Preprocessing-based DOA (Direction of Arrival) estimation method under mutual coupling and impulse noise

Abstract

The invention relates to the technical field of array signal processing, in particular to a preprocessing-based DOA (Direction of Arrival) estimation method under mutual coupling and impulse noise, which can meet the requirement on high real-time performance in practical application and does not need iterative solution, and is characterized by comprising the following steps of: 1, establishing a received signal model to obtain a received signal; 2, defining a piecewise function for compressing an abnormal amplitude value of a received signal to obtain a received data matrix, and keeping a small amplitude value by the piecewise function; step 3, performing DOA estimation by using the received data matrix; compared with the prior art, the method has the following advantages: (1) iterative solution is not needed, and the calculation amount is small; and (2) when the pulse noise characteristic index alpha is greater than or equal to 1, the success rate and the RMSE performance are superior to those of the existing DOA estimation algorithm and the like.

Description

Technical Field:

[0001] The present invention relates to the technical field of array signal processing, and particularly to a DOA estimation method based on preprocessing under mutual coupling and impulse noise that can meet the requirements of high real-time performance in practical applications and does not require iterative solution. Background Art:

[0002] Array signal processing is a technology that uses multiple sensors or receivers to capture, process, and analyze signals. As one of the important research directions of array signal processing, the main purpose of direction of arrival (DOA) estimation is to estimate the spatial angle parameters of several transmitted source signals from the received signals, and it is widely used in fields such as wireless communication, radar, sonar, and navigation. Since the 1980s, subspace algorithms represented by the multiple signal classification (MUSIC) algorithm and the estimating signal parameter via rotational invariance techniques (ESPRIT) have promoted the progress of DOA estimation theory due to their high-resolution characteristics.

[0003] However, the performance of the above algorithms depends on the accuracy of the array manifold. In an engineering environment, the array manifold will be affected by various factors, and the most important one is the mutual coupling effect. In practical applications, especially when working in the high-frequency band, in order to prevent array manifold ambiguity, the spacing between adjacent array elements is set very small. For an antenna array containing multiple array elements, each array element is affected not only by the electromagnetic field generated by its own current but also by the electromagnetic field generated by the currents on other array elements in the array, that is, the so-called mutual coupling effect is generated. The mutual coupling effect will cause the true array manifold to deviate from the ideal situation, resulting in a serious deterioration of the performance of traditional high-resolution algorithms.

[0004] To effectively complete DOA estimation under mutual coupling, many improved algorithms have emerged, including subspace algorithms and sparse reconstruction algorithms. However, these algorithms only consider the non-ideality of the array manifold and ignore the non-ideality of the noise that widely exists in practice. They often assume that the noise is Gaussian white noise.

[0005] As a typical non-Gaussian noise, impulse noise is very common in atmospheric environments, underwater environments, and various electromagnetic environments. For example, multi-user interference in wireless communication systems, interference introduced by marine transportation and biological activities in underwater acoustic communications, and noise generated by man-made equipment such as microwave ovens and electric motors are all impulse noise. Impulse noise is generally modeled using an Alpha stable distribution with a heavy tailing characteristic. The decay rate of the tail of its probability density function is slower than that of Gaussian noise, so it presents a stronger impulse characteristic in the time domain.

[0006] Faced with impulse noise with frequent large outliers, the performance of the mutual coupling DOA estimation algorithm based on the Gaussian white noise assumption is significantly reduced. For complex scenarios where mutual coupling and impulse noise coexist, there is still little existing research work. The few existing solutions often require multiple iterations and are computationally intensive. Summary of the invention:

[0007] In order to meet the requirement of high real-time performance in practical applications, the present invention proposes a DOA estimation method based on preprocessing under mutual coupling and impulse noise, which can be referred to as Nonlinear Mutual Coupling-MUSIC (NLMC-MUSIC) algorithm. The present invention is based on subspace theory and can effectively complete DOA estimation under the coexistence environment of mutual coupling and impulse noise.

[0008] The present invention is achieved by the following measures:

[0009] A DOA estimation method based on preprocessing under mutual coupling and impulse noise, characterized by comprising the following steps:

[0010] Step 1: Establish a receiving signal model to obtain the receiving signal;

[0011] Step 2: Define a piecewise function to compress the abnormal amplitude value of the received signal and obtain a received data matrix. The piecewise function also plays a role in maintaining small amplitude values.

[0012] Step 3: Use the received data matrix to perform DOA estimation.

