Extended Transformed Nested Arrays for DOA Estimation of Non-Circular Sources

Recently, direction of arrival (DOA) estimation for non-circular (NC) sources has attracted great attention in array signal processing since NC signals can provide more information to generate the virtual difference and sum (diff-sum) coarray. In this paper, we propose a generalized vectorized noncircular MUSIC (GVNCM) method, which can solve the underdetermined DOA estimation problem when the difference coarray and the sum coarray are discontinuous. Based on the GVNCM algorithm, an extended transformed nested array strategy is proposed by splitting the dense subarray of transformed nested array (TNA) into several parts and shifting them to the right to reduce the redundant elements between the difference coarray and the sum coarray. Following this strategy, two novel array configurations are developed. The analytical expressions of the new structures and the consecutive range of their diff-sum coarrays are provided. The two proposed arrays can achieve higher degrees of freedom and maximum number of detectable signals compared with TNA. In order to evaluate the underdetermined DOA estimation performance of NC signals, we derive the corresponding expression of Cramér-Rao bound. The effectiveness of the proposed arrays is verified through numerical simulations.


I. INTRODUCTION
Direction of arrival (DOA) estimation is an important topic in array signal processing field and plays a critical role in many applications such as radar, sonar, wireless communication and navigation [1]- [6]. It is well known that many algorithms, such as MUSIC [7] and ESPRIT [8], have been proposed to estimate DOAs. These methods can resolve up to N − 1 sources with an N -element uniform linear array (ULA).
The problem of detecting more sources than the number of sensors has attracted tremendous attentions. Toward this purpose, the concept of coarray was introduced to DOA estimation. The difference coarray of well-designed sparse array can enhance the degrees of freedom (DOFs) greatly. For example, minimum redundancy arrays (MRAs) [9] were constructed by maximizing the consecutive range of the resulting difference coarray for a given number of physical sensors. However, there is no closed form expressions for the MRAs, which brings many difficulties in practical applications. Two kinds of sparse arrays, i.e. nested arrays (NAs) [10] and coprime arrays (CAs) [11], have been proposed to solve this problem. Particularly, the nested array, consisting of one The associate editor coordinating the review of this manuscript and approving it for publication was Hasan S. Mir. dense and one sparse uniform linear subarray, can resolve O(N 2 ) sources with N physical sensors. Moreover, super nested arrays [12], [13] and augmented nested arrays [14], [15] were proposed to further enhance performance of DOA estimation by rearranging the dense array of nested array.
However, the forementioned research was only developed for circular signal. In fact, numerous non-circular (NC) signals, such as binary phase shift keying (BPSK) and pulse amplitude modulation (PAM) signals, have been widely applied in various communication systems. Different from circular signals, the elliptic covariance of NC signals is not equal to zero. Thus NC signals can provide more information to enhance DOFs and improve DOA estimation performance. Many DOA estimation approaches have been proposed for NC signals such as NC-MUSIC [16], NC-Root-MUSIC [17], NC-ESPRIT [18] and NC-Unitary-ESPRIT [19] method. For these methods, equivalent observed signal corresponding to physical array and its flipped array can be obtained by combining received signals and complex conjugate counterpart. These algorithms can detect at most 2N − 2 sources with N sensors. Recently, vectorized noncircular MUSIC (VNCM) algorithm [20] and reduced-dimension MUSIC algorithm with the sum-difference coarray (SD-RD-MUSIC) [21] have been developed to generate difference and sum (diff-sum) coarray by combining NC characteristic and the concept of coarray. The diff-sum coarray consists of conventional difference coarray, positive sum coarray and negative sum coarray. The aperture of diff-sum coarray can exceed two times of array aperture which could help to decrease the size of physical array.
Based on VNCM and SD-RD-MUSIC algorithm, some novel sparse array configurations have been presented to further enlarge the consecutive range of diff-sum coarray. In [22], the authors translated the two subarrays of NA separately and proposed a new sparse array called improved nested arrays with sum-difference coarray (INAwSDCA). Nevertheless, INAwSDCA need an additional sensor located at zero point as reference sensor. Transformed nested array (TNA) has been developed in [23] by exchanging two subarrays of NA. The forementioned two array geometries increase the number of DOFs by reducing the redundancy between the difference and the sum coarray. Unfortunately, the VNCM and the SD-RD-MUSIC algorithm ask for both the difference and the sum coarray continuous, which brings limitation to sparse array design.
In this paper, we improve VNCM method and propose a generalized vectorized noncircular MUSIC (GVNCM) algorithm which can deal with the case that diff-sum coarray has a long consecutive range but the difference and the sum coarray are discontinuous. Then, we introduce an extended transformed nested array (ETNA) strategy, which displaces some elements in the dense subarray of TNA to the right to increase the aperture of diff-sum coarray. A series of conditions are derived to guarantee the continuity of diff-sum coarray. According to this strategy, two novel array configurations, referred to as two-level extended transformed nested array (TwETNA) and three-level extended transformed nested array (ThETNA), are proposed. Compared with TNA, the proposed array structures can increase the available number of DOFs and the maximum number of detectable signals (MNDS). As an important performance criterion of parameter estimation, the Cramér-Rao bound (CRB) is derived for the underdetermined DOA estimation of NC signals.
The rest of the paper is organized as follows. Section II reviews the NC signal model and proposes the GVNCM algorithm. The extended transformed nested array strategy and two proposed array structures, i.e., TwETNA and ThETNA, are elaborated in Section III with analytical expressions of the continuous diff-sum coarray range. The new CRB expression is derived in Section IV. Simulation results are provided in Section V to demonstrate the improved performance of the proposed array geometries with the GVNCM algorithm used. Section VI concludes this paper.
Notations: Throughout the paper, we use upper-case letters in boldface to denote matrices (e.g., A), lower-case letters in boldface for vectors (e.g., a) and upper-case double line characters for sets (e.g., A). (·) * , (·) T , (·) H represent complex conjugation, transpose and conjugate transpose, respectively. E[·] is the statistical expectation operator and vec(·) is the vectorizing operator. The symbol ⊗ and respectively denote left Kronecker product and Khatri-Rao product. a, b denotes the set of integer {a, a + 1, . . . , b}.
[A] mn indicates (m, n)th element of A. I n denotes n × n identity matrix. 0 n and 0 m×n are zero matrices with dimension n × n and m × n, respectively.

