# An Angle Estimation Method for Monostatic MIMO Radar Based on RCC-FLOM Algorithm

Authors: Jian Gong; Huan Wang; Yiduo Guo
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

The performance of the angle estimation algorithm based on the two-order or higher order cumulants in the impact noise background will decline sharply. Therefore, it is necessary to study the new algorithm to estimate target angle in the impact noise background. In order to solve the angle estimation problem of coherent sources in the impulse noise background, a conjugate rotation invariant subspace algorithm based on reduced order fractional lower order covariance matrix is proposed. Use the reduced dimension lower order fraction covariance matrix to reduce the impulse noise influence. And according to the conjugate rotation invariant subspace, the coherent source is decohered. The Monte-Carlo experiments show that the proposed algorithm has the advantages of high estimation probability and low root mean square error in the case of low signal-to-noise ratio, compared with the existing FLOM-MUSIC algorithm and FLOM-Unitary ESPRIT algorithm.


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## Body

## 1. Introduction

The noise encountered in the practical application of Multiple-Input Multiple-Output (MIMO) radar often has certain impact characteristics, such as cosmic noise, atmospheric noise, and meteorological noise. In recent years, a large number of experimental data and simulation results [1–3] confirm that the impulse noise is consistent with the α stable distribution. Because the α stable distribution does not have two-order square or higher, this means that the performance of the angle estimation algorithm based on the two-order or higher order cumulants in the impact noise background will decline sharply. Therefore, it is necessary to study the new algorithm to estimate target angle in the impact noise background.Tsakalides [4–6] and Liu [7] estimate the direction of arrival using Multiple Signal Classification (MUSIC) algorithm based on covariance and fractional lower order moment respectively, but all of them need to search spectral peak, and the amount of operation and storage is very large. Lv Zejun [8] and He Jin [9] estimate the arrival angle based on covariance in the impact noise environment, but the above methods need to estimate the appropriate fractional lower order parameters. Li Li [10] uses the Maximum Correntropy Criterion (MCC) to derive the ductile parallel factor algorithm for the impact noise environment, but has not solved the coherent source problem.Based on the above research, we combine reduced dimension conjugate covariance (RCC) with fraction lower order matrix (FLOM) and then propose a RCC-FLOM algorithm for monostatic MIMO radar. The proposed algorithm can reduce the impulse noise influence and has the advantages of high estimation probability. The remainder of this paper is organized as follows. In Section2, the monostatic MIMO radar echo model is established. Then, the angle estimation method based on RCC-FLOM is proposed in Section 3. Some simulations are conducted to verify the performance of the proposed method in Section 4. Finally, we conclude the paper in Section 5.Notation:·T and ·H denote transpose and conjugate-transpose operators; ⊗ denotes the Kronecker product; E· denotes the expected value; ·∗ denotes the conjugate operator; ∑· denotes the sum operator.


## 2. Monostatic MIMO Radar Echo Model

Considering the monostatic MIMO radar as shown in Figure1, the transmitting and receiving antenna adopt M and N elements, respectively. The spacing of array elements is d=λ/2, λ is carrier wavelength. Assuming that the direction of arrival (DOA) and the direction of departure (DOD) both are θp. The echo signal processed by matching filtering as below,(1)yt=Aβt+ntwhere A=arθ1⊗atθ1,arθ2⊗atθ2,⋯,arθP⊗atθP, the transmit steering vector of thepth target is atθp=1,exp⁡-jπsin⁡θp,⋯,exp⁡-jπM-1sin⁡θpT, and the receive steering vector is arθp=1,exp⁡-jπsin⁡θp,⋯,exp⁡-jπN-1sin⁡θpT, β=ξ1ej2πfd1t,ξ2ej2πfd2t,⋯,ξPej2πfdPtT, ξp is the reflection coefficient of thepth target, fdp is the normalized Doppler frequency of thepth target, and nt representation of noisy column vectors.Figure 1
The structure of a monostatic MIMO radar.


