# A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

Authors: Wei Wang; Ming Zhang; Bobin Yao; Tongxing Zheng; Xiaojiao Fan
Journal: International Journal of Antennas and Propagation
 (2023)

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## Abstract

In this paper, we investigate the problem of the heavy computational burden of the direction-of-arrival (DoA) estimation for the noncircular (NC) signals. A novel low-complexity direction-of-arrival estimation algorithm for NC signals via subspace rotation technique (SRT) is proposed. The proposed algorithm divides the noise subspace matrix along its row direction into two submatrices, and the SRT is performed to get a new reduced-dimension noise subspace. Then, utilizing the separation of variables and the orthogonality between the reduced-dimension noise subspace and the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is derived to estimate DoAs. As the size of the block matrices of the noise subspace matrix has a great impact on the computational complexity of the spectral search, the optimal number of rows of the block matrices is determined. The proposed algorithm not only avoids the two-dimensional spectral search but also efficiently removes the redundancy computations in the one-dimensional spectral search. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation for the NC signals, especially in scenarios where large numbers of sensors are applied.


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## Body

## 1. Introduction

Direction-of-arrival (DoA) estimation has been widely used in various communication applications such as radar, sonar, target location, and wireless communication, which is an extremely important topic in the field of array signal processing [1–9]. High resolution methods for DoA estimation include multiple signal classification (MUSIC) [10], root-multiple signal classification (Root-MUSIC) [11], estimating signal parameters via rotational invariance techniques (ESPRIT) [12], and so forth. However, these traditional algorithms based on the premise of the narrow-band or wide-band sources have not exploited the properties of the incoming sources [13–16]. As the noncircular (NC) signals such as binary phase shift keying (BPSK) signals, amplitude modulation (AM) signals, and multiple amplitude shift keying (MASK) signals have been more and more widely utilized in practical communication, the NC property of incoming sources has been considered in DOA estimation [17–20]. The knowledge of the elliptic covariance matrix, which is not equal to zero, can be used to improve the estimation performance by increasing the dimension of the received signal matrix and extending the array aperture. A two-dimensional noncircular MUSIC (2D-NC-MUSIC) algorithm exploiting the NC property of the incoming sources was proposed in [21], which needs a two-dimensional spectral search, bringing huge computational complexity. In order to decrease the computational complexity of the 2D-NC-MUSIC algorithm, the noncircular MUSIC (NC-MUSIC) algorithm was also presented in [21] to convert the two-dimensional spectral search into the one-dimensional spectral search. A polynomial rooting NC-MUSIC algorithm proposed in [22] can replace the two-dimensional spectral search with the polynomial rooting so as to reduce the computational complexity of the 2D-NC-MUSIC algorithm to some extent. In [23–25], convex optimization was utilized to avoid the two-dimensional spectral search and greatly improves the computational effectively. However, the required one-dimensional spectral search still has the high computational cost. A SRTRD-NC-MUSIC algorithm was presented in [26], which enhances the computational efficiency by means of the conversion of the two-dimensional peak searching into the one-dimensional peak searching and the removal of some redundancy computations of the one-dimensional peak searching, but still remains redundancy computations in peak searching. Based on the sparsity of target azimuth information in the spatial domain, the methods proposed in [27, 28] use the spatial dictionary and the sparse signal reconstruction to finish the DoA estimation, which can reduce the computational complexity only on the premise of precisely compensating for the NC phase caused by transmission delay.In order to reduce the computational complexity of the 2D-NC-MUSIC algorithm, we improve the SRTRD-NC-MUSIC algorithm and propose a novel low-complexity DoA estimation algorithm called the SRTRD-p-NC-MUSIC algorithm, which can further avoid redundancy computations in spectral search. In this paper, we divide the noise subspace matrix along its row direction into two submatrices and the subspace rotation technique (SRT) [29] is used to construct a new reduced-dimension noise subspace. As the new reduced-dimension noise subspace is also orthogonal to the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is finally derived to estimate DOAs by means of the separation of variables. Considering that the size of the block matrices of the noise subspace matrix has a great impact on the dimensions of the new reduced-dimension noise subspace and the computational complexity of the spectral search, we have a discussion about the numbers of rows of the block matrices, and the optimal number of rows of the block matrices is derived. The SRTRD-p-NC-MUSIC algorithm can reduce the computational complexity to less than 3% as compared to the 2D-NC-MUSIC algorithm on the premise of ensuring the accuracy of DOA estimation. Especially, the efficiency advantage of the new algorithm is more obvious for the case that large numbers of sensors are required.


### 1.1. Notation

Throughout the paper, we utilize upper-case bold letters to denote matrices (e.g.,A), lower-case bold letters for vectors (e.g., a) and upper-case double-line characters for sets (e.g., ℂ). (·)∗, (·)T, (·)H, and (·)† represent complex conjugate, transpose, conjugate transpose, and pseudoinverse, respectively. E {·} is used to represent the statistical expectation. ‖.‖ and det (·), respectively, indicate the norm of a vector and the determinant of a matrix. diag {.} stands for a diagonal matrix composed of the embraced elements. Span (·) is the space spanned by the columns of a matrix, dim (·) is the dimension of a linear space and rank (·) is the rank of a matrix. IM represents a M × M identity matrix and 0M×N denotes a M × N matrix with all zero elements.


