# DOA and Noncircular Phase Estimation of Noncircular Signal via an Improved Noncircular Rotational Invariance Propagator Method

Authors: Xueqiang Chen; Chenghua Wang; Xiaofei Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We consider the computationally efficient direction-of-arrival (DOA) and noncircular (NC) phase estimation problem of noncircular signal for uniform linear array. The key idea is to apply the noncircular propagator method (NC-PM) which does not require eigenvalue decomposition (EVD) of the covariance matrix or singular value decomposition (SVD) of the received data. Noncircular rotational invariance propagator method (NC-RI-PM) avoids spectral peak searching in PM and can obtain the closed-form solution of DOA, so it has lower computational complexity. Animproved NC-RI-PM algorithm of noncircular signal for uniform linear array is proposed to estimate the elevation angles and noncircular phases with automatic pairing. We reconstruct the extended array output by combining the array output and its conjugated counterpart. Our algorithm fully uses the extended array elements in the improved propagator matrix to estimate the elevation angles and noncircular phases by utilizing the rotational invariance property between subarrays. Compared with NC-RI-PM, the proposed algorithm has better angle estimation performance and much lower computational load. The computational complexity of the proposed algorithm is analyzed. We also derive the variance of estimation error and Cramer-Rao bound (CRB) of noncircular signal for uniform linear array. Finally, simulation results are presented to demonstrate the effectiveness of our algorithm.


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## Body

## 1. Introduction

Over the last several decades, the problem of estimating the direction-of-arrival (DOA) of multiple sources in the field of array signal processing has received considerable attention [1–3]. A variety of DOA estimation algorithms have been developed and applied in many fields, such as mobile communication system, radio astronomy, sonar, and radar [4–13]. Although the maximum likelihood estimator provides the optimum parameter estimation performance, its computational complexity is extremely demanding [8–10]. Simpler but suboptimal solutions can be achieved by the subspace-based approaches, which rely on the decomposition of observation space into signal subspace and noise subspace. For example, multiple signals classification (MUSIC) method [11] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm [12, 13] are subspace-based DOA estimation algorithms both well known for their good angle estimation performance. However, conventional subspace techniques necessitate eigenvalue decomposition (EVD) of covariance matrix or singular value decomposition (SVD) of data matrix to estimate the signal and noise subspaces; thus, huge computation complexity will be involved, particularly when the number of sensors is large, such as the large towed arrays in sonar [14, 15]. It is well known that propagator method (PM) does not require EVD of covariance matrix or SVD of received data; thus, the computational load of PM algorithm can be significantly smaller [16]. But the spectral peak searching process is used in conventional PM algorithms [17–20]; in order to save the complexity, rotational invariance PM (RI-PM) algorithms are proposed [21–23], which avoid the spectral peak searching process and can obtain the closed-form solution of DOA.The binary phase shift keying (BPSK), amplitude modulation (AM), and unbalanced quadrature phase shift keying (UQPSK) modulated signals frequently used in communication systems are noncircular (NC) signals [24]. The statistical parameters of noncircular signal, such as first and second moments, are rotational variant. The noncircularity of signal is investigated to enhance the performance of angle estimation algorithm [25]. Many DOA estimation methods of noncircular signals have been reported, which contain NC-MUSIC algorithms [26–28], NC-ESPRIT algorithms [29–31], and noncircular parallel factor (NC-PARAFAC) algorithm [32]. These algorithms we mention above can estimate more sources and have better angle estimation performance by introducing the noncircularity of signal into the conventional DOA estimation algorithms.Many DOA estimation algorithms of noncircular signal based on PM have also been studied [33–38]. These noncircular propagator method (NC-PM) algorithms can be divided into two kinds, that is, the noncircular spectral peak searching PM algorithms and noncircular rotational invariance PM (NC-RI-PM) algorithms. In fact, Liu et al. have proposed NC-PM algorithm [34] and real-valued NC-PM algorithm [35]; unfortunately, the spectral peak searching process is involved, which generally requires highly computational complexity. Subsequently, Liu et al. proposed NC-root-PM algorithm [36, 37] by exploiting the uniform distributed characteristic of the array elements, but its complexity is still high. The NC-RI-PM algorithm has been proposed by Sun and Zhou in [38], which has much lower computational complexity than Liu’s algorithm and can obtain the closed-form solution of DOA. However, we find that the reconstructed array output can be optimized, and some array elements in the extended propagator matrix have not been used; thus, the angle estimation performance of NC-RI-PM algorithm can be enhanced further. It is worth noting that the rotational invariance PM (RI-PM) algorithms have also been studied in [39–42], but the noncircularity of signal has not been considered.The aim of this paper is to develop a computationally efficient parameter estimation algorithm of noncircular signal. We propose an improved noncircular rotational invariance PM (improved NC-RI-PM) algorithm of noncircular signal for uniform linear array, which combine the noncircularity of signal and rotational invariance property between subarrays. Compared with NC-RI-PM algorithm, the proposed algorithm has the following advantages: (1) It has better angle estimation performance than that of NC-RI-PM algorithm. (2) The proposed algorithm has much lower computational complexity. (3) It can estimate elevation angles and noncircular phases with automatic pairing.The reminder of this paper is organized as follows. In Section2, we describe the data model for uniform linear array. The improved NC-RI-PM algorithm is presented in Section 3, as well as the computational complexity analysis and comparison. Section 4 derives the variance of estimation error and Cramer-Rao bound (CRB) of the proposed algorithm. Simulation results are presented in Section 5, while the conclusions are shown in Section 6.Notations. ·-1,·∗,·T,·H, and ·+ denote inverse, conjugate, transpose, conjugate-transpose, and pseudoinverse operations, respectively. diagv stands for a diagonal matrix, whose diagonal element is the vector v. IK is K×K identity matrix. E· is the expectation operator. angle· means to get the phase. ⊙ denotes the Hadamard product. x^ means the estimated value of x. ·F denotes the Frobenius norm.