II. GENERALIZED VNCM ALGORITHM
A. DATA MODEL Assume Q far-field narrowband uncorrelated NC sources from direction θ q , q = 1, 2, . . . , Q impinging on an N-element linear antenna array whose sensors are located at P = {p 1 , p 2 , . . . , p N }d. Here, the unit inter-element spacing d is set as λ/2, where λ denotes the wavelength of incoming signals. The first sensor is reference sensor, i.e. p 1 = 0. In this paper, we also assume that the sources are strictly non-circular [24], [25] and wide-sense quasi-stationary [26], [27]. It has F frames with the frame length being L. Then, for the ith frame, the received data from N sensors can be expressed as where is the steering vector corresponding to the qth signal; A = [a(θ 1 ), a(θ 2 ), . . . , a(θ Q )] is manifold matrix; s i (t) = [s 1i (t), s 2i (t), . . . , s Qi (t)] T represents received data vector in the ith frame; n i (t) = [n 1i (t), n 2i (t), . . . , n Ni (t)] T denotes white circular Gaussian noise vector with zero mean and variance σ 2 n . According to the NC characteristic, NC signals can be expressed as whereŝ q (t) and φ q are the real-valued signal and NC phase, respectively. Substituting (3) into (1), we can obtain

B. GENERALIZED VNCM ALGORITHM
Conventional VNCM algorithm, based on the NC property and the coarray concept, can achieve more DOFs than NC MUSIC method. However, both the difference and the sum coarray need to be continuous in the VNCM method. Therefore, we propose a new method called generalized VNCM, which can deal with the case that the difference and the sum coarray are discontinuous.
Firstly, we combine the observed data vector and its complex conjugate counterpart to obtain the following extended data vector of the ith frame, T are extended manifold matrix and extended noise vector, respectively. From (5), the covariance matrix R yi of extended data vector y i (t) is obtained as where R si = diag{σ 2 1i , σ 2 2i , . . . , σ 2 Qi } is the covariance matrix ofŝ i (t), with σ 2 qi representing the input signal power of the qth source in the ith frame.
Vectorizing R yi , one can obtain the following vector: where In order to facilitate the analysis, we introduce a new matrix B 0 = [b 0 (θ 1 ), b 0 (θ 2 ), . . . , b 0 (θ Q )] which satisfies the following relationship [21]: and J ∈ C 4N 2 ×4N 2 is a permutation matrix, defined as Here, (10), we can see that b 0 (θ q ) consists of four parts: c 1 , c 2 , c 3 and c 4 . c 1 is the same as c 4 corresponding to difference coarray L D = {p 1 − p 2 |p 1 ∈ P, p 2 ∈ P}, c 2 and c 3 respectively correspond to negative sum coarray L − S = {−p 1 − p 2 |p 1 ∈ P, p 2 ∈ P} and positive sum coarray L + S = {p 1 + p 2 |p 1 ∈ P, p 2 ∈ P}. Therefore, b 0 (θ q ) can be expressed as an equivalent steering vector of diff-sum coarray with sensors located at is total sum coarray. Assume that the consecutive range of diff-sum coarray, known as virtual uniform linear array(VULA), is [−l c , l c ] which mainly determines the performance of subspace-based algorithm. In this paper, the number of DOFs is defined by the aperture of VULA, i.e., 2l c + 1. Hence, we can obtain the received signal vector of VULA by removing the repeated and discrete lags in (7) z where )×Q denotes the manifold matrix of VULA and e 0 is a column vector of all zeros except a 1 at the middle. The lags in z i might be generated by difference coarray, negative sum coarray or positive sum coarray. It is noteworthy that if a consecutive segment in VULA belongs to L − S or L + S , the steering vector of this segment respectively contains the NC phases e −j2φ q and e j2φ q .Assume the VULA can be divided into M consecutive segments, their position sets are P 1 , P 2 , . . . , P M . As shown in Fig.1, the positive half of the VULA is composed of difference coarray and positive sum coarray in turn, and the negative half alternately consists of difference coarray and negative sum coarray. Then, the steering vector b(θ q ) of VULA can be rewritten as where b m (θ q ) is the steering vector corresponding to P m and ψ mq contains NC phase of the mth segment. If the mth segment of VULA is generated by difference coarray, then ψ mq = 1. If this segment is generated by negative and positive sum coarray, ψ mq is equal to e −j2φ q and e j2φ q , respectively. Therefore, b(θ q ) can be further simplified as where B (θ q ) is only related to DOA parameter θ q .
Then, we calculate the mean of equivalent received vector z i in (12) as wherep = [σ 2 1 ,σ 2 2 , . . . ,σ 2 Q ] T is the expectation of p i ;σ 2 q is the average power of the qth signal in all F frames. Each row in p i is transformed into a real-valued stationary process with zero-mean by subtractingz from z ī Now, NC virtual data matrix can be obtained by combininḡ where P = [p 1 ,p 2 , . . . ,p F ] ∈ C Q×F is equivalent signal matrix.Then, use MUSIC algorithm to estimate DOAs of NC sources. Eigenvalue decomposition of the covariance matrix where U s is termed as signal subspace which contains eigenvectors associated with Q largest eigenvalues of R Z . U n is noise subspace which contains eigenvectors corresponding to remaining eigenvalues. By using the orthogonality of the signal subspace and the noise subspace, we can obtain the following relationship Substituting (14) into (19), we obtain is rank deficient at θ = θ q , q = 1, 2, . . . , Q. Therefore, the DOAs of NC signals can be estimated by the following estimator [20], [24], Searching θ over the range (−90 • , 90 • ], DOAs can be obtained from the peaks of f (θ ). It is worth noting that the number of column of U n should be greater than or equal to M since U H n B (θ ) ∈ C (2l c +1−Q)×M must be a full column rank matrix. Otherwise, V (θ ) is rank deficient regardless of θ. Thus, the maximum number of detectable signals of GVNCM method is 2l c + 1 − M . However, the VNCM algorithm developed in [20] only divided VULA into three parts, i.e., difference coarray, negative sum coarray and positive sum coarray. The MNDS of VNCM approach is 2l c − 2. Therefore, the VNCM can be regarded as a special case of the proposed GVNCM method. The major computational load involved in the GVNCM algorithm contains the construction of the covariance matrix eigen-decomposition and MUSIC spectrum search. Summing the above operations, the computational complexity for the GVNCM algorithm can be determined as