## 1.1. Notation

Throughout the paper, we utilize upper-case bold letters to denote matrices (e.g.,A), lower-case bold letters for vectors (e.g., a) and upper-case double-line characters for sets (e.g., ℂ). (·)∗, (·)T, (·)H, and (·)† represent complex conjugate, transpose, conjugate transpose, and pseudoinverse, respectively. E {·} is used to represent the statistical expectation. ‖.‖ and det (·), respectively, indicate the norm of a vector and the determinant of a matrix. diag {.} stands for a diagonal matrix composed of the embraced elements. Span (·) is the space spanned by the columns of a matrix, dim (·) is the dimension of a linear space and rank (·) is the rank of a matrix. IM represents a M × M identity matrix and 0M×N denotes a M × N matrix with all zero elements.


## 2. Data Model and 2D-NC-MUSIC Algorithm

### 2.1. Data Model

The noncircular signals have been extensively used in various modern communication systems. As the noncircular signals have the property that the pseudocovariance is nonzero, the effective array aperture can be extended by means of increasing the number of virtual sensors [17–20]. Considering the impact of the actual transmission channel, the received signals can be represented as(1)St=SRtejφ,where SR (t) is the real signal, and the phase φ is the sum of the initial phase of the signal and the additional phase of the transmission channel. Consider an M-element uniform linear array (ULA) with the interelement spacing d and select the leftmost sensor of the ULA as the reference one. K independent narrow-band far-field noncircular signals Sk (t) (k = 1, 2, … , K) are impinging on the ULA from the different directions θk (k = 1, 2, … , K), as shown in Figure 1. The received signals of the ULA can be expressed as(2)Xt=ASt+Nt=AΨSRt+Nt,where St=S1t,S2t,⋯,SKtT∈CK×1 is the noncircular incident signal vector, A=aθ1,aθ2,⋯,aθK∈CM×K is the manifold matrix with the steering vector aθk=1,ej2πdsinθk/λ,⋯,ej2πM−1dsinθk/λT (k = 1, 2, … , K) and λ is the wavelength of the carrier wave. Nt=n1t,n2t,⋯,nMtT∈CM×1 represents the white Gaussian noise vector with zero mean and variance σ2. SRt=SR,1t,SR,2t,⋯,SR,KtT∈RK×1 and Ψ=diagejφ1,ejφ2,⋯,ejφK with φk (k=1,2,⋯,K) being the noncircular phase of the kth incident signal.Figure 1
Array signal model.Considering that a noncircular signal has the characteristics ofE {S (t) ST (t)} ≠ 0 and E {S∗ (t) SH (t)} ≠ 0, we concatenate the array output and its conjugation to increase the number of available sensors [13–20]. So a new extended vector is obtained, as shown in the following:(3)Yt=XtX∗t=AΨA∗Ψ∗SRt+NtN∗t=BSRt+N0t,where B=AΨTAΨHT=bθ1,φ1,bθ2,φ2,⋯,bθK,φK and N0t=NTtNHtT.


### 2.2. 2D-NC-MUSIC Algorithm

The extended covariance matrixRY is expressed as(4)RY=EYtYHt.According to the eigenvalue decomposition (EVD) of the matrixRY, the label (4) can be further rewritten as(5)RY=∑i=1KλiviviH+∑i=K+12MλiviviH=VSΛSVSH+VNΛNVNH,where λ1≥λ2≥⋯≥λK>λK+1=λK+2=⋯=λ2M, ΛS=diagλ1,λ2,⋯,λK, ΛS=diagλK+1,λK+2,⋯,λ2M, and Vi (i = 1, 2, …, 2M) is the eigenvector corresponding to the eigenvalue λi (i = 1, 2, …, 2M). VS=v1,v2,⋯,vK and VN=vK+1,vK+2,⋯,v2M are the signal subspace and the noise subspace, respectively. In practice, we usually use the L snapshots of the received signal X (t) to estimate the extended covariance matrix RY as follows:(6)R^Y=1L∑t=1LYtYHt.Based on the orthogonality property between the signal subspace and the noise subspace, the two-dimensional NC-MUSIC function can be constructed to estimate the DOAs of the noncircular signals as follows [21]:(7)P2D−NC−MUSICθ,φ=1bHθ,φV^NV^NHbθ,φ,where(8)bθ,φ=ejφ,ejφ+2πdsinθ/λ,⋯,ejφ+2πM−1dsinθ/λ,e−jφ,e−jφ+2πdsinθ/λ,⋯,e−jφ+2πM−1dsinθ/λT.TheK maxima referring to source DoAs can be obtained by the two-dimensional spectral search. It is observed that although the 2D-NC-MUSIC algorithm can extend the effective array aperture and greatly improve the resolution of the DoA estimation, the expensive computation cost of two-dimension spectral search limits the practical application of the 2D-NC-MUSIC algorithm in the high-real-time processing scenarios [30–32].


## 2.1. Data Model

The noncircular signals have been extensively used in various modern communication systems. As the noncircular signals have the property that the pseudocovariance is nonzero, the effective array aperture can be extended by means of increasing the number of virtual sensors [17–20]. Considering the impact of the actual transmission channel, the received signals can be represented as(1)St=SRtejφ,where SR (t) is the real signal, and the phase φ is the sum of the initial phase of the signal and the additional phase of the transmission channel. Consider an M-element uniform linear array (ULA) with the interelement spacing d and select the leftmost sensor of the ULA as the reference one. K independent narrow-band far-field noncircular signals Sk (t) (k = 1, 2, … , K) are impinging on the ULA from the different directions θk (k = 1, 2, … , K), as shown in Figure 1. The received signals of the ULA can be expressed as(2)Xt=ASt+Nt=AΨSRt+Nt,where St=S1t,S2t,⋯,SKtT∈CK×1 is the noncircular incident signal vector, A=aθ1,aθ2,⋯,aθK∈CM×K is the manifold matrix with the steering vector aθk=1,ej2πdsinθk/λ,⋯,ej2πM−1dsinθk/λT (k = 1, 2, … , K) and λ is the wavelength of the carrier wave. Nt=n1t,n2t,⋯,nMtT∈CM×1 represents the white Gaussian noise vector with zero mean and variance σ2. SRt=SR,1t,SR,2t,⋯,SR,KtT∈RK×1 and Ψ=diagejφ1,ejφ2,⋯,ejφK with φk (k=1,2,⋯,K) being the noncircular phase of the kth incident signal.Figure 1
Array signal model.Considering that a noncircular signal has the characteristics ofE {S (t) ST (t)} ≠ 0 and E {S∗ (t) SH (t)} ≠ 0, we concatenate the array output and its conjugation to increase the number of available sensors [13–20]. So a new extended vector is obtained, as shown in the following:(3)Yt=XtX∗t=AΨA∗Ψ∗SRt+NtN∗t=BSRt+N0t,where B=AΨTAΨHT=bθ1,φ1,bθ2,φ2,⋯,bθK,φK and N0t=NTtNHtT.