## 3. Angle Estimation Based on RCC-FLOM

### 3.1. Reduced Dimension Lower Order Fraction Covariance Matrix

If defined,(2)arθp⊗atθp≜FbθpIn the form, bθp=1,exp⁡-jπsin⁡θp,⋯,exp⁡-jπM+N-1sin⁡θpT and F∈CMN×M+N-1 is a dimensionality reduction matrix, which can be expressed as(3)F=10⋯00⋯001⋯00⋯0⋮⋮⋯⋮⋮⋯⋮00⋯10⋯0M01⋯00⋯000100⋯0⋮⋮⋯⋮⋮⋯⋮00⋯01⋯0M⋮0⋯010⋯00⋯001⋯0⋮⋮⋯⋮⋮⋯⋮0⋯000⋯1M∈CMN×M+N-1Then the matrix A can be expressed as(4)A=FA′where A′=a′θ1,a′θ2,⋯,a′θP∈CM+N-1P, a′θp=1,ap,⋯,apI-1T, ap=ejπsin⁡θp, and it is a Vanermonde matrix. I is the virtual elements number after dimension reduction transformation, then I=M+N-1.According to formula (3), if formula W≜FHF is defined, there are(5)W=diag1,2,⋯,min⁡M,N,⋯,min⁡M,N︸M-N+1,⋯,2,1If both sides of formula (1) are multiplied by W-1FH, we can get(6)y′=A′β+n′where n′tl=W-1FHntl is a M+N-1 dimension additive noise vector.Whenn′ is impact noise, y′ has no more than two-order square. According to [6, 7], the FLOM of y′ can be constructed as(7)Cuv=Eyu′tyv′tp-2yv′∗t1<p<α≤2where yu′(t) and yv′(t) is the output of matched filtering after MIMO radar dimensionality reduction. Cuv is the u,v element of the fractional lower order covariance matrix C and satisfies 1≤u,v≤I.Substituting formula (6) in formula (7), Cuv can be expressed as(8)Cuv=∑v=1PAuv′Eβv∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗+Enu∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗It can be written as a matrix as follows:(9)C=A′ΛA′H+γIwhere Λuv=δuvEβv∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗ and γ=Enu∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗.After obtaining the reduced order fractional lower order covariance matrix formula (9), the eigenvalues of C according to [11, 12] are as follows:(10)C=ESΛSESH+ENΛNENHwhere ES is a subspace formed by eigenvectors corresponding to large eigenvalues, i.e., signal subspace, while EN is a subspace formed by eigenvectors corresponding to small eigenvalues, i.e., noise subspace.Under ideal conditions, the signal subspace and the noise subspace are orthogonal to each other; that is, the steering vector in the signal subspace is also orthogonal to the noise subspace as(11)a′HθEN=0The classical MUSIC algorithm is based on the above properties. However, considering the fact that the actual received data matrix is finite and there is noise,a′(θ) and EN can not be completely orthogonal; that is to say, formula (11) does not hold. Therefore, in fact, DOA is realized by minimum optimization search; that is,(12)θMUSIC=arg⁡minθ⁡a′HθENENHa′θSo the target angle can be estimated by the MUSIC algorithm’s spectral search formula such as(13)PMUSICθ=1a′HθENENHa′θ


### 3.2. Conjugate Rotation Invariant Subspace Algorithm

Constructing the lower order fractional lower order covariance matrix can only solve the problem of impulse noise. When the multipath effect and electronic interference are affected, the angle estimation of coherent sources is also needed. In this paper, the conjugate rotation invariant subspace algorithm (C-ESPRIT) is used to solve the problem of coherent source angle estimation. Taking the MIMO radar with 5 virtual elements as an example, the virtual space array constructed is shown in Figure2.Figure 2
Monostatic MIMO radar conjugate virtual array.The target echo data obtained by virtual subarray 1 can be expressed as follows:(14)ye1=y1y2y3⋯yPy2y3y4⋯yP+1⋮⋮⋮⋮yI-PyI-P+1yI-P+2⋯yI-1yI∗yI-1∗yI-2∗⋯yI-P+1∗⋮⋮⋮⋮yP+2∗yP+1∗yP∗⋯y3∗yP+1∗yP∗yP-1∗⋯y2∗2I-P×PAccording to formula (6), the matrix ye1 can be expressed as(15)ye1=RA′θ+ne1where(16)R=β1β2β3⋯βPβ1a1β2a2β3a3⋯βPaP⋮⋮⋮⋮β1a1I-P-1β2a2I-P-1β3a3I-P-1⋯βPaPI-P-1β1∗a1-I-1β2∗a2-I-1β3∗a3-I-1⋯βP∗aP-I-1⋮⋮⋮⋮β1∗a1-P+1β2∗a2-P+1β3∗a3-P+1⋯βP∗aP-P+1β1∗a1-Pβ2∗a2-Pβ3∗a3-P⋯βP∗aP-P2I-P×PThe target echo data obtained by virtual subarray 2 can be expressed as follows:(17)ye2=y2y3y4⋯yP+1y3y4y5⋯yP+2⋮⋮⋮⋮yI-P+1yI-P+2yI-P+3⋯yIyI-1∗yI-2∗yI-3∗⋯yI-P∗⋮⋮⋮⋮yP+1∗yP∗yP-1∗⋯y2∗yP∗yP-1∗yP-2∗⋯y1∗2I-P×PAccording to formula (6), the matrix ye2 can be expressed as(18)ye2=RΦA′θ+ne2where Φ is a rotation factor and its expression is Φ=diaga1,a2,⋯,aP.The matrixA′θ is a Vanermonde matrix, because the coherent source is located in different directions of space and linearly independent of each row, rankA′θ=P.When2I-P≥P, the R of the provable matrix is linearly independent; that is, rankR=P. Because of rankΦ=P, the rank of virtual subarray ye1 and ye2 is P. Therefore, the traditional TL-ESPRIT algorithm can be used to estimate P coherent sources.