[0013] In step 1 of the present invention, it is assumed that the receiving array is an M-element uniform linear array, the distance between adjacent elements is d, and the relationship between the element spacing and the signal wavelength λ is d=λ / 2. For the incident signal, it is assumed that the directions of all K far-field narrowband signals are different and unrelated to each other. At a certain time t, the signal received by the array is:

[0014] x(t)=As(t)+n(t) (1),

[0015] In the formula, x(t)=[x1(t),x2(t),...,x M(t)] Τ , representing the data received by each array element at time t; A = [a(θ1),..., a(θ K )] is the steering vector matrix, and a(θ k ) = [1, ε(θ k ),..., ε M-1 (θ k )] Τ is the steering vector, where ε(θ k ) = exp(-i2πdsin(θ k ) / λ), θ k ∈[-90°, 90°] is the angle between the incident trajectory of the k-th signal and the array normal; s(t) = [s1(t),..., s K (t)] Τ is the incident signal; n(t) = [n1(t), n2(t),..., n M (t)] Τ is the additive noise;

[0016] Considering the mutual coupling effect between array elements, assuming that the mutual coupling is only related to the distance between array elements, that is, the mutual coupling matrix has a banded Toeplitz structure:

[0017]

[0018] where c p is the complex mutual coupling coefficient between the m-th and the (m + p)-th array elements, p = 0, 1,..., P - 1, m = 1, 2,..., M, and the magnitude relationship between mutual coupling coefficients is |c P-1 | <... |c1| < |c0| = 1, that is, the mutual coupling effect weakens with the increase of distance. Here, only the influence of the (m - P + 1)-th to the (m - 1)-th array elements and the (m + 1)-th to the (m + P - 1)-th array elements on the m-th array element is considered, and the influence of array elements farther away is ignored. Therefore, P < M, c P = c P+1 =... = c M = 0;

[0019] Let the number of snapshots be T, then the sampled received data is:

[0020] X = CAS + N (3),

[0021] That is, the received data model is shown in Equation (3). In the equation, X = [x(0),..., x(T - 1)] is the sampled received data matrix, S = [s(0),..., s(T - 1)] is the sampled signal matrix, and N = [n(0),..., n(T - 1)] is the sampled noise matrix.

[0022] In step 2 of the present invention, a piecewise function is defined as shown in Equation (4):

[0023]

[0024] The piecewise function is used to compress the amplitude values of anomalies while keeping small amplitude values unchanged. Before performing DOA estimation, to unify the scale, the elements of the received data matrix are normalized as follows:

[0025]

[0026] In the formula, |·| represents taking the modulus of each element of the matrix while keeping the matrix dimension unchanged; vec(·) represents the vectorization operation; median(·) represents finding the median. Subsequently, for the element x normalized in the i-th row (i = 1,..., M) and j-th column (j = 1,..., T) of X i,j the following processing is performed:

[0027]

[0028] In the formula, g(|x i,j |) is the processing of the amplitude of x i,j , which can compress the large outliers introduced by impulse noise; sgn(x i,j ) is the processing of the phase of x i,j , and sgn(x i,j ) = x i,j / |x i,j |, so the phase information of x i,j is retained. Denote the received data matrix after the processing in (6) as X * .

[0029] Step 3 of the present invention is specifically as follows: In the ideal case where the noise is Gaussian white noise and the number of snapshots is infinite, the covariance matrix of the array received data is:

[0030] R X = E[XX Η (7),

[0031] Performing eigenvalue decomposition on R X yields:

[0032]

[0033] In the formula, and are the signal subspace and the noise subspace respectively; and are diagonal matrices related to the signal power and the noise power respectively;

[0034] According to the identity transformation method, the actual steering vector is written as:

[0035] a m (θ) = Ca(θ) = Γ(θ)α(θ) (9),

[0036] where Γ(θ) = blkdiag(Γ1(θ), Γ2(θ), Γ3(θ)) is a block diagonal matrix, and the elements on its block diagonal are:

[0037] Γ1(θ) = diag(1, ε(θ),..., ε P-2 (θ)) (10),

[0038] Γ2(θ) = [ε P-1 (θ), ε P (θ),..., ε M-P (θ)] Τ (11),

[0039] Γ3(θ) = diag(ε M-P+1 (θ), ε M-P+2 (θ),..., ε M-1 (θ)) (12),

[0040] The column vector α(θ) = [u1(θ),..., u P-1 (θ), v, w1(θ),..., w P-1 (θ)] Τ The elements of are defined as:

[0041]

[0042]

[0043]

[0044] where i = 1,.., P - 1;

[0045] According to the subspace theory, the steering vector of the incident signal is orthogonal to the noise subspace. Therefore, when there is mutual coupling, the actual steering vector given by Equation (9) has the following relationship with the noise subspace:

[0046]

[0047] Substituting Equation (9) into Equation (17), we get:

[0048] α Η Q(θ)α = 0 (17),

[0049] where is a positive semi - definite Hermitian matrix; Assume that the number of mutual coupling coefficients, the number of signals, and the number of array elements satisfy \(K\leq M - 2P+1\), that is the number of rows and columns of satisfies \(2P - 1\leq M - K\). Then, k is column full rank, and \(Q(\theta)\) is full rank if and only if \(\theta\) is equal to the signal incident angle, that is, \(\theta=\theta

[0050]

[0051] where \(det(\cdot)\) is the determinant operation, is calculated as and is obtained from the eigen - decomposition of / T.

[0052] Compared with the prior art, the present invention has the following advantages: (1) It does not require iterative solution and has a small computational amount; (2) When the impulse noise characteristic index \(\alpha\geq1\), both the success rate and the RMSE performance are better than those of the existing DOA estimation algorithms, etc. BRIEF DESCRIPTION OF THE DRAWINGS:

[0053] FIG Figure 1 is a success rate curve diagram under different generalized signal - to - noise ratios when the characteristic index is 0.5 in the embodiment of the present invention.

[0054] FIG Figure 2 is a success rate curve diagram under different generalized signal - to - noise ratios when the characteristic index is 1 in the embodiment of the present invention.

[0055] FIG Figure 3 is a success rate curve diagram under different generalized signal - to - noise ratios when the characteristic index is 1.5 in the embodiment of the present invention.

[0056] FIG Figure 4 is a success rate curve diagram under different generalized signal - to - noise ratios when the characteristic index is 2 in the embodiment of the present invention. FIG Figure 5 is a success rate curve diagram under different characteristic indices when the generalized signal - to - noise ratio is 3 dB in the embodiment of the present invention.

[0057] FIG Figure 6 is the success rate under different snapshot numbers in the embodiment of the present invention.

[0058] FIG Figure 7 is the RMSE diagram under different generalized signal - to - noise ratios when the characteristic index is 1 in the embodiment of the present invention. FIG Figure 8 is the RMSE diagram under different generalized signal - to - noise ratios when the characteristic index is 1.5 in the embodiment of the present invention.

[0059] Appendix Figure 9 This is the RMSE graph under different generalized signal-to-noise ratios when the characteristic index is 2 in the embodiment of the present invention. Appendix Figure 10 This is the RMSE graph under different numbers of snapshots in the embodiment of the present invention.

[0060] Appendix Figure 11 This is the schematic diagram of the running time under different numbers of snapshots in the embodiment of the present invention.

[0061] Appendix Figure 12 This is the partial schematic diagram of the running time under different numbers of snapshots in the embodiment of the present invention. Specific implementation manner:

[0062] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0063] Embodiment:

[0064] This example provides a DOA estimation method based on preprocessing under mutual coupling and impulse noise, which specifically includes the following steps:

[0065] First, perform signal modeling: Assume that the receiving array is a uniform linear array of M elements, and the distance between adjacent elements is d. The relationship between the element spacing and the signal wavelength λ is d = λ / 2. For the incident signals, assume that all K far-field narrowband signals have different directions and are mutually uncorrelated. At a certain moment t, the signal received by the array is:

[0066] x(t) = As(t) + n(t) (1),

[0067] where x(t) = [x1(t), x2(t),..., x M (t)] Τ , representing the data received by each element at time t; A = [a(θ1),..., a(θ K )] is the steering vector matrix, and a(θ k ) = [1, ε(θ k ),..., ε M-1 (θ k )] Τ is the steering vector, where ε(θ k ) = exp(-i2πdsin(θ k ) / λ), θ k ∈[-90°, 90°] is the angle between the incident trajectory of the kth signal and the normal of the array; s(t) = [s1(t),..., s K (t)] Τ is the incident signal; n(t) = [n1(t), n2(t),..., n M (t)]Τ is additive noise.