## 2.2. 2D-NC-MUSIC Algorithm

The extended covariance matrixRY is expressed as(4)RY=EYtYHt.According to the eigenvalue decomposition (EVD) of the matrixRY, the label (4) can be further rewritten as(5)RY=∑i=1KλiviviH+∑i=K+12MλiviviH=VSΛSVSH+VNΛNVNH,where λ1≥λ2≥⋯≥λK>λK+1=λK+2=⋯=λ2M, ΛS=diagλ1,λ2,⋯,λK, ΛS=diagλK+1,λK+2,⋯,λ2M, and Vi (i = 1, 2, …, 2M) is the eigenvector corresponding to the eigenvalue λi (i = 1, 2, …, 2M). VS=v1,v2,⋯,vK and VN=vK+1,vK+2,⋯,v2M are the signal subspace and the noise subspace, respectively. In practice, we usually use the L snapshots of the received signal X (t) to estimate the extended covariance matrix RY as follows:(6)R^Y=1L∑t=1LYtYHt.Based on the orthogonality property between the signal subspace and the noise subspace, the two-dimensional NC-MUSIC function can be constructed to estimate the DOAs of the noncircular signals as follows [21]:(7)P2D−NC−MUSICθ,φ=1bHθ,φV^NV^NHbθ,φ,where(8)bθ,φ=ejφ,ejφ+2πdsinθ/λ,⋯,ejφ+2πM−1dsinθ/λ,e−jφ,e−jφ+2πdsinθ/λ,⋯,e−jφ+2πM−1dsinθ/λT.TheK maxima referring to source DoAs can be obtained by the two-dimensional spectral search. It is observed that although the 2D-NC-MUSIC algorithm can extend the effective array aperture and greatly improve the resolution of the DoA estimation, the expensive computation cost of two-dimension spectral search limits the practical application of the 2D-NC-MUSIC algorithm in the high-real-time processing scenarios [30–32].


## 3. SRTRD-p-NC-MUSIC Algorithm

In this section, we propose a new algorithm to reduce the computational complexity of the 2D-NC-MUSIC algorithm. We first use the block matrix method and SRT to construct a reduced-dimensional noise space. Then, based on the orthogonality between the reduced-dimensional noise space and the space spanned by the columns of the extended manifold matrix, the separation of variables is utilized to convert the two-dimensional spectral search into the one-dimensional spectral search. Finally, because the number of rows in the block matrices has a great impact on the computational complexity of the proposed algorithm, we discuss about the selection of the number of rows in the block matrices and give the optimum value.


### 3.1. SRT and Reconstruction of the Noise Subspace

The noise subspaceVN∈C2M×2M−K can be divided along its row direction into two block matrices, as shown in the following:(9)VN=VN,12M−KM−pVN,22M−KM+p,where VN,1∈CM−p×2M−K and VN,2∈CM+p×2M−K. Obviously, rankVN=dimspanVN=2M−K, and if 1 ≤ p ≤ M − K, the matrix VN,2 is row full rank. According to the matrix theory [33], we know that the matrix VN,2 has a right inverse matrix VN,2† satisfying VN,2†VN,2†=IM+p. In order to construct the new noise space span Vnew, we use the matrix VN,2† to weight the column vectors of the noise space VN with the following form:(10)Vnew=VNVN,2†.The abovementioned label (10) means we rotate the space span VN by an angle in the 2M-dimensional space. Consider rankVnew≤minrankVN,rankVN,2†, then the new constructed space span Vnew is a subset of the space span (VN):(11)spanVnew⊆spanVN.As we know,spanVN⊥spanVS and spanVS=spanB, so the space span (B) is orthogonal to the reconstructed space span Vnew:(12)VnewHbθi,φi=0M+p×1,where i = 1,2, …, K.