## 2. Data Model

We assume that there areK uncorrelated narrowband signals impinging on a uniform linear array equipped with M antennas as shown in Figure 1, and M>K. We also assume that sources are far away from the array; thus, the incoming waves over the antennas are essentially planes. The noise is additive independent identically distributed Gaussian with zero mean and variance σ2, independent of signals. We denote the elevation angles of sources as θ=θ1,θ2,…,θK, where θk is the elevation angle of kth source, k=1,…,K.Figure 1
Illustration of the array geometry [38].From [38], the array output at time t can be modeled as (1)xt=Αst+nt,where Α=aθ1,…,aθK∈CM×K is the steering matrix, aθk=1,ejμk,…,ej(M-1)μkT, and μk=2πdcos⁡θk/λ, d is the distance between adjacent antennas, and λ is the wavelength. st=s1t,…,sKtT is the signal vector. nt=n1t,…,nMtT is the noise vector, and EntnHt=σ2IM, m=1,…,M.We just consider the signal of maximum noncircular rate in this paper. According to the noncircularity of signals,st can be denoted as (2)st=Ψs0t,where s0t∈RK×1. Ψ is a diagonal matrix; (3)Ψ=ejψ1/2⋱ejψK/2,where ψk is the noncircular phase of kth signal.


## 3. Angle Estimation Algorithm

In [38], Sun and Zhou have proposed the NC-RI-PM algorithm for uniform linear arrays which has much lower computational complexity than NC-PM algorithm based on spectral peak searching and can obtain the closed-form solution of DOA. However, we find that NC-RI-PM algorithm just uses part of the array elements in the extended propagator matrix, and the reconstructed array output data can be optimized. Thus, we propose an improved NC-RI-PM algorithm of noncircular signal for uniform linear array, which is presented as follows.


### 3.1. Improved NC-RI-PM Algorithm

According to (1), we construct the extended array output as (4)x-t=xtJMx∗t=A-st+n-t,where n-t=ntJMn∗t. JM is the permutation matrix, (5)JM=01⋰10,(6)A-=AAΦΨ-2,where Φ=Ω-M-1 is a diagonal matrix, and (7)Ω=ejμ1⋱ejμK.Consider that channel state information is constant during sampling process. The sample data of extended array output can be expressed as(8)X=A-S+N,where S=St1,…,StL and N=Nt1,…,NtL are the sample signal matrix and sample noise matrix, respectively, l=1,…,L, and L is the number of snapshots.We partition the extended steering matrixA- as (9)A-=A1A2,where A1∈CK×K is a nonsingular matrix, A2∈C2M-K×K. From [43], A2 is a linear transformation of A1, (10)PcHA1=A2,where Pc∈CK×2M-K is the improved propagator matrix.According to (4), the covariance matrix of the extended array output is [38](11)R=EXtXHt,where R∈C2M×2M is the covariance matrix.Partition the covariance matrixR as (12)R=R1,R2,where R1∈C2M×K, R2∈C2M×(2M-K).In the noiseless case, we can have(13)R1Pc=R2.Actually, there is always noise, and the propagator matrix can be estimated by the following minimization problem [43]:(14)JcsmPc=R2-R1PcF2,where ·F denotes the Frobenius norm. The estimate of Pc is via (15)P^c=R1HR1-1R1HR2.Then define(16)P=IKPcH,where P∈C2M×K. In the noiseless case, according to (9)-(10), we can have (17)PA1=A-.In the following, we estimate the elevation angles and noncircular phases via utilizing the rotational invariance property between subarrays. Partition the matrixP into two parts as (18)P=PxPy,where Px∈CM×K, Py∈CM×K.Define(19)Ja=100⋱⋮010M-1×M,(20)Jb=010⋮⋱001M-1×M,where Ja and Jb are the selective matrix.Let(21)Pa=JaPxJaPy,Pb=JbPxJbPy.According to (6)-(7) and (21), (22)PaA1Ω=PbA1.Define(23)Pr=Pa+Pb.According to (22)-(23), we can have (24)Pr=A1ΩA1-1.Perform the EVD ofPr, which can be written as (25)Pr=UΣUH,where U=u1,…,uK, Σ=diagε1,…,εK. The eigenvalues of Pr in Σ are corresponding to the diagonal elements of Ω, and the eigenvectors of Pr in U are estimates of the column vectors in matrix A1.Thus, the elevation angle ofkth source can be estimated by(26)θ^k=arccosλ·angleεk2πd,where θ^k is the estimate of θk.Next, we can also estimate the noncircular phases. According to (6) and (17)-(18), we can have (27)PyA1=PxA1Δ,where Δ=ΦΨ-2 is a diagonal matrix.Define(28)Pe=Px+Py.According to (27)-(28), we can get(29)Pe=A1ΔA1-1.ThenPe can be written via EVD as (30)Pe=VΛVH,where V=υ1,…,υK and Λ=ρ1,…,ρK. The eigenvalues of Pe in Λ corresponding to the diagonal elements of Δ and the eigenvectors of Pe in V are estimates of the column vectors in matrix A1.Note that the EVD ofPr and Pe are performed, respectively; we should consider the column and scale ambiguity between Λ and Σ before estimating the noncircular phases. According to (23) and (28), we can find that Pr and Pe are all computed from the matrix P, so replace Px and Py in (28) with (31)Px′=PxU,Py′=PyU.Thus, the estimated noncircular phases can be expressed as(32)ψ^k=-angleρk-2πdM-1cos⁡θ^kλ,where ψ^k is the estimate of kth noncircular phase.


