# Beamspace Unitary ESPRIT Algorithm for Angle Estimation in Bistatic MIMO Radar

Authors: Dang Xiaofang; Chen Baixiao; Yang Minglei; Zheng Guimei
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

The beamspace unitary ESPRIT (B-UESPRIT) algorithm for estimating the joint direction of arrival (DOA) and the direction of departure (DOD) in bistatic multiple-input multiple-output (MIMO) radar is proposed. The conjugate centrosymmetrized DFT matrix is utilized to retain the rotational invariance structure in the beamspace transformation for both the receiving array and the transmitting array. Then the real-valued unitary ESPRIT algorithm is used to estimate DODs and DOAs which have been paired automatically. The proposed algorithm does not require peak searching, presents low complexity, and provides a significant better performance compared to some existing methods, such as the element-space ESPRIT (E-ESPRIT) algorithm and the beamspace ESPRIT (B-ESPRIT) algorithm for bistatic MIMO radar. Simulation results are conducted to show these conclusions.


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) radar [1–6] is developed on the basis of MIMO communication theory, which has gained increasing attention and wide investigation in recent years. MIMO radar can emit orthogonal waveforms simultaneously through multiple antennas and also extract the orthogonal waveforms by using a bank of matched filters. Compared with the traditional phased array radars, a lot of potential advantages of MIMO radars, such as more degrees of freedom (DOFs) [2], better parameter identifiability [2], and higher angular estimation accuracy [3], increasingly appear. According to the configuration of transmitting/receiving arrays, MIMO radars can be grouped into two classes: the former is called statistical MIMO radar, where the transmitting and receiving antennas are widely spaced [1, 3, 4]. It aims at overcoming the radar cross section (RCS) scintillation effect which was encountered in radar systems by capitalizing on the spatial diversity [5]. The latter is known as monostatic MIMO radar or bistatic MIMO radar [2, 6–10], where the transmitting and receiving antennas are closely spaced. Monostatic or bistatic MIMO radar, which can form receiving beam and virtual transmitting beam jointly at the receiver [2], has many advantages, such as narrower beamwidth, lower sidelobes, higher angular resolution, and higher angular estimation accuracy [6, 11]. And this paper focuses on the bistatic MIMO radar.For the bistatic MIMO radar, one of the most important issues is to estimate the directions of departure and arrival of multiple targets from the received signals corrupted by noise. So far various approaches have been put forward. In [7], the two-dimensional (2D) Capon estimator is introduced to estimate the DOAs and the DODs of the targets in bistatic MIMO radar. To reduce the computational cost, a polynomial rooting estimator is introduced in [8]. The 2D multiple signal classification (MUSIC) algorithm and reduced-dimension MUSIC (RD-MUSIC) algorithm are discussed in [12, 13]. To alleviate the computational burden, the ESPRIT algorithm [14] is used for target direction estimation by utilizing the invariance property of both the transmitting array and the receiving array. Nevertheless, an additional matched pair is required in the ESPRIT algorithm. In order to solve this problem, a unitary ESPRIT algorithm for bistatic MIMO radar is proposed in [15], which fully exploits the real-valued rotational invariance equations of signal subspace to estimate DODs and DOAs that are paired automatically. When some a priori knowledge of the angle information of sources is known, the beamspace ESPRIT (B-ESPRIT) [16] algorithm is raised to reduce computational complexity; however, an additional pair matching is also needed and a degradation of the algorithm’s performance can be observed as well.To overcome the aforementioned problems, a beamspace unitary ESPRIT algorithm is developed to estimate DODs and DOAs of the targets in bistatic MIMO radar in this paper. First, the conjugate centrosymmetrized DFT matrix is employed to retain the rotational invariance structure in the beamspace transformation for both the receiving array and the transmitting array. Then the DODs and DOAs can be estimated in accordance with a new version of ESPRIT for the bistatic MIMO radar. The ESPRIT algorithm works in the beamspace and involves only real-valued computation from start to finish. Over the beamspace ESPRIT algorithm, the proposed algorithm has better performance that can be paired automatically for DODs and DOAs estimation. In some situation, the proposed algorithm also has a better performance over the element ESPRIT.And the structure of the rest of the paper will be organized as follows. The bistatic MIMO radar signal model is presented in Section2. The proposed beamspace unitary ESPRIT for DODs and DOAs estimation is proposed in Section 3. Section 4 gives the computational complexity analysis of the proposed algorithm, unitary ESPRIT, and beamspace ESPRIT; advantages and disadvantages of the proposed algorithm also have been discussed in Section 4. And simulation results are provided to verify the performance of the proposed algorithm in Section 5; the Cramer-Rao Bound (CRB) also has been derived in this section. Finally, Section 6 concludes this paper.