[0068] Next, consider the mutual coupling effect between array elements. To simplify the problem under study, assume that the mutual coupling is only related to the distance between array elements, that is, the mutual coupling matrix has a banded Toeplitz structure:

[0069]

[0070] where c p is the complex mutual coupling coefficient between the mth and the (m + p)th array elements, p = 0, 1,..., P - 1, m = 1, 2,..., M, and the magnitude relationship between the mutual coupling coefficients is |c P-1 | <... |c1| < |c0| = 1, that is, the mutual coupling effect decreases with the increase of distance. Here, only consider the influence of the (m - P + 1)th to the (m - 1)th array elements and the (m + 1)th to the (m + P - 1)th array elements on the mth array element, and ignore the influence of array elements farther away. Therefore, P < M, c P = c P+1 =... = c M = 0.

[0071] Let the number of snapshots be T, then the received data after sampling is:

[0072] X = CAS + N (3),

[0073] where X = [x(0),..., x(T - 1)] is the received data matrix after sampling, S = [s(0),..., s(T - 1)] is the signal matrix after sampling, and N = [n(0),..., n(T - 1)] is the noise matrix after sampling; the received data model is shown in Equation (3).

[0074] Thereafter, define a piecewise function:

[0075]

[0076] The piecewise function is used to compress the abnormal amplitude values and at the same time keep the small amplitude values. Before performing DOA estimation, in order to unify the scale, normalize the elements of the received data matrix:

[0077]

[0078] where |·| represents taking the modulus of each element of the matrix while keeping the matrix dimension unchanged; vec(·) represents the vectorization operation; median(·) represents finding the median. Subsequently, for the element x normalized in the ith row (i = 1,..., M) and the jth column (j = 1,..., T) of X i,j perform the following processing:

[0079]

[0080] wherein, g(|x i,j |) is the processing of the magnitude of x i,j , which can compress large outliers introduced by impulse noise; sgn(x i,j ) is the processing of the phase of x i,j , and sgn(x i,j ) = x i,j / |x i,j |, so the phase information of x i,j is retained;

[0081] Denote the received data matrix after being processed by (6) as X * , and the following uses X * to perform DOA estimation. In the ideal case where the noise is Gaussian white noise and the number of snapshots is infinite, the covariance matrix of the array received data is:

[0082] R X = E[XX Η (7),

[0083] Performing eigenvalue decomposition on R X gives:

[0084]

[0085] wherein, and are the signal subspace and the noise subspace respectively; and are diagonal matrices related to the signal power and the noise power respectively.

[0086] In this example, according to the identity transformation method (recorded in Liao B, Zhang Z - G, Chan S - C. DOA estimation and tracking of ULAs with mutual coupling[J]. IEEE Transactions on Aerospace and Electronic Systems, 2012, 48(1):891 - 905), the actual steering vector is written as:

[0087] a m (θ) = Ca(θ) = Γ(θ)α(θ) (9),

[0088] wherein, Γ(θ) = blkdiag(Γ1(θ), Γ2(θ), Γ3(θ)) is a block - diagonal matrix, and the elements on its block diagonal are:

[0089] Γ1(θ) = diag(1, ε(θ),..., ε P-2 (θ)) (10),

[0090] Γ2(θ) = [ε P-1 (θ), ε P (θ),..., ε M-P (θ)] Τ (11),

[0091] Γ3(θ) = diag(ε M-P+1 (θ), ε M-P+2 (θ),..., ε M-1 (θ)) (12),

[0092] The column vector α(θ) = [u1(θ),..., u P-1 (θ), v, w1(θ),..., w P-1 (θ)] Τ The elements of are defined as:

[0093]

[0094]

[0095]

[0096] where i = 1,.., P - 1.

[0097] According to subspace theory, the steering vector of the incident signal is orthogonal to the noise subspace. Therefore, when there is mutual coupling, the actual steering vector given by Equation (9) has the following relationship with the noise subspace:

[0098]

[0099] Substituting Equation (9) into Equation (17), we get:

[0100] α Η Q(θ)α = 0 (17),

[0101] where is a positive semi - definite Hermitian matrix; Assume that the number of mutual coupling coefficients, the number of signals, and the number of array elements satisfy K ≤ M - 2P + 1, that is The number of rows and columns of satisfies 2P - 1 ≤ M - K. Then, in most cases, is full column rank and Q(θ) is full rank. If and only if θ is equal to the signal incident angle, that is θ = θ kWhen \(k = 1,\cdots,K\), equation (17) holds. At this time, \(Q(\theta)\) is rank-deficient and its determinant is equal to zero. Therefore, the determinant of \(Q(\theta)\) can be used to plot the spatial spectrum.

