# DOA Estimation Method of Weak Signal under the Compound Background of Strong Interference and Colored Noise

Authors: Bin Lin; Guoping Hu; Hao Zhou; Guimei Zheng; Yuwei Song
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

The traditional algorithm performing direction of arrival (DOA) estimation under the background of strong interference and colored noise has the problems of low estimation accuracy and small measurement targets. Based on the construction of a fourth-order cumulant (FOC) matrix to suppress colored noise, this paper adopts the extended noise subspace (ENS) algorithm and the fixed projection blocking (FPB) algorithm to estimate the DOA of weak targets. Firstly, a FOC matrix of the received signal vector is established to curb the noise component, and the eigenvalue decomposition is performed. Then, two approaches of weak signal DOA estimation are proposed. One approach is to merge the space where the strong interference steering vector lies into the noise subspace to construct an extended noise subspace, and then, the multisignal classification (MUSIC) algorithm is used to obtain the DOA estimation of the weak signal on the basis of the extended noise subspace. Another approach is to build the orthogonal projection matrix of the interference subspace as the interference blocking matrix, and the receiving array signal is preprocessed, and on the basis of it, the eigen decomposition is performed again to obtain the DOA information of the weak signal. Both algorithms make breakthroughs in the aperture limitation of the traditional algorithm, effectively expand the aperture, and promote the accuracy of estimation. The simulation tests the effectiveness of the proposed method.


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## Body

## 1. Introduction

With the increasing intricacy of the electromagnetic environment, radar detection is interference by more and more electromagnetic, making it increasingly difficult for radar to detect targets [1, 2]. Strong electromagnetic interference will cover up the target's echo signal or overload the signal processor and receiver [3, 4]. When the power of the interference signal is greater than that of the echo signal, the received data is directly used in the estimation of the direction of arrival, and the false peak resulting from strong interference will be regarded as the peak of the source, which will give birth to the misinterpretation of the correct angle [5, 6]. Firstly, at present, one method to address this problem is signal separation, mainly including the Relaxation (Relax) algorithm [7] and the CLEAN algorithm [8]. This type of algorithm separates the output part of the array of all signals into multiple data blocks and then determines and removes the data blocks containing the interference signals according to the characteristics of the interference signal so as to achieve target recognition. However, this type of algorithm is extremely complex. Secondly, the jamming jam method (JJM) was proposed by Fang et al. and its extended application [9, 10]. The core concept is to use the angle information of the signal receiving matrix to establish an interference blocking matrix to eliminate the interference signal part of the received signal covariance matrix. However, the algorithm requires a precise prediction of the direction of interference. In addition, there is an Extended Noise Subspace algorithm [11]. The algorithm first constructs an extended noise subspace for strong interferences and noises, and then performs conventional DOA estimation. Dong et al. [12] deduced an interference blocking method based on modified projection. The interference direction is not a necessity in this approach, and it is of more practical use.However, the abovementioned strong interference rejection algorithms are characterized by low estimation accuracy and small measurement targets in the direction finding under the background of nonideal colored noise. According to related literature, the calculation of high-order cumulants is blind Gaussian which can achieve aperture expansion and provides a powerful tool in response to Gaussian colored noise. Literature [13, 14] accordingly proposed a variety of FOC-based algorithms, which have better angle estimation performance under Gaussian colored noise conditions. Nonetheless, there are still some problems in terms of angular resolution and complexity. Tufail and Ahmed [15] used the FOC and ESPRIT algorithm to propose DOA estimation based on the genetic algorithm (GA) and obtained the multiple invariant cumulant ESPRIT algorithm, which has a better angular resolution, but the problem of excessive complexity remains unsolved. Literature [16] used the real-valued sparse Bayesian learning method to transform the FOC matrix into a real-valued matrix and simplify the algorithm through unitary transformation. The abovementioned FOC algorithm has made certain progress in the field of array signal processing, but the research on the background of strong interference is scarce.To tackle the above problems, this paper puts forward the fourth-order cumulant-expanded noise subspace (FOC-ENS) algorithm and the fourth-order cumulant-fixed projection blocking (FOC-FPB) algorithm. One is to use the FOC matrix to substitute for the traditional covariance matrix. After eigenvalue decomposition, the space of the strong interference steering vector is incorporated into the noise subspace, and thus, the extended noise subspace is constructed. And the conventional MUSIC algorithm is used to obtain the DOA estimation of the weak signal on the basis of the extended noise subspace. The other method is to construct the orthogonal projection matrix of the strong interference signal subspace as the interference blocking matrix, and the received array signal is preprocessed. And the eigenvalue decomposition is performed again to obtain the noise subspace. Eventually, the MUSIC algorithm is used to search for spectral peaks to obtain the azimuth angle of the weak signal. Compared with traditional algorithms, this algorithm enjoys higher estimation accuracy under a variety of conditions.