### 3.3. The Steps of the RCC-FLOM Algorithm

Based on the above analysis, the operation steps of the RCC-FLOM algorithm can be summarized as follows.Step 1.
The echo signal of single pulse is matched filtered to get the signal represented by formula (5).Step 2.
Formula (6) is obtained for reducing the dimension of the signal represented by formula (5).Step 3.
According to formula (7), the fractional lower order covariance y′ of the reconstructed random variable Cuv is obtained.Step 4.
The fractional lower order covarianceCuv is rewritten into the matrix form shown in formula (9), which solves the problem of impulse noise.Step 5.
According to formula (15), the target echo data ye1 received by virtual subarray 1 is obtained.Step 6.
According to formula (18), the target echo data ye2 received by virtual subarray 2 is obtained.Step 7.
Estimated the angle ofP coherent sources based on the traditional TL-ESPRIT algorithm.For a more intuitive presentation, the flow chart is drawn in Figure3.Figure 3
Flow chart of the RCC-FLOM algorithm.


## 3.1. Reduced Dimension Lower Order Fraction Covariance Matrix

If defined,(2)arθp⊗atθp≜FbθpIn the form, bθp=1,exp⁡-jπsin⁡θp,⋯,exp⁡-jπM+N-1sin⁡θpT and F∈CMN×M+N-1 is a dimensionality reduction matrix, which can be expressed as(3)F=10⋯00⋯001⋯00⋯0⋮⋮⋯⋮⋮⋯⋮00⋯10⋯0M01⋯00⋯000100⋯0⋮⋮⋯⋮⋮⋯⋮00⋯01⋯0M⋮0⋯010⋯00⋯001⋯0⋮⋮⋯⋮⋮⋯⋮0⋯000⋯1M∈CMN×M+N-1Then the matrix A can be expressed as(4)A=FA′where A′=a′θ1,a′θ2,⋯,a′θP∈CM+N-1P, a′θp=1,ap,⋯,apI-1T, ap=ejπsin⁡θp, and it is a Vanermonde matrix. I is the virtual elements number after dimension reduction transformation, then I=M+N-1.According to formula (3), if formula W≜FHF is defined, there are(5)W=diag1,2,⋯,min⁡M,N,⋯,min⁡M,N︸M-N+1,⋯,2,1If both sides of formula (1) are multiplied by W-1FH, we can get(6)y′=A′β+n′where n′tl=W-1FHntl is a M+N-1 dimension additive noise vector.Whenn′ is impact noise, y′ has no more than two-order square. According to [6, 7], the FLOM of y′ can be constructed as(7)Cuv=Eyu′tyv′tp-2yv′∗t1<p<α≤2where yu′(t) and yv′(t) is the output of matched filtering after MIMO radar dimensionality reduction. Cuv is the u,v element of the fractional lower order covariance matrix C and satisfies 1≤u,v≤I.Substituting formula (6) in formula (7), Cuv can be expressed as(8)Cuv=∑v=1PAuv′Eβv∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗+Enu∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗It can be written as a matrix as follows:(9)C=A′ΛA′H+γIwhere Λuv=δuvEβv∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗ and γ=Enu∑q=1PAuv′βq+nv′p-2∑q=1PAuv′βq+nv′∗.After obtaining the reduced order fractional lower order covariance matrix formula (9), the eigenvalues of C according to [11, 12] are as follows:(10)C=ESΛSESH+ENΛNENHwhere ES is a subspace formed by eigenvectors corresponding to large eigenvalues, i.e., signal subspace, while EN is a subspace formed by eigenvectors corresponding to small eigenvalues, i.e., noise subspace.Under ideal conditions, the signal subspace and the noise subspace are orthogonal to each other; that is, the steering vector in the signal subspace is also orthogonal to the noise subspace as(11)a′HθEN=0The classical MUSIC algorithm is based on the above properties. However, considering the fact that the actual received data matrix is finite and there is noise,a′(θ) and EN can not be completely orthogonal; that is to say, formula (11) does not hold. Therefore, in fact, DOA is realized by minimum optimization search; that is,(12)θMUSIC=arg⁡minθ⁡a′HθENENHa′θSo the target angle can be estimated by the MUSIC algorithm’s spectral search formula such as(13)PMUSICθ=1a′HθENENHa′θ


