Real-Valued MUSIC for Efficient Direction Estimation ... - IEEE Xplore

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Real-Valued MUSIC for Efficient Direction Estimation With Arbitrary Array Geometries Feng-Gang Yan, Member, IEEE, Ming Jin, Shuai Liu, and Xiao-Lin Qiao

Abstract—Most of the existing methods for direction-of-arrival (DOA) estimation are based on numerical characteristics behind the entire array output covariance matrix (AOCM). Since the AOCM is generally a complex matrix, those approaches require tremendous complex computations accordingly. This paper addresses the problem of DOA estimation with real-valued computations by considering the real part of AOCM (R-AOCM) and the imaginary part of AOCM (I-AOCM) separately. It is shown that the null space of R-AOCM and that of I-AOCM are the same subspace, which coincides with the intersection of the original noise subspace and its conjugate subspace. Using such a mathematical fact, a novel real-valued MUSIC (RV-MUSIC) estimator with a real-valued subspace decomposition on only R-ACOM (or I-AOCM) instead of the entire ACOM is derived. Compared with most state-of-the-art unitary algorithms suitable for only centro-symmetric arrays (CSAs), the proposed technique can be used with arbitrary array geometries. Unlike conventional MUSIC with exhaustive spectral search, RV-MUSIC involves a limited search over only half of the total angular field-of-view with a real-valued noise subspace, and hence reduces the complexity by 75%. Theoretical performance analysis on the mean square error (MSE) and numerical simulations demonstrate that RV-MUSIC shows a very close accuracy to the standard MUSIC. Index Terms—Arbitrary array geometries, direction-of-arrival (DOA) estimation, real-valued MUSIC (RV-MUSIC), spectral search, subspace decomposition.

I. INTRODUCTION

I

N array signal processing, numerical results behind the entire array output covariance matrix (AOCM) are extensively derived by different algorithms to estimate the direction-of-arrival (DOA) of multiple narrow-band sources. For example, beamfroming approaches such as the conventional beamformer (CBF) [1] and Capon’s minimum variance distortionless response (MVDR) beamfomer [2] use the ACOM and its inverse to find signal DOAs, respectively. On the other hand, subspace-based methods including multiple signal classification (MUSIC) [3], estimation of signal parameters via Manuscript received March 30, 2013; revised July 03, 2013 and December 15, 2013; accepted December 15, 2013. Date of publication January 09, 2014; date of current version February 26, 2014. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jean Pierre Delmas. F.-G. Yan is with the Department of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: yfglion@gmail. com). M. Jin, S. Liu, and X.-L. Qiao are with the Department of Electronics and Information Engineering, Harbin Institute of Technology, Weihai 264209, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2014.2298384

rotational invariance techniques (ESPRIT) [4], root-MUSIC [5] and their derivations solve the problem of DOA estimation based on subspace decomposition which is usually accomplished by eigenvalue decomposition (EVD) or singular value decomposition (SVD) on the AOCM. Since the inputs of sensor arrays are generally complex-valued signals, AOCM is a complex matrix accordingly. Therefore, those approaches require complex-valued operations, which may be computationally expensive for real-time applications. To reduce the complexity, real-valued (or unitary) estimators including U-MUSIC [6], U-ESPRIT [7], U-root-MUSIC [8], unitary method of direction-of-arrival estimation (U-MODE) [9] and unitary matrix pencil (U-MP) [10] usually exploit unitary transformations [11] and forward/backword (FB) averaging [12] to transform AOCM to a real matrix, then estimate source DOAs with real-valued computations. Since one multiplication between two complex values generally requires four times that between two real ones, unitary methods can reduce about 75% computational burdens as compared to their complex-valued versions. Another outstanding advantage of unitary algorithms is that they also show improved accuracies as compared to complex-valued approaches. For example, it has been shown in [6] that U-MUSIC has optimal Hermitian per-symmetric estimator of AOCM in the sense of Euclidean distance, and hence outperforms the standard MUSIC. In [8], both theoretical analysis and real-world experiments with sonar- and ultrasonic-data have demonstrated that U-root-MUSIC shows better accuracies than the conventional root-MUSIC as well. Despite their increased estimation accuracies with reduced costs [13], almost all of the state-of-the-art unitary methods are based on centro-symmetrical arrays (CSAs), which severely limits their applications. For efficient DOA estimation with arbitrary array configurations, there have been several promising attempts to extend the concept of root-MUSIC to arbitrary array geometries. Techniques such as array interpolation (AI) [14] and beamspace transformation [15] have been developed to map the steering vector of a nonuniform array (NUA) to that of a uniform linear array (ULA) with Vandermonde structure, and then find DOAs via polynomial rooting. Another two recently reported approaches extending root-MUSIC to arbitrary arrays are the manifold separation technique (MST) [16] and Fourier-domain (FD) root-MUSIC [17]. For AI, it usually introduces mapping errors which may cause increased bias [18] and excess variance [19] while for MST and FD root-MUSIC, polynomial rooting with a sufficiently high order [17] is generally needed to warrant that the truncation errors are small, which may cause higher additional complexities than expected [20]. We have shown in [21] that the exhaustive spectral search involved in the conventional MUSIC can be compressed to a limited angular sector instead of the total angular field-of-view