## 2. Signal Model and FOC Vector Formulation

### 2.1. Signal Model

Assuming thatK far-field narrowband signals and J strong interference signals Siti=1,2,…,J,…,K+J are incident on a uniform linear array, the incident angle is θi, and if the first J is a strong interference signal, their power satisfies σ12>σ22>⋯>σJ2≫σJ+12>⋯>σK+J2. The distance d of the array elements is half of the signal wavelength. The array is composed of M array elements, each of which is omnidirectional. The received signal of the m-th array element of the array at time t can be expressed as(1)xmt=∑i=1K+Jsitexpjm−1πsinθi+nmt,where nmt is the Gaussian colored noise of the m-th element at time t and sit is the complex envelope of the i-th source.Assuming that the plane wave propagates along a straight line, the array response vector corresponding to the direction can be expressed as(2)aθi=1,exp−jπsinθi,…,exp−jπM−1sinθiT,among which, (·) T represents the transpose of the matrix.The direction matrix is defined as(3)Aθ=aθ1,aθ2,…,aθJ,…,aθK+J.This matrix is aM×K+J dimensional direction matrix.The vector output of the array element is expressed as(4)Xt=AθSt+Nt.In the above formula,St is the K+J×1 dimensional signal vector and Nt is the M×1 dimensional Gaussian colored noise vector.


### 2.2. Observation Model Based on FOC Vector

Compared with the second-order cumulant, the high-order cumulant can better characterize the signal characteristics and has the blind Gaussian property. Therefore, in the actual array signal processing, the fourth-order cumulant is frequently used for processing.Regarding a uniform linear array, according to the symmetric definition of the FOC of the zero-mean stable random process, the FOC of the array received data is(5)cumxk1,xk2,xk3∗,xk4∗=Exk1xk2xk3∗xk4∗−Exk1xk3∗Exk2xk4∗−Exk1xk4∗Exk2xk3∗−Exk1xk2Exk3∗xk4∗.In the above formula,⋅∗ represents conjugate and cum() represents cumulant.According to literature [13], it is easy to get(6)Cx=EX⊗X∗X⊗X∗H−EX⊗X∗EX⊗X∗H−.EXXH⊗EXXH∗In the above formula,⊗ represents the Kronecker product.Substituting the signal vector into the above formula can obtain the FOC matrix of the signal as follows:(7)Cs=ES⊗S∗S⊗S∗H−ES⊗S∗ES⊗S∗H−ESSH⊗ESSH∗,where the signal FOC matrix Cs is a K2×K2 complex matrix, and its i+1K+j row and k−1K+l column elements can be expressed as(8)cumsi,sj,sk∗,sl∗,∀i,j,k,l∈1,2,…,K.With respect to an independent signal source, according to the nature of the higher-order cumulant, the elements in the FOCCs of the signal are not zero if and only if i=j=k=l, and all other elements are zero, namely,(9)Cs=cumsi,sj,sk∗,sl∗≠0,i=j=k=l,=0,otherwise .Hence, there are onlyK nonzero elements in Cs, and they are situated at position k−1K+kk=1,2,…,K on the diagonal of Cs. Now, the rows and columns are deleted with all zero elements in Cs; Cs is reduced from the K2×K2 dimensional matrix to the K×K dimensional diagonal matrix. Then, the simplified signal FOC matrix can be expressed as(10)Cs=diagγ1,γ2,…,γk.In the above formula,γk=cumsk,sk,sk∗,sk∗.It can also be obtained that the FOC matrix of noise is(11)Cn=EN⊗N∗N⊗N∗H−EN⊗N∗⋅EN⊗N∗H−EN⋅NH⊗EN⋅NH∗.In addition, the array steering vector after array expansion according to the FOC defined above is(12)bθ=aθ⊗a∗θ.The expanded direction matrix is(13)Bθ=bθ1bθ2⋯bθk=aθ1⊗a∗θ1aθ2⊗a∗θ2⋯aθK⊗a∗θK.If each signal source is completely independent, the following formula holds [17]:(14)Cx=BθCsBHθ+Cn.Assuming that the noise is Gaussian noise, whether it is white noise or colored noise with unknown spectral characteristics, the theoretical definition by the above formulaCn should always be a zero matrix. Due to noise deviation from Gaussian or finite data length calculation accuracy, Cn is a matrix approaching 0.


