# Reduced-Dimension Noncircular-Capon Algorithm for DOA Estimation of Noncircular Signals

Authors: Weihua Lv; Huapu Sun; Xiaofei Zhang; Dazhuan Xu
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

The problem of the direction of arrival (DOA) estimation for the noncircular (NC) signals, which have been widely used in communications, is investigated. A reduced-dimension NC-Capon algorithm is proposed hereby for the DOA estimation of noncircular signals. The proposed algorithm, which only requires one-dimensional search, can avoid the high computational cost within the two-dimensional NC-Capon algorithm. The angle estimation performance of the proposed algorithm is much better than that of the conventional Capon algorithm and very close to that of the two-dimensional NC-Capon algorithm, which has a much higher complexity than the proposed algorithm. Furthermore, the proposed algorithm can be applied to arbitrary arrays and works well without estimating the noncircular phases. The simulation results verify the effectiveness and improvement of the proposed algorithm.


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## Body

## 1. Introduction

Direction of arrival (DOA) estimation is a hot topic in the array signal processing field and has been widely used in communication, radar, sonar, and medical image [1–4]. Classical DOA estimation algorithms include multiple signal classification (MUSIC) [5], estimation of signal parameters via rotational invariance technique (ESPRIT) [6–8], propagator method [9], and the Capon [10]. Besides, compressive sensing (CS) [11] and Bayesian compressive sensing (BCS) [12] have recently been used to solve the problem of DOA estimation, and they have an advantage of not requiring knowledge of the number of impinging signals.To improve the DOA estimation performance, the noncircular property of incoming signals has been considered in [13–22]. In wireless communications, the noncircular signals have been extensively used, for example, the binary phase shift keying, amplitude modulation, and unbalanced quadrature phase shift keying [22]. If Ebt=0, EbtbHt≠0, and EbtbTt≠0, then bt is a noncircular signal [13–15]. This statistics redundancy can be properly exploited to enhance the DOA estimation performance. In general, we use the array output and its conjugation to extend the data model and array aperture. A noncircular MUSIC (NC-MUSIC) algorithm was proposed in [14] for the DOA estimation of the noncircular signals. In order to avoid the peak search in NC-MUSIC, a polynomial rooting NC-MUSIC (NC-Root-MUSIC) was presented in [15]. NC-ESPRIT algorithms were proposed in [16, 17] for DOA estimation without spectrum search. Real-valued implementation of unitary ESPRIT (NC-Unitary-ESPRIT) for noncircular sources was presented in [18], and it has a low complexity. Besides, a noncircular propagator method (NC-PM) for direction estimation of noncircular signals was proposed in [19], which has better angle estimation performance than PM in [9]. Based on the parallel factor (PARAFAC) technique, a noncircular PARAFAC (NC-PARAFAC) algorithm was proposed in [20] to obtain the two-dimensional (2D) DOA estimation of the noncircular signals for arbitrarily spaced acoustic vector-sensor array. Moreover, a two-dimensional direction-finding for noncircular signals using two parallel linear arrays via the extended rank reduction algorithm was presented in [21].Many DOA estimation algorithms mentioned above require the prior knowledge of the number of sources or need to estimate the number of sources by information theory algorithm, matrix decomposition algorithm, smoothed rank algorithm, or Gerschgorin disks algorithm [23–26]. Notably, the Capon method can work well without information of the number of sources [10]. CS method and BCS method work well without estimating the number of the sources, but the noncircular property is not considered. By combining the Capon method and the noncircular property, a noncircular Capon (NC-Capon) algorithm was proposed in [27] for the DOA estimation of noncircular signals, but it needs an exhaustive two-dimensional (2D) search over the regions of both DOA and noncircular phase.In this work, we will propose a reduced-dimension NC-Capon (RD-NC-Capon) algorithm for the DOA estimation of noncircular signals. The proposed algorithm, which only requires one-dimensional search, can avoid the high computational cost within the two-dimensional NC-Capon (2D-NC-Capon) algorithm [27]. The angle estimation performance of the proposed algorithm is much better than that of the conventional Capon algorithm and very close to that of the two-dimensional NC-Capon algorithm. Moreover, the proposed algorithm can be applied to arbitrary arrays and works well without estimating the noncircular phases. Numerical simulations verify the improvement and effectiveness of the proposed algorithm.The remainder of this paper is structured as follows. Section2 introduces the data model. Section 3 proposes the RD-NC-Capon algorithm. Section 4 gives the performance analysis. In Section 5, simulation results are provided to show the effectiveness, while the conclusions are drawn in Section 6.Notations. Lowercase (capital) bold symbols denote vector (matrix). ·∗, ·T, ·H, ·-1, and ·+ denote complex conjugation, transpose, conjugate-transpose, inverse, and pseudoinverse operations, respectively. diagv stands for a diagonal matrix whose diagonal is a vector v. Re· is to get the real part of the complex. E· presents the statistical expectation. Pi,j denotes the (i,j) element of the matrix P. ⊕ is Hadamard product. IM stands for an M×M identity matrix and 0M×N is a zero matrix with M×N.