III. PROPOSED ARRAY CONFIGURATION
In this section, we first analyze the diff-sum coarray of TNA, which can reduce the redundancy between the difference coarray and the sum coarray by exchanging the position of dense ULA and sparse ULA of NA. Then, an extended transformed nested strategy is presented to further enlarge the aperture of diff-sum coarray. Based on this strategy, we propose two improved nested configurations, called TwETNA and ThETNA, which have low redundancy and can acquire high DOFs.
As both the difference and the sum coarray are mirror symmetric about the zero point, we only consider the positive part of them, namely positive difference coarray L + D and positive sum coarray L + S . The properties of the negative part of diff-sum coarray are the same as that of the positive part. For notational simplicity, we start from introducing a few set operations.
Definition 1: For two given integers sets A and B, one can define the following operations: A. THE EXTENDED TRANSFORMED NESTED STRATEGY As shown in Fig.2, conventional NA is constructed by two uniform linear subarrays. One is a dense ULA containing N 1 + 1 sensors with the inter-element spacing d, where d = λ/2 is unit spacing. The other is a sparse ULA consisting of N 2 sensors with the inter-element spacing (N 1 + 1)d. For convenience, in the following sections of this paper, all the locations are normalized by d. Therefore, the position set of NA can be expressed as Note that the two subarrays share the sensor at position N 1 . The total number of the sensors used in NA is N 1 + N 2 .
Based on the conclusion in [28], the positive difference coarray L + D of NA contains all the consecutive integers in the range 0, N 1 N 2 + N 2 − 1 . The consecutive range of the positive sum coarray L + S is 0, N 1 N 2 + N 1 + N 2 − 1 . From the above analysis, we can see that L + D is a subset of L + S . For the NA, it is clear that the difference coarray does not contribute to increasing the aperture of VULA. In [23], the authors introduced a transform concept to reduce the redundancy between the difference and the sum coarray of NA. Transformed nested array is a special nested configuration which can be generated by exchanging the positions of the two subarrays of NA. The location set of TNA can be  represented as Here, denote N 1 and N 2 as the sensor number of dense and sparse array. Similarly, P 1 and P 2 share the sensor located at (N 2 − 1)(N 1 + 1). The TNA configuration is depicted in Fig.3.
For the TNA, the positive difference set L + D consists of three parts: The positive sum set L + S also consists of three parts: Therefore, the L + D is continuous in the range 0, (N 2 − 1) Compared with the NA, the difference coarray of TNA has the same continuous range. However, the sum coarray of TNA is shifted to the right side along with the axis. As a consequence, the TNA can decrease the redundancy between the difference and the sum coarray and increase the number of available DOFs.