## 2.1. Signal Model

Assuming thatK far-field narrowband signals and J strong interference signals Siti=1,2,…,J,…,K+J are incident on a uniform linear array, the incident angle is θi, and if the first J is a strong interference signal, their power satisfies σ12>σ22>⋯>σJ2≫σJ+12>⋯>σK+J2. The distance d of the array elements is half of the signal wavelength. The array is composed of M array elements, each of which is omnidirectional. The received signal of the m-th array element of the array at time t can be expressed as(1)xmt=∑i=1K+Jsitexpjm−1πsinθi+nmt,where nmt is the Gaussian colored noise of the m-th element at time t and sit is the complex envelope of the i-th source.Assuming that the plane wave propagates along a straight line, the array response vector corresponding to the direction can be expressed as(2)aθi=1,exp−jπsinθi,…,exp−jπM−1sinθiT,among which, (·) T represents the transpose of the matrix.The direction matrix is defined as(3)Aθ=aθ1,aθ2,…,aθJ,…,aθK+J.This matrix is aM×K+J dimensional direction matrix.The vector output of the array element is expressed as(4)Xt=AθSt+Nt.In the above formula,St is the K+J×1 dimensional signal vector and Nt is the M×1 dimensional Gaussian colored noise vector.


## 2.2. Observation Model Based on FOC Vector

Compared with the second-order cumulant, the high-order cumulant can better characterize the signal characteristics and has the blind Gaussian property. Therefore, in the actual array signal processing, the fourth-order cumulant is frequently used for processing.Regarding a uniform linear array, according to the symmetric definition of the FOC of the zero-mean stable random process, the FOC of the array received data is(5)cumxk1,xk2,xk3∗,xk4∗=Exk1xk2xk3∗xk4∗−Exk1xk3∗Exk2xk4∗−Exk1xk4∗Exk2xk3∗−Exk1xk2Exk3∗xk4∗.In the above formula,⋅∗ represents conjugate and cum() represents cumulant.According to literature [13], it is easy to get(6)Cx=EX⊗X∗X⊗X∗H−EX⊗X∗EX⊗X∗H−.EXXH⊗EXXH∗In the above formula,⊗ represents the Kronecker product.Substituting the signal vector into the above formula can obtain the FOC matrix of the signal as follows:(7)Cs=ES⊗S∗S⊗S∗H−ES⊗S∗ES⊗S∗H−ESSH⊗ESSH∗,where the signal FOC matrix Cs is a K2×K2 complex matrix, and its i+1K+j row and k−1K+l column elements can be expressed as(8)cumsi,sj,sk∗,sl∗,∀i,j,k,l∈1,2,…,K.With respect to an independent signal source, according to the nature of the higher-order cumulant, the elements in the FOCCs of the signal are not zero if and only if i=j=k=l, and all other elements are zero, namely,(9)Cs=cumsi,sj,sk∗,sl∗≠0,i=j=k=l,=0,otherwise .Hence, there are onlyK nonzero elements in Cs, and they are situated at position k−1K+kk=1,2,…,K on the diagonal of Cs. Now, the rows and columns are deleted with all zero elements in Cs; Cs is reduced from the K2×K2 dimensional matrix to the K×K dimensional diagonal matrix. Then, the simplified signal FOC matrix can be expressed as(10)Cs=diagγ1,γ2,…,γk.In the above formula,γk=cumsk,sk,sk∗,sk∗.It can also be obtained that the FOC matrix of noise is(11)Cn=EN⊗N∗N⊗N∗H−EN⊗N∗⋅EN⊗N∗H−EN⋅NH⊗EN⋅NH∗.In addition, the array steering vector after array expansion according to the FOC defined above is(12)bθ=aθ⊗a∗θ.The expanded direction matrix is(13)Bθ=bθ1bθ2⋯bθk=aθ1⊗a∗θ1aθ2⊗a∗θ2⋯aθK⊗a∗θK.If each signal source is completely independent, the following formula holds [17]:(14)Cx=BθCsBHθ+Cn.Assuming that the noise is Gaussian noise, whether it is white noise or colored noise with unknown spectral characteristics, the theoretical definition by the above formulaCn should always be a zero matrix. Due to noise deviation from Gaussian or finite data length calculation accuracy, Cn is a matrix approaching 0.


## 3. DOA Estimation Based on FOC Vector

Eigenvalue decomposition is performed onCx, and its eigenvalues are arranged from large to small as λ1,λ2,…,λM2, and the corresponding eigenvector is e1,e2,…,eM2, where the eigenvectors corresponding to the J large eigenvalues of the matrix Cx are transformed into a fourth-order strong interference signal subspace [18–20]:(15)EJ=e1,e2,…,eJ.The eigenvectors corresponding to theK larger eigenvalues are transformed into a fourth-order signal subspace:(16)Es=eJ+1,eJ+2,…,eJ+K.The eigenvectors corresponding to the otherM2−K−J small eigenvalues are transformed into a fourth-order noise subspace:(17)EN=eJ+K+1,eJ+K+2,…,eM2.