### 3.2. Algorithm Steps

The implementation of the proposed algorithm with finite array output data is summarized in this section. According to (4), the covariance matrix of sample extended array output data can be expressed as (33)R^=1L∑l=1Lx-tlx-Htl.We show the major steps of the improved NC-RI-PM algorithm of noncircular signals for uniform linear arrays as follows.(1)
Construct the sample array output matrixX and corresponding covariance matrix R^.(2)
PartitionR^ into two parts as (12), and compute the propagator matrix Pc.(3)
ComputePr and Pe, and perform the EVD of Pr and Pe as (25) and (30).(4)
Estimate the elevation angles and noncircular phases via (26) and (32).


### 3.3. Differences between Our Works and NC-RI-PM Algorithm

NC-RI-PM algorithm [38] is simply reviewed for comparison in this section. According to (1), the extended subarray output in [38] is reconstructed as(34)x1t=JaxtJax∗t=B1st+n1t,x2t=JbxtJbx∗t=B2st+n2t,where B1=JaAJaA∗, B2=JbAJbA∗, n1t=JantJan∗t, and n2t=JbntJbn∗t. Ja and Jb are the selective matrices defined in (19) and (20), respectively.Define(35)yt=x1tx2t=Bst+nyt,where yt∈C4M×1, nyt=n1tn2t, B=B1B2, and B∈C4M-4×K.The sample array output ofyt can be expressed as Y=BS+Ny, where Y∈C4M-4×L, Ny∈C4M-4×L. Suppose that F^=(1/L)∑l=1LytlyHtl is the sample covariance matrix, which is partitioned as F^=F1,F2, where F1∈C4M-4×K and F2∈C4M-4×4M-4-K. Then the propagator matrix of NC-RI-PM algorithm can be computed by P-=F1HF1-1F1HF2, where P-∈CK×4M-4-K. Partition the propagator matrix P- as P-=P-1,P-2,P-3, where P-1∈CK×2M-2-K, P-2∈CK×K, and P-3∈CK×2M-2-K.Based on the rotational invariance property betweenP-1 and P-3, we can get (36)P-t=B11ΩB11-1,where yt∈C4M-4×1, P-t=P-1H+P-3H, B1=B11B12, and B11∈CK×K, B12∈C2M-2-K×K.Thus, the elevation angles can be estimated by performing the EVD ofP-t. With the above analysis, we summarize the differences between the improved NC-RI-PM algorithm and NC-RI-PM algorithm in the following.(1)
According to (34)-(35), yt∈C4M-4×1, there are 2M-2 duplicated data in Y. These duplicated data have no benefit to improve the angle estimation performance but increase the computational complexity. The dimension of extended covariance matrix F^ in NC-RI-PM algorithm is 4M-4×4M-4, whereas that of R^ in our algorithm is just 2M×2M. So our algorithm has much lower computational complexity than NC-RI-PM algorithm.(2)
According to (36), NC-RI-PM algorithm just uses part of the extended array elements in the propagator matrix P-, that is, the array elements in P-1 and P-3, but the array elements in P-2 have not been used, whereas the improved NC-RI-PM algorithm fully uses all the elements in the extended propagator matrix. Thus, the angle estimation performance of our algorithm is better than that of NC-RI-PM algorithm.(3)
The proposed algorithm can estimate elevation angles and noncircular phases with automatic pairing, while just the elevation angles are estimated in NC-RI-PM algorithm.Remark 1.
We assume that the number of sources is preknown, and it can be estimated by some methods shown in [44–46].Remark 2.
According to (23) and (28), the elevation angles and noncircular phases are all estimated from the matrix P, and the column and scale ambiguity can be solved by (31). Thus, the elevation angles and noncircular phases can be pairing automatically.