However, in the positive difference set of TNA, the D + (P 2 ) is completely contained in D(P 2 , P 1 ). In addition, in the positive sum set, the first half part of S(P 2 ), namely 2(N 2 − 1)(N 1 + 1) + 0, N 1 , is overlapped with S(P 2 , P 1 ). Moreover, the self difference set D + (P 2 ) and self sum set S(P 2 ) are redundant since P 2 is a dense ULA. For illustrative purpose, we consider a TNA with (N 1 , N 2 ) = (4, 4) as an example. The corresponding positive difference and positive sum coarray are shown in Fig.4. It is evident that D + (P 2 ) = 0, 4 , which only has five elements, is all contained in D(P 2 , P 1 ) = 0, 19 . Meanwhile, the first five elements of S(P 2 ) = 30, 38 , namely 30, 34 , are overlapped with S(P 2 , P 1 ) = 15, 34 . Therefore, TNA can be further improved to achieve higher DOFs by reducing the aforementioned redundancy.
We propose an extended transformed nested strategy which can effectively reduce the redundancy of TNA by moving part of the sensors in dense ULA to the right, where the shift interval should be integer multiple of N 1 + 1 units. Assume the elements in P 2 can be divided into two parts, i.e., P 2 = P 21 ∪P 22 . The elements in P 22 are shifted x(N 1 +1) units to the right, where 1 ≤ x ≤ N 2 − 1 and the resulting position set is expressed as P 22 = P 22 + x (N 1 + 1). Then, Thus, D(P 22 , P 1 ) ∪ D(P 21 , P 1 ) is continuous in the range 0, (N 2 + x − 1)(N 1 + 1) + N 1 , except the holes at set since N 2 , N 2 + x − 1 (N 1 + 1) ⊂ P 1 , and the holes at set Similarly, S(P 22 , P 1 ) ∪ S(P 21 , P 1 ) is continuous in the range (N 2 − 1)(N 1 + 1), (2N 2 + x − 2)(N 1 + 1) + N 1 except the holes at set and the holes at set since N 2 , N 2 + x − 1 (N 1 + 1) ⊂ P 1 . We can find that H D2 ⊂ S(P 21 , P 1 ), and H S1 ⊂ D(P 22 , P 1 ). The holes at set H D2 are filled by S(P 22 , P 1 ) ∪ S(P 21 , P 1 ), while the holes at set H S1 are filled by D(P 22 , P 1 ) ∪ D(P 21 , P 1 ). Therefore, the diff-sum coarray contains all consecutive integers in the range 0, (2N 2 + x − 2)(N 1 + 1) + N 1 except the holes at sets H D1 and H S2 . As a result, to guarantee the continuity of diff-sum coarray, we need to move the elements in dense ULA to satisfy: (1) the holes at set H D1 are covered by D + (P 21 ∪ P 22 ); (2) the holes at set H S2 are covered by S(P 21 ∪ P 22 ); (3) S(P 21 ∪ P 22 ) is extended as far as possible to enlarge the aperture of VULA.
For example, consider an extended TNA with (N 1 , N 2 ) = (4, 4), where P 21 = {15, 18, 19}, P 22 = {21, 22}. P 22 is generated by shifting P 22 = {16, 17} to the right. The cross difference set D(P 22 , P 1 ) ∪ D(P 21 , P 1 ) and cross sum set S(P 22 , P 1 ) ∪ S(P 21 , P 1 ) of this array are shown in Fig.5(b) and Fig.5(c), respectively. We can find that the difference set is continuous in the range 0, 24 except the holes at set H D1 = {1, 2} and H D2 = {20, 23, 24}. On the other hand, sum set is continuous in the range 15, 39 except the holes at set H S1 = {16, 17} and H S2 = {35, 38, 39}. It is clear that H D2 is a subset of S(P 21 , P 1 ) and H S1 is a subset of D(P 22 , P 1 ). Therefore, as long as the holes at set H D1 and H S2 can be covered, the continuous range of diff-sum coarray will be extended.

