# Low Complexity Direction and Doppler Frequency Estimation for Bistatic MIMO Radar in Spatial Colored Noise

Authors: Lingyun Xu; Guangbin Zhang; Xiaofei Zhang; Zongze Xu
Journal: International Journal of Antennas and Propagation
 (2014)

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## Abstract

We investigate the algorithm of direction and Doppler frequency estimation for bistatic multiple-input multiple-output (MIMO) radar in spatial colored noise. A novel method of joint estimation of direction and Doppler frequency in spatial colored noise based on propagator method (PM) for bistatic MIMO radar is discussed. Utilizing the cross-correlation matrix which is formed by the adjacent outputs of match filter in the time domain, the special matrix is constructed to eliminate the influence of spatial colored noise. The proposed algorithm provides lower computational complexity and has very close parameters estimation compared to estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm in high signal-to-noise ratio (SNR). It is applicable even if the transmitted waveforms are not orthogonal. The estimated parameters can be paired automatically and the Cramér-Rao Bound (CRB) is given in spatial colored noise. Simulation results confirm the effectiveness of the proposed  method.


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## Body

## 1. Introduction

Since multiple-input multiple-output (MIMO) radars use multiple antennas to simultaneously transmit diverse waveforms and utilize multiple antennas to receive the reflected signals, they have many potential advantages over conventional phased-array radars [1–4]. According to the array configuration, MIMO radar can be divided into two categories: the statistical MIMO radar and collocated MIMO radar [5–15]. The advantages of MIMO radar with collocated antennas have been studied extensively, which include improved direction performance and higher resolution [16], higher sensitivity or detection of moving targets, and increased degrees of freedom for transmit beamforming [17]. MIMO radar with widely separated antennas can capture the spatial diversity of the target’s radar cross-section (RCS) [18]. This spatial diversity provides radar system with the ability to improve target parameter estimation [19–22], high resolution target localization, and tracking performance [23, 24].Target direction estimation is a basic function of a radar system. Many advanced direction estimation algorithms for MIMO radar have been extensively discussed in the current literature which include ESPRIT algorithm, Capon algorithm, parallel factor (PARAFAC) algorithm, multiple signal classification (MUSIC) algorithm, and PM algorithm [25–33]. In [25, 29], ESPRIT algorithm exploited the invariance property of both the transmit array and the receive array for direction estimation in MIMO radar systems. Reference [30] derived a reduced-dimension multiple signal classification (MUSIC) algorithm for direction of departure (DOD) and direction of arrival (DOA) estimation. The algorithm, which only requires one-dimensional search, can avoid the high computational cost of the two-dimensional MUSIC algorithm. However, the mentioned algorithm above did not consider the Doppler frequency estimation, and the noises were assumed to be the Gaussian white noise. In [31], the ESPRIT method was used for DOD-DOA and Doppler frequency estimation which necessitates eigen decomposition of the sample covariance matrix. Huge computation will be involved where the large array size is required in applications. Yunhe [32] proposed the DOA matrix algorithm to estimate the DOD-DOA and Doppler frequency, but it cannot eliminate the influence of spatial colored noise. In this paper, we propose a low-complexity angle and Doppler frequency estimation algorithm which can reduce computational cost. It has very close parameters estimation performance compared to ESRPIT and DOA matrix algorithm in high SNR. And the proposed algorithm pairs the parameters automatically and eliminates the influence of the spatial colored noise. Simulation results illustrate performance of the proposed algorithm.The remainder of this paper is structured as follows. Section2 develops the data model for a bistatic MIMO radar system, and Section 3 proposes the proposed algorithm for angle and Doppler frequency estimation in MIMO radar. In Section 4, simulation results are presented to verify improvement for the proposed algorithm, while the conclusions are shown in Section 5.Notation. (·)*,  (·)T,  (·)H,  (·)-1,  (·)†, and vec⁡[·] denote complex conjugation, transpose, conjugate-transpose, inverse, Moore-Penrose inverse (pseudoinverse), and vectorization operator, respectively. ∥∥F presents Frobenius norm; IK is a K×K identity matrix; ⊗ represents Kronecker product; min⁡(·) is to get minimum elements of an array; angle(·) denotes the phase of a complex; row(·) denotes the operator that stacks the rows of a matrix in a column vector. Dn(·) is to take the nth row of the matrix to construct a diagonal matrix. E[·] is expectation operator.