### 3.4. Complexity Analysis

Regarding the major computational complexity, we just consider matrix complex multiplication operations. It is known that the complexity of computing the covariance matrixR^ with L snapshots is in the order of OM2L and that of the EVD of M×M dimension matrix is OM3. In this paper, computational complexity of computing the matrices R^, Pr, and Pe and EVD of Pr, Pe are analyzed for the improved NC-RI-PM algorithm. Table 1 presents the total computational complexity of the improved NC-RI-PM algorithm, NC-ESPRIT algorithm in [29], and NC-RI-PM algorithm in [38]. Figure 2 shows the simulation results of computational complexity comparison versus L and M, respectively. From Figure 2, we can find that our algorithm has much lower computational complexity than that of NC-RI-PM algorithm and has approximate but still lower complexity than NC-ESPRIT algorithm.Table 1
Computational complexity of the proposed algorithm, NC-RI-PM algorithm, and NC-ESPRIT algorithm.
Algorithm
Computational complexity
NC-RI-PM in [38]
O
16
M
-
1
2
K
+
L
+
10
K
2
M
-
1
NC-ESPRIT in [29]
O
6
M
-
1
K
2
+
2
K
3
+
8
M
3
+
4
M
2
L
Improved NC-RI-PM
O
7
M
K
2
+
6
K
2
K
-
1
+
4
M
2
L
+
KFigure 2
Complexity comparison versusL and M.


## 3.1. Improved NC-RI-PM Algorithm

According to (1), we construct the extended array output as (4)x-t=xtJMx∗t=A-st+n-t,where n-t=ntJMn∗t. JM is the permutation matrix, (5)JM=01⋰10,(6)A-=AAΦΨ-2,where Φ=Ω-M-1 is a diagonal matrix, and (7)Ω=ejμ1⋱ejμK.Consider that channel state information is constant during sampling process. The sample data of extended array output can be expressed as(8)X=A-S+N,where S=St1,…,StL and N=Nt1,…,NtL are the sample signal matrix and sample noise matrix, respectively, l=1,…,L, and L is the number of snapshots.We partition the extended steering matrixA- as (9)A-=A1A2,where A1∈CK×K is a nonsingular matrix, A2∈C2M-K×K. From [43], A2 is a linear transformation of A1, (10)PcHA1=A2,where Pc∈CK×2M-K is the improved propagator matrix.According to (4), the covariance matrix of the extended array output is [38](11)R=EXtXHt,where R∈C2M×2M is the covariance matrix.Partition the covariance matrixR as (12)R=R1,R2,where R1∈C2M×K, R2∈C2M×(2M-K).In the noiseless case, we can have(13)R1Pc=R2.Actually, there is always noise, and the propagator matrix can be estimated by the following minimization problem [43]:(14)JcsmPc=R2-R1PcF2,where ·F denotes the Frobenius norm. The estimate of Pc is via (15)P^c=R1HR1-1R1HR2.Then define(16)P=IKPcH,where P∈C2M×K. In the noiseless case, according to (9)-(10), we can have (17)PA1=A-.In the following, we estimate the elevation angles and noncircular phases via utilizing the rotational invariance property between subarrays. Partition the matrixP into two parts as (18)P=PxPy,where Px∈CM×K, Py∈CM×K.Define(19)Ja=100⋱⋮010M-1×M,(20)Jb=010⋮⋱001M-1×M,where Ja and Jb are the selective matrix.Let(21)Pa=JaPxJaPy,Pb=JbPxJbPy.According to (6)-(7) and (21), (22)PaA1Ω=PbA1.Define(23)Pr=Pa+Pb.According to (22)-(23), we can have (24)Pr=A1ΩA1-1.Perform the EVD ofPr, which can be written as (25)Pr=UΣUH,where U=u1,…,uK, Σ=diagε1,…,εK. The eigenvalues of Pr in Σ are corresponding to the diagonal elements of Ω, and the eigenvectors of Pr in U are estimates of the column vectors in matrix A1.Thus, the elevation angle ofkth source can be estimated by(26)θ^k=arccosλ·angleεk2πd,where θ^k is the estimate of θk.Next, we can also estimate the noncircular phases. According to (6) and (17)-(18), we can have (27)PyA1=PxA1Δ,where Δ=ΦΨ-2 is a diagonal matrix.Define(28)Pe=Px+Py.According to (27)-(28), we can get(29)Pe=A1ΔA1-1.ThenPe can be written via EVD as (30)Pe=VΛVH,where V=υ1,…,υK and Λ=ρ1,…,ρK. The eigenvalues of Pe in Λ corresponding to the diagonal elements of Δ and the eigenvectors of Pe in V are estimates of the column vectors in matrix A1.Note that the EVD ofPr and Pe are performed, respectively; we should consider the column and scale ambiguity between Λ and Σ before estimating the noncircular phases. According to (23) and (28), we can find that Pr and Pe are all computed from the matrix P, so replace Px and Py in (28) with (31)Px′=PxU,Py′=PyU.Thus, the estimated noncircular phases can be expressed as(32)ψ^k=-angleρk-2πdM-1cos⁡θ^kλ,where ψ^k is the estimate of kth noncircular phase.


## 3.2. Algorithm Steps

The implementation of the proposed algorithm with finite array output data is summarized in this section. According to (4), the covariance matrix of sample extended array output data can be expressed as (33)R^=1L∑l=1Lx-tlx-Htl.We show the major steps of the improved NC-RI-PM algorithm of noncircular signals for uniform linear arrays as follows.(1)
Construct the sample array output matrixX and corresponding covariance matrix R^.(2)
PartitionR^ into two parts as (12), and compute the propagator matrix Pc.(3)
ComputePr and Pe, and perform the EVD of Pr and Pe as (25) and (30).(4)
Estimate the elevation angles and noncircular phases via (26) and (32).