B. TWO-LEVEL EXTENDED TNA
Based on the aforementioned design ideology, we split dense ULA into two parts and redistribute one of them to satisfy the three continuity conditions. The resulting array geometry is named as two-level extended transformed nested array. The definition of the TwETNA is given by Definition 2 (Two-Level Extended Transformed Nested Array): Consider a TwETNA with N = N 1 + N 2 physical sensors. N 1 and N 2 are the number of sensors in dense and sparse subarray, respectively, satisfying N 1 ≥ 7 and N 2 ≥ 1. The TwETNA is specified by the integer set P TwETNA , defined by where P 1 = 0, N 2 − 1 (N 1 + 1), In Definition 2, P 1 represents the position set of sparse subarray. The sensors in the dense ULA of TNA are broken into two parts, namely P 2 = P 21 ∪P 22 . The position of sensors in P 21 remains unchanged. The elements in P 22 are moved (N 1 + 1) units to the right to form a new set P 22 . Based on the above description and explanation, the TwETNA possesses the following relationship: As shown in  (b) The VULA can be split into 3 parts. Based on the GVNCM algorithm, the MNDS of TwETNA is DOFs-3.
Proof: The proof is provided in Appendix A. Fig.7 shows the diff-sum coarray of the TwETNA configuration in Fig.6(b). One can find that in the positive region, the diff-sum coarray contains all the consecutive lags in the range 0, 64 , where 0, 26 belongs to difference coarray and 27, 64 belongs to sum coarray. Therefore, the whole diff-sum coarray consists of three parts due to its symmetry about zero point. As a result, the DOFs of this TwETNA is 129 and the MNDS is 126.