## 2. Signal Model of the Bistatic MIMO Radar

Considering a narrowband bistatic MIMO radar system with anM-element transmitting array and an N-element receiving array, we found that both are half-wave length spaced uniform linear arrays (Figure 1).Figure 1
The configuration of bistatic MIMO radar.At the transmitting site,M different orthogonal narrowband waveforms are emitted simultaneously. In each receiver, the echoes are processed for all of the transmitted waveforms. Assume that there are P uncorrelated targets in the same range bin, located in the far field of the array. The DOD and DOA of the pth target with respect to the transmitting array and the receiving array are denoted by θp and φp, respectively. Thus, the output of all the matched filters in receivers can be written as(1)Xt=ASt+Nt,where A=[a1,a2,…,ap] is an MN×P matrix, the column vectors ap, p=1,…,P are P steering vectors, which can be written as ap=ar(μp)⊗at(νp), is the combined steering vector of pth target, where ar(μp)=1ejμp⋯ejN-1μpT∈CN×1 and at(νp)=1ejνp⋯ejM-1νpT∈CM×1 are the receiving and transmitting steering vector of the pth target, and μp=πsinφp and νp=πsinθp, respectively; ⊗ denotes the Kronecker product; St=s1(t)s2(t)⋯sP(t)T is a column vector consisted of the amplitudes and phases of the P sources at time t; sp(t) is usually in the form of sp(t)=αpejωpt with αp being the complex amplitude and ωp the Doppler frequency of the pth target; N(t) is the additive noise, which is modeled as a zero-mean, spatially white Gaussian process with covariance matrix σn2IMN, where IMN denotes an MN×MN identity matrix.Note that the dimension ofX will be MN×J, and J is the number of time samples. When M and N are large, the computational load and time will be huge. To overcome this shortcoming, the beamspace unitary ESPRIT is proposed as follows.