## 2. Data Model

As shown in Figure1, we consider a linear array consisting of M omnidirectional sensors and select the first sensor as the referenced one. The vector d=d1,d2,…,dM denotes the displacement between the sensor and the referenced one, and d1=0. We assume that there are K far-field, narrow-band signals impinging on the linear array from different angles θkk=1,2,…,K. The received signal of the array can be expressed by vector xt [28]:(1)xt=Ast+nt,where A=aθ1,…,aθK∈CM×K with aθk=1,e-j2πd2sinθk/λ,…,e-j2πdMsinθk/λT∈CM×1 and λ being the wavelength. st∈CK×1 is the narrow-band noncircular signal vector and nt denotes the additive white Gaussian noise.Figure 1
Array geometry.As for the definition of noncircular signal, we consider that ifEst=0K×1, EstsHt≠0K×K, and EstsTt=0K×K, then st is a circular signal vector. Otherwise, if Est=0K×1, EstsHt≠0K×K, and EstsTt≠0K×K, then st is a noncircular signal vector [13–19]. Additionally, the vector of strictly second-order noncircular signals can be expressed as [13–16](2)st=Ψs0t,where s0t∈RK×1 and Ψ=diag⁡e-jϕ1,e-jϕ2,…,e-jϕK with ϕk being the noncircular phase of the kth signal. According to (2), we rewrite (1) as(3)xt=AΨs0t+nt.


## 3. DOA Estimation Algorithm

In this section, we will propose the RD-NC-Capon algorithm for the DOA estimation of noncircular signals. In order to double the array aperture, we will first give the extended data model by exploiting the noncircular property, which is a common method of expansion for noncircular signals and has been widely used in many noncircular DOA estimation algorithms [13–22]. Then we will talk about the conventional 2D-NC-Capon algorithm in Section 3.2, whereas the RD-NC-Capon algorithm will be proposed in Section 3.3.


### 3.1. Data Construction

When the noncircular signals impinge on the array, we use the array output and its conjugation to extend the data model [13–15]:(4)yt=xtx∗t=AΨA∗Ψ∗s0t+ntn∗t=Bs0t+n0t,where n0t=ntn∗t∈C2M×1. Consider B=AΨA∗Ψ∗∈C2M×K, and it can be written as (5)B=e-jϕ1e-jϕ2⋯e-jϕKe-j2πd2sinθ1/λ+ϕ1e-j2πd2sinθ2/λ+ϕ2⋯e-j2πd2sinθK/λ+ϕK⋮⋮⋱⋮e-j2πdMsinθ1/λ+ϕ1e-j2πdMsinθ2/λ+ϕ2⋯e-j2πdMsinθK/λ+ϕKejϕ1ejϕ2⋯ejϕKej2πd2sinθ1/λ+ϕ1ej2πd2sinθ2/λ+ϕ2⋯ej2πd2sinθK/λ+ϕK⋮⋮⋱⋮ej2πdMsinθ1/λ+ϕ1ej2πdMsinθ2/λ+ϕ2⋯ej2πdMsinθK/λ+ϕK.The covariance matrix of the extended data model can be expressed as R=Ey(t)yH(t).


### 3.2.
2D-NC-Capon Algorithm

The following 2D-NC-Capon function can be utilized to estimate the DOAs of the noncircular signals [27, 28](6)P2D-NC-Caponθ,ϕ=1bHθ,ϕR-1bθ,ϕ,where (7)bθ,ϕ=e-jϕe-j2πd2sinθ/λ+ϕ⋮e-j2πdMsinθ/λ+ϕejϕej2πd2sinθ/λ+ϕ⋮ej2πdMsinθ/λ+ϕ.The 2D-NC-Capon algorithm can obtainK local peak values of (6) by 2D search of θ and ϕ. Since the 2D-NC-Capon algorithm requires an exhaustive 2D search, this approach is normally inefficient due to its high computational cost. A reduced-dimension (RD) Capon algorithm was proposed in [29] for the angle estimation of bistatic multiple-input multiple-output radar. In the following subsections, we make reference to the RD idea from [29] and present a RD-NC-Capon algorithm for the DOA estimation of noncircular signals just through one-dimensional (1D) search.There are some differences between the work in [29] and our work. Firstly, [29] used the RD-Capon method for angle estimation in bistatic multiple-input multiple-output radar, while our work is to extend the RD-Capon idea and propose RD-NC-Capon for angle estimation of noncircular signals. Secondly, the noncircular property is not considered in [29], and we use the noncircular property of the sources to double the array aperture and enhance the angle estimation performance. Thirdly, the modeling in each work is different. The received signal in [26] can be used directly for RD processing, while a data model extension is required in our work.


