# Joint Angles and Mutual Coupling Estimation Algorithm for Bistatic MIMO Radar

Authors: Jianfeng Li; Xiaofei Zhang
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

We study the problem of angle estimation for a bistatic multiple-input multiple-output (MIMO) radar with unknown mutual coupling and proposed a joint algorithm for angles and mutual coupling estimation with the characteristics of uniform linear arrays and subspaces exploitation. We primarily obtain an initial estimate of DOA and DOD, then employ the local one-dimensional searching to estimate exactly DOA and DOD, and finally evaluate the parameters of mutual coupling coefficients via the estimated angles. Exploiting twice of the one-dimensional local searching, our method has much lower computational cost than the algorithm in (Liu and Liao (2012)), and automatically obtains the paired two-dimensional angle estimation. Slightly better performance for angle estimation has been achieved via our scheme in contrast to (Liu and Liao (2012)), while the two methods indicate very close performance of mutual coupling estimation. The simulation results verify the algorithmic effectiveness of our scheme.


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## Body

## 1. Introduction

Since multiple-input multiple-output (MIMO) radars exploit multiple antennas to manipulate parallel processing for transmitting diverse waveforms and receiving the reflected signals, several potential advantages over conventional phased-array radars such as angular diversity, flexible beampattern, and more degrees of freedom have been demonstrated in [1–4]. In contrast to other radar systems with a single transmit antenna, the additional degrees of freedom gained from MIMO radar has the capability of overcoming fading effect, enhancing spatial resolution, strengthening parameter identifiability, and improving target detection performance [5–8]. Several established angle estimation algorithms for MIMO radar comprise of the signal parameters estimation through the algorithms reliable on rotational invariance techniques (ESPRIT) [9], Capon [10], multiple signal classification (MUSIC) [11], parallel factor (PARAFAC) analysis [12], and adaptive PARAFAC [13], to name a few. However, most of the algorithms mentioned above take little concern for the problem of mutual coupling, which always exists in practical situations. The algorithm in the presence of unknown mutual coupling [14] for this practical situation achieves well estimation of angles and mutual coupling coefficients, while it is compulsory to perform twice of one-dimensional peak searching and additional pairing. Actually, the algorithmic complexity in [14] can be further reduced through the schemes of multidimensional data processing.In our study, we present a low-complexity joint algorithm for the angles and mutual coupling coefficients estimation in bistatic MIMO radars. By employing the transformation of special matrix, we combine the signal subspace and noise subspace of the received data together to achieve the joint estimation of DOD and DOA. The angles can be automatically obtained via two local one-dimensional searching without the knowledge of mutual coupling, whose coefficients can be easily estimated after the angle estimation.The remainder of this paper is structured as follows. Section2 develops the data model for a MIMO radar system with unknown mutual coupling; Section 3 establishes the joint algorithm for joint angles and mutual coupling estimation in addition with the complexity analysis and Cramér-Rao Bound (CRB). In Section 4, simulation results are presented to verify improvement for the proposed algorithm, while the final conclusions are summarized in Section 5.Notation
Bold lower (upper) case symbols denote column vectors or matrices;(·)T,(·)*, (·)H, (·)-1, and (·)† denote the operators of transpose, conjugate, conjugate-transpose, inverse, and pseudoinverse, respectively, diag(v) stands for diagonal matrix whose diagonal element is a vector v; Dn(·)denotes the operation of taking the nth row of the matrix to construct a diagonal matrix; IK is a K×K identity matrix; ⊗ and ⊙ represent the Kronecker product and Hadamard product, respectively; the real part of complex numbers is denoted by Re(·) and the minimum elements of an array are denoted by min(·); E[·] stands for the expectation operator; det(·) denotes the determinant of a matrix, vec(·) denotes an operator stacking the columns of a matrix on top of each other.


