithm is almost the same as that of global search or local search in coprime matrices, but the computational complexity is much lower.

[0047] 3. The algorithm can realize automatic pairing of two-dimensional angle estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1This is a flow chart for implementing the two-dimensional DOA estimation method of the coprime array reduced-dimensional Capon provided by the present invention.

[0049] Figure 2 This is the array structure model diagram of the coprime array.

[0050] Figure 3 This is a scatter plot of the direction of arrival estimated by the algorithm of the present invention.

[0051] Figure 4 The following is a comparison chart of the root mean square error of elevation angle under various algorithms.

[0052] Figure 5 Comparison chart of the root mean square error of azimuth under various algorithms.

[0053] Figure 6 Comparison of the root mean square error of elevation angle estimation using the proposed algorithm under different coprime array parameters.

[0054] Figure 7 Comparison of the root mean square error of azimuth angle estimation using the proposed algorithm under different coprime array parameters.

[0055] Figure 8 Comparison of the root mean square error of the elevation angle estimated by the proposed algorithm under different snapshot numbers.

[0056] Figure 9 Comparison of the root mean square error of the azimuth estimated by the proposed algorithm under different snapshot numbers. DETAILED DESCRIPTION

[0057] The present invention will be further explained below with reference to the accompanying drawings.

[0058] like Figure 1 The figure shows the implementation flow chart of the two-dimensional DOA estimation method of the coprime array dimensionality reduction Capon, which is composed of Figure 2 From the array structure, it can be seen that the coprime array can be divided into two uniform sub-arrays for consideration. First, the covariance matrix of the received signal is obtained, and then the two-dimensional spectrum function is reduced to one dimension. By using the linear relationship between the true and fuzzy DOA estimates, a one-dimensional partial spectrum search is performed on the local area. A fuzzy DOA estimate related to the theoretical DOA can be obtained, thereby significantly reducing the computational complexity. Finally, the true DOA estimate can be obtained based on the properties of the coprime array.

[0059] like Figure 3The figure shows the scatter plot results of DOA estimation using the proposed algorithm. In the simulation, it is assumed that the coprime array consists of two uniform planar subarrays with 5×5 and 4×4 elements, where d1=4λ / 2 and d2=5λ / 2. It is assumed that there are K=3 signal sources incident on the array from (θ1,φ1)=(10°,10°), (θ2,φ2)=(20°,20°), and (θ3,φ3)=(30°,30°). Figure 3 It can be seen that the two-dimensional DOA estimation results of the algorithm are concentrated near (10°, 10°), (20°, 20°) and (30°, 30°), which are consistent with the assumed source direction.

[0060] like Figure 4 and Figure 5 The figure shows the error comparison between the proposed Capon algorithm for dimensionality reduction in coprime arrays and other related algorithms. For a fair comparison, the uniform array design is arranged in a 5×8 array with a half-wavelength spacing, the same as the total number of elements in the coprime array, 40. The root mean square error is defined as Where C represents the number of Monte Carlo simulations, Denotes the estimated value of the kth incident angle in the cth trial, and C = 500. In this simulation, the two-dimensional DOA estimation performance of the proposed algorithm is compared with the traditional Capon algorithm for coprime arrays, the traditional Capon algorithm for uniform arrays, and the reduced-dimensional Capon algorithm, where the number of snapshots L = 500. Figure 4 and Figure 5 The results clearly demonstrate that the proposed algorithm achieves nearly identical DOA estimation performance to the local spectrum search method, but with significantly reduced computational complexity. Because the spacing of the coprime array is much larger than half a wavelength, the array aperture is increased, resulting in better DOA estimation performance compared to the Capon algorithm for uniform arrays.

[0061] like Figure 6 and Figure 7 The figure shows the DOA estimation performance comparison of coprime arrays with different M2 using the algorithm proposed in this invention, where M1=5, K=3 and L=500. Figure 6 and Figure 7 It can be seen from the figure that due to the effect of diversity gain, as M2 increases, the elevation and azimuth DOA estimation performance of this method is improved.

[0062] like Figure 8 and Figure 9 The following is a comparison of DOA estimation performance under different snapshot numbers L, where M1=5, M2=4 and K=3. Figure 8 and Figure 9 As can be seen from , as L increases, the DOA estimation performance becomes better, because the larger the number of snapshots, the more accurate the covariance matrix.

[0063] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as within the scope of protection of the present invention.

Claims

1. A two-dimensional DOA estimation method based on Capon dimensionality reduction of mutually prime arrays, characterized by: The following steps are involved: 1) Receive signals through a coprime array; 2) Divide the coprime array into two uniform sub-matrices i, and calculate the covariance matrix R of the received signal of the two uniform sub-matrices i respectively ix , get the two-dimensional Capon's spatial spectrum function, reduce the dimension of the two-dimensional Capon's spatial spectrum function, i = 1, 2; 3) Perform local spectrum peak search on the spatial spectrum function after dimensionality reduction to obtain v k The estimated value of v k The estimated value of is combined with the least squares method to obtain u k The estimated value of k=1,2,⋯,K; 4) Calculate all feasible parameter (u, v) pairs based on the phase difference between the true direction of arrival and the fuzzy direction of arrival, and then calculate the difference function d c And select K minimum values ​​to get the true DOA estimate 5) Based on the actual DOA estimate By calculating the elevation angle and azimuth angle of the source direction, we can obtain a two-dimensional DOA estimate.