### 3.3. The Proposed RD-NC-Capon Algorithm

Equation (7) can be denoted by(8)bθ,ϕ=aθe-jϕa∗θejϕ=aθ0M×10M×1a∗θe-jϕejϕ=Pθe0f,where 0M×1 is an M×1 zero matrix. Consider(9)aθ=1,e-j2πd2sinθ/λ,…,e-j2πdMsinθ/λT,Pθ=aθ0M×10M×1a∗θ,e0ϕ=e-jϕejϕ.Then we construct the following function:(10)Vθ,ϕ=1P2D-NC-Caponθ,ϕ.According to (6)–(10), we know that(11)Vθ,ϕ=e0HϕPHθR-1Pθe0ϕ.Obviously, V(θ,ϕ) can be expressed as(12)Vθ,ϕ=e-jϕejϕe0HϕPHθR-1Pθe0ϕ.Equation (12) can be denoted by(13)Vθ,ϕ=e-jϕe0HϕPHθe0ϕejϕ.We defineqϕ=e0ϕejϕ=1ej2ϕ and Qθ=PHθR-1Pθ. Equation (13) can be rewritten as(14)Vθ,ϕ=qHϕQθqϕ.As (14) is a quadratic optimization, we consider eliminating trivial solution qϕ=02×1 and add a constraint of eHqϕ=1, where e=1,0T. Then the optimization problem in (14) is reconstructed as follows:(15)minθ,ϕqHϕQθqϕs.t.eHqϕ=1.We construct the following cost function using Lagrange multiplier:(16)Lθ,ϕ=qHϕQθqϕ-ρeHqϕ-1,where ρ is a constant. If the derivative of (16) is set to zero, that is,(17)∂∂qϕLθ,ϕ=2Qθqϕ+ρe=0,then qϕ=-0.5ρQ-1θe. We define μ=-0.5ρ; then we get(18)q^ϕ=μQθ-1e.As eHqϕ=1 and μ=1/eHQθ-1e, we have(19)q^ϕ=Q-1θeeHQ-1θe.From (14) and (19), we have(20)θ^=arg⁡min1eHQ-1θe=arg⁡maxeHQ-1θe.As Qθ=PθHR-1Pθ, a new 1D search cost function is used for DOA estimation: (21)fNC-RD-Caponθ=eHPθHR-1Pθ-1e.In the finite sample case, the covariance matrix can be estimated as(22)R^=1L∑l=1LytlyHtl,where L denotes the number of snapshots. Then, the DOAs θkk=1,2,…,K can be obtained through the following search:(23)fRD-NC-Caponθ=eHPθHR^-1Pθ-1e.2D-NC-Capon algorithm obtains DOA estimation by 2D search ofθ and ϕ, whereas the proposed algorithm gets DOA estimation via 1D search according to the function in (23). The proposed RD-NC-Capon algorithm can work well without estimation of noncircular phases.The main steps of RD-NC-Capon algorithm are shown as follows.Step 1.
Construct the extended data model from the array output via (4).Step 2.
Compute the covariance matrixR^ of the extended data model via (22).Step 3.
Use the 1D spectrum search function to estimate DOAs via (23).


## 3.1. Data Construction

When the noncircular signals impinge on the array, we use the array output and its conjugation to extend the data model [13–15]:(4)yt=xtx∗t=AΨA∗Ψ∗s0t+ntn∗t=Bs0t+n0t,where n0t=ntn∗t∈C2M×1. Consider B=AΨA∗Ψ∗∈C2M×K, and it can be written as (5)B=e-jϕ1e-jϕ2⋯e-jϕKe-j2πd2sinθ1/λ+ϕ1e-j2πd2sinθ2/λ+ϕ2⋯e-j2πd2sinθK/λ+ϕK⋮⋮⋱⋮e-j2πdMsinθ1/λ+ϕ1e-j2πdMsinθ2/λ+ϕ2⋯e-j2πdMsinθK/λ+ϕKejϕ1ejϕ2⋯ejϕKej2πd2sinθ1/λ+ϕ1ej2πd2sinθ2/λ+ϕ2⋯ej2πd2sinθK/λ+ϕK⋮⋮⋱⋮ej2πdMsinθ1/λ+ϕ1ej2πdMsinθ2/λ+ϕ2⋯ej2πdMsinθK/λ+ϕK.The covariance matrix of the extended data model can be expressed as R=Ey(t)yH(t).