## 2. Data Model

We consider a MIMO radar system equipped withM transmit antennas and N receive antennas, both of the antenna arrays are permuted as the uniform linear array (ULA), where the arrays of transmitter and receiver have been collocated in the y-axis with the interelement distance being half-wavelength [12]. The transmitted M-encoded pulse signals are orthogonal to each other; denote the mth signal as sm=[sm(1),sm(2),…,sm(L)]T, where m=1,2,…,M, and L is the length of the signal. We assume that there are K uncorrelated targets, θk,ϕk are direction of departure (DOD) and direction of arrival (DOA) of the kth target with respect to the transmit array normal and the receive array normal, respectively. Assume the transmit and receive steering vectors are ãt(θk)andãr(ϕk), and hence, the signal arrive at the kth target is written as ãtT(θk)S, k=1,2,…,K, where S=[s1,s2,…,sM]T∈ℂM×L. Assume also that the reflection coefficients βk(t) and the phase shift fk are constant between pulse intervals, and the reflected signals of different targets are independent.The receive data of theN receive antennas of the tth pulse can be denoted as(1a)Y(t)=∑k=1Kβk(t)ej2πfktãr(ϕk)ãtT(θk)S+W(t),
where Y(t)∈ℂN×L and W stand for the N×L receive data matrix and the noise matrix with identical determinants, respectively.When each signal received by theN antennas has been processed by a number of M match filters [15], we obtain an M×N data matrix, which is equivalent to [Y(t)SH]T. After iterative vectorization for the input data, we finally construct the output of the tth pulse as(1b)x(t)=vec⁡([Y(t)SH]T).
Due to the orthogonality of the transmit signals towards each other, SSH=I is satisfied for the unitary matrix S. Hence, we have(1c)x(t)=vec⁡(∑k=1Kβk(t)ej2πfktãt(θk)ãrT(ϕk))+vec⁡([WSH]T)=∑k=1Kβk(t)ej2πfktvec⁡(ãt(θk)ãrT(ϕk))+n(t)=[ãr(ϕ1)⊗ãt(θ1),…,ãr(ϕK)⊗ãt(θK)]b(t)+n(t),where b(t)=[b1(t),b2(t),…,bK(t)]T∈ℂK×1, bk(t)=βk(t)ej2πfkt; n(t) is an MN×1 Gaussian white noise vector; ãr(ϕk)=Crar(ϕk); ãt(θk)=Ctat(θk). It follows that Cr∈ℂN×N and Ct∈ℂM×M represent the matrices containing the mutual coupling coefficients. According to the characteristics of ULA, if the elements have the same separation distance, the mutual coupling coefficients will be the same, so this will lead to two Toeplitz matrices as shown below.(2)Cr=[c1r⋯cnr0⋯0c2rc1r⋯cnr0⋯0⋮cnr⋯c1r⋯cnr0⋯00cnr⋯c1r⋯cnr0⋯0⋮0⋯0cnr⋯c1r],Ct=[c1t⋯cmt0⋯0c2tc1t⋯cmt0⋯0⋮cmt⋯c1t⋯cmt0⋯00cmt⋯c1t⋯cmt0⋯0⋮0⋯0cmt⋯c1t],
where cpr(p=1⋯n) and cqt(q=1⋯m) stand for the mutual coupling coefficients. ar(ϕk) and at(θk) represent the receive and the transmit steering vectors for thekth target, respectively. It is rewritten that(3)ar(ϕk)=[1,exp⁡(-jπsin⁡ϕk),…,exp⁡(-j(N-1)πsin⁡ϕk)]T,at(θk)=[1,exp⁡(-jπsin⁡θk),…,exp⁡(-j(M-1)πsin⁡θk)]T.
Define Ã=[ãr(ϕ1)⊗ãt(θ1),…,ãr(ϕK)⊗ãt(θK)], which can be denoted by
(4)Ã=[ÃTD1(ÃR)ÃTD2(ÃR)⋮ÃTDN(ÃR)],
where ÃT=[ãt(θ1),…,ãt(θK)]∈ℂM×K, ÃR=[ãr(ϕ1),…,ãr(ϕK)]∈ℂN×K. There exists a transformation matrix H corresponding to the finite number of row interchanged operations such that(5)F̃=HÃ=[ãt(θ1)⊗ãr(ϕ1),…,ãt(θK)⊗ãr(ϕK)].
And the matrix F̃ can be denoted by(6)F̃=[ÃRD1(ÃT)ÃRD2(ÃT)⋮ÃRDM(ÃT)].
By collecting J samples and representing the received data as X=[x(1),x(2),…,x(J)], the transmitted signal matrix can now be generalized as(7)X=ÃB+N,
where N∈ℂMN×Jstands for the noise matrix, B=[b(1),b(2),…,b(J)]∈ℂK×J.The covariance matrix of the received signalR=E[x(t)xH(t)] can be decomposed as(8)R=EsDsEsH+EnDnEnH,
where Ds denotes a K×K diagonal matrix formed by K largest eigenvalues and Dn denotes a diagonal matrix formed by the rest MN-K smaller eigenvalues. Es and En represent the signal subspace and noise subspace, respectively, of which Es stands for the eigenvectors corresponding to the K largest eigenvalues En consists of the rest eigenvectors. In case of noise absence, the equation between Es and Ã can be formulated as(9)Es=ÃT,
where T represents a nonsingular K×K matrix.