## 2. Data Model

We consider a narrowband bistatic MIMO radar system withM-element transmit antennas and N-element receive antennas, both of which are half-wavelength spaced uniform linear arrays. The transmit antennas transmit M orthogonal coded signals. sm=[sm(1),sm(2),…,sm(P)]T∈CP×1,  m=1,2,…,M denotes the sampled baseband coded signal of the mth transmit antenna with one repetition interval with P being the length of the transmitted code sequence, so the transmit signals can be expressed as S=[s1,s2,…,sM]T. We also assume that there are a number of K far-field independent targets; ar(θk) and at(ϕk) are the receive steering vector and transmit steering vector for θk(DOA) and ϕk(DOD) of the kth target, so the arrival signal of the kth target is atT(ϕk)S. The received array through reflections of the target can be expressed as
(1)Y(t)=∑k=1Kβkej2πfdk(t-1)/fsar(θk)atT(ϕk)S+W(t),
where βk and  fdk denote the radar cross-section (RCS) fading coefficient and Doppler frequency of the kth target. fs is the pulse repeat frequency. Due to the steering vector of ULA, we have ar(θk)=[1,e-jπsinθk,…,e-jπ(N-1)sinθk]T,  at(ϕk)=[1,e-jπsinϕk,…,e-jπ(M-1)sinϕk]T.w
(
t
)
∈
C
N
×
P denotes a Gaussian noise of zeros mean with unknown covariance matrix Qw. Matching the received data with the signal (1/P)SH, we obtain
(2)x(t)=Aη(t)+n(t),
where A=[a-1,a-2,…,a-K] is an MN×K matrix composed of the K steering vectors, and a-k=ar(θk)⊗a-t(ϕk) is the Kronecker product of the receive and the transmit steering vectors for the kth target. a-t(ϕk)=RsTat(ϕk), owing to Rs=E[SSH/P]=IM; then a-t(ϕk)=at(ϕk).  η(t)=[η1(t),η2(t),…,ηK(t)]T,  ηk(t)=Pβkej2πfdk(t-1)/fs,  k=1,2,…,K.  n(t)=vec⁡[W(t)SH/P].The covariance matrix ofn(t) is as follows:
(3)E[n(i)nH(j)]=E{vec⁡[W(i)SH]vec⁡H[W(j)SH]}P=E{[S*⊗IN][vec⁡(W(i))vec⁡H(W(j))][ST⊗IN]}P={IM⊗Qw,i=j0,i≠j.