## 3.2.
2D-NC-Capon Algorithm

The following 2D-NC-Capon function can be utilized to estimate the DOAs of the noncircular signals [27, 28](6)P2D-NC-Caponθ,ϕ=1bHθ,ϕR-1bθ,ϕ,where (7)bθ,ϕ=e-jϕe-j2πd2sinθ/λ+ϕ⋮e-j2πdMsinθ/λ+ϕejϕej2πd2sinθ/λ+ϕ⋮ej2πdMsinθ/λ+ϕ.The 2D-NC-Capon algorithm can obtainK local peak values of (6) by 2D search of θ and ϕ. Since the 2D-NC-Capon algorithm requires an exhaustive 2D search, this approach is normally inefficient due to its high computational cost. A reduced-dimension (RD) Capon algorithm was proposed in [29] for the angle estimation of bistatic multiple-input multiple-output radar. In the following subsections, we make reference to the RD idea from [29] and present a RD-NC-Capon algorithm for the DOA estimation of noncircular signals just through one-dimensional (1D) search.There are some differences between the work in [29] and our work. Firstly, [29] used the RD-Capon method for angle estimation in bistatic multiple-input multiple-output radar, while our work is to extend the RD-Capon idea and propose RD-NC-Capon for angle estimation of noncircular signals. Secondly, the noncircular property is not considered in [29], and we use the noncircular property of the sources to double the array aperture and enhance the angle estimation performance. Thirdly, the modeling in each work is different. The received signal in [26] can be used directly for RD processing, while a data model extension is required in our work.


## 3.3. The Proposed RD-NC-Capon Algorithm

Equation (7) can be denoted by(8)bθ,ϕ=aθe-jϕa∗θejϕ=aθ0M×10M×1a∗θe-jϕejϕ=Pθe0f,where 0M×1 is an M×1 zero matrix. Consider(9)aθ=1,e-j2πd2sinθ/λ,…,e-j2πdMsinθ/λT,Pθ=aθ0M×10M×1a∗θ,e0ϕ=e-jϕejϕ.Then we construct the following function:(10)Vθ,ϕ=1P2D-NC-Caponθ,ϕ.According to (6)–(10), we know that(11)Vθ,ϕ=e0HϕPHθR-1Pθe0ϕ.Obviously, V(θ,ϕ) can be expressed as(12)Vθ,ϕ=e-jϕejϕe0HϕPHθR-1Pθe0ϕ.Equation (12) can be denoted by(13)Vθ,ϕ=e-jϕe0HϕPHθe0ϕejϕ.We defineqϕ=e0ϕejϕ=1ej2ϕ and Qθ=PHθR-1Pθ. Equation (13) can be rewritten as(14)Vθ,ϕ=qHϕQθqϕ.As (14) is a quadratic optimization, we consider eliminating trivial solution qϕ=02×1 and add a constraint of eHqϕ=1, where e=1,0T. Then the optimization problem in (14) is reconstructed as follows:(15)minθ,ϕqHϕQθqϕs.t.eHqϕ=1.We construct the following cost function using Lagrange multiplier:(16)Lθ,ϕ=qHϕQθqϕ-ρeHqϕ-1,where ρ is a constant. If the derivative of (16) is set to zero, that is,(17)∂∂qϕLθ,ϕ=2Qθqϕ+ρe=0,then qϕ=-0.5ρQ-1θe. We define μ=-0.5ρ; then we get(18)q^ϕ=μQθ-1e.As eHqϕ=1 and μ=1/eHQθ-1e, we have(19)q^ϕ=Q-1θeeHQ-1θe.From (14) and (19), we have(20)θ^=arg⁡min1eHQ-1θe=arg⁡maxeHQ-1θe.As Qθ=PθHR-1Pθ, a new 1D search cost function is used for DOA estimation: (21)fNC-RD-Caponθ=eHPθHR-1Pθ-1e.In the finite sample case, the covariance matrix can be estimated as(22)R^=1L∑l=1LytlyHtl,where L denotes the number of snapshots. Then, the DOAs θkk=1,2,…,K can be obtained through the following search:(23)fRD-NC-Caponθ=eHPθHR^-1Pθ-1e.2D-NC-Capon algorithm obtains DOA estimation by 2D search ofθ and ϕ, whereas the proposed algorithm gets DOA estimation via 1D search according to the function in (23). The proposed RD-NC-Capon algorithm can work well without estimation of noncircular phases.The main steps of RD-NC-Capon algorithm are shown as follows.Step 1.
Construct the extended data model from the array output via (4).Step 2.
Compute the covariance matrixR^ of the extended data model via (22).Step 3.
Use the 1D spectrum search function to estimate DOAs via (23).