### 3.1. Expanded Noise Subspace DOA Estimation

Taking the MUISC algorithm as an example, the spatial spectrum value of a certain angle is the reciprocal of the projection modulus of the steering vector in direction to the projection space, that is, the noise subspace [21–23]. The peak value represents that the steering vector at this angle projected in the noise subspace is smaller than its surrounding angle. The direction steering vector is merged into the noise subspace to form an expanded noise subspace [24]. When the expanded noise subspace is used for spectrum estimation, there must be no peak in this direction, and the steering vector of the weak signal will be in the expanded noise subspace. The projection of the steering vector of the weak signal on the extended noise subspace is a small nonzero value, and its reciprocal will generate a larger peak so that strong interference can be curbed, and the DOA of a weak signal can be estimated [11].Accordingly, the feature vector of the interference signal can be incorporated into the noise subspace to form an extended noise subspace, namely,(18)EJN=e1,e2,…,eJ,eJ+K+1,…,eM2.Based on the interference-noise subspace, since the weak signal steering vector and the interference signal steering vector are often not orthogonal, in order to ensure the correct formation of the weak signal peak, the steering vector is transformed as follows:(19)cθ=I−EJEJHbθ,where bθ is the steering vector after the expansion of the FOC.The conventional MUSIC method is adopted to search for spectral peaks and estimate the DOA of the weak signal:(20)P1θ=cHθcθcHθEJNEJNHcθ.In the above formula, since the total signal subspace is orthogonal to the noise subspace, there isEJHEN=0. If θi is the incident angle of the weak signal, then(21)ENI−EJEJHaθi=ENaθi−0=0.So the weak signal steering vector is orthogonal to the interference-noise subspace after transformation by equation (20).Finally, the angle corresponding to the maximum point obtained through the spatial spectrum is the incident direction of the weak signal.According to the above analysis, the FOC-ENS algorithm is summarized, as shown in Table1.Table 1
FOC-ENS algorithm basic steps.
FOC-ENS algorithmStep 1: estimate the data covariance matrixR of the array from the output vector Xt of the array elementStep 2: perform eigen decomposition on the covariance matrixCx and arrange the eigenvalues in descending order so as to obtain interference subspace EJ, signal subspace Es, and noise subspace ENStep 3: take a set of orthogonal basesEJ determined by interference in the total signal subspace and merge it into the noise subspace EN to form the interference-noise subspace EJNStep 4: based on the interference-noise subspaceEJN, perform a spectral peak search on P1θ to estimate the DOA of the weak signal


### 3.2. Modified Projection Blocking Method DOA Estimation

After the division of the strong interference signal subspace, signal subspace, and noise subspace according (after formulas (15)–(17)) to the subspace theory, a modified projection blocking matrix G orthogonal to the strong interference signal subspace is established [12] as follows:(22)G=I−EJEJHEJ−1EJ.The global subspace is performed to get(23)D=GE=GEJ,ES,EN=0J,DK+1,DK+2,…DM2.That is, the matrixD after the modified projection blocking transformation is merely related to the weak signal and noise and has nothing to do with the interference, achieving the suppression of the strong interference. Therefore, in order to avoid interference, the transformed data vector should be(24)Y=GCx.The data covariance matrix after modified projection transformation is(25)RY=GERsEHGH+σ2GGH.After the modified projection transformation, the strong interference is blocked, thereby eliminating the influence of the strong interference on the weak signal. Then, eigenvalue decomposition is performed on the receiving matrix again to obtain a new signal subspaceES′ and noise subspace EN′. At this time, ES′ is only correlated with the weak signal, EN′ orthogonal to the steering vector in the direction of the weak signal, and the spatial spectrum function is used as follows [25–27]:(26)P2θ=1bHθEN′EN′Hbθ.Finally, the azimuth angle of the weak signal can be estimated with the help of the spectral peak search.According to the above analysis, the algorithm of this paper is summarized, as depicted in Table2.Table 2
FOC-FPB algorithm basic steps.
FOC-FPB algorithmStep 1 and Step 2 are the same as the FOC-ENS algorithmStep 3: construct a modified projection matrixG orthogonal to the interference steering vectorStep 4: preprocess the received data vector to getYStep 5: similarly duplicate Step 2 to do the second eigenvalue decomposition to get the noise subspaceEN′Step 6: perform a spectral peak search onP2θ and estimate the DOA of the weak signal