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by a compressed MUSIC (C-MUSIC) estimator which exploits the multiple orthogonality between the steering vector and the intersection of a noise-like subspace cluster (NLSC), in which both tasks of subspace decomposition and spectral search are accomplished by complex-valued computations. Thus, estimating source DOAs via real-valued computations with arbitrary array geometries still remains unresolved. In this paper, we present a novel real-valued MUSIC (RVMUSIC) estimator which finds signal DOAs by spectral search over only half of the total angular field-of-view with a realvalued noise subspace, which can be used with no dependence on array structures. The developed method also shows increased accuracies with reduced computational burdens as compared to C-MUSIC since the former can be taken as a real-valued version of the latter. Another newly developed result in the present work shows that the real part of AOCM (R-AOCM) and the imaginary part of AOCM (I-AOCM) can be used separately for DOA estimates, whereas attentions used to be paid on the entire AOCM by most existent methods. Throughout the paper, matrices and vectors are denoted by upper- and lower-boldface letters, respectively. Complex- and real-matrices are denoted by single-bar- and double-bar-upper boldface letters, respectively. In addition, the mathematical notations are denoted as follows zero matrix (vector); identity matrix; th column of matrix th row of matrix

II. PRELIMINARIES A. Signal Model and Subspace Decomposition A standard signal model for DOA estimation using a linear omnidirectional sensors is given by [3] array composed of

(3) where is the numbers of sources, is the number of snapshots. is the matrix of the signal direction vectors, is the angle set of unknown source DOAs, and

(4) is the steering vector, is center wavelength, is the coordinate of sensor , and , and are the array output-, source waveform-, and additive Gaussian white noise (AGWN)-vectors, respectively. The EVD of the entire AOCM (5)

; can be expressed as

;

intersection;

(6)

direct sum;

where is source covariance matrix, and are the so-called signal- and noise-subspace matrices, respectively. For practical situations, the theoretical AOCM in (5) is unavailable, and it is usually estimated by

conjugation; transpose; Hermitian transpose;

(7)

round down to integer; Frobenius norm;

Thus, the subspace decomposition is in fact given by

Kronecker delta;

(8)

inner product;

The subspace decomposition can also be obtained by performing SVD on a direct-data matrix

mathematical expectation; real part of the embraced matrix;

(9)

imaginary part of the embraced matrix;

as follows

rank of the embraced matrix. , For column-spaces of

and denote the null- and , respectively, given by

(10)

(1)

given by (9) is perturbed In a noise environment, the idea , where by AGWN as is the matrix of AGWN. Therefore, the SVD of is given by

(2)

(11) Noting that

where

,

are unknown coefficients.

, we obtain from (8) and (11) that (12)

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which implies that the subspace decomposition by the EVD of is equivalent to that by the SVD of [21], [22]. B. Related Works Using some facts and , the conventional MUSIC algorithm [3] suggests to estimate source DOAs by spectral search as follows (13) One of the most important advantages of the MUSIC algorithm is its easy implementation with arbitrary array configurations. However, since MUSIC involves a tremendous spectral search step, it is computationally expensive for real-time applications. We have proposed another C-MUSIC estimator, which exploited NLSC (see [21] for detailed illustrations) to limit the exhaustive spectral search of MUSIC to a small sector by (14)

Remark 1: Assumption A1 is a standard restriction in the literature on most DOA estimators including MUSIC [3]. For non-coherent signals, this assumption can be easily satisfied. Assumption A2 is commonly referred to as the so-called rankambiguity restriction on array geometry (see [23], [24] and the references therein), which must be satisfied in practical applications. It is worth noting that this assumption is a sufficient condition for estimating signal DOAs without are linearly ambiguity since vectors independent implied guarantees that , vectors are linearly independent. Although assumption A3 is too strict for conventional MUSIC, in which only is required, it is to be shown shortly that this assumption allows a significant reduction on computational complexity as compared to MUSIC. B. Physical Analysis on AOCM, R-AOCM and I-AOCM For ideal

and

, we see clearly from (5) and (9) that

where is the intersection of NLSC, is the compression time and is a small angular sector. For MUSIC and C-MUSIC, both subspace decomposition and spectral search require complex-valued computations. To realize real-valued computations, the U-MUSIC [6] technique transforms the standard MUSIC to a real-valued function as (20)

(15) where and are real vector and matrix, respectively. Unfortunately, U-MUSIC can be used with only CSAs, which severely limits its applications.