## 3. Direction and Doppler Frequency Estimation Algorithm for MIMO Radar

### 3.1. The Proposed Algorithm Description

We assume thatat(ϕk) and ar(θk) are constant for L samples and defineX as X=[x(1),x(2),…,x(L)], and we assume that the number of snapshots is L. Let
(4)X1=[x(1),x(2),…,x(L-1)]=Aη1+N1,X2=[x(2),x(3),…,x(L)]=Aη2+N2,
where η1=[η(1),η(2),…,η(L-1)],  η2=[η(2),η(3),…,η(L)],  N1=[n(1),n(2),…,n(L-1)], and N2=[n(2),n(3),…,n(L)]. Equation (3) shows that the cross-covariance matrix of noises is 0. This characteristic will be utilized in this paper to improve the estimate performance.Note thatE[N2N1H]=0, η2=𝚽η1,  𝚽=diag⁡[ej2πfd1/fs,ej2πfd2/fs,…,ej2πfdK/fs]. This indicates that the rotation factor 𝚽 is generated by adjacent outputs of match filters.The covariance matrix ofX1 and X2 can be written as follows:
(5)Rx=E[X2X1H]=A𝚽RηAH+E[N2N1H]=A𝚽RηAH,
where Rη=E[η1η1H]. For the independent targets, Rη should be a diagonal matrix; then we have the relationship Rη𝚽H=𝚽HRη. Owing to E[N2N1H]=0,  Rx can eliminate the influence of spatial colored noise. A new matrix R is constructed by utilizing (5):
(6)R=[RxHRx]=[A𝚽HA𝚽]RηAH=[A𝚽-1A𝚽]︸BRηAH=BRηAH.
The propagator method relies on the partition of the matrix B [34]. B can be denoted by B=[B1B2], where B1∈CK×K is the full rank matrix; B2∈C(2MN-K)×K. The propagator V is a unique linear operator which can be written as VHB1=B2. Similarly partitioning received data matrix R into two submatrices R1 and R2 with dimensions K×MN and (2MN-K)×MN, respectively. Then the unique linear operation holds between R1 and R2:
(7)VHR1=R2.
An estimation matrix V can be obtained by minimizing the cost function: J(V)=min⁡∥R2-VHR1∥F2. The optimal solution is given by V=(R1R1H)-1R1R2H. Define a new matrix V~H=[IKVH], where IK is the identity matrix. Combining V~ and (7), we obtain
(8)V~HR1=[R1R2]=[B1B2]RηAH=[A𝚽-1A𝚽]RηAH=BRηAH.
Rewrite (8) as
(9)V~H=BRηAHR1†︸T=BT,
where R1† is the pseudoinverse of R1 and T is a nonsingular matrix. From (9), the columns in V~ span the same signal subspace as the column vectors in B. So the signal subspace can be obtained by avoiding the estimation and eigen decomposition of the sample covariance matrix. The matrix V~H can be partitioned into two submatrices V~H=[V~1V~2]. According to (9), we can get
(10)[V~1V~2]=[A𝚽-1A𝚽]︸BT.
Notice that 𝚽=𝚽-1𝚽2; from (10) we can obtain
(11)𝚽2=TV~1†V~2T-1,(12)A𝚽=V~2T-1. Equation (11) shows that the main diagonal elements of 𝚽2 are equal to the eigen values obtained via the eigen decomposition of V~1†V~2 with T the corresponding eigenvectors. Thus the Doppler frequency of the kth target can be calculated as
(13)f^k=angle(φk)fs4π,
where φk is the kth diagonal element of 𝚽2. The A^ can be calculated from (12), because the 𝚽 is the diagonal matrix, and it cannot affect the estimation of DOD and DOA using the least square method from the matrix A^. We note that the pairing is automatically obtained because the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors. The matrix A^ can be also denoted by
(14)A^=A^R∘A^T=[A^TD1(A^R)A^TD2(A^R)⋮A^TDN(A^R)]=[A^TA^T𝚽^r⋮A^T𝚽^rN-1],
where A^T=[a^t(ϕ1),a^t(ϕ2),…,a^t(ϕK)] and A^R=[a^r(θ1),a^r(θ2),…,a^r(θK)] are the transmit and receive direction matrices, respectively:
(15)𝚽^r=diag⁡[exp⁡⁡(-jπsin⁡θ^1),exp⁡⁡(-jπsin⁡θ^2),…,exp⁡⁡(-jπsin⁡θ^K)].
There exists an MN×MN transformation matrix C corresponding to the finite number of row interchanged operations such that
(16)F^=CA^=A^T∘A^R=[A^RD1(A^T)A^RD2(A^T)⋮A^RDM(A^T)]=[A^RA^R𝚽^t⋮A^R𝚽^tM-1],
where
(17)𝚽^t=diag⁡[exp⁡(-jπsinϕ^1),exp⁡(-jπsinϕ^2),…,exp⁡(-jπsinϕ^K)].We define(18)Pb=[(A^T𝚽r)T,(A^T𝚽r2)T,…,(A^T𝚽rN-1)T]T,Pa=[(A^T)T,(A^T𝚽r)T,…,(A^T𝚽rN-2)T]T.
Owing to Pa𝚽^r=Pb, 𝚽^r=Pa†Pb, then we get the estimation of DOA:
(19)θ^k=sin-1(angle(pk)π),
where pk is the kth diagonal element of the matrix 𝚽^r.We also define(20)Bb=[(A^R𝚽t)T,(A^R𝚽t2)T,…,(A^R𝚽tN-1)T]T,Ba=[(A^R)T,(A^R𝚽t)T,…,(A^R𝚽tN-2)T]T,
Owing to Ba𝚽^t=Bb,  𝚽^t=Ba†Bb, we can also get the estimation of DOD:
(21)ϕ^k=sin-1(angle(λk)π).
where λk is the kth diagonal element of the matrix 𝚽^t.Now we show the major steps of the proposed algorithm as follows.(1)
Compute the covariance matrix of the received data through (5).(2)
Estimate the propagatorV from (7).(3)
Compute the Doppler frequency according to (10), (11), and (13).(4)
Estimate matrixA^ from (12); then use the least square method to estimate the DOD and DOA according to (14)~(21).Remark 1.
In fact completely orthogonal signals cannot be found in reality, if we consider the transmitted nonorthogonal signals; that is,a-t(ϕk)≠at(ϕk), and the output signal can be expressed as x(t)=A~η(t)+n(t),   where A~=[a~1,a~2,…,a~K] and a~k=ar(θk)⊗a-t(ϕk). Define A~T=[a-t(ϕ1),a-t(ϕ2),…,a-t(ϕK)],  A~T=RsAT,  A~=AR∘A~T=R-s[ATT,(AT𝚽r)T,…,(AT𝚽rN-1)T]T,   where R-s=diag⁡[Rs,Rs,…,Rs]. The estimation of A~ can be denoted as A^=A~Π=R-sAΠ,  Π-1=Π. Estimating the matrix A~ from (12), then multiplying(R-s)-1, we get A-=AΠ, using the least square method to estimate the DOD and DOA according to (14)~(21).