## 3.1. Expanded Noise Subspace DOA Estimation

Taking the MUISC algorithm as an example, the spatial spectrum value of a certain angle is the reciprocal of the projection modulus of the steering vector in direction to the projection space, that is, the noise subspace [21–23]. The peak value represents that the steering vector at this angle projected in the noise subspace is smaller than its surrounding angle. The direction steering vector is merged into the noise subspace to form an expanded noise subspace [24]. When the expanded noise subspace is used for spectrum estimation, there must be no peak in this direction, and the steering vector of the weak signal will be in the expanded noise subspace. The projection of the steering vector of the weak signal on the extended noise subspace is a small nonzero value, and its reciprocal will generate a larger peak so that strong interference can be curbed, and the DOA of a weak signal can be estimated [11].Accordingly, the feature vector of the interference signal can be incorporated into the noise subspace to form an extended noise subspace, namely,(18)EJN=e1,e2,…,eJ,eJ+K+1,…,eM2.Based on the interference-noise subspace, since the weak signal steering vector and the interference signal steering vector are often not orthogonal, in order to ensure the correct formation of the weak signal peak, the steering vector is transformed as follows:(19)cθ=I−EJEJHbθ,where bθ is the steering vector after the expansion of the FOC.The conventional MUSIC method is adopted to search for spectral peaks and estimate the DOA of the weak signal:(20)P1θ=cHθcθcHθEJNEJNHcθ.In the above formula, since the total signal subspace is orthogonal to the noise subspace, there isEJHEN=0. If θi is the incident angle of the weak signal, then(21)ENI−EJEJHaθi=ENaθi−0=0.So the weak signal steering vector is orthogonal to the interference-noise subspace after transformation by equation (20).Finally, the angle corresponding to the maximum point obtained through the spatial spectrum is the incident direction of the weak signal.According to the above analysis, the FOC-ENS algorithm is summarized, as shown in Table1.Table 1
FOC-ENS algorithm basic steps.
FOC-ENS algorithmStep 1: estimate the data covariance matrixR of the array from the output vector Xt of the array elementStep 2: perform eigen decomposition on the covariance matrixCx and arrange the eigenvalues in descending order so as to obtain interference subspace EJ, signal subspace Es, and noise subspace ENStep 3: take a set of orthogonal basesEJ determined by interference in the total signal subspace and merge it into the noise subspace EN to form the interference-noise subspace EJNStep 4: based on the interference-noise subspaceEJN, perform a spectral peak search on P1θ to estimate the DOA of the weak signal


## 3.2. Modified Projection Blocking Method DOA Estimation

After the division of the strong interference signal subspace, signal subspace, and noise subspace according (after formulas (15)–(17)) to the subspace theory, a modified projection blocking matrix G orthogonal to the strong interference signal subspace is established [12] as follows:(22)G=I−EJEJHEJ−1EJ.The global subspace is performed to get(23)D=GE=GEJ,ES,EN=0J,DK+1,DK+2,…DM2.That is, the matrixD after the modified projection blocking transformation is merely related to the weak signal and noise and has nothing to do with the interference, achieving the suppression of the strong interference. Therefore, in order to avoid interference, the transformed data vector should be(24)Y=GCx.The data covariance matrix after modified projection transformation is(25)RY=GERsEHGH+σ2GGH.After the modified projection transformation, the strong interference is blocked, thereby eliminating the influence of the strong interference on the weak signal. Then, eigenvalue decomposition is performed on the receiving matrix again to obtain a new signal subspaceES′ and noise subspace EN′. At this time, ES′ is only correlated with the weak signal, EN′ orthogonal to the steering vector in the direction of the weak signal, and the spatial spectrum function is used as follows [25–27]:(26)P2θ=1bHθEN′EN′Hbθ.Finally, the azimuth angle of the weak signal can be estimated with the help of the spectral peak search.According to the above analysis, the algorithm of this paper is summarized, as depicted in Table2.Table 2
FOC-FPB algorithm basic steps.
FOC-FPB algorithmStep 1 and Step 2 are the same as the FOC-ENS algorithmStep 3: construct a modified projection matrixG orthogonal to the interference steering vectorStep 4: preprocess the received data vector to getYStep 5: similarly duplicate Step 2 to do the second eigenvalue decomposition to get the noise subspaceEN′Step 6: perform a spectral peak search onP2θ and estimate the DOA of the weak signal


## 4. Simulation Results and Analysis

Suppose the signal-to-noise ratio of the signal is defined as10log10σk2/σn2, the interference-to-signal ratio is defined as 10log10σj2/σk2, and the interference-to-noise ratio is defined as 10log10σj2/σn2. Among them, σk2 is the power of the k-th signal, σj2 is the power of the strong interference signal, and σn2 is the noise power.The root mean square error (RMSE) is(27)RMSE=1KN∑i=1N∑k=1Kθi−θik2.Among them,N is the number of Monte-Carlo experiments, K is the number of weak signals, and θi and θik are the true value and estimated value of the azimuth angle of the i-th signal in the k-th experiment.