[0102] In the actual situation where the noise is impulse noise and the number of snapshots is limited, the spatial spectrum is given by the following formula:

[0103]

[0104] where \(\det(\cdot)\) is the determinant operation, is calculated as while is obtained from the eigen-decomposition of \(\mathbf{R}_{xx} / T\).

[0105] Next, from the two aspects of success rate and RMSE, the subspace-based NLMC-MUSIC algorithm proposed in this example is compared with several existing algorithms to illustrate its superiority.

[0106] The comparison algorithms are as follows: for the subspace algorithm for mutual coupling (recorded in Tian Y, Wang R, Chen H, et al. Real-Valued DOA Estimation Utilizing Enhanced Covariance Matrix With Unknown Mutual Coupling[J]. IEEE Communications Letters, 2022, 26(4): 912-6), hereinafter referred to as "ECM-MUSIC" (i.e., Enhanced Covariance Matrix-MUSIC, the enhanced covariance matrix MUSIC algorithm); for the subspace algorithm for impulsive noise (recorded in Cheng L, Chao J, Ding S, et al. A Robust Direction of Arrival Estimation Method in Impulsive Noise; proceedings of the 2022 4th International Conference on Intelligent Control, Measurement and Signal Processing(ICMSP), F, 2022[C]. IEEE), hereinafter referred to as "tanh-MUSIC"; for the sparse algorithm for mutual coupling and impulsive noise (recorded in Zhang J, Qiu T, Luan S. Robust Sparse Representation for DOA Estimation With Unknown Mutual Coupling Under Impulsive Noise[J]. IEEE Communications Letters, 2020, 24(7): 1455-8), hereinafter referred to as "Cauchy IHT".

[0107] The basic parameter settings during simulation are as follows: the receiving array is a uniform linear array with 16 array elements, and the distance between adjacent array elements is λ / 2, where λ is the signal wavelength; the mutual coupling coefficients are set to 0.386 4 - 0.377 6i, 0.2171 - 0.190 0i, 0.100 0 + 0.170 6i. The relative intensity of impulsive noise is measured by the generalized signal to noise ratio (GSNR), and its definition is:

[0108]

[0109] where, ω sdenotes the signal average power, and γ denotes the noise dispersion coefficient. For the impulse noise sample n l = Re(n l ) + iIm(n l ), l = 1, ..., L. Assume that the real part Re(n l ) and the imaginary part Im(n l ) of the sample satisfy independent and identically distributed, and both follow a symmetric Alpha-stable distribution with symmetric parameter β = 0, location parameter δ = 0, and characteristic exponent α ∈ [0.5, 2]. In particular, when α = 2, the symmetric Alpha-stable distribution degenerates into a Gaussian distribution, that is, the symmetric Alpha-stable distribution noise degenerates into Gaussian noise.

[0110] First, verify the effectiveness of the proposed method from the perspective of the success rate. The success rate is defined as the ratio of the number of successful times to the total number of experiments. In any experiment, if the DOA estimation value and the true value satisfy the following relationship, then this estimation is considered successful:

[0111]

[0112] where, and θ k denote the estimated value and the true value of the DOA of the k-th signal respectively. When comparing the success rates, assume that the two incident signals come from θ1 = -15° and θ2 = 20° respectively; the search grids of the sparse class algorithm and the subspace class algorithm are both set to {-90°, -89°,..., 89°, 90°}.

[0113] Figures 1 to 4 The variation of the success rates of different algorithms with the generalized signal-to-noise ratio is given. The noise characteristic exponent α is fixed at 0.5, 1, 1.5, and 2 respectively, the number of snapshots is fixed at 200, and at the same time, the number of Monte Carlo experiments is set to 400. Among them, Figures 1 to 3Results under impulsive noise: As can be seen from the figure, the performance of ECM-MUSIC proposed based on the Gaussian noise assumption degrades severely, especially when the impulsive characteristics of the noise are strong (α = 0.5 and α = 1). The success rate of the algorithm is almost zero at various signal-to-noise ratios. Cauchy IHT can achieve a relatively high success rate, but when the generalized signal-to-noise ratio is low, the algorithm performance deteriorates significantly. For the proposed NLMC-MUSIC algorithm, when α = 0.5 and the generalized signal-to-noise ratio is less than 6 dB, its success rate is lower than that of Cauchy IHT. However, in the cases of α = 1 and α = 1.5, the success rate of the NLMC-MUSIC algorithm is better than that of Cauchy IHT, especially when the generalized signal-to-noise ratio is less than 3 dB. In the strong impulsive environments of α = 0.5 and α = 1, the performance of the tanh-MUSIC algorithm is poor, and especially when α = 0.5, the algorithm completely fails. When α = 1.5, the impulsive intensity weakens, and the success rate of tanh-MUSIC can reach nearly 100% after the generalized signal-to-noise ratio is greater than 3 dB, but when the generalized signal-to-noise ratio is lower than 3 dB, its performance is inferior to that of NLMC-MUSIC.