## 3. Proposed Method for DOD and DOA Estimation

In a radar application, the operation of reducing dimension could be facilitated in beamspace when a priori information on the general angular locations of the signal arrivals presents. In this case, by utilizing the beamspace transform matrix, beams, which involve the sector of interest, would be formed, thereby reducing computational complexity. And if there is no a priori information, one may apply angle estimation algorithm via parallel processing to each of the number of sets of successive overlapped sectors, which will also reduce computational complexity.To retain the rotational invariance structure in the beamspace transformation, in this paper, the conjugate centrosymmetrized DFT matrix is applied as the beamspace transformed matrix. LetWrH be the receiving beamspace transformed matrix and let WtH be the transmitting beamspace transformed matrix. The mth row of WrH and WtH is formulated as [17](2)wr,mH=ejN-1/2m2π/N·1,e-jm2π/N,…,e-jN-1m2π/N,m=0,…,N-1wt,mH=ejM-1/2m2π/M·1,e-jm2π/M,…,e-jM-1m2π/M,m=0,…,M-1.Both WrH and WtH are conjugate centrosymmetrized matrixes and their mth row vector represents a DFT beam steered at the spatial frequency μ=m(2π/N) and ν=m(2π/M), respectively. An important property of the Kronecker operator that will prove useful throughout the transformation is(3)A⊗BC⊗D=AC⊗BD,where A is an m×n matrix, which is not the matrix A mentioned above in (1). B is a p×q matrix, which is also not the matrix B mentioned below in (5). C is an n×r matrix and D is a q×s matrix.The receiving beamspace manifold is defined asBr=WrHAr and the transmitting beamspace manifold is defined as Bt=WtHAt, respectively. Then the final beamspace manifold can be written as(4)B=WHA=Br⊗Bt=WrHAr⊗WtHAt.According to property (3), the equation WrHAr⊗WtHAt=WrH⊗WtHAr⊗At holds. Thus, the final beamspace transformed matrix is defined as (5)WH=WrH⊗WtH.W is an KtKr×MN matrix, where Kt and Kr are the number of transmitting beam and receiving beam, respectively. A new beamspace received signal is defined as(6)Yt=WHASt+N-t=ΒSt+N-t,where N-t=WHN(t) is the noise of beamspace.Next the rotational invariance structure in the beamspace will be examined. Considering the receiving beamspace manifoldBr=[br(μ1),…,br(μP)], the pth component of Br is br(μp)=[br,0(μp),…,br,N-1(μp)]T. And the mth elements of br(μp) are(7)br,mμp=wr,mHarμ=sin⁡N/2μp-m(2π/N)sin⁡1/2μp-m(2π/N).Comparing (7) with br,m+1(μp), it is observed that the numerator of br,m+1(μp) is the negative of that of br,m(μp). Then the two adjacent components of br(μp) are related as [18](8)sin⁡12μp-m2πNbr,mμp+sin⁡12μp-m+12πNbr,m+1μp=0.Trigonometric equations lead to(9)tan⁡μp2m+1πNcos⁡mπNbr,mμp+cos⁡m+1πNbr,m+1μp=sin⁡mπNbr,mμp+sin⁡m+1πNbr,m+1μp.Due to the fact that the beams with indices m=0 and m=N-1 are steered at the spatial frequencies μp,0=0 and μp,N-1-2π=N-1(2π/N)-2π=-2π/N, where the two beams are physically adjacent to each other. First, we observe that(10)bNμp=sin⁡N/2μp-N2π/Nsin⁡1/2μp-N2π/N=sin⁡N/2μp-Nπsin⁡1/2μp-π=-1Nsin⁡N/2μp-sin⁡1/2μp=-1N-1·b0μp.Then, the first and the last elements of br(μp) are related by setting m=N-1 in (9), and there is(11)tan⁡μp2cos⁡N-1πNbr,N-1μp+cos⁡π-1N-1br,0μpN-1πN=sin⁡N-1πNbr,N-1μp+sin⁡π-1N-1br,0μp.All N equations (0≤m≤N-1) lead to an invariance relationship for the br(μp) as follows:(12)tan⁡μp2Γ1brμp=Γ2brμp,where Γ1 and Γ2 are the two selection matrices defined as (13)Γ1=1cosπN00⋯000cosπNcos2πN0⋯0000cos2πNcos3πN⋯00⋮⋮⋮⋮·⋮⋮0000⋯cosN-2πNcosN-1πN-1N000⋯0cosN-1πN,Γ2=0sinπN00⋯000sinπNsin2πN0⋯0000sin2πNsin3πN⋯00⋮⋮⋮⋮·⋮⋮0000⋯sinN-2πNsinN-1πN0000⋯0sinN-1πN.When some a priori knowledge of the angle information of sources is known, a reduced dimension processing can be achieved by applying a subset of row vectors defined in (4) to the data matrix X. Hence, only those subblocks of the selection matrices Γ1 and Γ2 which are correlated with the corresponding components of br(μp) will be used.WithP targets, (12) translates into the receiving beamspace matrix relation:(14)Γ1BrΩμ=Γ2Br,where Ωμ=diagtanμ1/2,…,tanμP/2 is a real-valued diagonal matrix whose diagonal elements contain the desired DOA information. Similar to the relation of receiving beamspace matrix, the relation of transmit beamspace matrix is(15)Γ3BtΩν=Γ4Bt,where Γ3 and Γ4 are defined similar to (13) with N replaced by M and Ων=diagtanν1/2,…,tanνP/2 is a real-valued diagonal matrix whose diagonal elements contain the desired DOD information.According to the use of the property of the Kronecker operator in (3), we find that the whole beamspace manifold B satisfies(16)Γμ1BΩμ=Γμ2BΓν1BΩν=Γν2B,where Γμ1=Γ1⊗IM and Γμ2=Γ2⊗IM are the receiving beamspace selection matrices and Γν1=IN⊗Γ3 and Γν2=IN⊗Γ4 are the transmitting beamspace selection matrices, respectively.Looking back on the beamspace received signalY(t) defined in (5), the proper NM×P matrix of signal subspace for the proposed algorithm can be formed by P “largest” left singular vectors of the real-valued matrix ReY,ImY. As we know, the signal subspace Es may be spanned by B, which is expressed as(17)Es=BT,where T is an unknown P×P real-valued matrix. Substituting B=EsT-1 into (16) yields (18)Γμ1EsΨμ=Γμ2Es,Γν1EsΨν=Γν2Es,where Ψμ=T-1ΩμT and Ψν=T-1ΩνT. Equation (18) can be solved by the least squares (LS) or the total least squares (TLS) algorithm. Note the fact that all of the quantities of Ψμ and Ψν are real-valued. Automatic pairing of the spatial frequency μ and ν estimation can be obtained by decomposing Ψμ+jΨν as follows:(19)Ψμ+jΨν=T-1Ωμ+jΩνT.Hence, the real and imaginary parts of the eigenvalues Ωμ+jΩν are the estimation of μp,νp, p=1,…,P. Then the DODs and DOAs of targets can be derived as follows:(20)θ^p=arcsin⁡2arctan⁡Ωvpπ,p=1,…,P,φ^p=arcsin⁡2arctan⁡Ωμpπ,p=1,…,P.The beamspace unitary ESPRIT algorithm based on this development is summarized below.(1)
To form the beamspace transformed matrixWH=WrH⊗WtH, where WH∈KtKr×MN.(2)
To transform the receiving dataX(t) into beamspace Y=WHX, where Y∈KtKr×J, J is the number of time samples.(3)
To compute the SVD ofReY,ImY. The P dominant left singular vectors will be called Es∈KtKr×P. And if P is not known as the a priori information, the number of sources P should be estimated [19].(4a)
To computeΨμ by solving the overdetermined system of equation Γμ1EsΨμ=Γμ2Es via the least squares or the total least squares algorithm. The selection matrices Γμ1=Γ1⊗IM and Γμ2=Γ2⊗IM, where Γ1 and Γ2 are defined in (13). When one has the a priori information on the general angular locations of the signal arrivals, only the appropriate subblocks of Γ1 and Γ2 are employed.(4b)
To computeΨν by solving the overdetermined system of equation Γν1EsΨν=Γν2Es by the least squares or the total least squares algorithm. The selection matrices are Γν1=IN⊗Γ3 and Γν2=IN⊗Γ4.(5)
To compute the eigendecomposition of matrixΨμ+jΨν. Then P largest eigenvalues λp,p=1,…,P could be obtained as λp=Ωvp+jΩμp, where Ωvp and Ωμp are the real and imaginary parts of λp, respectively.(6)
To compute the DODs and DOAs as the solution toθ^p=arcsin⁡2arctan⁡Ωvp/π, p=1,…,P and φ^p=arcsin⁡2arctan⁡Ωμp/π, p=1,…,P which are paired automatically.