C. THREE-LEVEL EXTENDED TNA
The second kind of improved TNA is named as three-level extended transformed nested array. The definition of the ThETNA is given by Definition 3 (Three-Level Extended Transformed Nested Array): For two integers N 1 and N 2 satisfying N 1 ≥ 13 and N 2 ≥ 1, the location set of ThETNA can be expressed as where P 1 = 0, N 2 − 1 (N 1 + 1), where P 1 denotes the location of sparse ULA. The sensors in dense ULA are divided into three parts, i.e., P 2 = P 21 ∪ P 22 ∪ P 23 . P 21 keeps its position. The elements in P 22 are translated (N 1 +1) units to the right to generate P 22 . P 23 can be obtained by shifting P 23 2(N 1 +1) units to the right. Similarly, the ThETNA has a corresponding relationship    (13, 3). The dense ULA in TNA is split into three sets: P 21 , P 22 and P 23 . The blue and green arrows respectively illustrate how sensors migrate from P 22 and P 23 to P 22 and P 23 .
The VULA can be divided into 7 parts. Based on the GVNCM method, the MNDS of ThETNA is DOFs-7.
Proof: The proof is provided in Appendix B. It is worth to note that there are several ways to split the VULA of ThETNA. We choose a way with the least segments to maximize the number of detectable signals.
For illustrative purpose, we consider the diff-sum coarray of the ThETNA configuration shown in Fig.8(b) as an example. It can be seen in Fig.9 that the positive part of diff-sum coarray is continuous in the range 0, 114 , where 0, 41 and {47} are generated by difference coarray, 42, 46 and 48, 114 are generated by sum coarray. Therefore, the whole diff-sum coarray consists of seven parts. Accordingly, the DOFs is 229 and the MNDS is 222 for this ThETNA.

IV. Cramér-Rao BOUND
CRB is an important tool to evaluate the performance of DOA estimation as it presents a lower bound on the variances of unbiased estimates of parameters [29], [30]. In this section, we derive the CRB of the underdetermined DOA estimation problem for non-circular sources.
Proof: The proof is provided in Appendix C. According to Proposition 3, the CRB(θ ) is a function of the position of physical array and its diff-sum coarray, the number of frame F, the frame length L, the normalized DOAsθ, the NC phases φ 1 , . . . , φ Q and the SNR of incoming signals σ 2 1 /σ 2 n , . . . ,σ 2 Q /σ 2 n . The proof of the fact CRB(θ ) depends on the SNR rather than the value ofσ 2 q , q = 1, 2, .., Q and σ 2 n can be given as follow: If we changeσ 2 q and σ 2 n to ρσ 2 q and ρσ 2 n , then R y = ρR y ,P = ρP.
Therefore, it is easy to obtain M θ = Mθ . M o can be represented as As a result, CRB(θ ) is related to SNR but not to the value of power.
In the following section, we calculate the CRB of normalized DOA for the two proposed arrays to evaluate the estimation performance.