### 3.2. Construction of the Reduced-Dimension Spectrum Function

Based on the orthogonality property between the reconstructed noise subspaceVnew and the column space spanned by the matrix B, the two-dimensional spectrum function is defined according to label (13) [27]:(13)P2Dθ,φ=1fθ,φ=1bHθ,φVnewVnewHbθ,φ.The estimation of two-dimensional DOAs of noncircular signals can be acquired by searching the minimum offθ,φ. Based on the separation of variables, bθ,φ is represented as(14)bθ,φ=aθejφa∗θe−jφ=A˜θEφ,where(15)A˜θ=aθ0M×10M×1a∗θ,Eφ=ejφe−jφT.The label (13) can be further rewritten as(16)P2Dθ,φ=1fθ,φ=1EHφA˜HθVnewVnewHA˜θEφ.SinceA˜HθVnewVnewHA˜θ is a Hermitian matrix and fθ,φ is a nonnegative-definite quadratic form, the minimum of fθ,φ can be determined by the minimum eigenvalue of the matrix A˜HθVnewVnewHA˜θ. When θ is one of the real DOAs of the received signals, the minimum eigenvalue of the matrix A˜HθVnewVnewHA˜θ is zero. Then the minimum of the determinant of the matrix A˜HθVnewVnewHA˜θ is also zero [31]:(17)detA˜HθiVnewVnewHA˜θi=0.We construct a new reduced-dimension spectrum function as follows:(18)PSRTRD−p−NC−MUSICθ=1detA˜HθVnewVnewHA˜θ.By only one-dimensional peak searching, the DOAs of the noncircular signals can be obtained by searchingK minimum values of (18).Compared with the 2D-NC-MUSIC algorithm, the proposed algorithm can not only reduce the dimension of the noise subspace, but also need only one-dimensional peak searching to obtain the estimation of the DOAs of the received signals. The following analysis will show that the proposed SRTRD-p-NC-MUSIC algorithm can significantly reduce the computational complexity of the 2D-NC-MUSIC algorithm and dramatically improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation.


### 3.3. Simplification of the SRTRD-p-NC-MUSIC Algorithm

Substituting label (9) into label (10), label (10) can be further represented as(19)Vnew=VNVN,2†=VN,1VN,2VN,2†=VN,1VN,2†IM+p=VrotM+pM−pIM+p,with Vnew ∈ℂ(M−P)×(M+P). Then, we can divide the matrix A˜Hθ∈C2×2M into block matrices as follows:(20)where(21)a˜θ=1,ej2πdsinθ/λ,⋯,ej2πM−p−1dsinθ/λT∈CM−p×1,e0θ=ej2πM−pdsinθ/λ,⋯,ej2πM−1dsinθ/λT∈Cp×1,e1θ=e0Hθ01×M∈C1×M+p,e2θ=01×paTθ∈C1×M+p.Based on label (19) and label (20), we can obtain(22)A˜HθVnew=a˜Hθe1θ01×M−pe2θVrotIM+p=a˜HθVrot+e1θe2θ.Furthermore, we substitute label (22) into label (18) to get(23)A˜HVnewVnewHA˜=a˜HVrot+e1e2VrotHa˜+e1He2H=a˜HVrot2+2Rea˜HVrote1H+e12a˜HVrote2H+e1e2He2VrotHa˜+e2e1He22=a˜HVrot2+2Rea˜HVrote1H+pa˜HVrote2He2VrotHa˜M,where the tag θ representing the angle is omitted for notational convenience. Then, the proposed SRTRD-NC-MUSIC algorithm can be further simplified, as shown in the following:(24)PSRTRD−p−NC−MUSICθ=1Mηθ2+2Reηθe1Hθ+p−ηθe2Hθ2,where ηθ=a˜HθVrot. From label (24), we can see that the computational complexity of the proposed SRTRD-NC-MUSIC algorithm is greatly affected by the selection of p because of the relation between the value of p and the dimensions of ηθ, a˜θ, Vrot, and e1θ. The following analysis shows that when the value of p is optimal, the proposed SRTRD-NC-MUSIC algorithm can dramatically improve the computational efficiency as compared to the 2D-NC-MUSIC algorithm.


### 3.4. Optimal Selection ofp

In this section, we further discuss about the selection of the value ofp to dramatically improve the computational efficiency. Note that the complexities are given in terms of complex-valued flops [26–29]. For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the EVD of the covariance matrix RY can use the fast subspace decomposition (FSD) technique [34], and O (4M2K) flops are required to use the FSD technique of the covariance matrix RY. Compared with the 2D-NC-MUSIC, we need to additionally construct the new noise subspace Vnew and calculate the matrix VN,2† for the SRTRD-p-NC-MUSIC. As we know, the dimensions of the matrix Vnew and the matrix VN,2† are 2M × (2M − K) and (M + p) × (2M − K), respectively. Then, the construction of the matrix Vnew needs Ο [(M − p) (M + p) (2M − K)] + O [2M (2M − K)] flops [29].For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the spectral search step is needed to estimate the DOAs of the received signals. Assuming there areJ points within the search range [−90° and 90°] of the incident angle θ and the search range [0, 2π] of the noncircular phase φ, respectively, the 2D-NC-MUSIC requires O [J2 (2M + 1) (2M − K)] flops to finish the spectral search. For the SRTRD-p-NC-MUSIC, it has to compute η2, ηθe1Hθ, and ηθe2Hθ2 for each search point. Then, the cost of computing η2, ηθe1Hθ, and ηθe2Hθ2 are O [(M − p+1) (M + p)] flops, O ((M − p + 1) p) flops, and O ((M − p + 1) M + 1) flops, respectively. Consequently, the spectral search of the SRTRD-p-NC-MUSIC costs O [J (−2p2 + 2p + 2 M2 + 2M + 1)] flops.Above all, we can obtain the number of total flops of the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC with the following form:(25)C2D−NC−MUSIC=Ο4M2K+ΟJ22M+12M−K,CSRTRD−p−NC−MUSICp=Ο4M2K+ΟM−pM+p2M−K+Ο2M2M−K+ΟJ−2p2+2p+2M2+2M+1.As it usually satisfiesJ >> M > K in practice, the label (25) can be further approximated as follows:(26)C2D−NC−MUSIC≈ΟJ22M+12M−K,CSRTRD−p−NC−MUSICp≈ΟJ−2p2+2p+2M2+2M+1.Obviously, the selection of the value ofp depends on the following optimization problem:(27)minpCSRTRD−p−NC−MUSICp,s.t.1≤p≤M−K,As shown in Figure2, the label (26) is a quadratic function of p which is monotone decreasing if p>0.5. Considering that p satisfies 1 ≤ p ≤ M−K, we can get CSRTRD−p−NC−MUSICM−K≤CSRTRD−p−NC−MUSIC1 since the number of sensors and the number of sources usually satisfy M > K. Therefore, we can further get the optimal choice of p, which is represented as(28)popt=M−K.Figure 2
Analysis of the value ofCSRTRD−p−NC−MUSICp.