## 4. Computational Complexity Analysis

In Table1, the beamspace unitary ESPRIT is compared against the unitary ESPRIT and the beamspace ESPRIT algorithm in terms of computational complexity. For the element ESPRIT and the unitary ESPRIT algorithm, the computational complexity of eigendecomposition is O(M3N3), which is very heavy. Unlike the element ESPRIT and the unitary ESPRIT algorithm, the beamspace ESPRIT and the beamspace unitary ESPRIT algorithm transform the original data vector into several lower-dimensional beamspaces. Since the data processing of each beamspace is independent, it can be parallel processed. If only Kt<M transmitting beams and Kr<N receiving beams are selected, the computational complexity could be reduced from O(M3N3) to O(Kt3Kr3). It is noted that the computational saving is quite significant. Comparing the beamspace unitary ESPRIT with the beamspace ESPRIT algorithm, the proposed algorithm involves only real-valued computation from start to finish. So the computational complexity of beamspace unitary ESPRIT is slightly lower than the beamspace ESPRIT algorithm. And the beamspace unitary ESPRIT algorithm can estimate DOAs and DODs that are paired automatically, which need to be paired additionally in the beamspace ESPRIT algorithm, to save computational complexity.Table 1
Comparison of computational complexities of the beamspace unitary ESPRIT, the beamspace ESPRIT, and the unitary ESPRIT.
Algorithms
Beamspace transformation
Subspace decomposition
DOA and DOD estimation
The unitary ESPRIT
None
O
(
M
3
N
3
)
O
(
2
3
P
3
)
The beamspace ESPRIT
O
(
J
K
t
K
r
M
N
)
O
(
K
t
3
K
r
3
)
O
(
2
3
P
3
)
The beamspace unitary ESPRIT
O
(
J
K
t
K
r
M
N
)
O
(
K
t
3
K
r
3
)
O
(
P
3
)