## 3.3. Differences between Our Works and NC-RI-PM Algorithm

NC-RI-PM algorithm [38] is simply reviewed for comparison in this section. According to (1), the extended subarray output in [38] is reconstructed as(34)x1t=JaxtJax∗t=B1st+n1t,x2t=JbxtJbx∗t=B2st+n2t,where B1=JaAJaA∗, B2=JbAJbA∗, n1t=JantJan∗t, and n2t=JbntJbn∗t. Ja and Jb are the selective matrices defined in (19) and (20), respectively.Define(35)yt=x1tx2t=Bst+nyt,where yt∈C4M×1, nyt=n1tn2t, B=B1B2, and B∈C4M-4×K.The sample array output ofyt can be expressed as Y=BS+Ny, where Y∈C4M-4×L, Ny∈C4M-4×L. Suppose that F^=(1/L)∑l=1LytlyHtl is the sample covariance matrix, which is partitioned as F^=F1,F2, where F1∈C4M-4×K and F2∈C4M-4×4M-4-K. Then the propagator matrix of NC-RI-PM algorithm can be computed by P-=F1HF1-1F1HF2, where P-∈CK×4M-4-K. Partition the propagator matrix P- as P-=P-1,P-2,P-3, where P-1∈CK×2M-2-K, P-2∈CK×K, and P-3∈CK×2M-2-K.Based on the rotational invariance property betweenP-1 and P-3, we can get (36)P-t=B11ΩB11-1,where yt∈C4M-4×1, P-t=P-1H+P-3H, B1=B11B12, and B11∈CK×K, B12∈C2M-2-K×K.Thus, the elevation angles can be estimated by performing the EVD ofP-t. With the above analysis, we summarize the differences between the improved NC-RI-PM algorithm and NC-RI-PM algorithm in the following.(1)
According to (34)-(35), yt∈C4M-4×1, there are 2M-2 duplicated data in Y. These duplicated data have no benefit to improve the angle estimation performance but increase the computational complexity. The dimension of extended covariance matrix F^ in NC-RI-PM algorithm is 4M-4×4M-4, whereas that of R^ in our algorithm is just 2M×2M. So our algorithm has much lower computational complexity than NC-RI-PM algorithm.(2)
According to (36), NC-RI-PM algorithm just uses part of the extended array elements in the propagator matrix P-, that is, the array elements in P-1 and P-3, but the array elements in P-2 have not been used, whereas the improved NC-RI-PM algorithm fully uses all the elements in the extended propagator matrix. Thus, the angle estimation performance of our algorithm is better than that of NC-RI-PM algorithm.(3)
The proposed algorithm can estimate elevation angles and noncircular phases with automatic pairing, while just the elevation angles are estimated in NC-RI-PM algorithm.Remark 1.
We assume that the number of sources is preknown, and it can be estimated by some methods shown in [44–46].Remark 2.
According to (23) and (28), the elevation angles and noncircular phases are all estimated from the matrix P, and the column and scale ambiguity can be solved by (31). Thus, the elevation angles and noncircular phases can be pairing automatically.


## 3.4. Complexity Analysis

Regarding the major computational complexity, we just consider matrix complex multiplication operations. It is known that the complexity of computing the covariance matrixR^ with L snapshots is in the order of OM2L and that of the EVD of M×M dimension matrix is OM3. In this paper, computational complexity of computing the matrices R^, Pr, and Pe and EVD of Pr, Pe are analyzed for the improved NC-RI-PM algorithm. Table 1 presents the total computational complexity of the improved NC-RI-PM algorithm, NC-ESPRIT algorithm in [29], and NC-RI-PM algorithm in [38]. Figure 2 shows the simulation results of computational complexity comparison versus L and M, respectively. From Figure 2, we can find that our algorithm has much lower computational complexity than that of NC-RI-PM algorithm and has approximate but still lower complexity than NC-ESPRIT algorithm.Table 1
Computational complexity of the proposed algorithm, NC-RI-PM algorithm, and NC-ESPRIT algorithm.
Algorithm
Computational complexity
NC-RI-PM in [38]
O
16
M
-
1
2
K
+
L
+
10
K
2
M
-
1
NC-ESPRIT in [29]
O
6
M
-
1
K
2
+
2
K
3
+
8
M
3
+
4
M
2
L
Improved NC-RI-PM
O
7
M
K
2
+
6
K
2
K
-
1
+
4
M
2
L
+
KFigure 2
Complexity comparison versusL and M.