Hence, R-AOCM and I-AOCM can be expressed as (21) (22)

III. THE PROPOSED ALGORITHM A. Basic Assumptions The following basic assumptions on the data model (3) are considered to hold throughout the paper. is of full rank such that A1. Matrix (16) A2. The array steering vectors associated with any dif, , i.e, ferent angles , , are linearly independent. In other words, the following equation (17) holds if and only if (18) where , are unknown scalars. A3. The number of signal DOAs is smaller than half that of sensors such that (19)

respectively. Noting that is composed of all the snapshots of received datum, it must contain the entire information regarding source DOAs, which is separately stored in and . From such a physical point of view, and can be taken as two basic factors that are both necessary and hence, must be used simultaneously for DOA estimates. This means that using only or may fail to find source DOAs since half of the original information is lost. However, this doesn’t mean that exploiting both and always warrants success in DOA estimates. Intuitively, due to the presences of both and in (21) and (22), the information contained in either R-AOCM or I-AOCM is enough for finding source DOAs since both the two basic factors are not lost in R-AOCM or I-AOCM. Hence, it could be predicted that R-AOCM and I-AOCM can be used separately, which hence gives a significant reduction on complexity for computationally efficient DOA estimations. Moreover, with the above intuitive explanations, differences between the accuracies of DOA estimators based on the entire AOCM and of those based on only R-AOCM or only I-AOCM can be also predicted. Observing that R-AOCM contains the two second-order terms and without the cross term while on the contrary, I-AOCM contains only the cross term without

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the two second-order terms, estimators using the entire AOCM may slightly outperform those using only R-AOCM or only I-AOCM. In the sections to follow, we shall show in detail how to find source DOAs by using R-AOCM and I-AOCM separately.

which implies that is a symmetrical real-valued matrix, whose EVD and SVD must require only real-valued computations [6]. Therefore, we must have

C. Subspace Decompositions on R-AOCM and I-AOCM

must According to (24), it is clear that the dimension of (see [21], where is also similar to be the intersection of NLSC with two angular sectors). Hence, the EVD (or SVD) of can be written as

Let us consider the intersection of the original noise subspace and the conjugate noise subspace , which is given by

(29)

(30) (23) is a subset of , it also contains a part of Since . Therefore, we can use instead the vectors of of to estimate source DOAs. An important advantage of over is that the at both the true former has a double orthogonality to DOAs and their mirror directions simultaneously, i.e.,

and stand for the signal- and the where subscripts noise-subspace, respectively, and and are two real diagonal matrices composed of the significant- and the zero-eigenvalues of , respectively. Since , it is clear that the column space of must equal to the direct sum of the original signaland the conjugate signal-subspace. In other words, we have

(24)

(31)

holds for arbitrary linear This is because array geometries. Such a double orthogonality can help us limit the exhaustive spectral search to only half of the total angular field-of-view, and hence reduces a significant computational complexity as compared to the standard MUSIC. Another outstanding advantage of over is that the orthogonal basis of the former can be computed by subspace decompositions on R-AOCM and I-AOCM with only real-valued computations while that of the latter resulted from the EVD of the entire AOCM usually requires complex-valued computations. To see clearly about this, we give the following theorem which reveals the relationship between , the null space of R-AOCM and that of I-AOCM. Theorem 1: Under assumptions A1 A3, we have

(32) is an antisymmetrical real-valued which implies that matrix. According to matrix theory [25], the SVD of must require only real-valued computations (note that the EVD of may involve complex-valued computations [25]). Therefore, we can write the SVD of as follows (33)

(25)

where again, subscripts and denote the signal- and the noisesubspace, respectively, and and are two real diagonal matrices composed of the significant- and the zero-singular values of , respectively. Using the facts [21] and [25] gives

(26)

(34)

, we ,

offer an orthogonal basis for the null Hence, the columns of space of . According to theorem 1, matrix can be further taken as an orthogonal basis of equivalently, and we have

(27)

(35)

where

Proof: See Appendix A. It follows directly from theorem 1 that must have , which means that and we further have

can be On the other hand, the orthogonal basis of also computed by the subspace on equivalently. Noting that , we have

Equation (27) can be identified as the characteristic one for the real-valued matrix . Therefore, is an eigenvalue of and is the eigenvector associated with . Since is , matrix can be computed by an arbitrary vector of the subspace decomposition on accordingly. Noting that , we have (28)

Thus, the SVD of

can be rewritten as (36)

, and matrix can also be computed by the Thus, with only real-valued computations. SVD of The above analysis reveals the relationship among subspace decompositions on AOCM, conjugate AOCM, R-AOCM and I-AOCM, which is shown in Fig. 1 for clear illustrations.

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By introducing two electronic angles (39) is transformed into

as follows

Fig. 1. Relationship among subspace decompositions on AOCM, conjugate AOCM, R-AOCM and I-AOCM.