### 3.5. Analysis of Computational Complexity

Whenp = popt, we substitute label (28) into label (26) and the computational complexity of the proposed SRTRD-p-NC-MUSIC can be further written, as shown in the following:(29)CSRTRD−p−NC−MUSICp=ΟJ4MK+4M+1−2K2−2K.Figure3 illustrates the computational complexity comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC [21], RD-NC-MUSIC [23], and SRTRD-NC-MUSIC [26] with different numbers of sensors and different numbers of sources, where J = 1000 is considered. The RD-NC-MUSIC and the SRTRD-NC-MUSIC require ΟJ2M−K4M+2+M+2 flops [23] and ΟJ2M2+2M+1 flops [26], respectively.Figure 3
Comparison of computational complexities with a different number of sensors and a different number of sources.From Figure3, we can see that the computational complexities of different algorithms increase with the number of sensors. Compared with the 2D-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can reduce the computational cost dramatically. Especially, the efficiency advantage of the proposed SRTRD-p-NC-MUSIC is more obvious in scenarios where the large numbers of sensors are applied. Moreover, the proposed SRTRD-p-NC-MUSIC can reduce the computational complexity to about 20% as compared to the RD-NC-MUSIC. Compared with the SRTRD-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can enhance the computational efficiency significantly when the large numbers of sensors are applied. In addition, unlike the 2D-NC-MUSIC and the RD-NC-MUSIC, the computational complexity of the proposed SRTRD-NC-MUSIC increases as the number of sources increases.


### 3.6. Summary of the Proposed Algorithm

The main steps of the proposed SRTRD-p-NC-MUSIC are shown as follows:Step 1. Estimate the extended covariance matrixRY according to label (6).Step 2. Perform the eigenvalue decomposition of the matrixRY according to label (5) to acquire the noise subspace VN.Step 3. Obtain the optimal choice ofP according to label (28). Divide the matrix VN according to label (9) to obtain VN,2.Step 4. Construct the new noise subspaceVnew according to label (19). Divide the matrix A˜Hθ according to label (20).Step 5. Construct the one-dimensional spectral search functionPSRTRD−p−NC−MUSICθ according to label (24). Estimate the DoAs of the noncircular signals by searching K minimum values of label (24).


## 3.1. SRT and Reconstruction of the Noise Subspace

The noise subspaceVN∈C2M×2M−K can be divided along its row direction into two block matrices, as shown in the following:(9)VN=VN,12M−KM−pVN,22M−KM+p,where VN,1∈CM−p×2M−K and VN,2∈CM+p×2M−K. Obviously, rankVN=dimspanVN=2M−K, and if 1 ≤ p ≤ M − K, the matrix VN,2 is row full rank. According to the matrix theory [33], we know that the matrix VN,2 has a right inverse matrix VN,2† satisfying VN,2†VN,2†=IM+p. In order to construct the new noise space span Vnew, we use the matrix VN,2† to weight the column vectors of the noise space VN with the following form:(10)Vnew=VNVN,2†.The abovementioned label (10) means we rotate the space span VN by an angle in the 2M-dimensional space. Consider rankVnew≤minrankVN,rankVN,2†, then the new constructed space span Vnew is a subset of the space span (VN):(11)spanVnew⊆spanVN.As we know,spanVN⊥spanVS and spanVS=spanB, so the space span (B) is orthogonal to the reconstructed space span Vnew:(12)VnewHbθi,φi=0M+p×1,where i = 1,2, …, K.


## 3.2. Construction of the Reduced-Dimension Spectrum Function

Based on the orthogonality property between the reconstructed noise subspaceVnew and the column space spanned by the matrix B, the two-dimensional spectrum function is defined according to label (13) [27]:(13)P2Dθ,φ=1fθ,φ=1bHθ,φVnewVnewHbθ,φ.The estimation of two-dimensional DOAs of noncircular signals can be acquired by searching the minimum offθ,φ. Based on the separation of variables, bθ,φ is represented as(14)bθ,φ=aθejφa∗θe−jφ=A˜θEφ,where(15)A˜θ=aθ0M×10M×1a∗θ,Eφ=ejφe−jφT.The label (13) can be further rewritten as(16)P2Dθ,φ=1fθ,φ=1EHφA˜HθVnewVnewHA˜θEφ.SinceA˜HθVnewVnewHA˜θ is a Hermitian matrix and fθ,φ is a nonnegative-definite quadratic form, the minimum of fθ,φ can be determined by the minimum eigenvalue of the matrix A˜HθVnewVnewHA˜θ. When θ is one of the real DOAs of the received signals, the minimum eigenvalue of the matrix A˜HθVnewVnewHA˜θ is zero. Then the minimum of the determinant of the matrix A˜HθVnewVnewHA˜θ is also zero [31]:(17)detA˜HθiVnewVnewHA˜θi=0.We construct a new reduced-dimension spectrum function as follows:(18)PSRTRD−p−NC−MUSICθ=1detA˜HθVnewVnewHA˜θ.By only one-dimensional peak searching, the DOAs of the noncircular signals can be obtained by searchingK minimum values of (18).Compared with the 2D-NC-MUSIC algorithm, the proposed algorithm can not only reduce the dimension of the noise subspace, but also need only one-dimensional peak searching to obtain the estimation of the DOAs of the received signals. The following analysis will show that the proposed SRTRD-p-NC-MUSIC algorithm can significantly reduce the computational complexity of the 2D-NC-MUSIC algorithm and dramatically improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation.