[0114] Figure 4 Results under Gaussian noise (α = 2): When the generalized signal-to-noise ratio is less than 3 dB, there is still an obvious performance degradation for Cauchy IHT, and its success rate is lower than that of ECM-MUSIC. While ECM-MUSIC and tanh-MUSIC can achieve a success rate close to 100% under Gaussian noise with different signal-to-noise ratios, which is not only better than Cauchy IHT but also surpasses the ECM-MUSIC algorithm proposed based on Gaussian noise.

[0115] Figure 5 The variation of the success rate of different algorithms with the noise characteristic index α is given, with the generalized signal-to-noise ratio fixed at 3 dB and the number of snapshots fixed at 200. This figure shows that as the characteristic index decreases (the impulsive characteristics become stronger), the ECM-MUSIC proposed based on Gaussian noise starts to show an obvious performance degradation first. Subsequently, the tanh-MUSIC shows an obvious performance degradation after α < 1.3, while the proposed NLMC-MUSIC shows an obvious performance degradation only after α < 0.7. Although the success rate of NLMC-MUSIC is lower than that of Cauchy IHT under strong impulses with α ≤ 0.6, when α > 0.7, the success rate of NLMC-MUSIC is higher than that of Cauchy IHT.

[0116] Figure 9The success rates of different algorithms are given as a function of the number of snapshots, with the noise characteristic exponent α fixed at 1.5 and the generalized signal-to-noise ratio fixed at 3 dB. As can be seen from the figure, ECM-MUSIC cannot effectively handle impulsive noise. As the number of snapshots increases, the change in the success rate of the algorithm is unstable and generally low; the proposed NLMC-MUSIC and tanh-MUSIC belong to subspace algorithms. Their success rates are lower than that of the sparse Cauchy IHT when the number of snapshots is less than 100. However, as the number of snapshots increases, the success rate of NLMC-MUSIC quickly reaches 100%, while the success rate of tanh-MUSIC fluctuates slightly.

[0117] The above simulation experiments have demonstrated the effectiveness of the proposed NLMC-MUSIC algorithm from the perspective of the success rate. To further verify the accuracy of the DOA estimation results of the algorithm, the RMSEs of different algorithms are compared below. The definition of RMSE is as follows:

[0118]

[0119] where T is the number of Monte Carlo experiments; K is the number of spatial signal sources; is the estimated DOA value of the k-th signal source measured in the t-th Monte Carlo experiment; θ k is the actual incident angle of the k-th signal source.

[0120] When comparing the RMSEs, it is assumed that the incident signals come from θ1 = 20°; the search grids of the sparse algorithms and subspace algorithms range from 0° to 40°, with an interval of 0.2°. At the same time, the number of Monte Carlo experiments is set to 400.

[0121] Figures 7 to 9 The RMSEs of different algorithms are given as a function of the generalized signal-to-noise ratio, with the noise characteristic exponents α fixed at 1, 1.5, and 2 respectively, and the number of snapshots fixed at 200. Among them, Figures 7 to 8 represents the results under impulsive noise. As can be seen from the figure, the ECM-MUSIC algorithm proposed based on the Gaussian noise assumption is close to failure; Cauchy IHT can achieve a relatively low RMSE when the generalized signal-to-noise ratio is greater than 1 dB, but as the generalized signal-to-noise ratio decreases, the performance of the algorithm deteriorates severely; while the RMSE of the proposed algorithm is generally lower than that of the three comparison algorithms, and the performance degradation only begins when the generalized signal-to-noise ratio is lower than -4 dB.