## 4. Error Analysis and CRB

### 4.1. Error Analysis

This section analyses the variance of estimation error for the improved NC-RI-PM algorithm. We assume that(37)R^=R+∂R,where ∂R is the estimation error of covariance matrix.According to (12), we can have(38)R^1=R1+∂R1,R^2=R2+∂R2,where ∂R1 and ∂R2 are the estimation error matrix of R1 and R2, respectively.Combine (15) and (38); the estimate of the propagator is (39)P^c=R1+∂R1HR1+∂R1-1R1+∂R1H·R2+∂R2.Define(40)P^c=Pc+∂Pc,where ∂Pc=R1HR1-1R1H∂R2-∂R1Pc is the estimation error of the propagator matrix.Thus, the estimation error of matrixP is (41)∂P=0K∂PcH.According to (16)–(18) and (41), (42)Px+∂PxPy+∂PyA1=A^,where ∂Px and ∂Py are the estimation error of matrices Px and Py, respectively. A^ is the estimate of the steering matrix A-.According to (21) and (42),(43)P^a=JaPx+∂PxJaPy+∂Py,P^b=JbPx+∂PxJbPy+∂Py.DefineP^a=Pa+∂Pa, P^b=Pb+∂Pb. According to [43], we can obtain the estimate of matrix Pr as (44)P^r=A1-1IK+Pa+∂Pb-∂PaΩA1.Letε^k denote the kth eigenvalues of P^r, which can be written as (45)ε^k=εk+∂εk,where ∂εk is the estimation error of εk, ∂εk=εkekTPa+∂Pb-∂Paek, and ek is a unit column vector, ek=0,…,1,…,0.Based on the first-order Taylor series expansion, the variance of estimation error ofkth elevation angle can be expressed as (46)E∂θk2=12πcos⁡θkE∂εk2-ReE∂εk2εk∗2.


### 4.2. Cramer-Rao Bound

We derive the CRB of noncircular signal for uniform linear array of the proposed algorithm in this section. The parameters needed to be estimated can be denoted as [32](47)ξ=θ1,…,θK,SRTt1,…,SRTtL,SITt1,…,SITtL,σ2T,where SRtl and SItl denote the real and imaginary parts of Stl, respectively.According to (8), the extended array output with L snapshots can be rewritten as (48)Z=XTt1,…,XTtL,XHt1,…,XHtLT.The meanμ and covariance Γ  of Z are (49)μ=A-St1⋮A-StLA-∗S∗t1⋮A-∗S∗tL,(50)Γ=σ2I0⋱0σ2I.From [47], the i,j element of the CRB matrix P can be expressed as (51)P-1ij=tr⁡Γ-1Γi′Γ-1Γj′+2Reμi′∗Γ-1μj′,where μi′ and Γi′ denote the first-order derivative of μ and Γ with respect to the ith element of ζ, respectively.For the covariance matrixΓ is just related to σ2 and the first term of (51) can be ignored. The i,j element of CRB matrix P can be simplified as (52)P-1ij=2Reμi′∗Γ-1μj′.According to (48) and (52), (53)∂μ∂θk=∂A-∂θkSt1⋮∂A-∂θkStL∂A-∗∂θkS∗t1⋮∂A-∗∂θkS∗tL=dkθSkt1⋮dkθSktLdkθ∗Sk∗t1⋮dkθ∗Sk∗tL,k=1,…,Kl=1,…,L,where Sktl is the kth element of Stl, dkθ=∂a-k/∂θk, and a-k is the kth column vector of extended direction matrix A-.Define(54)Δ=d1θS1t1⋯dKθSKt1⋮⋮⋮d1θS1tL⋯dKθSKtL,G=A-0⋱0A-,s=St1⋮StL.Then we can have(55)μ=GsG∗s∗,∂μ∂sRT=GG∗,∂μ∂sIT=jG-jG∗.According to (49) and (54)-(55), the first-order derivative of μ with respect to ζ is(56)∂μ∂ζT=ΔGjG0Δ∗G∗-jG∗0.Combine (50) with (56); (52) can be rewritten as (57)2Re∂μ∗∂ζΓ-1∂μ∂ζT=J000,where(58)J=2σ2ReΔHΔTGHGT-jGHjGTΔGjGΔ∗G∗-jG∗.Let(59)B≜GHG-1GHΔ,F≜I00-BRI0-BI0I,where BR and BI are the real and imaginary parts of B, respectively.According to (56) and (59), we can demonstrate that(60)ΔGjGF=Δ-GBGjG=ΠG⊥ΔGjG,Δ∗G∗-jG∗F=Δ∗-G∗B∗G∗-jG∗=ΠG∗⊥Δ∗G∗-jG∗,where ΠG⊥=I-GGHG-1GH and GHΠG⊥=0. Consider (61)FTJF=2σ2·ReFHΔHΔTGHGT-jGHjGTΔGjGΔ∗G∗-jG∗F=2σ2·ReΔHΠG⊥ΔTΠG⊥GHGT-jGHjGTΠG⊥ΔGjGΠG∗⊥Δ∗G∗-jG∗=4σ2ReΔHΠG⊥Δ000GHGjGHG0-jGHGGHG.We just consider the elements related to the angles. According to (61), we can have J-1 as(62)J-1=FFTJF-1FT=σ24I00-BRI0-BI0I·ReΔHΠG⊥Δ000κκ0κκ·I00-BRI0-BI0I=σ24ReΔHΠG⊥Δ-1κκκκκκκκ,where κ denotes the parts unrelated to the elevation angles.Until now, we obtain the CRB matrix as follows:(63)CRB=σ24ReΔHΠG⊥Δ-1.After further simplification, the CRB matrix can be rewritten as(64)CRB=σ24LReDHΠA-⊥D⊙PST-1,where D=∂a-1/∂θ1,…,∂a-K/∂θK, ΠA-⊥=I2M-A-A-HA--1A-H, PST=(1/L)∑l=1LStlStlH, and σ2 is the power of noise. ⊙ denotes the Hadamard product.