## 3.3. Simplification of the SRTRD-p-NC-MUSIC Algorithm

Substituting label (9) into label (10), label (10) can be further represented as(19)Vnew=VNVN,2†=VN,1VN,2VN,2†=VN,1VN,2†IM+p=VrotM+pM−pIM+p,with Vnew ∈ℂ(M−P)×(M+P). Then, we can divide the matrix A˜Hθ∈C2×2M into block matrices as follows:(20)where(21)a˜θ=1,ej2πdsinθ/λ,⋯,ej2πM−p−1dsinθ/λT∈CM−p×1,e0θ=ej2πM−pdsinθ/λ,⋯,ej2πM−1dsinθ/λT∈Cp×1,e1θ=e0Hθ01×M∈C1×M+p,e2θ=01×paTθ∈C1×M+p.Based on label (19) and label (20), we can obtain(22)A˜HθVnew=a˜Hθe1θ01×M−pe2θVrotIM+p=a˜HθVrot+e1θe2θ.Furthermore, we substitute label (22) into label (18) to get(23)A˜HVnewVnewHA˜=a˜HVrot+e1e2VrotHa˜+e1He2H=a˜HVrot2+2Rea˜HVrote1H+e12a˜HVrote2H+e1e2He2VrotHa˜+e2e1He22=a˜HVrot2+2Rea˜HVrote1H+pa˜HVrote2He2VrotHa˜M,where the tag θ representing the angle is omitted for notational convenience. Then, the proposed SRTRD-NC-MUSIC algorithm can be further simplified, as shown in the following:(24)PSRTRD−p−NC−MUSICθ=1Mηθ2+2Reηθe1Hθ+p−ηθe2Hθ2,where ηθ=a˜HθVrot. From label (24), we can see that the computational complexity of the proposed SRTRD-NC-MUSIC algorithm is greatly affected by the selection of p because of the relation between the value of p and the dimensions of ηθ, a˜θ, Vrot, and e1θ. The following analysis shows that when the value of p is optimal, the proposed SRTRD-NC-MUSIC algorithm can dramatically improve the computational efficiency as compared to the 2D-NC-MUSIC algorithm.


## 3.4. Optimal Selection ofp

In this section, we further discuss about the selection of the value ofp to dramatically improve the computational efficiency. Note that the complexities are given in terms of complex-valued flops [26–29]. For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the EVD of the covariance matrix RY can use the fast subspace decomposition (FSD) technique [34], and O (4M2K) flops are required to use the FSD technique of the covariance matrix RY. Compared with the 2D-NC-MUSIC, we need to additionally construct the new noise subspace Vnew and calculate the matrix VN,2† for the SRTRD-p-NC-MUSIC. As we know, the dimensions of the matrix Vnew and the matrix VN,2† are 2M × (2M − K) and (M + p) × (2M − K), respectively. Then, the construction of the matrix Vnew needs Ο [(M − p) (M + p) (2M − K)] + O [2M (2M − K)] flops [29].For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the spectral search step is needed to estimate the DOAs of the received signals. Assuming there areJ points within the search range [−90° and 90°] of the incident angle θ and the search range [0, 2π] of the noncircular phase φ, respectively, the 2D-NC-MUSIC requires O [J2 (2M + 1) (2M − K)] flops to finish the spectral search. For the SRTRD-p-NC-MUSIC, it has to compute η2, ηθe1Hθ, and ηθe2Hθ2 for each search point. Then, the cost of computing η2, ηθe1Hθ, and ηθe2Hθ2 are O [(M − p+1) (M + p)] flops, O ((M − p + 1) p) flops, and O ((M − p + 1) M + 1) flops, respectively. Consequently, the spectral search of the SRTRD-p-NC-MUSIC costs O [J (−2p2 + 2p + 2 M2 + 2M + 1)] flops.Above all, we can obtain the number of total flops of the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC with the following form:(25)C2D−NC−MUSIC=Ο4M2K+ΟJ22M+12M−K,CSRTRD−p−NC−MUSICp=Ο4M2K+ΟM−pM+p2M−K+Ο2M2M−K+ΟJ−2p2+2p+2M2+2M+1.As it usually satisfiesJ >> M > K in practice, the label (25) can be further approximated as follows:(26)C2D−NC−MUSIC≈ΟJ22M+12M−K,CSRTRD−p−NC−MUSICp≈ΟJ−2p2+2p+2M2+2M+1.Obviously, the selection of the value ofp depends on the following optimization problem:(27)minpCSRTRD−p−NC−MUSICp,s.t.1≤p≤M−K,As shown in Figure2, the label (26) is a quadratic function of p which is monotone decreasing if p>0.5. Considering that p satisfies 1 ≤ p ≤ M−K, we can get CSRTRD−p−NC−MUSICM−K≤CSRTRD−p−NC−MUSIC1 since the number of sensors and the number of sources usually satisfy M > K. Therefore, we can further get the optimal choice of p, which is represented as(28)popt=M−K.Figure 2
Analysis of the value ofCSRTRD−p−NC−MUSICp.