### 4.1. Advantages and Disadvantages of the Proposed Algorithm

The proposed algorithm has the following advantages.(1)
The proposed algorithm involves only real-valued computation after the initial transformation of beamspace.(2)
The proposed algorithm has a low complexity for the fact that peak searching is not required. And when a priori information is known, only several beams encompassing the sector of interest need to be formed, thereby yielding further reduced computational complexity.(3)
The proposed algorithm can obtain automatically paired DOD and DOA angle estimations.(4)
The proposed algorithm has a better angle estimation performance than B-ESPRIT algorithm which has been explained in the following section.The proposed algorithm also has the following disadvantages.(1)
The proposed algorithm shows that, due to the dramatic reduction in computational complexity, the performance degradation would appear.(2)
The proposed algorithm is sensitive to array errors. The rotational invariance structure will be damaged in the presence of array errors.


## 4.1. Advantages and Disadvantages of the Proposed Algorithm

The proposed algorithm has the following advantages.(1)
The proposed algorithm involves only real-valued computation after the initial transformation of beamspace.(2)
The proposed algorithm has a low complexity for the fact that peak searching is not required. And when a priori information is known, only several beams encompassing the sector of interest need to be formed, thereby yielding further reduced computational complexity.(3)
The proposed algorithm can obtain automatically paired DOD and DOA angle estimations.(4)
The proposed algorithm has a better angle estimation performance than B-ESPRIT algorithm which has been explained in the following section.The proposed algorithm also has the following disadvantages.(1)
The proposed algorithm shows that, due to the dramatic reduction in computational complexity, the performance degradation would appear.(2)
The proposed algorithm is sensitive to array errors. The rotational invariance structure will be damaged in the presence of array errors.