## 4. Performance Analysis

In this section, we first analyze the computational complexity of the proposed algorithm and then derive the Cramér-Rao bound (CRB) of DOA estimation.


### 4.1. Complexity Analysis

For the proposed algorithm, computing the covariance matrix requiresO(4M2L), inversion operation of the covariance matrix needs O(8M3), and a spectrum searching wants O8M2+M. The main computational complexity of the proposed RD-NC-Capon algorithm is O4M2L+8M3+8M2+Mn, where n is the number of searches, whereas the 2D-NC-Capon algorithm needs O(4M2L+8M3+(4M2+2M)n2). Thus, the RD-NC-Capon algorithm has a much lower computational complexity than the 2D-NC-Capon algorithm. Figure 2 shows the complexity comparison versus the snapshots L, where K=3, n=6000, and M=8 are considered. Figure 3 presents the running time comparison, where we compare the proposed algorithm against the 2D-NC-Capon algorithm. From Figures 2 and 3, we find that the proposed algorithm has a much lower complexity than the 2D-NC-Capon algorithm.Figure 2
The complexity comparison.Figure 3
The running time comparison.


### 4.2. Cramér-Rao Bound

In the finite sample case, the data model can be rewritten as(24)Y=BS0+N0,where Y=y(t1),…,y(tL) and S0=s0t1,…,s0tL. B∈C2M×K is the same as that shown in (5).We assume that the signal is deterministic, and then the estimation parameter vector is expressed as(25)ζ=θ1,…,θK,ϕ1,…,ϕK,s0Tt1,…,s0TtL,σ2T,where s0Ttl is the lth column of S0. σ2 is the noise power.According to (24), we have (26)y=vecY=vecBS0+vecN0,where vec(·) is to convert a matrix into a vector.The expected valueμ and the covariance matrix Γ of y are(27)μ=Bst1⋮BstL=GS,Γ=σ2I2M0⋱0σ2I2M,where (28)G=B0⋱0B,S=s0t1⋮s0tL.According to [22, 30, 31], we know that the (i,j) element of the CRB matrix (Pcr) can be expressed as(29)Pcr-1ij=trΓ-1Γi′Γ-1Γj′+2Reμi′HΓ-1μj′,where Γi′, Γj′ and μi′, μj′ are the derivatives of Γ and μ on the ith or jth element of ζ, respectively. tr[·] is to obtain the trace of a matrix. Since the covariance matrix is just related to σ2, the first part of (29) is zero. Then(30)Pcr-1ij=2Reμi′HΓ-1μj′.We have(31)∂μ∂θk=∂B∂θks0t1⋮∂B∂θks0tL=dkθs0kt1⋮dkθs0ktL,k=1,…,K,∂μ∂ϕk=∂B∂ϕks0t1⋮∂B∂ϕks0tL=dkϕs0kt1⋮dkϕs0ktL,k=1,…,K,where s0kt is the kth element of s0t, dkθ=∂bk/∂θk, and dkϕ=∂bk/∂ϕk with bk being the kth column of B.We define(32)Δ=d1θs01t1⋯dKθs0Kt1d1ϕs01t1⋯dKϕs0Kt1⋮⋮⋮⋮d1θs01tL⋯dKθs0KtLd1ϕs01tL⋯dKϕs0KtL.As μ=GS, (33)∂μ∂ST=G.Now we have(34)∂μ∂ζT=Δ,G,02ML×1.According to (34),(35)2Re∂μH∂ζΓ-1∂μ∂ζT=J0KL+2K×101×KL+2K0,where J=2/σ2ReΔHGHΔG.Define(36)Q=GHG-1GHΔ,F=I2K02K×KL-QIKL,where QR and QI are the real and imaginary parts of Q, respectively.It can be demonstrated that(37)ΔGF=Δ-GQG=ΠG⊥ΔG,where ΠG⊥=I-GGHG-1GH and GHΠG⊥=0. Then we have(38)FTJF=2σ2ReFHΔHGHΔGF.According to (37), we know that(39)FHΔHGHΔGF=ΔHΠG⊥GHΠG⊥ΔG=ΔHΠG⊥Δ02K×KL0KL×2KGHG.Hence, combining (37)-(38), FTJF can be rewritten as(40)FTJF=2σ2ReΔHΠG⊥Δ02K×KL0KL×2KGHG.It is obvious that J-1=FFTJF-1FT. Then J-1 is written as(41)J-1=σ22I2K02K×KL-QIKLReΔHΠG⊥Δ02K×KL0KL×2KGHG-1I2K-QT02K×KLIKL.Furthermore,(42)J-1=σ22ReΔHΠG⊥Δ-1κκκ,where κ denotes the part we are not concerned about.Till now, we can give the CRB matrix [30, 31]:(43)CRB=σ22ReΔHΠG⊥Δ-1.After further simplification, we rewrite the CRB matrix as(44)CRB=σ22LReDHΠB⊥D⊕R^s-1,where σ2 is noise power, R^s=1/L∑l=1Ls0tls0Htl, and ΠB⊥=I2M-B(BHB)-1BH. Consider D=[d1θ,d2θ,…,dKθ,d1ϕ,d2ϕ,…,dKϕ], dkθ=∂bk/∂θk, and dkϕ=∂bk/∂ϕk with bk being the kth column of B.