(40) Consequently, we have , with which the 2-D steering vector also shows a double orthogonality to the real-valued noise matrix . Thus, we can similarly exploit the 2-D-RV-MUSIC estimator. Noting that the definitions of and in (39) can be changed with array geometries, RV-MUSIC has no dependence on array configurations. Detailed steps for implementing the proposed RV-MUSIC algorithm are summarized in Algorithm 1.

Fig. 2. 2-D array on the X-Y plane, which is composed of tional sensors with arbitrary array geometries.

omnidirec-

Algorithm 1 The proposed RV-MUSIC algorithm Require:

D. The Proposed Method With computed by the EVD of (or by the SVD of ), the proposed RV-MUSIC estimator is given by

:

1: Initiation: 2: for

;

snapshots of array output vector. ,

,

;

.

do ;

3: 4: end for (37) According to (24), the minima of over only half of total angular field-of-view are either the true DOAs or their images. Because the steering vector is orthogonal to at only the true incident angles, the original noise subspace responding to the true DOAs are much larger than those associated with symmetrical mirror DOAs. Moreover, as the number of the true DOAs, i.e., , is known in advance, we can use the standard MUSIC to select the true DOAs among the candidate angles by minimizing such that estimation ambiguity is avoided. Although using the standard MUSIC to exclude the symmetrical mirror DOAs means that there is an additional EVD step on involved in the proposed estimator. However, the complexity of this step is substantially lower than that of spectral search [17], [21] since we only need to compute the product for at most spectral points. Remark 2: The proposed RV-MUSIC algorithm can be directly extended to estimate the two-dimensional (2-D) signal with ardirections bitrary plane array geometries. To demonstrate this clearly, we take the plane array in Fig. 2 for example, with which the 2-D steering vector is given by

(38)

5: Compute

, (or

);

6: for each

(or

) do ,

7:

;

8: end for 9: for 10:

then

else if

then ;

else

15: 16:

do ;

13: 14:

;

if

11: 12:

;

; end if

17: end for 18: return : a estimated angle set of the

source DOAs.

Comparisons of the primary real-valued computational complexities of various algorithms are shown in Table I, where stands for the total sample points of the standard MUSIC spa. The common term tial spectrum over included in all the six algorithms gives the cost for computing the subspace decomposition on a real-valued matrix of dimensions by using the fast subspace decomposition (FSD)

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Proof: See Appendix B. According to theorem 2 where is similar to also computed by the SVD of equivalently. Since and , the SVD of can be written as

TABLE I COMPARISONS OF REAL-VALUED COMPUTATIONAL COMPLEXITY

,

can be

(44) where and are composed of the singular vectors of associated with the non-zeroand zero-singular values respectively. With a similar . proof to that of theorem 2, it can be easily shown that Therefore, is given by the following theorem. Theorem 3: Assume the elements of matrix are random variables with zero means, then the perturbation of at high SNR can be expressed by a linear function of matrix as follows (45) technique [26]. Another term involved in MVDR [2] gives the complexity of computing the inverse of [25]. Since the spectral search of the proposed RV-MUSIC is involved over (or ) with , it only requires for computing its null spectrum. On the other hand, the standard MUSIC uses to compute its null spectrum over , it costs for the spectral search step. [17], it is observed from Table I that Noting that about 75% computational complexities are reduced by the proposed estimator as compared to MUSIC. Hence, RV-MUSIC shows a similar efficiency to the U-MUSIC [6] algorithm. It is also seen from Table I that RV-MUSIC has much lower complexities than C-MUSIC [21] with two angular sectors since complex-valued computations are required in C-MUSIC for both subspace decomposition and spectral search.

where (46) Proof: See Appendix C. Using the result of lemma 1 and that of theorem 3, a closed-form MSE expression for DOA estimation by the proposed RV-MUSIC estimator is given by the following theorem. Theorem 4: Assume the elements of matrix are random variables with zero means and variance , then the MSE for the estimation of incident angles by RV-MUSIC at high SNR is given by (47) where

IV. PERFORMANCE ANALYSIS

(48)

In this section, we use the theory of subspace perturbation and Taylor’s expansion series to derive a closed-from expression for the mean square error (MSE) of direction estimation by the proposed RV-MUSIC estimator base on a high signal-to-noise ratio (SNR) assumption. Under such an assumption, the perturbation of is given by the following lemma [22]. Lemma 1: Assume the elements of matrix are random variables with zero means, then the perturbation of at high SNR can be expressed by a linear function of matrix as follows

(49)

(41) As shown in Section III that can be computed by the EVD (or by the SVD of ). However, it is difficult to of establish the perturbation of based on the EVD of (or based on the SVD of ). To derive an expression for , we give another theorem as follows. Theorem 2: Using the original signal subspace matrix to define a real matrix as follows (42) Then, we have (43)