## 4.1. Error Analysis

This section analyses the variance of estimation error for the improved NC-RI-PM algorithm. We assume that(37)R^=R+∂R,where ∂R is the estimation error of covariance matrix.According to (12), we can have(38)R^1=R1+∂R1,R^2=R2+∂R2,where ∂R1 and ∂R2 are the estimation error matrix of R1 and R2, respectively.Combine (15) and (38); the estimate of the propagator is (39)P^c=R1+∂R1HR1+∂R1-1R1+∂R1H·R2+∂R2.Define(40)P^c=Pc+∂Pc,where ∂Pc=R1HR1-1R1H∂R2-∂R1Pc is the estimation error of the propagator matrix.Thus, the estimation error of matrixP is (41)∂P=0K∂PcH.According to (16)–(18) and (41), (42)Px+∂PxPy+∂PyA1=A^,where ∂Px and ∂Py are the estimation error of matrices Px and Py, respectively. A^ is the estimate of the steering matrix A-.According to (21) and (42),(43)P^a=JaPx+∂PxJaPy+∂Py,P^b=JbPx+∂PxJbPy+∂Py.DefineP^a=Pa+∂Pa, P^b=Pb+∂Pb. According to [43], we can obtain the estimate of matrix Pr as (44)P^r=A1-1IK+Pa+∂Pb-∂PaΩA1.Letε^k denote the kth eigenvalues of P^r, which can be written as (45)ε^k=εk+∂εk,where ∂εk is the estimation error of εk, ∂εk=εkekTPa+∂Pb-∂Paek, and ek is a unit column vector, ek=0,…,1,…,0.Based on the first-order Taylor series expansion, the variance of estimation error ofkth elevation angle can be expressed as (46)E∂θk2=12πcos⁡θkE∂εk2-ReE∂εk2εk∗2.


## 4.2. Cramer-Rao Bound

We derive the CRB of noncircular signal for uniform linear array of the proposed algorithm in this section. The parameters needed to be estimated can be denoted as [32](47)ξ=θ1,…,θK,SRTt1,…,SRTtL,SITt1,…,SITtL,σ2T,where SRtl and SItl denote the real and imaginary parts of Stl, respectively.According to (8), the extended array output with L snapshots can be rewritten as (48)Z=XTt1,…,XTtL,XHt1,…,XHtLT.The meanμ and covariance Γ  of Z are (49)μ=A-St1⋮A-StLA-∗S∗t1⋮A-∗S∗tL,(50)Γ=σ2I0⋱0σ2I.From [47], the i,j element of the CRB matrix P can be expressed as (51)P-1ij=tr⁡Γ-1Γi′Γ-1Γj′+2Reμi′∗Γ-1μj′,where μi′ and Γi′ denote the first-order derivative of μ and Γ with respect to the ith element of ζ, respectively.For the covariance matrixΓ is just related to σ2 and the first term of (51) can be ignored. The i,j element of CRB matrix P can be simplified as (52)P-1ij=2Reμi′∗Γ-1μj′.According to (48) and (52), (53)∂μ∂θk=∂A-∂θkSt1⋮∂A-∂θkStL∂A-∗∂θkS∗t1⋮∂A-∗∂θkS∗tL=dkθSkt1⋮dkθSktLdkθ∗Sk∗t1⋮dkθ∗Sk∗tL,k=1,…,Kl=1,…,L,where Sktl is the kth element of Stl, dkθ=∂a-k/∂θk, and a-k is the kth column vector of extended direction matrix A-.Define(54)Δ=d1θS1t1⋯dKθSKt1⋮⋮⋮d1θS1tL⋯dKθSKtL,G=A-0⋱0A-,s=St1⋮StL.Then we can have(55)μ=GsG∗s∗,∂μ∂sRT=GG∗,∂μ∂sIT=jG-jG∗.According to (49) and (54)-(55), the first-order derivative of μ with respect to ζ is(56)∂μ∂ζT=ΔGjG0Δ∗G∗-jG∗0.Combine (50) with (56); (52) can be rewritten as (57)2Re∂μ∗∂ζΓ-1∂μ∂ζT=J000,where(58)J=2σ2ReΔHΔTGHGT-jGHjGTΔGjGΔ∗G∗-jG∗.Let(59)B≜GHG-1GHΔ,F≜I00-BRI0-BI0I,where BR and BI are the real and imaginary parts of B, respectively.According to (56) and (59), we can demonstrate that(60)ΔGjGF=Δ-GBGjG=ΠG⊥ΔGjG,Δ∗G∗-jG∗F=Δ∗-G∗B∗G∗-jG∗=ΠG∗⊥Δ∗G∗-jG∗,where ΠG⊥=I-GGHG-1GH and GHΠG⊥=0. Consider (61)FTJF=2σ2·ReFHΔHΔTGHGT-jGHjGTΔGjGΔ∗G∗-jG∗F=2σ2·ReΔHΠG⊥ΔTΠG⊥GHGT-jGHjGTΠG⊥ΔGjGΠG∗⊥Δ∗G∗-jG∗=4σ2ReΔHΠG⊥Δ000GHGjGHG0-jGHGGHG.We just consider the elements related to the angles. According to (61), we can have J-1 as(62)J-1=FFTJF-1FT=σ24I00-BRI0-BI0I·ReΔHΠG⊥Δ000κκ0κκ·I00-BRI0-BI0I=σ24ReΔHΠG⊥Δ-1κκκκκκκκ,where κ denotes the parts unrelated to the elevation angles.Until now, we obtain the CRB matrix as follows:(63)CRB=σ24ReΔHΠG⊥Δ-1.After further simplification, the CRB matrix can be rewritten as(64)CRB=σ24LReDHΠA-⊥D⊙PST-1,where D=∂a-1/∂θ1,…,∂a-K/∂θK, ΠA-⊥=I2M-A-A-HA--1A-H, PST=(1/L)∑l=1LStlStlH, and σ2 is the power of noise. ⊙ denotes the Hadamard product.