### 3.2. Complexity Analysis

In contrast to ESPRIT algorithm [31], our algorithm has a low computational load; the main computational cost of our algorithm is the estimation of the matrix V~H, which takes O(2M2N2K+MNK2+K3); the total computational complexity of our algorithm is O[M2N2(L-1)+2M2N2K+2MNK2+2K3+M(N-1)K2+N(M-1)K2], while ESPRIT requires O(4(L-1)M2N2+8M3N3+2K3+M(N-1)K2+N(M-1)K2) in the eigen decomposition of the covariance matrices. DOA matrix algorithm [32] requires O(2(L-1)M2N2+2M3N3+M(N-1)K2+N(M-1)K2). Figure 1 shows the complexity comparison with M=9, N=9, K=3, and different L. From Figure 1 we find that our algorithm has much lower computational load than ESPRIT algorithm and DOA matrix algorithm. The computational cost (CPU time) of the proposed algorithm is 0.053029 s, while ESPRIT algorithm and DOA matrix algorithm need 0.210587 s and 0.121686 s, respectively, at M=9, N=9, K=3, and L=100 with CPU frequency 2.20 GHz.Figure 1
Complexity comparison withM=9, N=9, K=3, and different L.


### 3.3. Discussion

(1)
From Figure1 we find that our algorithm has much lower computational load than ESPRIT algorithm and DOA matrix algorithm. ESPRIT algorithm employs either eigen-value decomposition (EVD) of cross-correlation matrix or singular value. Using the techniques, the computational complexity is very high. Reference [34] has shown the propagator method (PM) for array signal processing to estimate DOA of incident signals without eigen-value decomposition of cross-correlation matrix of the received data. In our proposed algorithm, propagator V is a linear operator which can easily be extracted from the data matrix R. But the construction of matrix V~ leads the proposed algorithm’s performance to degrade in low SNR. So the proposed algorithm has very close parameters estimation to ESPRIT algorithm and DOA matrix algorithm in high SNR.
(2)
Since the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors, it can achieve automatically paired estimation of angles and Doppler frequencies.
(3)
The proposed algorithm can eliminate the effect of the spatial colored noise since the new matrix is constructed by (5) and (6).