### 4.1. Spatial Spectrum Estimation

Experiment 1 sets the incident angles of 3 target signals at −30°, 0°, and 30°, respectively; the incident angles of 2 strong interference signals are −60° and 60°; the number of array elements is 10; the signal-to-noise ratio (SNR) is 10 dB; the signal-to-interference ratio (SIR) is 30 dB; the number of snapshots is 200. The advantages of the proposed algorithm in DOA estimation accuracy are analyzed. Figure1 illuminates the spatial spectrum curves of the conventional FOC-MUSIC algorithm, JJM algorithm, FOC-ENS algorithm, and FOC-FPB algorithm.Figure 1
Comparison of spatial spectrum curves.Figure1 illustrates that the conventional FOC-MUSIC algorithm is inclined to regard the false peak formed by the strong interference signal as the peak of the real target, thus failing to estimate the arrival angle of the target. Although the JJM algorithm can suppress strong interference signals, it has a mediocre effect on the estimation of the target's arrival angle, and it requires precise prediction of the direction of the strong interference signal. The FOC-ENS algorithm can better estimate the direction of arrival of the target. However, the direction of strong interference will exist during the estimation of small spikes, which will affect DOA estimation under certain conditions. The FOC-FPB algorithm can determine the target wave arrival angle best.


### 4.2. Comparative Analysis of Errors of Different Algorithms

Experiment 2 sets the number of array elements at 10, the SIR is 30 dB, and the number of Monte Carlo experiments is 500. The errors of different algorithms in the Monte Carlo simulation experiment are also analyzed. Figure2(a) illuminates that under the condition of SNR = 10 dB, the number of snapshots varies from 50 to 500 in step of 50 and the variation curves of the RMSE of the JJM algorithm, ENS algorithm, FPB algorithm, FOC-ENS algorithm, and FOC-FPB algorithm with the number of snapshots. Figure 2(b) shows the curve of the RMSE of the five algorithms changing from −10 dB to 10 dB in step of 2 dB under the condition of 200 snapshots.Figure 2
RMSE error analyses of different algorithms. (a) Relationship between RMSE and the number of snapshots. (b) Relationship between RMSE and SNR.
(a)(b)It can be inferred from Figure2 that the RMSE of these methods dwindles as the SNR and the number of snapshots grow. In a composite background, when the number of snapshots and the SNR climb to a certain extent, the RMSE of several algorithms tends to be stable. Due to the suppression of spatial colored noise, the FOC-ENS algorithm and the FOC-FPB algorithm are better than other algorithms in low SNR.


### 4.3. Comparative Analysis of Errors under Different Numbers of Interference Sources

Experiment 3 sets the number of array elements at 10, the SIR is 30 dB, the number of snapshots is 200, and the number of Monte Carlo experiments is 500. The RMSE of the proposed algorithm under different numbers of interference sources is analyzed. Figure3(a) shows the SNR of the FOC-ENS algorithm from −10 dB to 10 dB in step of 2 dB and the curves of the RMSE with the SNR under the different numbers of interference sources. Figure 3(b) depicts the FOC-FPB algorithm SNR changes from −10 dB to 10 dB in step of 2 dB, reflecting the variation curves of the RMSE with the SNR under the different numbers of interference sources.Figure 3
The statistical performance of the algorithm varying with SNR under different numbers of interference sources. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)As presented in Figure3, the two algorithms can suppress interference and accurately estimate weak signals in a composite background. It effectively addresses the defect that the traditional algorithm cannot accurately estimate the weak signal under the background of strong interference and colored noise. As the number of interference sources increases, the direction finding performance of the proposed algorithm is slightly worse. The reason for the poor performance may be that the increase in the number of interferences gives rise to the leakage of the interference subspace, which affects the noise subspace.


### 4.4. Comparative Analysis of Errors under Different Noise Backgrounds

Experiment 4 sets the number of array elements at 10, the SIR is 30 dB, the number of snapshots is 200, and the number of Monte Carlo experiments is 500. The RMSE under different noise backgrounds is analyzed. Figure4(a) shows the SNR of the FOC-ENS algorithm from −10 dB to 10 dB in step of 2 dB and the curves of the RMSE versus SNR under the two noise backgrounds. Figure 4(b) illuminates the FOC-FPB algorithm SNR changes from −10 dB to 10 dB in step of 2 dB, providing the curves of the RMSE with the SNR under different backgrounds.Figure 4
The statistical performance of different noise background algorithms varying with SNR. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)Due to the unique blind Gaussian type of the FOC, the performance of this method under colored noise is similar to that under white noise.


### 4.5. Algorithm Aperture Expansion Effect

Experiment 5 analyses the multitarget direction finding performance of the proposed algorithm. Figure5(a) shows the simulation of the FOC-ENS algorithm. It sets the incidence angle of 12 target signals to be uniformly distributed from −50° to 60°, and the incidence angle of 2 strong interference signals to be 70° and 80°, and other settings are the same as in Section 4.1. Figure 5(b) is the FOC-FPB simulation. It sets the incident angle of 11 target signals to be uniformly distributed from −40° to 60°, and the incident angle of 2 strong interference signals to be −60° and −50°.Figure 5
Multitarget direction [1] finding effect. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)Owing to the aperture expansion characteristics of the FOC algorithm, the combined ENS algorithm and the FPB algorithm do not experience aperture loss, and the two improved algorithms perform well in multiobjective situations.