[0122] Figure 9Results for Gaussian noise (α = 2): When the generalized signal-to-noise ratio is less than 1 dB, there is a significant performance degradation in Cauchy IHT; as the generalized signal-to-noise ratio decreases, although the RMSE of ECM-MUSIC does not increase significantly, the overall RMSE is relatively high within the given range of the generalized signal-to-noise ratio. The proposed NLMC-MUSIC algorithm and tanh-MUSIC algorithm can maintain a low RMSE under Gaussian noise with different signal-to-noise ratios. However, as the generalized signal-to-noise ratio increases, the RMSE of NLMC-MUSIC gradually becomes lower than that of tanh-MUSIC.

[0123] Figure 10 The variation of RMSE with the number of snapshots for different algorithms is given, where the noise characteristic exponent α is fixed at 1.5 and the generalized signal-to-noise ratio is fixed at 3 dB. As can be seen from the figure, the RMSE of ECM-MUSIC is at a relatively high level for different numbers of snapshots. For the set target angles and grid sizes, this algorithm proposed based on the Gaussian noise assumption cannot effectively complete DOA estimation; as the number of snapshots increases, the RMSE of the proposed algorithms gradually decreases and approaches the RMSE of Cauchy IHT, while the RMSE of tanh-MUSIC does not show a downward trend with the increase in the number of snapshots and remains at about 0.4.

[0124] Next, the average running times of the proposed NLMC-MUSIC algorithm and the existing Cauchy IHT algorithm are compared. When comparing the average running times, it is assumed that the two incident signals come from θ1 = -15° and θ2 = 20° respectively; the search grids of both the sparse type algorithm and the subspace type algorithm are set to {-90°, -89°,..., 89°, 90°}. At the same time, the number of Monte Carlo experiments is set to 400. Figures 11 to 12 The variation of the average running time with the number of snapshots for different algorithms is given, where the noise characteristic exponent α is fixed at 1.5 and the generalized signal-to-noise ratio is fixed at 10 dB. As can be seen from the figure, the average running time of the NLMC-MUSIC algorithm is much less than that of Cauchy IHT, and the time consumption does not increase significantly with the increase in the number of snapshots, showing significant advantages.

[0125] The present invention proposes a subspace type DOA estimation algorithm to solve the DOA estimation problem in the complex scenario where mutual coupling and impulsive noise coexist. The proposed algorithm uses an identity transformation to eliminate the influence of the mutual coupling effect, and compresses the outliers in the received data through a custom nonlinear function, thereby effectively suppressing impulsive noise. The simulation results show that the proposed algorithm has strong robustness. When the impulsive noise characteristic exponent α ≥ 1, both the success rate and RMSE performance are better than those of the existing DOA estimation algorithms. In addition, the proposed algorithm does not require iterative solution, so it is significantly superior to the existing DOA estimation algorithms based on sparse reconstruction under mutual coupling and impulsive noise in terms of operation speed.

Claims

1. A DOA estimation method based on preprocessing under mutual coupling and impulse noise, characterized in that It includes the following steps: Step 1: Establish a received signal model to obtain the received signal; Step 2: Define a piecewise function to compress the abnormal amplitude values of the received signal to obtain a received data matrix, and the piecewise function also plays a role in maintaining small amplitude values; Step 3: Use the received data matrix for DOA estimation.

2. The DOA estimation method based on preprocessing under mutual coupling and impulse noise according to claim 1, wherein In Step 1, assume that the receiving array is a uniform linear array with M array elements, the distance between adjacent array elements is d, and the relationship between the array element spacing and the signal wavelength λ is d = λ / 2. For the incident signal, assume that the directions of all K far-field narrowband signals are different and mutually uncorrelated. At a certain moment t, the signal received by the array is: x(t) = As(t) + n(t) (1), where \(x(t)=[x_1(t),x_2(t),\cdots,x M (t)] Τ \) represents the data received by each array element at time \(t\); \(A = [a(\theta_1),\cdots,a(\theta K )]\) is the steering vector matrix, and \(a(\theta k ) = [1,\varepsilon(\theta k ),\cdots,\varepsilon M-1 (\theta k )] Τ \) is the steering vector, where \(\varepsilon(\theta k )=\exp(-i2\pi d\sin(\theta k )) / \lambda\), \(\theta k \in[-90^{\circ},90^{\circ}]\) is the angle between the incident trajectory of the \(k\)-th signal and the array normal; \(s(t)=[s_1(t),\cdots,s K (t)] Τ \) is the incident signal; \(n(t)=[n_1(t),n_2(t),\cdots,n M (t)] Τ \) is the additive noise; Considering the mutual coupling effect between array elements, assume that the mutual coupling is only related to the distance between array elements, that is, the mutual coupling matrix has a banded Toeplitz structure: where c p is the complex mutual coupling coefficient between the m-th and the (m + p)-th array elements, p = 0, 1, ..., P - 1, m = 1, 2, ..., M, and the magnitude relationship between the mutual coupling coefficients is |c P-1 | < ... |c1| < |c0| = 1, that is, the mutual coupling effect decreases with the increase of distance. Here, only the influence of the (m - P + 1)-th to the (m - 1)-th array elements and the (m + 1)-th to the (m + P - 1)-th array elements on the m-th array element are considered, and the influence of array elements with a farther distance is ignored. Therefore, P < M, c P = c P+1 =... = c M = 0; Let the number of snapshots be T, then the sampled received data is: X = CAS + N (3), That is, the received data model is as shown in Equation (3). In the formula, X = [x(0),..., x(T-1)] is the sampled received data matrix, S = [s(0),..., s(T-1)] is the sampled signal matrix, and N = [n(0),..., n(T-1)] is the sampled noise matrix.