## 5. Simulation Results

In this section, some numerical examples are presented to assess the effectiveness of the proposed method. In all simulations, 2000 Monte Carlo trials are performed. The RMSE is defined as(21)RMSEt=1P∑p=1P11,000∑l=11,000θ^p,l-θp2,RMSEr=1P∑p=1P11,000∑l=11,000φ^p,l-φp2,RMSE=1P∑p=1P11,000∑l=11,000θ^p,l-θp2+φ^p,l-φp2,which are employed as the performance metric, where θp and φp denote the DOD and DOA of the pth source and θ^p,l and φ^p,l are the estimation of θp and φp in the lth Monte Carlo trail, respectively. J denotes the number of snapshots. According to [16], the CRB for the angle estimation in bistatic MIMO radar is derived as follows:(22)CRBθ,φ=σ22KReDHΠB⊥D⊙P^SST-1,where D=[∂b1/∂θ1,∂b2/∂θ2,…,∂bP/∂θP,∂b1/∂φ1, ∂b2/∂φ2, …, ∂bP/∂φP] with bp being the pth column of B, ΠB⊥=IMN-B(BHB)-1BH, P^SST=P^SP^SP^SP^S, P^S=(1/J)∑t=1JS(t)SH(t), σ2 stands for the noise variance, and J is the number of time samples.Assuming that there areP=3 sources, whose angles are θ1,φ1=30°,-30°, θ2,φ2=20°,-40°, and θ3,φ3=40°,-36°. For the B-ESPRIT and the proposed algorithm, there are five beams are formed, which involve the sector of interest.Figures2 and 3 depict the DOD and DOA estimation results of the proposed algorithm for the three sources in bistatic MIMO radar with M=8, N=6, J=64, SNR=15dB, and 20 dB, respectively. It shows that the DODs and DOAs of sources can be clearly observed, and the performance will be improved as the SNR increases.Figure 2
Angle estimation results withSNR=15dB.Figure 3
Angle estimation results withSNR=20dB.The proposed algorithm is compared against the beamspace ESPRIT algorithm, the element ESPRIT algorithm, the unitary ESPRIT, and CRB. And in this Simulation, a new source located atθ4,φ4=36°,-36° is added. Figures 4 and 5 present angle estimation performance comparison with M=8, N=6, and J=64. It can be found that the angle estimation performance of proposed algorithm is better than the B-ESPRIT algorithm because unitary ESPRIT doubles the number of data samples effectively [20]. It is observed that the proposed algorithm performs slightly worse than the U-ESPRIT algorithm does in the DOD estimation in despite of the dramatic reduction in computational complexity (where the number of transmit antenna is M=8 and the number of transmit beam is Lt=5<M). And the proposed algorithm and U-ESPRIT algorithm both have similar estimation performance of DOA due to the fact that the number of receiving beam is nearly equal to the number of transmitting antenna (where Lr=5 and N=6). Compared with the E-ESPRIT algorithm, although the beamspace unitary ESPRIT algorithm is a unitary algorithm, its performance will also be degraded because of the dramatic reduction in computational complexity. When the degradation is not large enough, the proposed algorithm also has a better performance over the element ESPRIT.Figure 4
DOD estimation performance comparison.Figure 5
DOA estimation performance comparison.Figures6 and 7 present the angle estimation performance of the proposed algorithm with different J, where M=8, N=6. The proposed algorithm is compared with the unitary ESPRIT algorithm. It illustrates that the angle estimation performance of our algorithm is improved as the number of snapshots increases. Meanwhile, it shows that performance of our algorithm is slightly worse than the unitary ESPRIT algorithm when the computational complexity is reduced all at once.Figure 6
RMSE of DOD versus SNR for 3 sources with different values of sample.Figure 7
RMSE of DOA versus SNR for 3 sources with different values of sample.In Figure8, values of transmitting array antennas M=6,8 and 10 (N=6,J=64) are examined for the SNR support. From Figure 8, it can be seen that the angle estimation performance of proposed algorithm will gradually increase with the increasing transmitting antenna number.Figure 8
Total RMSE versus SNR for 3 sources with different values ofM.Figure9 gives the resolution performance of the proposed algorithm, the beamspace ESPRIT algorithm, the element ESPRIT algorithm, and the unitary ESPRIT algorithm. Considering a scenario with two uncorrelated sources of equal power, the angle parameters are θ1=φ1=-δ/2 and θ2=φ2=δ/2, and J=50, SNR=20dB. Then the angular source separation is parameterized by [21](23)2δ=θ1-θ22+φ1-φ22.The two sources are considered as resolved if the estimation of angle is close to the true angle parameter; that is,(24)θ^1-θ12+φ^1-φ12<δ2∩θ^2-θ22+φ^2-φ22<δ2.It can be seen that the probability of resolution of proposed algorithm is better than both the beamspace ESPRIT algorithm and the element ESPRIT algorithm but slightly worse than the unitary ESPRIT algorithm.Figure 9
Probability of resolution versus angular separationδ, at SNR=20dB, J=50.Figure10 depicts an evaluation of the computational complexity using TIC and TOC instructions that can serve for calculating the runtime of an algorithm in MATLAB. It can be seen that when all the beams are formed, the runtime of proposed algorithm is smaller than the U-ESPRIT algorithm in the case of the number of sensors is larger than 10. And when only five beams are formed, the runtime of proposed algorithm is much smaller than the U-ESPRIT algorithm. Also, the runtime of the B-UESPRIT is smaller than the B-ESPRIT algorithm, which is still in the case of only five beams are formed. It is observed that as the number of sensors increases, the runtime of the proposed algorithm increases slowly. The reason is that the computational complexities and runtime of proposed algorithm are influenced by the number of beams formed, which is still five.Figure 10
Runtime comparison against number of sensorsM=N.


## 6. Conclusion

In this paper, a beamspace unitary ESPRIT is developed to estimate angles of the targets in bistatic MIMO radar. The conjugate centrosymmetrized DFT matrix is applied to transform the received data into beamspace. Then the invariance property of the transmitting beam and the receiving beam is exploited, respectively, to calculate the DODs and the DOAs of targets. Unlike the beamspace ESPRIT, the B-UESPRIT involves only real-valued computation from beginning to end. Therefore, a reduction of the computational complexity is obtained, which is demonstrated by the analysis of computational complexity and the runtime of algorithm. Furthermore, the simulation results prove that the B-UESPRIT requires no matched pair but possesses better angle estimation performance than the E-ESPRIT and the B-ESPRIT. Additionally, the CRB has been derived to analyze the performance.



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