### 4.3. Discussions

Remark 1.
The strictly second-order noncircular signals are considered in the paper. The proposed algorithm works well for the general noncircular signals. When we consider the general noncircular signals, the source vector can be denoted by(45)st=diag⁡ρ1e-jϕ1,ρ2e-jϕ2,…,ρKe-jϕKs0t=Ψρs0t,where s0t∈RK×1, Ψ=diag⁡e-jϕ1,e-jϕ2,…,e-jϕK, and ρ=diag⁡(ρ1,ρ2,…,ρK)∈RK×K with ρk being the noncircularity coefficient of the kth source. The extended signal is(46)yt=xtx∗t=Bρs0t+n0t.We also estimate DOAs of the general noncircular signals in a similar way.Remark 2.
The proposed algorithm works well for the uniform circular array (UCA). We assumeaθ,φ is the direction vector of the UCA with DOA being (θ,φ). 3D-NC-Capon function can be utilized to estimate the 2D-DOA of the noncircular signals for UCA. 2D-DOA estimation function of the proposed algorithm is (47)fNC-RD-Caponθ,φ=eHPθ,φHR^-1Pθ,φ-1e,where (48)Pθ,φ=aθ,φ0M×10M×1a∗θ,φ.


## 4.1. Complexity Analysis

For the proposed algorithm, computing the covariance matrix requiresO(4M2L), inversion operation of the covariance matrix needs O(8M3), and a spectrum searching wants O8M2+M. The main computational complexity of the proposed RD-NC-Capon algorithm is O4M2L+8M3+8M2+Mn, where n is the number of searches, whereas the 2D-NC-Capon algorithm needs O(4M2L+8M3+(4M2+2M)n2). Thus, the RD-NC-Capon algorithm has a much lower computational complexity than the 2D-NC-Capon algorithm. Figure 2 shows the complexity comparison versus the snapshots L, where K=3, n=6000, and M=8 are considered. Figure 3 presents the running time comparison, where we compare the proposed algorithm against the 2D-NC-Capon algorithm. From Figures 2 and 3, we find that the proposed algorithm has a much lower complexity than the 2D-NC-Capon algorithm.Figure 2
The complexity comparison.Figure 3
The running time comparison.