3. A DOA estimation method based on preprocessing under mutual coupling and impulse noise according to claim 1, characterized in that In Step 2, define the piecewise function as Equation (4): The piecewise function is used to compress the abnormal amplitude values and also plays a role in maintaining small amplitude values. Before performing DOA estimation, in order to unify the scale, normalize the elements of the received data matrix: where |·| represents taking the modulus of each element in the matrix while keeping the matrix dimension unchanged; vec(·) represents the vectorization operation; median(·) represents finding the median. Subsequently, for the element x normalized in the i-th row (i = 1,..., M) and j-th column (j = 1,..., T) of X i,j the following processing is performed: where g(|x i,j |) is the processing of the amplitude of x i,j , which can compress large outliers introduced by impulse noise; sgn(x i,j ) is the processing of the phase of x i,j , and sgn(x i,j ) = x i,j / |x i,j |, so the phase information of x i,j is retained. Denote the received data matrix after the processing in (6) as X * .

4. A DOA estimation method based on preprocessing under mutual coupling and impulse noise according to claim 1, characterized in that Step 3 is specifically: In the ideal case where the noise is Gaussian white noise and the number of snapshots is infinite, the covariance matrix of the array received data is: R X = E[XX Η (7), For R X Performing eigen decomposition gives: wherein, and are the signal subspace and the noise subspace respectively; and are diagonal matrices related to the signal power and the noise power respectively; According to the identity transformation method, the actual steering vector is written as: a m (θ) = Ca(θ) = Γ(θ)α(θ) (9), In the formula, Γ(θ) = blkdiag(Γ1(θ), Γ2(θ), Γ3(θ)) is a block diagonal matrix, and the elements on its block diagonal are: Γ1(θ) = diag(1, ε(θ),..., ε P-2 (θ)) (10), Γ2(θ) = [ε P-1 (θ), ε P (θ),..., ε M-P (θ)] Τ (11), Γ3(θ) = diag(ε M-P+1 (θ), ε M-P+2 (θ),..., ε M-1 (θ)) (12), The column vector α(θ) = [u1(θ),..., u P-1 (θ), v, w1(θ),..., w P-1 (θ)] Τ is defined as follows: In the formula, i = 1,.., P-1; According to the subspace theory, the steering vector of the incident signal is orthogonal to the noise subspace. Therefore, when there is mutual coupling, the relationship between the actual steering vector given by Equation (9) and the noise subspace is as follows: Substitute Equation (9) into Equation (17) to get: α Η Q(θ)α = 0 (17), wherein, is a positive semi - definite Hermitian matrix; Assume that the number of mutual coupling coefficients, the number of signals, and the number of array elements satisfy \(K\leq M - 2P+1\), that is the number of rows and columns of satisfies \(2P - 1\leq M - K\), then is full column rank, \(Q(\theta)\) is full rank, if and only if \(\theta\) is equal to the signal incident angle, that is \(\theta=\theta\) k , \(k = 1,\cdots,K\), Equation (17) holds. At this time, \(Q(\theta)\) is rank - deficient and its determinant is equal to zero. In the actual case where the noise is impulse noise and the number of snapshots is limited, the spatial spectrum is given by the following formula: where det(·) represents the determinant operation, is calculated as and is obtained from the eigen - decomposition of ​

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