## 3. Joint Angles and Mutual Coupling Estimation Algorithm

### 3.1. Angles Estimation

According to the MUSIC method, the joint estimation of DOD and DOA can be achieved by the peak searching function as follows:(10)F(θ,ϕ,Cr,Ct)=1Q(θ,ϕ,Cr,Ct)=1[ãr(ϕ)⊗ãt(θ)]HEnEnH[ãr(ϕ)⊗ãt(θ)].
However, the method does not work if we cannot obtain Cr and Ct. Even if the mutual coupling coefficients is already known, a two-dimensional searching is still required resulting into a considerably heavy computation burden.With respect to the (4) and (9), the signal subspace matrix is represented by(11a)Es=[ÃTD1(ÃR)ÃTD2(ÃR)⋮ÃTDN(ÃR)]T.
For which, it is partitioned as(11b)Es=[Es1Es2⋮EsN],where Esl∈ℂM×K, l=1,…,N.Taking thenth and (n+1)th part into account, now we have(12a)Esn=ÃTdiag⁡(CnrAR)T,(12b)Es(n+1)=ÃTdiag⁡(Cn+1rAR)T,where Cnr is the nth row of Cr. When n satisfies n≤N/2, it is found that Cn+1r is the right shift of Cnr, and hence,(13)diag⁡(Cn+1rAR)=diag⁡(CnrAR)Φr,
where Φr=diag(exp[-jπsinϕ1],…,exp[-jπsinϕK]). Equation (12b) can be rewritten as(14)Es(n+1)=EsnT-1ΦrT.The diagonal elements ofΦr are the eigenvalues of Esn†Es(n+1), and the corresponding eigenvectors are displayed as T̂-1. In case of noise absence, T̂-1=T-1Π, where Π is a column permutation matrix with Π-1=ΠT. The estimation of Φr is Φ̂r=ΠTΦrΠ, and the DOA estimation can be obtained via(15)ϕ̂kini=sin-1(-angle(ûk)π),
where ûk is the kth diagonal element of Φ̂r.The following matrixEc stands for the column transformation of the direction matrix F̃:(16a)Ec=HEsT̂-1=F̃Π.
This matrix Ec can now be similarly partitioned as(16b)Ec=[Ec1Ec2⋮EcM],
where Ecl∈ℂN×K, l=1,…,M. Considering the mth and (m+1)th part of Ec, by the similar method in equations from (11a) and (11b) to (13), the iterative matrix updates for Ecm now become(16c)Ec(m+1)=EcmΠTΦtΠ,
where Φt=diag(exp[-jπsinθ1],…,exp[-jπsinθK]). We assume that v̂k is the kth diagonal element of Ecm†Ec(m+1), then the angle estimates can be achieved by(16d)θ̂kini=sin-1(-angle(v̂k)π).Therefore, the initial estimations of DOA and DOD have been obtained with the characteristics of automatic pairing.However, the estimation obtained by (15) and (16d) has low accuracy and bias, for it just employs partial information of the antennas, the more accurate estimation of the angles will be obtained below.Referring to [14, Lemma 1], the estimation for the transmitted signals and their paired response can be obtained by
(17)ãt(θ)=Ctãt(θ)=Tt(θ)ct,ãr(ϕ)=Crãr(ϕ)=Tr(ϕ)cr,