## 3.2. Conjugate Rotation Invariant Subspace Algorithm

Constructing the lower order fractional lower order covariance matrix can only solve the problem of impulse noise. When the multipath effect and electronic interference are affected, the angle estimation of coherent sources is also needed. In this paper, the conjugate rotation invariant subspace algorithm (C-ESPRIT) is used to solve the problem of coherent source angle estimation. Taking the MIMO radar with 5 virtual elements as an example, the virtual space array constructed is shown in Figure2.Figure 2
Monostatic MIMO radar conjugate virtual array.The target echo data obtained by virtual subarray 1 can be expressed as follows:(14)ye1=y1y2y3⋯yPy2y3y4⋯yP+1⋮⋮⋮⋮yI-PyI-P+1yI-P+2⋯yI-1yI∗yI-1∗yI-2∗⋯yI-P+1∗⋮⋮⋮⋮yP+2∗yP+1∗yP∗⋯y3∗yP+1∗yP∗yP-1∗⋯y2∗2I-P×PAccording to formula (6), the matrix ye1 can be expressed as(15)ye1=RA′θ+ne1where(16)R=β1β2β3⋯βPβ1a1β2a2β3a3⋯βPaP⋮⋮⋮⋮β1a1I-P-1β2a2I-P-1β3a3I-P-1⋯βPaPI-P-1β1∗a1-I-1β2∗a2-I-1β3∗a3-I-1⋯βP∗aP-I-1⋮⋮⋮⋮β1∗a1-P+1β2∗a2-P+1β3∗a3-P+1⋯βP∗aP-P+1β1∗a1-Pβ2∗a2-Pβ3∗a3-P⋯βP∗aP-P2I-P×PThe target echo data obtained by virtual subarray 2 can be expressed as follows:(17)ye2=y2y3y4⋯yP+1y3y4y5⋯yP+2⋮⋮⋮⋮yI-P+1yI-P+2yI-P+3⋯yIyI-1∗yI-2∗yI-3∗⋯yI-P∗⋮⋮⋮⋮yP+1∗yP∗yP-1∗⋯y2∗yP∗yP-1∗yP-2∗⋯y1∗2I-P×PAccording to formula (6), the matrix ye2 can be expressed as(18)ye2=RΦA′θ+ne2where Φ is a rotation factor and its expression is Φ=diaga1,a2,⋯,aP.The matrixA′θ is a Vanermonde matrix, because the coherent source is located in different directions of space and linearly independent of each row, rankA′θ=P.When2I-P≥P, the R of the provable matrix is linearly independent; that is, rankR=P. Because of rankΦ=P, the rank of virtual subarray ye1 and ye2 is P. Therefore, the traditional TL-ESPRIT algorithm can be used to estimate P coherent sources.


## 3.3. The Steps of the RCC-FLOM Algorithm

Based on the above analysis, the operation steps of the RCC-FLOM algorithm can be summarized as follows.Step 1.
The echo signal of single pulse is matched filtered to get the signal represented by formula (5).Step 2.
Formula (6) is obtained for reducing the dimension of the signal represented by formula (5).Step 3.
According to formula (7), the fractional lower order covariance y′ of the reconstructed random variable Cuv is obtained.Step 4.
The fractional lower order covarianceCuv is rewritten into the matrix form shown in formula (9), which solves the problem of impulse noise.Step 5.
According to formula (15), the target echo data ye1 received by virtual subarray 1 is obtained.Step 6.
According to formula (18), the target echo data ye2 received by virtual subarray 2 is obtained.Step 7.
Estimated the angle ofP coherent sources based on the traditional TL-ESPRIT algorithm.For a more intuitive presentation, the flow chart is drawn in Figure3.Figure 3
Flow chart of the RCC-FLOM algorithm.