### 4.6. Time Complexity Analysis

Supposing that the number of sampling points isN, and the number of angle searches is Nθ. The calculation amount of FOC-ENS and FOC-FPB algorithms mentioned in this paper is mainly the construction of the cumulant matrix, eigenvalue decomposition, and spectral peak search. The calculation amount is O9NM4+M6+NθM4 and O9NM4+2M6+NθM4, respectively. Compared with the JJM and ENS algorithms with complexity ONM4+M6+NθM4 and the FPB algorithm with complexity ONM4+2M6+NθM4, the complexity is slightly higher.Firstly, the calculation time of the two improved algorithms is simulated, and the number of target signals and the number of interference sources are set.K represents the number of target, and J represents the number of interference sources. Figures 6(a) and 6(b) show the simulation time comparison of FOC-ENS and FOC-FPB algorithms at different target numbers and interference sources, respectively. Then, K = 8 and J = 2 are set, and they are compared with traditional algorithms. Figure 6(c) shows the simulation time comparison of different algorithms.Figure 6
Comparison of simulation time. (a) Number of different targets. (b) Number of different interference sources. (c) Different algorithms.
(a)(b)(c)It can be seen from Figure6 that the FOC-ENS and FOC-FPB algorithms construct FOC matrix, so the calculation time is slightly longer than that of the JJM, ENS, and FPB algorithms. FOC-ENS algorithm simulation time is shorter than the FOC-FPB algorithm in that the FOC-FPB algorithm has one more feature decomposition than the FOC-ENS algorithm, which enhances the complexity to a certain extent.


## 4.1. Spatial Spectrum Estimation

Experiment 1 sets the incident angles of 3 target signals at −30°, 0°, and 30°, respectively; the incident angles of 2 strong interference signals are −60° and 60°; the number of array elements is 10; the signal-to-noise ratio (SNR) is 10 dB; the signal-to-interference ratio (SIR) is 30 dB; the number of snapshots is 200. The advantages of the proposed algorithm in DOA estimation accuracy are analyzed. Figure1 illuminates the spatial spectrum curves of the conventional FOC-MUSIC algorithm, JJM algorithm, FOC-ENS algorithm, and FOC-FPB algorithm.Figure 1
Comparison of spatial spectrum curves.Figure1 illustrates that the conventional FOC-MUSIC algorithm is inclined to regard the false peak formed by the strong interference signal as the peak of the real target, thus failing to estimate the arrival angle of the target. Although the JJM algorithm can suppress strong interference signals, it has a mediocre effect on the estimation of the target's arrival angle, and it requires precise prediction of the direction of the strong interference signal. The FOC-ENS algorithm can better estimate the direction of arrival of the target. However, the direction of strong interference will exist during the estimation of small spikes, which will affect DOA estimation under certain conditions. The FOC-FPB algorithm can determine the target wave arrival angle best.


## 4.2. Comparative Analysis of Errors of Different Algorithms

Experiment 2 sets the number of array elements at 10, the SIR is 30 dB, and the number of Monte Carlo experiments is 500. The errors of different algorithms in the Monte Carlo simulation experiment are also analyzed. Figure2(a) illuminates that under the condition of SNR = 10 dB, the number of snapshots varies from 50 to 500 in step of 50 and the variation curves of the RMSE of the JJM algorithm, ENS algorithm, FPB algorithm, FOC-ENS algorithm, and FOC-FPB algorithm with the number of snapshots. Figure 2(b) shows the curve of the RMSE of the five algorithms changing from −10 dB to 10 dB in step of 2 dB under the condition of 200 snapshots.Figure 2
RMSE error analyses of different algorithms. (a) Relationship between RMSE and the number of snapshots. (b) Relationship between RMSE and SNR.
(a)(b)It can be inferred from Figure2 that the RMSE of these methods dwindles as the SNR and the number of snapshots grow. In a composite background, when the number of snapshots and the SNR climb to a certain extent, the RMSE of several algorithms tends to be stable. Due to the suppression of spatial colored noise, the FOC-ENS algorithm and the FOC-FPB algorithm are better than other algorithms in low SNR.