wherect=[c1t,…,cmt], cr=[c1r,…,cnr],Tt(θ)=Tt1(θ)+Tt2(θ), and the nonzero elements calculated from Tt1(θ)∈ℂM×m and Tt2(θ)∈ℂM×m can now be written as(18)[Tt1]ij=(at(θ))i+j-1,i+j≤M+1,[Tt2]ij=(at(θ))i-j+1,i≥j≥2.
It is shown for Q(θ,ϕ,Cr,Ct)that(19)Q(θ,ϕ,Cr,Ct)=(cr⊗ct)HQM(θ,ϕ)(cr⊗ct),
where QM(θ,ϕ)=(Tr(ϕ)⊗Tt(θ))HEnEnH(Tr(ϕ)⊗Tt(θ)).Sincecr⊗ct≠0, and when MN-K≥mn, we construct the function to search for the estimations of DOA and DOD [16] as follows:(20)(θ̂,ϕ̂)=argmax⁡θ,ϕ1det⁡(QM(θ,ϕ)).
Equation (20) requires a two-dimensional searching, which has a high complexity. Considering the initial estimation of DOD and DOA obtained in (15) and (16a), (16b), (16c), and (16d), we can use two one-dimensional searching to obtain more accurate estimation of DOD and DOA(21)θ̂k=argmax⁡θ∈[θ̂kini-Δθ,θ̂kini+Δθ]1det⁡(QM(θ,ϕ̂kini)).
Throughout the local searching for θ within [θ̂kini-Δθ,θ̂kini+Δθ], where Δθ is a small value, we obtain more accurate estimation of the DOD θ̂k. Similarly, ϕk can be also estimated from ϕ within [ϕ̂kini-Δϕ,ϕ̂kini+Δϕ], where Δϕ is a small value. Hence we have(22)ϕ̂k=argmax⁡ϕ∈[ϕ̂kini-Δϕ,ϕ̂kini+Δϕ]1det⁡(QM(θ̂k,ϕ)),
where θ̂k is estimation of θk via (21).


### 3.2. Mutual Coupling Estimation

According to (19), the estimates of crt=cr⊗ct will be obtained from the minimum eigenvector of (1/K)∑k=1KQM(θ̂k,ϕ̂k). After normalizing crt(crt(1)=1), and reshaping it to an m×n matrix Crt, we have(23)Crt=ctcrT.
Then we obtain the estimation of cr and ct from the singular value decomposition of Crt. Let us assume Crt=USVH and the largest left singular vector is ut, while the largest right singular vector is vr, it is easy to construct that(24)ĉt=utut(1),ĉr=(vrvr(1))*.
Till now, we have achieved the proposal of our scheme for the joint angle and mutual coupling estimation in MIMO-radar. We show the major algorithmic steps as follows.(1)
Estimate the covariance matrix of the received data throughR̂=∑t=1Jx(t)xH(t).(2)
Perform eigenvalue decomposition ofR̂ to get Ês and ÊN, and partition the matrix Es to get Es1,Es2,…,EsN. It follows to obtain ϕ̂kini from the eigenvalues of Esn†Es(n+1) and T̂ from the corresponding eigenvectors.(3)
ComputeEc, and get the initial estimation of the DOD θ̂kini.(4)
Obtain the estimation of DODθ̂k through one-dimensional searching via (21) while keeping ϕkini fixed.(5)
Obtain the estimation of DOAϕ̂k through one-dimensional searching via (22) while keeping θ̂k fixed.(6)
Obtain the estimation ofCrt from the minimum eigenvector of (1/K)∑k=1KQM(θ̂k,ϕ̂k), and step further to get the estimation of cr and ct from the singular value decomposition of Crt via (24).