## 4. Computer Simulation Results

Assuming that the MIMO radar has 5 transmitting elements and 6 receiving elements, the number of virtual elements after dimension reduction is 10. The receiving and transmitting antennas are equal spaced uniform linear arrays with a spacing of0.5λ. And we use MATLAB software to simulate the following experiments in this paper. The generalized signal-to-noise ratio (GSNR) is the ratio of average power to dispersion coefficient in simulation as follows:(19)GSNR=10lg⁡1γN∑t=1Nst2where N is the number of accumulated pulses. If α=2, the generalized signal-to-noise ratio is the same as the ordinary signal-to-noise ratio.Experiment 1.
Angle estimation of coherent sources under different impulse noise environments
Under the background of impact noise, the direction of arrival of 3 equal power coherent sources are 30°, 10°, −20°, respectively. The impulse noise ofSαS distribution is α=1.5, and the generalized signal-to-noise ratio is 5 dB. Figures 4(a), 4(b), and 4(c), respectively, simulate the angle estimation of ESPRIT algorithm, FLOM-Unitary ESPRIT algorithm proposed in the paper [13], and RCC-FLOM algorithm in 50 simulation experiments.
The simulation results of Figure4 can be seen as follows: In the background of impulse noise, the coherent sources’ direction of arrival can not be estimated accurately by using the ESPRIT algorithm. It is shown that the conventional subspace based algorithm can not suppress impulse noise and coherent decoherence when applied to monostatic MIMO radar. The RCC-FLOM algorithm proposed in this paper can accurately estimate the direction of arrival of coherent sources in the background of impulsive noise. The simulation results verify the effectiveness of the proposed algorithm.Figure 4
Angle estimation algorithms of coherent sources under impulse noise.
(a)
The estimation effect of ESPRIT algorithm (b)
The estimation effect of FLOM-Unitary ESPRIT algorithm (c)
The estimation effect of RCC-FLOM algorithmExperiment 2.
Comparison of statistical performance between the proposed RCC-FLOM algorithm and other algorithms
The mean square error (RSME) of DOA estimation is defined as(20)RMSEθ=1Lm∑lm=1Lmθ^p-θp2where Lm represents the number of Monte-Carlo experiments and θp and θ^p represent the true angle and estimated angle, respectively.
Under the background of impact noise, the direction of arrival of 2 equal power coherent sources is 30° and −20°, respectively. The impulse noise ofSαS distribution is α=1.5. Figures 5(a) and 5(b), respectively, simulate the statistical performance of FLOM-MUSIC algorithm [11] and RCC-FLOM algorithm in 50 Monte-Carlo experiments.
The simulation results of Figure5 can be seen as follows: Under the impact noise background, the RCC-FLOM algorithm proposed in this paper is superior to the FLOM-MUSIC algorithm proposed in the literature [11] in terms of the success probability and the root mean square error. With the increase of the generalized signal-to-noise ratio, the success probability of the two algorithms is gradually increased and the root mean square error is gradually reduced. That is, the performance of the angle estimation is increased with the increase of the generalized signal-to-noise ratio. When the SNR is high to a certain extent, the performance of the two algorithms is basically the same.Figure 5
Performance comparison of different algorithms under impact noise.
(a)
Success probability (b)
Root mean square error


## 5. Conclusions

In this chapter, the MIMO radar angle estimation problem of coherent sources in the impulse noise background is studied. A conjugate rotation invariant subspace (RCC-FLOM) algorithm based on reduced order fractional lower order covariance matrix is proposed. The influence of impulse noise is reduced by using the method of constructing lower order fractional lower order covariance matrix. Then the coherent source is decohered according to the conjugate rotation invariant subspace method. It solves the problem that the two or four order statistical models can not effectively estimate the target angle under the impact noise background. The 50 Monte-Carlo experiments show that the proposed algorithm has the advantages of high estimation probability and low root mean square error in the case of low signal-to-noise ratio, compared with the existing FLOM-MUSIC algorithm.



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