## 4.2. Cramér-Rao Bound

In the finite sample case, the data model can be rewritten as(24)Y=BS0+N0,where Y=y(t1),…,y(tL) and S0=s0t1,…,s0tL. B∈C2M×K is the same as that shown in (5).We assume that the signal is deterministic, and then the estimation parameter vector is expressed as(25)ζ=θ1,…,θK,ϕ1,…,ϕK,s0Tt1,…,s0TtL,σ2T,where s0Ttl is the lth column of S0. σ2 is the noise power.According to (24), we have (26)y=vecY=vecBS0+vecN0,where vec(·) is to convert a matrix into a vector.The expected valueμ and the covariance matrix Γ of y are(27)μ=Bst1⋮BstL=GS,Γ=σ2I2M0⋱0σ2I2M,where (28)G=B0⋱0B,S=s0t1⋮s0tL.According to [22, 30, 31], we know that the (i,j) element of the CRB matrix (Pcr) can be expressed as(29)Pcr-1ij=trΓ-1Γi′Γ-1Γj′+2Reμi′HΓ-1μj′,where Γi′, Γj′ and μi′, μj′ are the derivatives of Γ and μ on the ith or jth element of ζ, respectively. tr[·] is to obtain the trace of a matrix. Since the covariance matrix is just related to σ2, the first part of (29) is zero. Then(30)Pcr-1ij=2Reμi′HΓ-1μj′.We have(31)∂μ∂θk=∂B∂θks0t1⋮∂B∂θks0tL=dkθs0kt1⋮dkθs0ktL,k=1,…,K,∂μ∂ϕk=∂B∂ϕks0t1⋮∂B∂ϕks0tL=dkϕs0kt1⋮dkϕs0ktL,k=1,…,K,where s0kt is the kth element of s0t, dkθ=∂bk/∂θk, and dkϕ=∂bk/∂ϕk with bk being the kth column of B.We define(32)Δ=d1θs01t1⋯dKθs0Kt1d1ϕs01t1⋯dKϕs0Kt1⋮⋮⋮⋮d1θs01tL⋯dKθs0KtLd1ϕs01tL⋯dKϕs0KtL.As μ=GS, (33)∂μ∂ST=G.Now we have(34)∂μ∂ζT=Δ,G,02ML×1.According to (34),(35)2Re∂μH∂ζΓ-1∂μ∂ζT=J0KL+2K×101×KL+2K0,where J=2/σ2ReΔHGHΔG.Define(36)Q=GHG-1GHΔ,F=I2K02K×KL-QIKL,where QR and QI are the real and imaginary parts of Q, respectively.It can be demonstrated that(37)ΔGF=Δ-GQG=ΠG⊥ΔG,where ΠG⊥=I-GGHG-1GH and GHΠG⊥=0. Then we have(38)FTJF=2σ2ReFHΔHGHΔGF.According to (37), we know that(39)FHΔHGHΔGF=ΔHΠG⊥GHΠG⊥ΔG=ΔHΠG⊥Δ02K×KL0KL×2KGHG.Hence, combining (37)-(38), FTJF can be rewritten as(40)FTJF=2σ2ReΔHΠG⊥Δ02K×KL0KL×2KGHG.It is obvious that J-1=FFTJF-1FT. Then J-1 is written as(41)J-1=σ22I2K02K×KL-QIKLReΔHΠG⊥Δ02K×KL0KL×2KGHG-1I2K-QT02K×KLIKL.Furthermore,(42)J-1=σ22ReΔHΠG⊥Δ-1κκκ,where κ denotes the part we are not concerned about.Till now, we can give the CRB matrix [30, 31]:(43)CRB=σ22ReΔHΠG⊥Δ-1.After further simplification, we rewrite the CRB matrix as(44)CRB=σ22LReDHΠB⊥D⊕R^s-1,where σ2 is noise power, R^s=1/L∑l=1Ls0tls0Htl, and ΠB⊥=I2M-B(BHB)-1BH. Consider D=[d1θ,d2θ,…,dKθ,d1ϕ,d2ϕ,…,dKϕ], dkθ=∂bk/∂θk, and dkϕ=∂bk/∂ϕk with bk being the kth column of B.


## 4.3. Discussions

Remark 1.
The strictly second-order noncircular signals are considered in the paper. The proposed algorithm works well for the general noncircular signals. When we consider the general noncircular signals, the source vector can be denoted by(45)st=diag⁡ρ1e-jϕ1,ρ2e-jϕ2,…,ρKe-jϕKs0t=Ψρs0t,where s0t∈RK×1, Ψ=diag⁡e-jϕ1,e-jϕ2,…,e-jϕK, and ρ=diag⁡(ρ1,ρ2,…,ρK)∈RK×K with ρk being the noncircularity coefficient of the kth source. The extended signal is(46)yt=xtx∗t=Bρs0t+n0t.We also estimate DOAs of the general noncircular signals in a similar way.Remark 2.
The proposed algorithm works well for the uniform circular array (UCA). We assumeaθ,φ is the direction vector of the UCA with DOA being (θ,φ). 3D-NC-Capon function can be utilized to estimate the 2D-DOA of the noncircular signals for UCA. 2D-DOA estimation function of the proposed algorithm is (47)fNC-RD-Caponθ,φ=eHPθ,φHR^-1Pθ,φ-1e,where (48)Pθ,φ=aθ,φ0M×10M×1a∗θ,φ.