### 3.3. Complexity Analysis and Cramer-Rao Bound (CRB)

The proposed algorithm has lower complexity than the algorithm in [14]. Our scheme costs O{M2N2J+M3N3+2K2M+3K3+2K2N+K(2N+2) + 2n1K[MNmn+MNmn(MN-K) + (mn)! + (MN-K)m2n2]+m3n3+min(n2m,m2n)}, where n1 is the number of steps within the local searching range, while the algorithm in [14] requires O{M2N2J+M3N3+n2[M2Nn(MN-K) + (MN-K)M2n2+2(Mn)!+N2Mm(MN-K) + N2m2(MN-K)]+K2[MNmn+MNmn(MN-K) + (mn)!+(MN-K)m2n2]+m3n3+min(n2m,m2n)}, where n2 is the number of steps within the global searching range, and n2≫n1.According to [17], we derive the CRB for the joint angles estimation and mutual coupling algorithm in bistatic MIMO radar as follows:(25)CRB=σ22{Re[DHΠG⊥D]}-1,
where σ2stands for the covariance of the noise. 								(26)D=[∂u∂θ1,…,∂u∂θK,∂u∂ϕ1,…,∂u∂ϕK,∂u∂c2t,…,∂u∂cmt,∂u∂c2r…,∂u∂cnr]
with
(27)u=[Ãb(1)⋮Ãb(J)],ΠG⊥=[ΠÃ⊥0⋱0ΠÃ⊥],
with
(28)ΠÃ⊥=IMN-Ã(ÃHÃ)-1ÃH.


### 3.4. Advantages of the Proposed Algorithm

By compares and contrasts to [14], the algorithmic advantage of our approach can be briefly summarized as follows:(1)
Our approach can achieve the estimation for automatically paired two-dimensional angles, while the algorithm in [14] requires additional pairing.(2)
The proposed algorithm only requires the local one-dimensional searching and hence has much lower complexity, while the algorithm in [14] urges two global one-dimensional searching.(3)
For the performance of angle and mutual coupling estimation, slightly improvement can be verified in our method than the one in [14].(4)
By fully exploiting both the signal subspace and the noise subspace, our method displays more validity and effectiveness, while the algorithm in [14] just employs the noise subspace.