V. SIMULATION RESULTS
In this section, a series of numerical examples are provided to illustrate the superiority of the proposed arrays. DsCAMpS [20], CA [31], TNA-1 and TNA-2 [23] are selected as comparison arrays. These six array configurations can achieve high DOFs as they are all designed for the DOA estimation of non-circular sources.

A. DOFs AND MNDS COMPARSION
The DOFs and MNDS of different configurations are listed in Table 1. Moreover, for a given number of  sensors N , the approximate maximum DOFs are also provided in Table 1.It is evident that the two coprime arrays can achieve O( N 2 2 ) DOFs and four nested arrays possess O(N 2 ) DOFs. TNA-1, as shown in Fig.2, has the smallest number of DOFs among the four nested configurations. As an improved structure of TNA-1, TNA-2 can increase 2N 1 DOFs. The two kinds of ETNAs possess higher number of DOFs than TNA-2. In addition, the MNDS of these six arrays also have the same conclusion.

C. MEAN SQUARE ERROR
In this subsection, we conduct 500 Monte Carlo experiments to further evaluate the accuracy of DOA estimation. The performance metric is the mean square error (MSE) of the estimated normalized DOAs, which is defined as whereθ q denotes the real normalized DOA of the qth source, θ q (i) is the estimate ofθ q for the ith trial, i = 1, . . . , 500.
In all the experiments, we consider Q = 51 narrowband uncorrelated NC signals uniformly distributed between −60 • and 60 • . In Fig.11, we evaluate the MSE performance as  It is observed that the DOA estimation performance of all the configurations improves with the increase of the input SNR, the number of frames and the frame length. We can see that the estimation error of DsCAMpS and CA is much larger than four nested configurations, because their DOFs are only about half of the latter. Among the four nested structures, the MSE results of the TwETNA and ThETNA perform better than TNA-1 and TNA-2 due to the larger continuous range of their diff-sum coarray. Furthermore, the performance of the ThETNA slightly outperforms TwETNA.The results of the three simulations indicate that the accuracy of DOA estimation is proportional to the available number of DOFs, which verifies the superiority of the proposed arrays.

D. SPATIAL RESOLUTION PERFORMANCE
This simulation examines the performance of DOA estimation when dealing with two closely spaced sources. In this  example, we consider two sources located at 10 • and 10+ • , where represents the inter-source spacing. The comparison of spatial resolution performance of these six configurations as a function of inter-source spacing is presented in Fig.14, where SNR = 0 dB, F = L = 200. We can see that DsCAMpS, CA, TNA-1 and TNA-2 have a 100% probability to identity the two signals when the inter-source spacing is not less than 0.18 • , 0.16 • , 0.1 • and 0.09 • , respectively. However, TwETNA and ThETNA can distinguish the two sources accurately when the inter-source spacing is greater than or equal to 0.08 • . Clearly the proposed arrays can obtain better spatial resolution than other four configurations due to higher DOFs capacity and larger coarray aperture.

E. CRB
In this subsection, we use CRB as a performance criterion to present the lower bounds of DOA estimation error of the six configurations. All simulation parameters are the same as those of the first experiment in subsection C. Fig.15 shows the relationship between the CRB and SNR with F = L = 200. It can be seen that if SNR < 20 dB, CRBs decrease with the increase of the SNR. If SNR ≥ 20 dB, CRBs are almost unchanged when the SNR increases. In addition, the ThETNA can acquire the lowest CRB among all the configurations, which is consistent with the theoretical results.