## 5. Simulation Results

The Monte Carlo simulations are adopted to evaluate the angle estimation performance of the proposed algorithm. We define the root-mean square error (RMSE) as(65)RMSE=1K∑k=1K1N∑n=1Nθ^k,n-θk2,where θ^k,n is the estimate of θk of the nth Monte Carlo trial and N=300 is the number of simulation loops. In the following simulations except Figures 6 and 8, we assume that there are K=3 sources, which are located at angles of θ1,θ2,θ3=10∘,30∘,50∘, respectively. The corresponding noncircular phases are ψ1,ψ2,ψ3=20∘,40∘,60∘, respectively. We also assume that all sources have the same symbol duration and the same input signal-to-noise ratio (SNR), and the distance between adjacent antennas is equivalent to half of the wavelength, d=λ/2.Figure3 presents angle estimation results of elevation angles and noncircular phases of the proposed algorithm. M=8 and L=300 are used in the simulations, while SNR = 5 dB is used for elevation angles and SNR = 20 dB is used for noncircular phases. From Figure 3, we can see that the elevation angles and noncircular phases can be clearly observed.Figure 3
Angle estimation result over Monte Carlo simulations.Figure4 shows the angle estimation performance comparison among the proposed algorithm, NC-RI-PM algorithm [38], NC-ESPRIT algorithm [29], and CRB of noncircular signals for uniform linear array. M=6 and L=200 are used. It is indicated in Figure 4 that the angle performance of the improved NC-RI-PM algorithm is better than NC-RI-PM algorithm, since the proposed algorithm fully uses all the elements in the extended propagator matrix. The angle estimation performance of our algorithm is close to that of NC-ESPRIT algorithm. Because the noncircular phases have not been estimated in NC-RI-PM algorithm and NC-ESPRIT algorithm, we just consider elevation angles in Figure 4 for angle performances comparison.Figure 4
Comparison of angle estimation performances.Figure5 shows angle estimation performance of the proposed algorithm with L=300 and different values of M. It is indicated in Figure 5 that the increasing of M will lead to the improvement of angle estimation performance of the proposed algorithm.Figure 5
Angle estimation performances with different values ofM.Figure 6
Angle estimation performances with different values ofL.Figure6 depicts angle estimation performance of the proposed algorithm with M=8 and different values of L. From Figure 6, the angle estimation performance of the proposed algorithm is enhanced with the number of snapshots increasing.Figure7 presents angle estimation performance of the proposed algorithm with M=8, L=300, and different values of K. From Figure 7, we find that the angle performance of our algorithm degrades with the number of sources increasing.Figure 7
Angle estimation performances with different values ofK.Figure 8
Angle estimation of closely spaced sources with SNR = 15 dB.Figure8 displays the simulation result of the proposed algorithm with two closely spaced sources. We assume that the two close sources are located at angles of θ1,θ2=30∘,32∘, and the corresponding noncircular phases are ψ1,ψ2=20∘,40∘, respectively. M=8, L=300, and SNR = 15 dB are used in the simulation. It implies in Figure 8 that our algorithm works well when two sources are closely spaced.


## 6. Conclusions

In this paper, we have proposed an improved NC-RI-PM algorithm of noncircular signals for uniform linear array. The proposed algorithm has the following advantages: (1) It has much lower computational complexity, particularly when the ratio of the number of antennas to the number of sources is large. (2) It has better angle estimation performance than NC-RI-PM algorithm by constructing an improved propagator matrix, which fully uses all the array elements in the extended propagator matrix. (3) It can estimate elevation angles and noncircular phases with automatic pairing. (4) The maximum number of sources estimated by our algorithm is two times of that of PM algorithm via utilizing the noncircularity of signal.We also analyze the computational complexity of the proposed algorithm. The error analysis of the proposed algorithm and CRB of noncircular signals for uniform linear array are also derived. It is well known that CRB expresses a lower bound on the variance of an unbiased estimator, which can be used to compare the angle performance of different algorithms. From Figure4, we can see that the RMSE of the improved NC-RI-PM algorithm is much closer to CRB compared with that of NC-RI-PM algorithm. It is clearly indicated that the angle performance of the improved NC-RI-PM algorithm is better than that of NC-RI-PM algorithm, since the proposed algorithm fully uses all the elements in the extended propagator matrix. Finally, the angle estimation performance and computational complexity of the proposed algorithm are evaluated by computer simulations. Simulation results illustrate the effectiveness of the proposed algorithm in a variety of scenarios, particularly when the sources are closely spaced. Therefore, the proposed algorithm can be regarded not only as an improvement of the work in [38] but also as an improved DOA and noncircular phase estimation algorithm.



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*Source: 235173-2015-04-29.xml*