## 3.1. Angles Estimation

According to the MUSIC method, the joint estimation of DOD and DOA can be achieved by the peak searching function as follows:(10)F(θ,ϕ,Cr,Ct)=1Q(θ,ϕ,Cr,Ct)=1[ãr(ϕ)⊗ãt(θ)]HEnEnH[ãr(ϕ)⊗ãt(θ)].
However, the method does not work if we cannot obtain Cr and Ct. Even if the mutual coupling coefficients is already known, a two-dimensional searching is still required resulting into a considerably heavy computation burden.With respect to the (4) and (9), the signal subspace matrix is represented by(11a)Es=[ÃTD1(ÃR)ÃTD2(ÃR)⋮ÃTDN(ÃR)]T.
For which, it is partitioned as(11b)Es=[Es1Es2⋮EsN],where Esl∈ℂM×K, l=1,…,N.Taking thenth and (n+1)th part into account, now we have(12a)Esn=ÃTdiag⁡(CnrAR)T,(12b)Es(n+1)=ÃTdiag⁡(Cn+1rAR)T,where Cnr is the nth row of Cr. When n satisfies n≤N/2, it is found that Cn+1r is the right shift of Cnr, and hence,(13)diag⁡(Cn+1rAR)=diag⁡(CnrAR)Φr,
where Φr=diag(exp[-jπsinϕ1],…,exp[-jπsinϕK]). Equation (12b) can be rewritten as(14)Es(n+1)=EsnT-1ΦrT.The diagonal elements ofΦr are the eigenvalues of Esn†Es(n+1), and the corresponding eigenvectors are displayed as T̂-1. In case of noise absence, T̂-1=T-1Π, where Π is a column permutation matrix with Π-1=ΠT. The estimation of Φr is Φ̂r=ΠTΦrΠ, and the DOA estimation can be obtained via(15)ϕ̂kini=sin-1(-angle(ûk)π),
where ûk is the kth diagonal element of Φ̂r.The following matrixEc stands for the column transformation of the direction matrix F̃:(16a)Ec=HEsT̂-1=F̃Π.
This matrix Ec can now be similarly partitioned as(16b)Ec=[Ec1Ec2⋮EcM],
where Ecl∈ℂN×K, l=1,…,M. Considering the mth and (m+1)th part of Ec, by the similar method in equations from (11a) and (11b) to (13), the iterative matrix updates for Ecm now become(16c)Ec(m+1)=EcmΠTΦtΠ,
where Φt=diag(exp[-jπsinθ1],…,exp[-jπsinθK]). We assume that v̂k is the kth diagonal element of Ecm†Ec(m+1), then the angle estimates can be achieved by(16d)θ̂kini=sin-1(-angle(v̂k)π).Therefore, the initial estimations of DOA and DOD have been obtained with the characteristics of automatic pairing.However, the estimation obtained by (15) and (16d) has low accuracy and bias, for it just employs partial information of the antennas, the more accurate estimation of the angles will be obtained below.Referring to [14, Lemma 1], the estimation for the transmitted signals and their paired response can be obtained by
(17)ãt(θ)=Ctãt(θ)=Tt(θ)ct,ãr(ϕ)=Crãr(ϕ)=Tr(ϕ)cr,
wherect=[c1t,…,cmt], cr=[c1r,…,cnr],Tt(θ)=Tt1(θ)+Tt2(θ), and the nonzero elements calculated from Tt1(θ)∈ℂM×m and Tt2(θ)∈ℂM×m can now be written as(18)[Tt1]ij=(at(θ))i+j-1,i+j≤M+1,[Tt2]ij=(at(θ))i-j+1,i≥j≥2.
It is shown for Q(θ,ϕ,Cr,Ct)that(19)Q(θ,ϕ,Cr,Ct)=(cr⊗ct)HQM(θ,ϕ)(cr⊗ct),
where QM(θ,ϕ)=(Tr(ϕ)⊗Tt(θ))HEnEnH(Tr(ϕ)⊗Tt(θ)).Sincecr⊗ct≠0, and when MN-K≥mn, we construct the function to search for the estimations of DOA and DOD [16] as follows:(20)(θ̂,ϕ̂)=argmax⁡θ,ϕ1det⁡(QM(θ,ϕ)).
Equation (20) requires a two-dimensional searching, which has a high complexity. Considering the initial estimation of DOD and DOA obtained in (15) and (16a), (16b), (16c), and (16d), we can use two one-dimensional searching to obtain more accurate estimation of DOD and DOA(21)θ̂k=argmax⁡θ∈[θ̂kini-Δθ,θ̂kini+Δθ]1det⁡(QM(θ,ϕ̂kini)).
Throughout the local searching for θ within [θ̂kini-Δθ,θ̂kini+Δθ], where Δθ is a small value, we obtain more accurate estimation of the DOD θ̂k. Similarly, ϕk can be also estimated from ϕ within [ϕ̂kini-Δϕ,ϕ̂kini+Δϕ], where Δϕ is a small value. Hence we have(22)ϕ̂k=argmax⁡ϕ∈[ϕ̂kini-Δϕ,ϕ̂kini+Δϕ]1det⁡(QM(θ̂k,ϕ)),
where θ̂k is estimation of θk via (21).