### 3.4. The Cramér-Rao Bound (CRB)

In this section, we derive CRB of parameter estimation for MIMO radar and rewrite the received data as(22)Z=row(X)=K(fd,ϕ,θ)β+W,
where K(fd,ϕ,θ)=[K1,K2,…,KK],  Kk=af(fdk)⊗ak,  af(fdk)=[1,ej2πfdk/fs,…,ej2πfdk(L-1)/fs]T,  β=[Pβ1,…,PβK]T,  and W represents the noise vector. The fisher information matrix (FIM) with respect to ϕ=[ϕ1,ϕ2,…,ϕK],  θ=[θ1,θ2,…,θK], and fd=[fd1,fd2,…,fdK] can be calculated as follows [31, 35]:
(23)F=[F11F12F13F21F22F23F31F32F33],
where
(24)F11=2Re[ΓH(Kθ′)HΠHQ-1ΠKθ′Γ],F12=2Re[ΓH(Kθ′)HΠHQ-1ΠKϕ′Γ],F13=2Re[ΓH(Kθ′)HΠHQ-1ΠKf′Γ],F21=2Re[ΓH(Kϕ′)HΠHQ-1ΠKθ′Γ],F22=2Re[ΓH(Kϕ′)HΠHQ-1ΠKϕ′Γ],F23=2Re[ΓH(Kϕ′)HΠHQ-1ΠKf′Γ],F31=2Re[ΓH(Kf′)HΠHQ-1ΠKθ′Γ],F32=2Re[ΓH(Kf′)HΠHQ-1ΠKϕ′Γ],F33=2Re[ΓH(Kf′)HΠHQ-1ΠKf′Γ],
And  Γ=diag⁡(β) denotes the diagonal matrix constructed by the vector β:
(25)Q=IML⊗Qw,Π=IMN-K(KHQ-1K)-1KHQ-1,Kθ′=[af(fd1)⊗[at(ϕ1)⊗∂ar(θ1)∂θ1],…,af(fdK)⊗[at(ϕK)⊗∂ar(θK)∂θK]],Kϕ′=[af(fd1)⊗[∂at(ϕ1)∂ϕ1⊗ar(θ1)],…,af(fdK)⊗[∂at(ϕK)∂ϕK⊗ar(θK)]]Kf′=[∂af(fd1)∂fd1⊗[at(ϕ1)⊗ar(θ1)],…,∂af(fdK)∂fdK⊗[at(ϕK)⊗ar(θK)]]
Then the CRB matrix is
(26)CRB=F-1.