## 3.5. Analysis of Computational Complexity

Whenp = popt, we substitute label (28) into label (26) and the computational complexity of the proposed SRTRD-p-NC-MUSIC can be further written, as shown in the following:(29)CSRTRD−p−NC−MUSICp=ΟJ4MK+4M+1−2K2−2K.Figure3 illustrates the computational complexity comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC [21], RD-NC-MUSIC [23], and SRTRD-NC-MUSIC [26] with different numbers of sensors and different numbers of sources, where J = 1000 is considered. The RD-NC-MUSIC and the SRTRD-NC-MUSIC require ΟJ2M−K4M+2+M+2 flops [23] and ΟJ2M2+2M+1 flops [26], respectively.Figure 3
Comparison of computational complexities with a different number of sensors and a different number of sources.From Figure3, we can see that the computational complexities of different algorithms increase with the number of sensors. Compared with the 2D-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can reduce the computational cost dramatically. Especially, the efficiency advantage of the proposed SRTRD-p-NC-MUSIC is more obvious in scenarios where the large numbers of sensors are applied. Moreover, the proposed SRTRD-p-NC-MUSIC can reduce the computational complexity to about 20% as compared to the RD-NC-MUSIC. Compared with the SRTRD-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can enhance the computational efficiency significantly when the large numbers of sensors are applied. In addition, unlike the 2D-NC-MUSIC and the RD-NC-MUSIC, the computational complexity of the proposed SRTRD-NC-MUSIC increases as the number of sources increases.


## 3.6. Summary of the Proposed Algorithm

The main steps of the proposed SRTRD-p-NC-MUSIC are shown as follows:Step 1. Estimate the extended covariance matrixRY according to label (6).Step 2. Perform the eigenvalue decomposition of the matrixRY according to label (5) to acquire the noise subspace VN.Step 3. Obtain the optimal choice ofP according to label (28). Divide the matrix VN according to label (9) to obtain VN,2.Step 4. Construct the new noise subspaceVnew according to label (19). Divide the matrix A˜Hθ according to label (20).Step 5. Construct the one-dimensional spectral search functionPSRTRD−p−NC−MUSICθ according to label (24). Estimate the DoAs of the noncircular signals by searching K minimum values of label (24).