## 3.2. Mutual Coupling Estimation

According to (19), the estimates of crt=cr⊗ct will be obtained from the minimum eigenvector of (1/K)∑k=1KQM(θ̂k,ϕ̂k). After normalizing crt(crt(1)=1), and reshaping it to an m×n matrix Crt, we have(23)Crt=ctcrT.
Then we obtain the estimation of cr and ct from the singular value decomposition of Crt. Let us assume Crt=USVH and the largest left singular vector is ut, while the largest right singular vector is vr, it is easy to construct that(24)ĉt=utut(1),ĉr=(vrvr(1))*.
Till now, we have achieved the proposal of our scheme for the joint angle and mutual coupling estimation in MIMO-radar. We show the major algorithmic steps as follows.(1)
Estimate the covariance matrix of the received data throughR̂=∑t=1Jx(t)xH(t).(2)
Perform eigenvalue decomposition ofR̂ to get Ês and ÊN, and partition the matrix Es to get Es1,Es2,…,EsN. It follows to obtain ϕ̂kini from the eigenvalues of Esn†Es(n+1) and T̂ from the corresponding eigenvectors.(3)
ComputeEc, and get the initial estimation of the DOD θ̂kini.(4)
Obtain the estimation of DODθ̂k through one-dimensional searching via (21) while keeping ϕkini fixed.(5)
Obtain the estimation of DOAϕ̂k through one-dimensional searching via (22) while keeping θ̂k fixed.(6)
Obtain the estimation ofCrt from the minimum eigenvector of (1/K)∑k=1KQM(θ̂k,ϕ̂k), and step further to get the estimation of cr and ct from the singular value decomposition of Crt via (24).


## 3.3. Complexity Analysis and Cramer-Rao Bound (CRB)

The proposed algorithm has lower complexity than the algorithm in [14]. Our scheme costs O{M2N2J+M3N3+2K2M+3K3+2K2N+K(2N+2) + 2n1K[MNmn+MNmn(MN-K) + (mn)! + (MN-K)m2n2]+m3n3+min(n2m,m2n)}, where n1 is the number of steps within the local searching range, while the algorithm in [14] requires O{M2N2J+M3N3+n2[M2Nn(MN-K) + (MN-K)M2n2+2(Mn)!+N2Mm(MN-K) + N2m2(MN-K)]+K2[MNmn+MNmn(MN-K) + (mn)!+(MN-K)m2n2]+m3n3+min(n2m,m2n)}, where n2 is the number of steps within the global searching range, and n2≫n1.According to [17], we derive the CRB for the joint angles estimation and mutual coupling algorithm in bistatic MIMO radar as follows:(25)CRB=σ22{Re[DHΠG⊥D]}-1,
where σ2stands for the covariance of the noise. 								(26)D=[∂u∂θ1,…,∂u∂θK,∂u∂ϕ1,…,∂u∂ϕK,∂u∂c2t,…,∂u∂cmt,∂u∂c2r…,∂u∂cnr]
with
(27)u=[Ãb(1)⋮Ãb(J)],ΠG⊥=[ΠÃ⊥0⋱0ΠÃ⊥],
with
(28)ΠÃ⊥=IMN-Ã(ÃHÃ)-1ÃH.


## 3.4. Advantages of the Proposed Algorithm

By compares and contrasts to [14], the algorithmic advantage of our approach can be briefly summarized as follows:(1)
Our approach can achieve the estimation for automatically paired two-dimensional angles, while the algorithm in [14] requires additional pairing.(2)
The proposed algorithm only requires the local one-dimensional searching and hence has much lower complexity, while the algorithm in [14] urges two global one-dimensional searching.(3)
For the performance of angle and mutual coupling estimation, slightly improvement can be verified in our method than the one in [14].(4)
By fully exploiting both the signal subspace and the noise subspace, our method displays more validity and effectiveness, while the algorithm in [14] just employs the noise subspace.