(50) denoting the first-derivative of with with respect to . Proof: See Appendix D. V. SIMULATIONS Computer simulations with 500 independent Monte Carlo trials are conducted to assess the performance of the proposed estimator and to verify the derived MSE expression of DOA estimates by RV-MUSIC. For the performance comparison, the C-MUSIC algorithm with and angular sectors [21] and the unconditional Cramér-Rao Lower Bound (CRLB) given in [28] are also applied for references. For all search-based algorithms, a coarse gird 1 was firstly used to get candidate peaks, and a fine one 0.0053 was secondly applied around the candidate peaks for final DOA estimates. First, we use a ULA of half-wavelength spaced sensors to compare the root MSEs (RMSEs) of the proposed RV-MUSIC estimator with those of different algorithms, including ESPRIT, conventional MUSIC, Capon’s MVDR, rootMUSIC, U-MUSIC [6], and C-MUSIC with sectors.

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Fig. 3. RMSE for sources at and

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versus the SNR, where are used on a ULA of

and sensors.

Fig. 4. RMSE for versus the numbers of snapshots, where the and sources at and are used on sensors. a ULA of

Fig. 3 demonstrates the RMSEs against the SNR, where the SNR varies over a wide range from 20 dB to 30 dB. It can be concluded from the figure that U-MUSIC, despite its limited applications for only CSAs, is the most accurate one among the presented seven algorithms, which shows a RMSE closest to the CRLB. It can be also seen from the figure that the standard MUSIC, root-MUSIC, C-MUSIC with angular sectors as well as the proposed method show similar performances to each other, which is much better than ESPRIT and Capon’s method. Noting that the accuracy of C-MUSIC decreases as increases and C-MUSIC with angular sectors reduces the complexity by only 50% [21] while RV-MUSIC saves that by about 75%, the proposed method hence shows improved accuracy with reduced complexity as compared to the C-MUSIC technique. To see more clearly the performance of the new approach, Fig. 4 plots RMSEs of different algorithms as functions of the

Fig. 5. Experimental- and theoretical-RMSEs for , sources at and where

versus the SNR, are used on ULAs.

Fig. 6. Experimental- and theoretical-RMSEs for versus the num, sources at and bers of snapshots, where are used on ULAs.

number of snapshots, where the number of snapshots varies over to . It can be seen again from a wide range from Fig. 4 that the proposed technique performs similarly to the standard MUSIC and to C-MUSIC with angular sectors. It can be also seen that RV-MUSIC shows much better performances than ESPRIT and Capon’s MVDR beamformer, especially in scenarios with small numbers of snapshots. Fig. 5 and Fig. 6 compare RMSEs of DOA estimation by the proposed RV-MUSIC with different numbers of antennas, in terms of both based on the experimental- and the theoreticalresults given by (47). We can observe clearly from the figurers that the simulated results and the analytic expectations agree with each other closely when , which verifies the theoretical analysis in Section IV. Fig. 7 and Fig. 8 plot the RMSEs as functions of the SNR and as those of the number of snapshots, respectively, where the number of sources varies from to the its maximum

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Fig. 7. RMSE for sensors is used.

versus the SNR, where

Fig. 8. RMSE for and a ULA of

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and a ULA of

versus the numbers of snapshots, where the sensors is used.

. As seen clearly from the figures that the differences between the RMSEs of RV-MUSIC and those of MUSIC increase as increases. This are is mainly caused by the fact that the dimensions of whiles those of are , thus the difference between the former and the latter increases as increases, and the relationship between RV-MUSIC and MUSIC is similar to that between C-MUSIC and MUSIC [21]. Second, we examine the performance of the proposed approach with NUAs, where ESPRIT, U-MUSIC and the conventional root-MUSIC cannot be exploited for DOA estimates any more. We use the minimum-redundancy linear arrays (MRLAs) [29] to compare Capon’s MDVR beamformer, the standard MUSIC and C-MUSIC with as well as angular sectors. In the simulation, the unconditional CRLB is also applied for comparison reference. In Fig. 9, we fix the number of snapshots as and display RMSEs of different algorithms as functions of the SNR while in Fig. 10, we fix the , and plot RMSEs

Fig. 9. RMSE for sources at and

versus the SNR, where are used on a MRLA of

and sensors.

Fig. 10. RMSE for versus the numbers of snapshots, where the and sources at and are used on a sensors. MRLA of

as functions of the numbers of snapshot, where the amounts of snapshot varies from to . We see from Fig. 9 that the five techniques show very close accuracies to each other. More Specifically, the standard MUSIC and the proposed method slightly outperform MDVR and C-MUSIC with sectors, especially for small numbers of snapshots. On the other hand, C-MUSIC with angular sectors shows close performances to the standard MUSIC as well as to Capon’s MVDR and the proposed method. From Fig. 10, it is seen clearly that in scenarios of small amounts of snapshots, our method has much better accuracies than the MVDR beamformer and C-MUSIC with angular sectors while for large numbers of snapshots, the proposed method shows a similar performance to C-MUSIC with sectors. or sector involves a Noting that C-MUSIC with higher complexity than RV-MUSIC [21], the proposed method provides an improved performance-to-complexity tradeoff as compared to the C-MUSIC technique.