## 3.1. The Proposed Algorithm Description

We assume thatat(ϕk) and ar(θk) are constant for L samples and defineX as X=[x(1),x(2),…,x(L)], and we assume that the number of snapshots is L. Let
(4)X1=[x(1),x(2),…,x(L-1)]=Aη1+N1,X2=[x(2),x(3),…,x(L)]=Aη2+N2,
where η1=[η(1),η(2),…,η(L-1)],  η2=[η(2),η(3),…,η(L)],  N1=[n(1),n(2),…,n(L-1)], and N2=[n(2),n(3),…,n(L)]. Equation (3) shows that the cross-covariance matrix of noises is 0. This characteristic will be utilized in this paper to improve the estimate performance.Note thatE[N2N1H]=0, η2=𝚽η1,  𝚽=diag⁡[ej2πfd1/fs,ej2πfd2/fs,…,ej2πfdK/fs]. This indicates that the rotation factor 𝚽 is generated by adjacent outputs of match filters.The covariance matrix ofX1 and X2 can be written as follows:
(5)Rx=E[X2X1H]=A𝚽RηAH+E[N2N1H]=A𝚽RηAH,
where Rη=E[η1η1H]. For the independent targets, Rη should be a diagonal matrix; then we have the relationship Rη𝚽H=𝚽HRη. Owing to E[N2N1H]=0,  Rx can eliminate the influence of spatial colored noise. A new matrix R is constructed by utilizing (5):
(6)R=[RxHRx]=[A𝚽HA𝚽]RηAH=[A𝚽-1A𝚽]︸BRηAH=BRηAH.
The propagator method relies on the partition of the matrix B [34]. B can be denoted by B=[B1B2], where B1∈CK×K is the full rank matrix; B2∈C(2MN-K)×K. The propagator V is a unique linear operator which can be written as VHB1=B2. Similarly partitioning received data matrix R into two submatrices R1 and R2 with dimensions K×MN and (2MN-K)×MN, respectively. Then the unique linear operation holds between R1 and R2:
(7)VHR1=R2.
An estimation matrix V can be obtained by minimizing the cost function: J(V)=min⁡∥R2-VHR1∥F2. The optimal solution is given by V=(R1R1H)-1R1R2H. Define a new matrix V~H=[IKVH], where IK is the identity matrix. Combining V~ and (7), we obtain
(8)V~HR1=[R1R2]=[B1B2]RηAH=[A𝚽-1A𝚽]RηAH=BRηAH.
Rewrite (8) as
(9)V~H=BRηAHR1†︸T=BT,
where R1† is the pseudoinverse of R1 and T is a nonsingular matrix. From (9), the columns in V~ span the same signal subspace as the column vectors in B. So the signal subspace can be obtained by avoiding the estimation and eigen decomposition of the sample covariance matrix. The matrix V~H can be partitioned into two submatrices V~H=[V~1V~2]. According to (9), we can get
(10)[V~1V~2]=[A𝚽-1A𝚽]︸BT.
Notice that 𝚽=𝚽-1𝚽2; from (10) we can obtain
(11)𝚽2=TV~1†V~2T-1,(12)A𝚽=V~2T-1. Equation (11) shows that the main diagonal elements of 𝚽2 are equal to the eigen values obtained via the eigen decomposition of V~1†V~2 with T the corresponding eigenvectors. Thus the Doppler frequency of the kth target can be calculated as
(13)f^k=angle(φk)fs4π,
where φk is the kth diagonal element of 𝚽2. The A^ can be calculated from (12), because the 𝚽 is the diagonal matrix, and it cannot affect the estimation of DOD and DOA using the least square method from the matrix A^. We note that the pairing is automatically obtained because the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors. The matrix A^ can be also denoted by
(14)A^=A^R∘A^T=[A^TD1(A^R)A^TD2(A^R)⋮A^TDN(A^R)]=[A^TA^T𝚽^r⋮A^T𝚽^rN-1],
where A^T=[a^t(ϕ1),a^t(ϕ2),…,a^t(ϕK)] and A^R=[a^r(θ1),a^r(θ2),…,a^r(θK)] are the transmit and receive direction matrices, respectively:
(15)𝚽^r=diag⁡[exp⁡⁡(-jπsin⁡θ^1),exp⁡⁡(-jπsin⁡θ^2),…,exp⁡⁡(-jπsin⁡θ^K)].
There exists an MN×MN transformation matrix C corresponding to the finite number of row interchanged operations such that
(16)F^=CA^=A^T∘A^R=[A^RD1(A^T)A^RD2(A^T)⋮A^RDM(A^T)]=[A^RA^R𝚽^t⋮A^R𝚽^tM-1],
where
(17)𝚽^t=diag⁡[exp⁡(-jπsinϕ^1),exp⁡(-jπsinϕ^2),…,exp⁡(-jπsinϕ^K)].We define(18)Pb=[(A^T𝚽r)T,(A^T𝚽r2)T,…,(A^T𝚽rN-1)T]T,Pa=[(A^T)T,(A^T𝚽r)T,…,(A^T𝚽rN-2)T]T.
Owing to Pa𝚽^r=Pb, 𝚽^r=Pa†Pb, then we get the estimation of DOA:
(19)θ^k=sin-1(angle(pk)π),
where pk is the kth diagonal element of the matrix 𝚽^r.We also define(20)Bb=[(A^R𝚽t)T,(A^R𝚽t2)T,…,(A^R𝚽tN-1)T]T,Ba=[(A^R)T,(A^R𝚽t)T,…,(A^R𝚽tN-2)T]T,
Owing to Ba𝚽^t=Bb,  𝚽^t=Ba†Bb, we can also get the estimation of DOD:
(21)ϕ^k=sin-1(angle(λk)π).
where λk is the kth diagonal element of the matrix 𝚽^t.Now we show the major steps of the proposed algorithm as follows.(1)
Compute the covariance matrix of the received data through (5).(2)
Estimate the propagatorV from (7).(3)
Compute the Doppler frequency according to (10), (11), and (13).(4)
Estimate matrixA^ from (12); then use the least square method to estimate the DOD and DOA according to (14)~(21).Remark 1.
In fact completely orthogonal signals cannot be found in reality, if we consider the transmitted nonorthogonal signals; that is,a-t(ϕk)≠at(ϕk), and the output signal can be expressed as x(t)=A~η(t)+n(t),   where A~=[a~1,a~2,…,a~K] and a~k=ar(θk)⊗a-t(ϕk). Define A~T=[a-t(ϕ1),a-t(ϕ2),…,a-t(ϕK)],  A~T=RsAT,  A~=AR∘A~T=R-s[ATT,(AT𝚽r)T,…,(AT𝚽rN-1)T]T,   where R-s=diag⁡[Rs,Rs,…,Rs]. The estimation of A~ can be denoted as A^=A~Π=R-sAΠ,  Π-1=Π. Estimating the matrix A~ from (12), then multiplying(R-s)-1, we get A-=AΠ, using the least square method to estimate the DOD and DOA according to (14)~(21).