2. The two-dimensional DOA estimation method of the coprime array reduced-dimensional Capon method according to claim 1, characterized in that: The implementation process of step 2) is: 2.1) Assume that there are K independent narrowband sources in the space domain from {(θ k ,φ k )} incident on the coprime array, where θ k and φ k are the elevation and azimuth angles of the kth source, respectively, and the parameter u is defined k = sinθ k cosφ k , v k = sinθ k sinφ k , divide the coprime array into two uniform sub-matrices i, the number of elements of the uniform sub-matrix is ​​M i ×M i , M i is the number of array elements in each row or column of the ith sub-array, and the received signal of the ith sub-array is expressed as X i =A i S+N i , where X i =[x i (1),x i (2),…,x i (L)], xi(L) is the received signal of the sub-array under the L-th snapshot number, S = [s1, s2, ..., s K ] T is the source signal matrix, s k =[s k (1),s k (2),…,s k (L)],s k (L) represents the source of the Lth snapshot number, L is the snapshot number, The mean is 0 and the variance is σ 2 Additive Gaussian white noise, is the direction matrix of the i-th submatrix: a xi (u k ) and a yi (v k ) are the direction vectors of the x-direction and y-direction of the i-th sub-matrix: has xi (u k )=[1,exp(-j2πd i u k / λ),…,exp(-j2π(M i -1)d i u k / λ)] T , and yi (in k )=[1,exp(-j2πd i in k / λ),…,exp(-j2π(M i -1)d i in k / λ)] T ; 2.2) The received signal matrix X obtained in step 2.1) i =[x i (1),x i (2),…,x i (L)], find its covariance matrix pass Make an estimate; where: d i represents the element spacing of the i-th sub-array; and it satisfies d i =M j λ / 2,M j is the number of elements in each row of another sub-matrix; 2.3) According to step 2.2) the covariance matrix R is obtained ix , the spatial spectrum function of the two-dimensional Capon is: Wherein the parameters u=sinθcosφ,v=sinθsinφ,the direction vector has xi (u)=[1,exp(-j2πd i u / λ),…,exp(-j2π(M i -1)d i u / λ)] T , and yi (v)=[1,exp(-j2πd i v / λ),…,exp(-j2π(M i -1)d i v / λ)] T ; Define the function V i (u,v) is: or In the formula Indicates M i The unit matrix of order; 2.4) Refactored to Under the conditions That is u and v when the minimum value is obtained, Construct the cost function: to a xi (u) Derivative: again Then there is: a xi (u)=μQ -1 (v)e1, where μ is a constant, Then we get:

3. The two-dimensional DOA estimation method of the coprime array reduced-dimensional Capon as claimed in claim 2, characterized in that: The implementation process of step 3) is: 3.1) It is known that the phase difference between the true direction of arrival and the ambiguous direction of arrival is Then we have: 2πd i (u p -u a ) / λ=2z ui p, 2πd i (v p -v a ) / λ=2z vi p; Where the true direction of arrival elevation angle u p = sinθ p cosφ p , fuzzy direction of arrival elevation angle u a = sinθ a cosφ a , the true direction of arrival azimuth v p = sinθ p sinφ p , fuzzy direction of arrival azimuth v a = sinθ a sinφ a , is an integer, Then we have: where \(z\) ui \(\in(-M\) j , M\) j ), \(z\) vi \(\in(-M\) j / 2, M\) j / 2)\), \(i, j\in\{1, 2\}, i\neq j\); Therefore, for the i-th submatrix, directly in the length of Perform spectrum peak search on any sector area; 3.2) In v∈[0,2 / M j ] parameter v in the region k Estimated value of Perform a one-dimensional local search and find K peaks of is the maximum value of the (1, 1)th element, the largest K peaks Corresponding to v k = sinθ k sinφ k ; 3.3) According to Get K vectors a xi The phase of each element in the direction vector is: g k =-angle(a xi (u k ))=u k q=[0.2πd i you K / λ,…,2πd i (M i -1)u K / l] T ; Where q=[0,2πd i / λ,…,2πd i (M i -1) / λ] T , angle(·) means taking the phase of each element in the complex matrix; Solve u using the least squares method k , the least squares criterion is: where ||·|| F represents the Frobenius norm, M represents all 1s i dimensional column vector, is an unknown parameter vector, c k0 is the parameter error estimate, the least squares result get , get a preliminary estimate 4. The two-dimensional DOA estimation method of the coprime array reduced-dimensional Capon method according to claim 3, characterized in that: The implementation process of step 4) is: repeat steps 2) and 3) to calculate all other feasible preliminary estimates, and use (u 1m , v 1m ) and (u 2n , v 2n ) represent the mth and nth preliminary estimates using two sub-matrices, respectively, and calculate the difference function d c value: Select d c Get the initial estimate when K minimum values ​​are obtained, where K = 2, and average the two initial estimates to get the true DOA estimate. in and There are two sub-matrices such that d c A preliminary estimate when obtaining the K minimum values.

5. The two-dimensional DOA estimation method of the coprime array dimensionality reduction Capon as claimed in claim 4, characterized in that: In step 5): The elevation angle in the direction of the source is: The azimuth of the source direction is: Among them, j is a parameter.

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