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Fig. 11. Experimental- and theoretical-RMSEs for , sources at and where MRLAs.

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versus the SNR, are used on

Fig. 12. Experimental- and theoretical-RMSEs for versus the num, sources at and bers of snapshots, where are used on MRLAs.

To verify the theoretical MSE expression of DOA estimates by RV-MUSIC with MRLAs, Fig. 11 and Fig. 12 compare the simulated RMSEs with those computed by (47) with different numbers of antennas. It is seen clearly from the two figures that there is a close match between the simulated results and their theoretical expectations, especially for , which verifies again the theoretical analysis in Section IV. Next, we use a ULA composed of half-wavelength spaced sensors to compare resolution probabilities of different algorithms in Fig. 13 and Fig. 14, including the conventional MUSIC, U-MUSIC, C-MUSIC with sector and the proposed method. In the simulation, we use closely-spaced and , which are said to be successsources at fully resolved if and only if [30] (51)

Fig. 13. Resolution probability against the SNR, where and closely-spaced sources at and are used on a ULA with sensors.

Fig. 14. Resolution probability against the number of snapshots, where the and closely-spaced sources at and are used on a ULA with sensors.

It is observed that U-MUSIC, as addressed in [6], has the highest resolution among the four estimators. On the other hand, for low SNRs and small numbers of snapshots, the proposed method shows an improved resolution for two closely-spaced sources as compared to the standard MUSIC. It can be also concluded from the two figures that C-MUSIC with sectors shows a little better resolution than RV-MUSIC. Finally, we use a ULA to plot the simulation times of DOA estimates by different algorithms as functions of the number of sensors in Fig. 15. The simulated results are given by a PC with Intel(R) Core(TM) Duo T5870 2.0 GHz CPU and 1GB RAM by running the Matlab codes in the same environment. It can be seen from Fig. 15 that RV-MUSIC is the most efficient method among the five algorithms with a simulation time being about 4 times lower than that of MUSIC.

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which is equivalent to (A.5) According to (1) and (A.5), the columns of must belong to . Thus, it follows directly from (2) that (A.6) On the other hand, assume that have . Using (5),

, then we can be rewritten as

(A.7) Fig. 15. Simulation time against the number of sensors, where , the , sources at and are used on a ULA, are sampled by points with a grid 0.1 . and

VI. CONCLUSIONS We have proposed a novel efficient RV-MUSIC algorithm for DOA estimation with arbitrary array geometries in this paper, which exploits the subspace decomposition on only R-AOCM (or I-AOCM) instead of the entire AOCM. This is based on a newly developed result showing that the null space of R-AOCM and that of I-AOCM in fact equal to the intersection of the original noise subspace and its conjugate subspace. Theoretical analysis and simulation results demonstrate that RV-MUSIC shows a similar performance to the standard MUSIC while its complexity is about four times lower than that of MUSIC. Compared with C-MUSIC which can be taken as a complex-valued version of RV-MUSIC, the real-valued computations in RV-MUSIC also lead to increased accuracy with reduced computations in some scenarios. Future research includes to develop unitary estimators with arbitrary arrays as well as with fewer sensors. APPENDIX A PROOF OF THEOREM 1 Since and have These facts together with lead to

Therefore, we have (A.8) where and . Expanding and as weighted sums of the columns of and , respectively, (A.8) can be rewritten as (A.9) is used, and and are the -th where the fact element of and , respectively. According to assumption A2, , (noting (A.9) holds if and only if that ), which means that (A.10) is invertible, it follows from (A.10) that , and hence , , and we have implies that

Because

. This (A.11)

Combining (A.6) and (A.11), we finally have , we must . and

(A.12) By subtracting (A.3) from (A.2), it can be similarly proved that , which completes the proof.

(A.1) Now, postmultiplying the left- and the right-side of (5) by well as using (A.1), which gives

APPENDIX B PROOF OF THEOREM 2

as

It is beneficial to rewrite

as follows

(A.2) Similarly, we also obtain the following equation by postmultiplying the both sides of the conjugate version of (5) by (A.3) Adding (A.2) and (A.3), we have (A.4)

(B.1) Now, assume

, then we have . Thus, we further have , which leads to , and we finally have This implies that

and and .

(B.2)

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On the other hand, suppose

, then we have , which is equivalent to . Taking norm for the both sides of

the above equation gives . Because the projections of onto and and tively, we have Therefore, which gives , Hence, , and we finally have

and

Combining (B.2) and (B.3) gives completes the proof.

For the sake of notational simplicity, let us define

are , respec. , . Thus, .