VI. CONCLUSION
In this paper, we have considered the sparse array design issue for the DOA estimation of non-circular signals based on the GVNCM algorithm. An extended transformed nested strategy was proposed by dividing the dense subarray of TNA into several parts, which can be shifted to the right to construct a more sparse array. Based on this strategy, we developed two kinds of configurations named as TwETNA and ThETNA, which can provide higher DOFs and better estimation performance compared to TNA. The closed-form expressions of physical sensor position and continuous diff-sum coarray range were derived for any given number of elements. We also derived the CRB expression of the underdetermined DOA estimation for non-circular signals as a performance criterion. Simulation results demonstrated the effectiveness of the proposed array configurations.

APPENDIX A PROOF OF PROPOSITION 1
(a) As both the difference and the sum coarray are symmetrical with the zero point as the center, we just prove the continuity of positive part of diff-sum coarray.
Therefore, the diff-sum coarry of TwETNA contains all the consecutive integers in the range −l c , l c , where (b) According to the conclusion in property (a), it is clear that the VULA of TwETNA can be divided into three consecutive segments: Thus, based on the GVNCM algorithm, the maximum number of detectable signals of TwETNA is DOFs-3.

APPENDIX B PROOF OF PROPOSITION 2
(a) The range 0, N 1 can be covered by the self difference set of dense subarray D + (P 21 ∪ P 22 ∪ P 23 ), which consists of following six parts: Therefore, the range 0, N 1 ⊂ D + (P 21 ∪ P 22 ∪ P 23 ) when N 1 ≥ 13.

APPENDIX C PROOF OF PROPOSITION 3
In Section II, we assume that NC sources is quasi-stationary with F frames and the frame length is L. Then, the extended data vector of all F frames can be represented as whereÂ andn(t) is given in (5). The covariance matrix of y(t) is given by where R s = E[s(t)s H (t)] = diag{σ 2 1 ,σ 2 2 , . . . ,σ 2 Q },σ 2 q is the average power of the qth signal in all F frames. By vectoring R y , we can obtain wherep = [σ 2 1 ,σ 2 2 , . . . ,σ 2 Q ] T , the definition of B and 1 2N is the same as in (7).
For the signal model (41), the (m, n)th element of the Fisher information matrix (FIM) can be represented as [30], [32], [33] [ where Mθ and M o are the partial derivatives of z corresponding to normalized DOAsθ and other 2Q + 1 parameters, respectively. The expression of FIM is given by Then, the CRB ofθ can be expressed as In order to obtain a CRB expression more suitable for NC signals, we need to make a concrete analysis of Mθ and M o . Firstly, the permutation matrix J, as defined in (11), brings the following two relationships B = JB 0 , where B 0 is defined in (10), 1 N = vec(I N ). We can see that B 0 consists of four parts, respectively corresponding to difference coarray, negative sum coarray, positive sum coarray and difference coarray. It is useful to introduce a set of matrix F 1 , F 2 , F 3 and F 4 to express each part of B 0 as follow: a * (θ q ) ⊗ a(θ q ) = F 1 a D (θ q ), a * (θ q ) ⊗ a * (θ q )e −j2φ q = F 2 a − S (θ q ), a(θ q ) ⊗ a(θ q )e j2φ q = F 3 a + S (θ q ), a(θ q ) ⊗ a * (θ q ) = F 4 a D (θ q ), where a D (θ q ), a − S (θ q ) and a + S (θ q ) are the steering vectors of difference coarray, negative sum coarray and positive sum coarray, respectively. The matrix F 1 , F 2 , F 3 , F 4 are given by [30] [ [I D − (l)] n 1 ,n 2 = 1, if − n 1 + n 2 = l 0, otherwise, n 1 , n 2 ∈ P. Therefore, we can rewrite B 0 as where A D , A − S and A + S are manifold matrices corresponding to a D (θ q ), a − S (θ q ) and a + S (θ q ), respectively. F = block-diag{F 1 F 2 F 3 F 4 } is a block diagonal matrix, Combining (47) and (56) yields where