## 5. Simulation Results

This section uses Monte Carlo simulations to assess the DOA estimation performance of the proposed algorithm. We define the root mean square error (RMSE) of DOA as(49)RMSE=1K∑k=1K11000∑n=11000θ^k,n-θk2with θk being the accurate angle of the kth signal and θ^k,n being the estimate of θk of the nth Monte Carlo trial. For Figures 4 and 6–9, we assume that there are 3 (K=3) noncircular signals impinging on the array with angles being [10∘,20∘,30∘] and the noncircular phases being [10∘,30∘,50∘].Figure 4
The spectrum search results of RD-NC-Capon algorithm (SNR=20 dB, M=8, L=200, and K=3).Figure4 shows the spectrum search result of the proposed algorithm for the nonuniform linear array with d=[0,0.45λ,0.9λ,1.3λ,1.78λ,2.2λ,2.64λ,3.1λ], where λ is the wavelength. In this simulation, signal-to-noise ratio (SNR)=20 dB, M=8, and L=200 are considered. From Figure 4, we find that the proposed algorithm can work well.Figure5 presents the spectrum search result of the proposed algorithm for the nonuniform linear array with d=[0,0.45λ,0.9λ,1.3λ,1.78λ,2.2λ,2.64λ,3.1λ]. There are 9 (K=9) noncircular signals impinging on array with angles being [-40∘,-30∘,-20∘,-10∘,0∘,10∘,20∘,30∘,40∘] and the noncircular phases being [-40∘,-30∘,-20∘,-10∘,0∘,10∘,20∘,30∘,40∘]. SNR=20 dB, M=8, and L=200 are adopted for the simulation. From Figure 5, we find that proposed algorithm works well even when there are more sources than sensors, while the conventional Capon algorithm fails to work in this condition. The Capon algorithm with M elements can identify M-1 impinging signals. The proposed algorithm uses the noncircular property to double the number of resolvable sources.Figure 5
The spectrum search results of RD-NC-Capon algorithm (SNR=20 dB, M=8, L=200, and K=9).Figure 6
The DOA estimation performance comparison for nonuniform linear array (M=8, L=200, and K=3).Figure 7
The DOA estimation performance comparison for uniform linear array (M=8, L=200, and K=3).Figure 8
The DOA estimation performance with different numbers of antennas (L=200 and K=3).Figure 9
The DOA estimation performance with different number of snapshots (M=8 and K=3).We compare the proposed algorithm against other DOA estimation algorithms. Figure6 indicates the DOA estimation performance comparison among the Capon algorithm, the 2D-NC-Capon algorithm, and the proposed algorithm with d=[0,0.45λ,0.9λ,1.3λ,1.78λ,2.2λ,2.64λ,3.1λ], M=8, L=200, and K=3. ESPRIT and NC-ESPRIT cannot work for nonuniform linear array, because they require the invariance shift property. Therefore, ESPRIT and NC-ESPRIT are not considered in Figure 6. Figure 7 shows the DOA estimation performance comparison among the ESPRIT algorithm, the NC-ESPRIT algorithm, the Capon algorithm, the 2D-NC-Capon algorithm, and the proposed algorithm for a uniform linear array with d=[0,0.5λ,1.0λ,1.5λ,2.0λ,2.5λ,3.0λ,3.5λ]. From Figures 6 and 7, we find that the 2D-NC-Capon algorithm and the proposed algorithm have better angle estimation performance than the Capon algorithm. The reason is that the two NC-Capon algorithms make full use of the noncircular property of the signals, and the array aperture is extended twice consequently. The proposed algorithm has very close DOA estimation performance to the 2D-NC-Capon algorithm, which has a much higher complexity.Figure8 illustrates the DOA estimation performance of the proposed algorithm with different number of antennas (M). It is clearly shown that the angle estimation performance of the proposed algorithm is gradually improved with the number of antennas increasing. Multiple antennas improve the DOA estimation performance because of antenna diversity gain.Figure9 presents the DOA estimation performance of the proposed algorithm with different snapshots. In this simulation, M=8, K=3, and d=[0,0.45λ,0.9λ,1.3λ,1.78λ,2.2λ,2.64λ,3.1λ] are considered. From Figure 9, it is indicated that the DOA estimation performance becomes better with the snapshot increasing. The reason is that increasing the amount of sampled data makes the covariance matrix more accurate.Figure10 shows the spectrum search result of the proposed algorithm for UCA with SNR=20 dB, M=8, and L=200. The radii of the UCA are 0.7λ, and there are 3 noncircular signals impinging on UCA with angles being (10∘,25∘), (20∘,15∘), and (30∘,35∘). From Figure 10, we find that the proposed algorithm works well.Figure 10
The spectrum search result of the proposed algorithm for the UCA.


## 6. Conclusion

In this paper, we have proposed the RD-NC-Capon algorithm for the DOA estimation of noncircular signals. Compared to the conventional Capon algorithm, the proposed RD-NC-Capon algorithm has better estimation performance and can detect more sources by exploiting the noncircular property. Different from the 2D-NC-Capon algorithm, the proposed algorithm achieves DOA estimation with only a 1D search. It not only has much lower computational complexity but also has an angle estimation performance close to that of the 2D-NC-Capon algorithm, which has a much higher complexity than the proposed algorithm. Moreover, the proposed algorithm can be applied to arbitrary arrays and works well without estimating noncircular phases. Numerical simulation results verify the effectiveness and improvement of the proposed algorithm.



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*Source: 348636-2015-12-27.xml*