(D.1) (D.2) (D.3)

(B.3)

Using (45) and the zero means of additive noise, it can be easily proven with a high SNR assumption that

, which

(D.4)

APPENDIX C PROOF OF THEOREM 3 It follows from (B.1) that the perturbation of

APPENDIX D PROOF OF THEOREM 4

is given by

is a consistent estimate for at high Hence, SNR, and we can obtain the second-order approximation of the derivative of about the true value as follows (see [21], [27], and references therein)

(C.1) and are where the second-order terms neglected. Since is obtained by the SVD of , it can be similarly concluded from Lemma 1 that at high SNR can as follows be expressed by a linear function of (C.2)

(D.5) where higher-order terms are neglected, and the first- and second-order derivatives of with respect to are denoted by and , respectively. By performing a forward derivation, it can be shown that

Inserting (C.1) into (C.2) as well as using (41), we have

(D.6) (C.3)

is the intersection of and As must have and Thus, the first term of (C.3) can be simplified as

Inserting (C.8) into (D.6) and using the notations in (D.3) gives

, we . (C.4)

Similarly, the third term of (C.3) is given by (C.5) Therefore, (C.3) can be simplified as (C.6) Noting that onto

and and

(D.7) Noting that lows from (D.7) that

and

, it fol-

are the orthogonal projections , respectively, we must have (C.7)

Inserting (C.7) into (C.6),

is finally given by

(D.8) (C.8)

which completes the proof.

Now, consider the first term of (D.8). Expanding and as weighted sums of the columns and the rows

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of , respectively, and using the zero means of AGWN noise

Substituting (D.9)–(D.13) into (D.8) as well as using the fact , we finally obtain (D.14) which completes the proof. ACKNOWLEDGMENT

(D.9) where and are the th and th element of and , respectively; and and are the th row and the th column of , respectively. In a similar way, we can prove that (D.10) According to (D.9) and (D.10), it is clear that the first term of (D.8) equals to zero. Now, let us consider the second term of (D.8). Since the variance of noise is , we can similarly write that

(D.11)

(D.12)

(D.13) where

and (because ) are used in the last step of (D.13).

The authors would like to thank the anonymous reviewers for their many insightful comments and suggestions, which helped improve the quality and readability of this paper. REFERENCES [1] J. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag., vol. 13, no. 3, pp. 67–94, Jul. 1996. [2] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE, vol. 57, pp. 1408–1418, Aug. 1987. [3] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [4] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [5] B. D. Rao and K. V. S. Hari, “Performance analysis of root-MUSIC,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp. 1939–1949, Dec. 1989. [6] K. C. Huarng and C. C. Yeh, “A unitary transformation method for angle-of-arrival estiamtion,” IEEE Trans. Signal Process., vol. 39, pp. 975–977, April 1991. [7] Z. Guimei, C. Baixiao, and Y. Minglei, “Unitary ESPRIT algorithm for bistatic MIMO radar,” Electron. Lett., vol. 48, no. 3, pp. 179–181, Feb. 2012. [8] M. Pesavento, A. B. Gershman, and M. Haardt, “Unitary root-MUSIC with a real-valued eigendecomposition: A theoretical and experimental performance study,” IEEE Trans. Signal Process., vol. 48, no. 5, pp. 1306–1314, May 2000. [9] A. B. Gershman and P. Stoica, “On unitary and forward-backward MODE,” Digit. Signal Process., vol. 9, no. 2, pp. 67–75, Feb. 1999. [10] N. Yilmazer, J. Koh, and T. K. Sarkar, “Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction of arrival,” IEEE Trans. Signal Process., vol. 54, pp. 175–181, Jan. 2006. [11] M. Haardt and F. Romer, “Enhancements of unitary ESPRIT for noncircular sources,” in Proc. ICASSP, Feb. 2004, pp. 101–104. [12] D. A. Linebarger, R. D. DeGroat, and E. M. Dowling, “Efficient direction-finding methods employing forward-backward averaging,” IEEE Trans. Signal Process., vol. 42, no. 8, pp. 2136–2145, Aug. 1994. [13] M. Haardt and J. A. Nossek, “Unitary ESPRIT: How to obtain increased estimation accuracy with a reduced computational burden,” IEEE Trans. Signal Process., vol. 43, no. 5, pp. 1232–1242, May 1995. [14] B. Friedlander, “The root-MUSIC algorithm for direction finding with interpolated arrays,” Signal Process., vol. 30, pp. 15–29, 1993. [15] C. P. Mathews and M. D. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2395–2407, Sep. 1994. [16] F. Belloni, A. Richter, and V. Koivunen, “DoA estimation via manifold separation for arbitrary array structures,” IEEE Trans. Signal Process., vol. 55, no. 10, pp. 4800–4810, Oct. 2007. [17] M. Rbsamen and A. B. Gershman, “Direction-of-arrival estimation for nonuniform sensor arrays: From manifold separation to Fourier domain MUSIC Methods,” IEEE Trans. Signal Process., vol. 57, pp. 588–599, Feb. 2009. [18] P. Hyberg, M. Jansson, and B. Ottersten, “Array interpolation and bias reduction,” IEEE Trans. Signal Process., vol. 52, no. 10, pp. 2711–2720, Oct. 2004. [19] P. Hyberg, M. Jansson, and B. Ottersten, “Array interpolation andDOA MSE reduction,” IEEE Trans. Signal Process., vol. 53, no. 12, pp. 4464–4471, Dec. 2005. [20] J. Zhuang, W. Li, and A. Manikas, “Fast root-MUSIC for arbitrary arrays,” Electron. Lett., vol. 46, no. 2, Feb. 2010. [21] F. G. Yan, M. Jin, and X. L. Qiao, “Low-complexity DOA estimation based on compressed MUSIC and its performance analysis,” IEEE Trans. Signal Process., vol. 61, no. 8, pp. 1915–1930, Apr. 2013.