## 4.3. Comparative Analysis of Errors under Different Numbers of Interference Sources

Experiment 3 sets the number of array elements at 10, the SIR is 30 dB, the number of snapshots is 200, and the number of Monte Carlo experiments is 500. The RMSE of the proposed algorithm under different numbers of interference sources is analyzed. Figure3(a) shows the SNR of the FOC-ENS algorithm from −10 dB to 10 dB in step of 2 dB and the curves of the RMSE with the SNR under the different numbers of interference sources. Figure 3(b) depicts the FOC-FPB algorithm SNR changes from −10 dB to 10 dB in step of 2 dB, reflecting the variation curves of the RMSE with the SNR under the different numbers of interference sources.Figure 3
The statistical performance of the algorithm varying with SNR under different numbers of interference sources. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)As presented in Figure3, the two algorithms can suppress interference and accurately estimate weak signals in a composite background. It effectively addresses the defect that the traditional algorithm cannot accurately estimate the weak signal under the background of strong interference and colored noise. As the number of interference sources increases, the direction finding performance of the proposed algorithm is slightly worse. The reason for the poor performance may be that the increase in the number of interferences gives rise to the leakage of the interference subspace, which affects the noise subspace.


## 4.4. Comparative Analysis of Errors under Different Noise Backgrounds

Experiment 4 sets the number of array elements at 10, the SIR is 30 dB, the number of snapshots is 200, and the number of Monte Carlo experiments is 500. The RMSE under different noise backgrounds is analyzed. Figure4(a) shows the SNR of the FOC-ENS algorithm from −10 dB to 10 dB in step of 2 dB and the curves of the RMSE versus SNR under the two noise backgrounds. Figure 4(b) illuminates the FOC-FPB algorithm SNR changes from −10 dB to 10 dB in step of 2 dB, providing the curves of the RMSE with the SNR under different backgrounds.Figure 4
The statistical performance of different noise background algorithms varying with SNR. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)Due to the unique blind Gaussian type of the FOC, the performance of this method under colored noise is similar to that under white noise.


## 4.5. Algorithm Aperture Expansion Effect

Experiment 5 analyses the multitarget direction finding performance of the proposed algorithm. Figure5(a) shows the simulation of the FOC-ENS algorithm. It sets the incidence angle of 12 target signals to be uniformly distributed from −50° to 60°, and the incidence angle of 2 strong interference signals to be 70° and 80°, and other settings are the same as in Section 4.1. Figure 5(b) is the FOC-FPB simulation. It sets the incident angle of 11 target signals to be uniformly distributed from −40° to 60°, and the incident angle of 2 strong interference signals to be −60° and −50°.Figure 5
Multitarget direction [1] finding effect. (a) FOC-ENS algorithm. (b) FOC-FPB algorithm.
(a)(b)Owing to the aperture expansion characteristics of the FOC algorithm, the combined ENS algorithm and the FPB algorithm do not experience aperture loss, and the two improved algorithms perform well in multiobjective situations.


## 4.6. Time Complexity Analysis

Supposing that the number of sampling points isN, and the number of angle searches is Nθ. The calculation amount of FOC-ENS and FOC-FPB algorithms mentioned in this paper is mainly the construction of the cumulant matrix, eigenvalue decomposition, and spectral peak search. The calculation amount is O9NM4+M6+NθM4 and O9NM4+2M6+NθM4, respectively. Compared with the JJM and ENS algorithms with complexity ONM4+M6+NθM4 and the FPB algorithm with complexity ONM4+2M6+NθM4, the complexity is slightly higher.Firstly, the calculation time of the two improved algorithms is simulated, and the number of target signals and the number of interference sources are set.K represents the number of target, and J represents the number of interference sources. Figures 6(a) and 6(b) show the simulation time comparison of FOC-ENS and FOC-FPB algorithms at different target numbers and interference sources, respectively. Then, K = 8 and J = 2 are set, and they are compared with traditional algorithms. Figure 6(c) shows the simulation time comparison of different algorithms.Figure 6
Comparison of simulation time. (a) Number of different targets. (b) Number of different interference sources. (c) Different algorithms.
(a)(b)(c)It can be seen from Figure6 that the FOC-ENS and FOC-FPB algorithms construct FOC matrix, so the calculation time is slightly longer than that of the JJM, ENS, and FPB algorithms. FOC-ENS algorithm simulation time is shorter than the FOC-FPB algorithm in that the FOC-FPB algorithm has one more feature decomposition than the FOC-ENS algorithm, which enhances the complexity to a certain extent.


## 5. Conclusion

To address the direction finding problem under the compound background of strong interference and colored noise, this paper proposes two algorithms based on FOC. Both algorithms need to construct a FOC matrix by receiving signal vectors to suppress noise components and then perform eigenvalue decomposition on the matrix and divide the subspace in accordance with the eigenvalues. The FOC-ENS algorithm merges the space where the strong interference steering vector is located into the noise subspace to construct an extended noise subspace for spectral peak search. The FOC-FPB algorithm constructs the orthogonal projection matrix of the strong interference signal subspace as the interference blocking matrix, uses the matrix to pre-transform the global subspace, and then eigen decomposes again to obtain the target DOA. Compared with the traditional algorithm, the algorithm proposed in this paper has a smaller RMSE, and there is no array aperture loss. It functions in a complex and volatile electromagnetic environment.



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*Source: 3949988-2022-02-04.xml*