## 3.2. Complexity Analysis

In contrast to ESPRIT algorithm [31], our algorithm has a low computational load; the main computational cost of our algorithm is the estimation of the matrix V~H, which takes O(2M2N2K+MNK2+K3); the total computational complexity of our algorithm is O[M2N2(L-1)+2M2N2K+2MNK2+2K3+M(N-1)K2+N(M-1)K2], while ESPRIT requires O(4(L-1)M2N2+8M3N3+2K3+M(N-1)K2+N(M-1)K2) in the eigen decomposition of the covariance matrices. DOA matrix algorithm [32] requires O(2(L-1)M2N2+2M3N3+M(N-1)K2+N(M-1)K2). Figure 1 shows the complexity comparison with M=9, N=9, K=3, and different L. From Figure 1 we find that our algorithm has much lower computational load than ESPRIT algorithm and DOA matrix algorithm. The computational cost (CPU time) of the proposed algorithm is 0.053029 s, while ESPRIT algorithm and DOA matrix algorithm need 0.210587 s and 0.121686 s, respectively, at M=9, N=9, K=3, and L=100 with CPU frequency 2.20 GHz.Figure 1
Complexity comparison withM=9, N=9, K=3, and different L.


## 3.3. Discussion

(1)
From Figure1 we find that our algorithm has much lower computational load than ESPRIT algorithm and DOA matrix algorithm. ESPRIT algorithm employs either eigen-value decomposition (EVD) of cross-correlation matrix or singular value. Using the techniques, the computational complexity is very high. Reference [34] has shown the propagator method (PM) for array signal processing to estimate DOA of incident signals without eigen-value decomposition of cross-correlation matrix of the received data. In our proposed algorithm, propagator V is a linear operator which can easily be extracted from the data matrix R. But the construction of matrix V~ leads the proposed algorithm’s performance to degrade in low SNR. So the proposed algorithm has very close parameters estimation to ESPRIT algorithm and DOA matrix algorithm in high SNR.
(2)
Since the DOA-DODs and Doppler frequencies are given through the corresponding eigenvectors, it can achieve automatically paired estimation of angles and Doppler frequencies.
(3)
The proposed algorithm can eliminate the effect of the spatial colored noise since the new matrix is constructed by (5) and (6).


## 3.4. The Cramér-Rao Bound (CRB)

In this section, we derive CRB of parameter estimation for MIMO radar and rewrite the received data as(22)Z=row(X)=K(fd,ϕ,θ)β+W,
where K(fd,ϕ,θ)=[K1,K2,…,KK],  Kk=af(fdk)⊗ak,  af(fdk)=[1,ej2πfdk/fs,…,ej2πfdk(L-1)/fs]T,  β=[Pβ1,…,PβK]T,  and W represents the noise vector. The fisher information matrix (FIM) with respect to ϕ=[ϕ1,ϕ2,…,ϕK],  θ=[θ1,θ2,…,θK], and fd=[fd1,fd2,…,fdK] can be calculated as follows [31, 35]:
(23)F=[F11F12F13F21F22F23F31F32F33],
where
(24)F11=2Re[ΓH(Kθ′)HΠHQ-1ΠKθ′Γ],F12=2Re[ΓH(Kθ′)HΠHQ-1ΠKϕ′Γ],F13=2Re[ΓH(Kθ′)HΠHQ-1ΠKf′Γ],F21=2Re[ΓH(Kϕ′)HΠHQ-1ΠKθ′Γ],F22=2Re[ΓH(Kϕ′)HΠHQ-1ΠKϕ′Γ],F23=2Re[ΓH(Kϕ′)HΠHQ-1ΠKf′Γ],F31=2Re[ΓH(Kf′)HΠHQ-1ΠKθ′Γ],F32=2Re[ΓH(Kf′)HΠHQ-1ΠKϕ′Γ],F33=2Re[ΓH(Kf′)HΠHQ-1ΠKf′Γ],
And  Γ=diag⁡(β) denotes the diagonal matrix constructed by the vector β:
(25)Q=IML⊗Qw,Π=IMN-K(KHQ-1K)-1KHQ-1,Kθ′=[af(fd1)⊗[at(ϕ1)⊗∂ar(θ1)∂θ1],…,af(fdK)⊗[at(ϕK)⊗∂ar(θK)∂θK]],Kϕ′=[af(fd1)⊗[∂at(ϕ1)∂ϕ1⊗ar(θ1)],…,af(fdK)⊗[∂at(ϕK)∂ϕK⊗ar(θK)]]Kf′=[∂af(fd1)∂fd1⊗[at(ϕ1)⊗ar(θ1)],…,∂af(fdK)∂fdK⊗[at(ϕK)⊗ar(θK)]]
Then the CRB matrix is
(26)CRB=F-1.