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[22] F. Li, H. Liu, and R. J. Vaccaro, “Performance analysis for DOA estimation algorithms: Unification, simplification, and observations,” IEEE Trans. Aerosp. Electron. Syst., vol. 29, no. 4, pp. 1170–1184, Oct. 1993. [23] C. Proukakis and A. Manikas, “Study of ambiguities of linear arrays,” in Proc. ICASSP, Apr. 1994, vol. 4, pp. 549–552. [24] K. C. Tan and Z. Goh, “A detailed derivation of arrays free of higher rank ambiguities,” IEEE Trans. Signal Process., vol. 44, no. 2, pp. 351–359, Feb. 1996. [25] G. H. Golub and C. H. Van Loan, Matirx Computations. Baltimore, MD, USA: Johns Hopkins Univ. Press, 1996. [26] Xu and Kailath, “Fast subspace decomposition,” IEEE Trans. Signal Process., vol. 42, no. 3, pp. 539–551, Mar. 1994. [27] P. Stoica and T. Soderstrom, “Statistical analysis of a subspace method for bearing estimation without eigendecomposition,” Proc. Inst. Electr. Eng., vol. 139, no. 4, pt. F, pp. 301–305, 1992. [28] P. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, pp. 1783–1795, Oct. 1990. [29] C. Chambers et al., “Temporal and spatial sampling influence on the estimates of superimposed narrowband signals: When less can mean more,” IEEE Trans. Signal Process., vol. 44, no. 12, pp. 3085–3098, 2004. [30] Q. T. Zhang, “Probability of resolution of the MUSIC algorithm,” IEEE Trans. Signal Process., vol. 43, no. 4, pp. 978–987, Apr. 1994.

Feng-Gang Yan (S’11–M’14) received the B.E., M.S., and Ph.D. degrees in information and communication engineering from Xi’an Jiaotong University, Xi’an, the Graduate School of Chinese Science of Academic, Beijing, and Harbin Institute of Technology (HIT), Harbin, in 2005, 2008, and 2013, respectively. From July 2008 to March 2011, he was a Research Associate of the Fifth Research Institute of China Aerospace Science and Technology Corporation (CASC), where his research was mainly focused on the processing of remote sensing images. Since October 2013, he is a Teacher with the Department of Electronics Information Engineering, HIT, Weihai, China. His current research interests include array signal processing and statistical performance analysis.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 6, MARCH 15, 2014

Ming Jin received the B.E., M.S., and Ph.D. degrees in information and communication engineering from Harbin Institute of Technology (HIT), Harbin, China, in 1990, 1998, and 2004, respectively. From 1998 to 2004, he was with the Department of Electronics Information Engineering, HIT. Since 2006, he has been a Professor of The School of Information and Electricity Engineering, HIT, Weihai, China. His current interests are in the areas of array signal processing, parallel signal processing, and radar polarimetry.

Shuai Liu was born in Heilongjiang Province, China, in 1980. He received the B.E. and M.S. degrees from Northwestern Polytechnical University China, in 2002 and 2005, respectively, and received the Ph.D degree in information and communication engineering from Harbin Institute of Technology (HIT), China, in 2013. Since 2013, he has been an Associate Professor of The School of Information and Electricity Engineering, HIT, Weihai, China. His current interests are in the area of conformal array and polarization sensitive array signal processing.

Xiao-Lin Qiao was born in the Inner Mongolia Autonomous Region, China, in June 1948. He received the B.E., M.S., and Ph.D. degrees in information and communication engineering from Harbin Institute of Technology (HIT), Harbin, China, in 1976, 1983, and 1991, respectively. Dr. Qiao was with the Department of Electronics Information Engineering of HIT from 1983 to 1993. Since 1994, he has been a Professor of The School of Information and Electricity Engineering, HIT. During 1994–2011, he was the President of HIT, Weihai, China. In the past 15 years, he has authored and co-authored nearly 70 publications. His research interests are in the areas of signal processing, wireless communication, special radar, parallel signal processing, and radar polarimetry.