## 4. Simulation Results

In this section, we will demonstrate the effectiveness of the proposed SRTRD-p-NC-MUSIC, compared with the 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC. For all simulations, a ULA with spacingd = 0.5λ is used. In all experiments except for Figure 4, we set the number of sensors M = 8. We assume that the sources are narrow-band BPSK signals with equal power in the presence of the additive white Gaussian noise. All the simulation results are averaged via 3000 Monte Carlo runs. The Root Mean Square Error (RMSE) is defined as(30)RMSE=1KNm∑k=1K∑i=1Nmθk,i−θ^k,i2,where Nm is the total number of the Monte Carlo simulations, and θk,i, θ^k,i are the real value and estimated result of the kth source in the ith simulation, respectively. For the signals incoming from two similar directions, it is considered as the successful estimation when satisfying(31)Pθ1+Pθ22>Pθ1+θ22,where P (·) is the spectrum function, and θ1, θ2 are the incoming directions of the signals, respectively [29].Figure 4
Performance comparison with a different number of sensors, SNR = 10 dB andL = 500.Figure5 shows the DoA estimation results of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC, where K = 3 with θ = {−45°, −15°, 5°} and the snapshots L = 1000. The signal-to-noise ratio (SNR) is set to 5 dB. From Figure 5, the proposed SRTRD-p-NC-MUSIC can resolve all sources sufficiently well. Compared with the 2D-NC-MUSIC, the spatial spectrum of the SRTRD-p-NC-MUSIC has a sharper peak. In addition, the SRTRD-p-NC-MUSIC performs close to the 2D-NC-MUSIC and improves the computational efficiency significantly.Figure 5
Spatial spectrum of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC, SNR = 5 dB andL = 1000.Figures4, 6, and 7 indicate the RMSE performance comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC, where K = 2 sources with θ = {−15°, 20°}. In Figure 6, the number of snapshots is fixed at 500, and the SNR is varied from −10 dB to 20 dB. It is clearly shown that the RMSEs of all algorithms in Figure 6 decrease as the SNR increases. Furthermore, the SRTRD-p-NC-MUSIC shows close RMSEs to the 2D-NC-MUSIC, and it has the better RMSE performance than the other three algorithms. In Figure 7, the SNR is fixed at 10 dB, and the number of snapshots is varied. From Figure 7, when the number of snapshots increases, the extended covariance matrix can be estimated more accurately, and the RMSEs of all algorithms in Figure 7 decrease. At the same time, the SRTRD-p-NC-MUSIC can achieve a better RMSE performance, close to that of the 2D-NC-MUSIC, than the other three algorithms. In Figure 4, the number of sensors is varied from 5 to 15, where SNR = 10 dB and L = 500. It is indicated that the RMSE performance of all algorithms in Figure 4 becomes better with increasing the number of sensors because the increasing number of sensors expands the aperture of the ULA. In addition, the SRTRD-p-NC-MUSIC still has a closer RMSE performance to the 2D-NC-MUSIC in comparison with the other three algorithms. Obviously, it can be seen clearly that the computational efficiency of the SRTRD-p-NC-MUSIC is improved without sacrificing too much estimation accuracy.Figure 6
Performance comparison with different SNRs,L = 500.Figure 7
Performance comparison with a different number of snapshots, SNR = 10 dB.In Sections3.4 and 3.5, we have analysed how the choice of the value of p affects the computational complexity of the proposed SRTRD-p-NC-MUSIC. In order to investigate the estimation accuracy of the proposed SRTRD-p-NC-MUSIC, in Figure 8, we further compare the RMSE performance of the proposed algorithm with different choices of the value of p, where K = 2 sources with θ = {−15°, 20°}. In Figure 8(a), we set the number of snapshots to be 500 and the SNRs from −10 dB to 20 dB. It can be seen that as the SNR increases, the RMSEs of all choices of the value of p decrease. In Figure 8(b), we set the SNR to be 10 dB, and the number of snapshots is varied. It is observed that the RMSEs of all choices of the value of p decrease with increasing the number of snapshots. In addition, from Figure 8(a) and Figure 8(b), we can see that when the choice of the value of p is closer to the optimum popt, the RMSE performance of the proposed algorithm is much better. This is because as the value of p is closer to the optimum popt, the orthogonality between the reconstructed noise subspace and the space spanned by the columns of the extended manifold matrix is more improved [29, 35].Figure 8
Performance comparison of the proposed SRTRD-p-NC-MUSIC with different choices of the value ofp. (a) RMSEs with different SNRs, L = 500. (b) RMSEs with a different number of snapshots, SNR = 10 dB.
(a)(b)In Figures9 and 10, there are two signals impinging upon the ULA from the directions θ1 = 10° and θ2 = 11°. Here, the number of the snapshots is 500. Figure 9 illustrates the DoA resolution performance versus different SNRs, where two sources are located at similar directions. It is can be observed that these four algorithms in Figure 9 have a 100% resolution probability to distinguish the two sources when the SNR is higher than 5 dB. However, the SRTRD-p-NC-MUSIC can obtain much better spatial resolution than the other three algorithms when the SNR is lower than 0 dB. This is because the reconstruction of the noise subspace improves the orthogonality between the signal subspace and the noise subspace and the dimension of the reconstructed noise subspace is reduced significantly, which leads to a sharper peak of the spatial spectrum [29, 35]. In Figure 10, the DoA estimation results of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC are depicted for 10 simulations, where SNR = 5 dB. We can see that the spectral peak of the SRTRD-p-NC-MUSIC is sharper than that of the 2D-NC-MUSIC, and the SRTRD-p-NC-MUSIC achieves a better performance than the 2D-NC-MUSIC when distinguishing two signals from similar directions. The simulation results in Figure 10 also verify the experimental results in Figure 9.Figure 9
Resolution probability comparison with different SNRs,L = 500.Figure 10
Spatial spectrum of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC for 10 simulations, SNR = 5 dB andL = 500.Figure11 depicts the resolution probability comparison versus different intersource spacing ∆θ among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC, where SNR = 0 dB and the snapshots L = 500. It is considered that two sources are located at 20° and (20+Δθ)°, where Δθ represents the intersource spacing. We can see that when the intersource spacing Δθ is not less than 1.8°, these four algorithms have a 100% resolution probability to identify the two sources. Especially, the proposed SRTRD-p-NC-MUSIC can distinguish the two sources accurately when the intersource spacing Δθ is greater than 1°. Compared with the other three algorithms, the SRTRD-p-NC-MUSIC can achieve much better spatial resolution when the intersource spacing is less than 1° due to the better orthogonality between the signal subspace and the noise subspace and the sharper peak of the spatial spectrum [29, 35], which is also verified by the simulation results in Figure 10.Figure 11
Resolution probability comparison versus different intersource spacing Δθ, SNR = 0 dB and L = 500.Figure12 indicates the running time comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC versus different number of sensors. The simulation parameters are set as SNR = 10 dB, L = 500, K = 2 and θ = {−10°, 25°}. The searching interval is fixed at 0.05°. The running times in Figure 12 are given by running the MATLAB codes on a PC with an Intel (R) Core (TM) i5-5300U processor, a 2.3 GHz CPU, and 2 GB RAM in the same environment. It can be observed that the SRTRD-p-NC-MUSIC costs much lower running time than the other three algorithms. Specifically, as the number of sensors increases, the running times of all algorithms in Figure 12 increase. Simultaneously, the efficiency advantage of the SRTRD-p-NC-MUSIC is more obvious with increasing the number of sensors, which are consistent with the aforementioned computational complexity analysis. In Figure 12(a), compared with the 2D-NC-MUSIC, the SRTRD-p-NC-MUSIC only needs a one-dimensional peak searching, and furthermore, the reconstruction of the noise space efficiently avoids the redundancy computations in the one-dimensional peak searching due to the rank deficiency of the noise subspace. Therefore, the SRTRD-p-NC-MUSIC can reduce the computational complexity to less than 3% as compared to the 2D-NC-MUSIC. In Figure 12(b), the dimension of the reconstructed noise space of the SRTRD-p-NC-MUSIC is descended and the redundancy computations of the one-dimensional peak searching of the SRTRD-p-NC-MUSIC are removed, and so, the computational complexity of the SRTRD-p-NC-MUSIC is reduced to less than 20% as compared to the RD-NC-MUSIC. In Figure 12(c), compared with the SRTRD-NC-MUSIC, the SRTRD-p-NC-MUSIC can more completely remove the redundancy computations in the one-dimensional peak searching according to a novel block matrix method. Consequently, from Figure 12(c), we can see that the computational complexity of the SRTRD-p-NC-MUSIC is lower than that of the SRTRD-NC-MUSIC.Figure 12
Running time comparison with a different number of sensors, SNR = 10 dB andL = 500.
(a)(b)(c)


## 5. Conclusions

In this paper, we have proposed the SRTRD-p-NC-MUSIC algorithm for the DoA estimation of NC signals in order to reduce the computational complexity of the conventional 2D-NC-MUSIC algorithm. The proposed SRTRD-p-NC-MUSIC algorithm not only converts the two-dimensional peak searching into the one-dimensional peak searching, but also utilizes the block matrix method and SRT to efficiently avoid the redundancy computations in the one-dimensional peak searching due to the rank deficiency of the noise subspace. Compared with the 2D-NC-MUSIC algorithm, the proposed SRTRD-p-NC-MUSIC algorithm can dramatically improve the computational efficiency on the premise of ensuring the accuracy of DOA estimation, especially in scenarios where large numbers of sensors are applied.



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*Source: 8622428-2023-03-21.xml*