## 4. Simulation Results

We present the Monte Carlo simulations to assess the parameter estimation performance of our algorithm. Define root mean squared error (RMSE) as(1/K)∑k=1K(1/1000)∑n=11000[(ϕ^k,n-ϕk)2+(θ^k,n-θk)2], where θ^k,n is the estimate of DOA θk of the nth Monte Carlo trial and ϕ^k,n is the estimate of DOD ϕk of the nth Monte Carlo trial. Define RMSE of Doppler frequency as (1/K)∑k=1K(1/1000)∑n=11000(f^k,n-fk)2, where f^k,n is the estimate of Doppler frequency fk of the nth Monte Carlo trial. Note that M, N, L, and K are the number of transmit antennas, the receive antennas, the snapshots of the targets, and the number of the targets, respectively. We assume that there are K=3 independent targets, which are located at angles (θ1,ϕ1)=(10°,15°), (θ2,ϕ2)=(20°,25°), and (θ3,ϕ3)=(30°,35°), respectively. The Doppler frequencies of the three targets are 1000 Hz, 1500 Hz, and 2100 Hz, respectively, and their RCS are given by β1=β2=β3=1. The pulse repeat frequency fs is 10 KHz for a M=9 and N=9 bistatic MIMO radar. The (m,n)th element of the unknown noise covariance matrix Qw is 0.9|m-n|ejπ(m-n)/2. Figure 2 shows the estimation results with 100 Monte Carlo trials at L=100. As seen in Figure 2, the DODs, DOAs, and Doppler frequencies of all three targets are correctly paired and well localised.Angle and Doppler frequency estimation at SNR = 10 dB.
(a)
(b)Figures3 and 4 show the angle and Doppler frequency estimation performance comparison with M=9, N=9, K=3, and L=100, where we compare our algorithm with ESPRIT algorithm [31], DOA matrix algorithm [32], and CRB. It is indicated in Figures 3 and 4 that our algorithm has very close parameter estimation performance to ESPRIT algorithm and DOA matrix algorithm in high SNR condition.Figure 3
Angle estimation comparison performance.Figure 4
Doppler frequency estimation comparison.The simulation of Figures5 and 6 investigates the performance of our proposed algorithm under different number of targets K. The number of targets is set as 2 and 4. Our proposed algorithm has the different performance under different K, as shown in Figures 5 and 6 where M=9, N=9, and L=100 are considered. From Figures 5 and 6 we find that angle estimation performance of our proposed algorithm degrades with K increasing.Figure 5
Angle estimation with different values ofK.Figure 6
Doppler frequency estimation with different values ofK.Figure7 presents parameter estimation performance with three closely spaced targets, where M=8, N=8, and L=100 are considered. The closely spaced targets are located at angles (θ1,ϕ1)=(15°,18°), (θ2,ϕ2)=(18°,20°), and  (θ3,ϕ3)=(20°,25°). The Doppler frequencies of the three targets are 1200 Hz, 1500 Hz, and 1800 Hz, respectively. From Figure 7, we find that our algorithm works well in the case of closely spaced targets.Angle and Doppler frequency estimation at SNR = 12 dB.
(a)
(b)


## 5. Conclusion

We have presented a low-complexity angle and Doppler frequency estimation based on propagator method for MIMO radar in spatial colored noise. The proposed algorithm can obtain automatically paired transmit and receive angle estimations in the MIMO radar and eliminate the influence of the spatial colored noise. Furthermore, it provides lower computational complexity and has close parameters estimation compared to ESPRIT algorithm and DOA matrix algorithm in high SNR. It is applicable even if the transmitted waveforms are not orthogonal.



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*Source: 358563-2014-04-06.xml*