## 4. Simulation Results

The Root mean square error (RMSE) of the angle and the RMSE of the mutual coupling coefficients [12] can be defined as(29)RMSEa=1K∑k=1K11000∑l=11000[(ϕ̂k,l-ϕk)2+(θ̂k,l-θk)2],RMSEc=11000∑l=11000[‖ĉr,l-cr‖F2+‖ĉt,l-ct‖F2],
where θ̂k,l is the estimate of DOD, θkof the lth Monte Carlo trial, ϕ̂k,l is the estimate of DOA ϕk of the lth Monte Carlo trial, ĉr,lis the estimate of cr of the lth Monte Carlo trial, and ĉt,lis the estimate of ct of the lth Monte Carlo trial. We assume that there are K=3 targets with their angles correspondingly distributed as (θ1,ϕ1)=(50∘,-20∘), (θ2,ϕ2)=(-50∘,20∘), (θ1,ϕ1)=(10∘,40∘). ct=[1,0.1174+0.0577j], and cr=[1,-0.0121-0.1029j].Figure1 depicts the estimation result of the proposed algorithm for all three targets with M=8, N=10, J=200, and SNR = 10 dB. It is explicit to show that the angles can be clearly observed.Figure 1
Angle estimation performance of the proposed algorithm with SNR = 10 dB.We compare the proposed algorithm against the algorithm in [14] and CRB. Figures 2 and 3 present the comparison of the algorithms. From Figures 2 and 3, we can find that the angle estimation performance of the proposed algorithm is slightly better than the algorithm in [14], while our approach has very close performance for mutual coupling estimation to that of [14]. The reasons are (1) the method in [14] estimates the DOA and DOD separately, while the proposed considers the two angle’s information simultaneously. (2) The proposed obtains an initial estimation firstly, which provides more information of the angles in estimation.Angle and mutual coupling estimation performance comparison (M=8,N=10,J=200,K=3).
(a)(b)Angle and mutual coupling estimation performance comparison (M=8,N=6,J=150,K=2).
(a)(b)Figures4 and 5 present the estimation performance of the proposed algorithm with J=150, K=2, and variable proportion of M to N. From Figures 4 and 5, the angle estimation performance of the proposed algorithm is improved with the increment of receive/transmit antenna numbers. Due to the diversity gain, multiple receive/transmit antennas improve angle estimation performance [12]. Meanwhile, it can be verified that the mutual coupling estimation performance of our method is improved with the decrement of receive/transmit antenna numbers. The dimensions of the mutual coupling matrix enlarge in condition that the number of receive/transmit antennas is increasing.Angle estimation performance withN=6 and different values of M.
(a)(b)Angle estimation performance withM=8 and different values of N.
(a)(b)Figure6 presents angle estimation performance of our scheme with M=8, N=6, K=2 and different values of J. It is indicated in Figure 6 that the angle estimation and mutual coupling estimation performance of the proposed algorithm is improved with J increasing. When J increases, we obtain a more accurate covariance matrix, and thereafter gain a more accurate subspace, which leads to more accurate angle estimation.Angle estimation performance with different values ofJ.
(a)(b)


## 5. Conclusion

In this paper, we have presented a low-complexity algorithm for joint angles and mutual coupling estimation in bistatic MIMO radar with uniform linear arrays (ULAs). This method first obtains an initial estimate of DOA and DOD, then employs the local one-dimensional searching to achieve exact estimates for DOA and DOD, and finally estimates mutual coupling coefficients via the estimated angles. Automatically paired two-dimensional angle estimation can be also performed by our proposed algorithm. We have reduced the algorithmic complexity in comparison with that of [14] and established better performance of angle estimation.



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*Source: 921878-2012-06-13.xml*