Low-Complexity DOA Estimation Based on Compressed MUSIC and

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 8, APRIL 15, 2013

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Low-Complexity DOA Estimation Based on Compressed MUSIC and Its Performance Analysis Fenggang Yan, Student Member, IEEE, Ming Jin, and Xiaolin Qiao

Abstract—This paper presents a new computationally efficient method for direction-of-arrival (DOA) estimation with arbitrary arrays. The total angular field-of-view is first divided into several small sectors and the original noise subspace exploited by the multiple signal classification (MUSIC) algorithm is mapped from one sector to the other sectors by a Hadarmard product transformation. This transformation gives a new noise-like subspace cluster (NLSC), whose intersection is found to be simultaneously orthogonal to the steering vectors associated with the true DOAs and several virtual DOAs. Based on such a multiple orthogonality, a novel compressed MUSIC (C-MUSIC) spatial spectrum at hand is derived. Unlike MUSIC with tremendous spectral search, C-MUSIC involves a limited search over only one sector, and hence it is computationally very attractive. To obtain the intersection of NLSC for more than two sectors, a low-complexity method is also proposed in the present work, which shows advantages over the existing alternative projection method (APM) and singular value decomposition (SVD) techniques. Furthermore, the mean square errors (MSEs) of the proposed estimator is derived. Simulation results illustrate that C-MUSIC trades-off MSEs by complexity and resolution as compared to the standard MUSIC efficiently. Index Terms—Alternative projection method (APM), compressed multiple signal classification (C-MUSIC), direction-of-arrival (DOA) estimation, noise-like subspace cluster (NLSC), performance analysis, singular value decomposition (SVD).

I. INTRODUCTION

E

STIMATING the direction-of-arrival (DOA) of multiple narrow-band sources is of great interest in many applications including radar, sonar, passive location and wireless communication. Over several decades, subspace-based approaches such as multiple signal classification (MUSIC) [1], maximum-likelihood (ML) [2], subspace fitting [3], estimation of signal parameters via rotational invariance techniques (ESPRIT) [4] and Min-Norm [5] have received considerable attentions. Among these methods, the MUSIC algorithm offering a much better resolution than the conventional beamformers [6], [7] is one of the most popular techniques. The primary advantage of the MUSIC algorithm is its easy implementation [8] with no dependence on array configurations. However, as

Manuscript received May 25, 2012; revised September 08, 2012 and November 28, 2012; accepted January 08, 2013. Date of publication January 28, 2013; date of current version March 21, 2013. 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 and X. L. Qiao are with the Department of Electronics and Information Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TSP.2013.2243442

the conventional MUSIC involves a tremendous computation burden, it is prohibitively expensive when real-time processing is required. The high computational complexity of the MUSIC algorithm is mainly caused by an involved subspace decomposition step and a spectral search step. As subspace decomposition is usually accomplished by eigenvalue decomposition (EVD) or singular value decomposition (SVD), many efficient methods without EVD or SVD are reported [9]–[12] to reduce the complexity of MUSIC. Nevertheless, the computation of subspace decomposition can be reduced to by using the fast subspace decomposition (FSD) technique [13], where is the number of sensors and is the number of sources. On the other hand, all the spectral points with a large amount have to be computed in the spectral search step. As a rule, we have , which means that the complexity of spectral search is in fact substantially heavier than that of subspace decomposition [14], especially when a two-dimensional (2-D) search or a linear search with a fine gird is required. Therefore, avoiding the spectral search step or limiting the range for this spectral search becomes the key to reducing the complexity of MUSIC, and numerous modifications of the standard MUSIC algorithm have been presented from this point of view. The root-MUSIC algorithm [15] is a well-known search-free method, which exploits polynomial rooting instead of spectral search. Although root-MUSIC has an improved computational load and threshold performances as compared to the standard MUSIC [16], it is only applicable to Uniform Linear Arrays (ULAs). Another distinguished technique for DOA estimation without spectral search is ESPRIT. Unfortunately, the array geometry in ESPRIT is required to be shift-invariant. Although there are also numerous useful extensions of root-MUSIC and ESPRIT to circular arrays [17], [18] and other more general classes of array geometries [19]–[21], the array structures used in these papers are still rather specific. Recently, promising attempts to extend the concept of root-MUSIC to more general classes of array configurations have been made by manifold separation technique (MST) [22] and fourier-domain (FD) root-MUSIC [14] methods. Since these approaches usually use a polynomial with a sufficiently high order to warrant that the truncation errors are small, the computations for finding the roots of this polynomial may be higher than expected [14]. Moreover, the problem of efficient 2-D DOA estimation has not been addressed by MST and FD root-MUSIC. In this paper, we present a new computationally efficient algorithm for DOA estimation with no dependence on array configurations. The new method focuses on reducing the complexity of the standard MUSIC by limiting its spectral search to a small angular sector instead of the total angular field-of-view since the computational burden of subspace decomposition is

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substantially lower as compared to spectral search. The new method is referred to as the compressed MUSIC (C-MUSIC) and, it is somewhat similar to the beamspace MUSIC [6] and sector-focussed [23] approaches. However, different from them, C-MUSIC does not require any prior information regarding source locations. The essence of the proposed algorithm is to substitute the intersection of a transformed noise-like subspace cluster (NLSC) for the original noise subspace of the standard MUSIC to construct a new spatial spectrum. Unfortunately, the existing methods for computing the intersection of several subspaces such as the alternative projection method (APM) [24]–[26] and SVD-based method [27] have drawbacks. For example, APM is only suitable for several closed subspaces despite its heavy computational complexity; although the method based on SVD given in [27] is more efficient than APM, it is only applicable for the special case of two subspaces. In order to get the intersection of NLSC without these drawbacks, a new method for more than two subspaces with low complexity is also proposed in the present work. As it is to be shown that the steering vectors associated with the true DOAs and several virtual DOAs are orthogonal to the intersection of NLSC simultaneously, C-MUSIC can generate several virtual sources for each true source, which are uniformly distributed over the total angular field-of-view. On the other hand, since the steering vector lies in the signal subspace only at the directions of true DOAs and the number of the true sources is known in advance, the true DOAs can be easily selected among the observed candidate DOAs by minimizing the product of the steering vector and the original noise subspace of MUSIC. Consequently, the true DOAs can be found by spectral search only over one small angular sector instead of the total angular field-of-view. This means that the computational load of spectral search is reduced significantly by the proposed algorithm. Moreover, C-MUSIC also has a dimension-reduced noise subspace as compared to MUSIC and it hence requires lower complexity than MUSIC to compute each spectral point. Therefore, the total complexity of the proposed approach is much lower than the standard MUSIC. II. THE STANDARD MUSIC ALGORITHM

paper to denote matrices and vectors whose elements are all zeros. The -th column of is given by (3) where , observed by the

is the phase of the -th waveform -th sensor, which is given by (4)

where is the signal center wavelength, and is the coordinate of the -th sensor. array covariance matrix [1] can be written as The (5) is the source covariance matrix. where The EVD of the exact covariance matrix is given by (6) Then and are called the signal- and noisesubspace, respectively. In practical situations, the theoretical array covariance matrix given in (5) is unavailable and it is usually estimated by (7) Thus, the true signal- and noisesubspace matrices then estimated by the EVD of as follows

(1) where is transpose, is the signal vector, is the steering vector matrix and is the additive white Gaussian noise (AWGN) vector, whose elements are usually assumed to be Gaussian random variables with zero means and variance as follows

Here, transpose,

are (8)

Take

snapshots of the noise-free array output to form a matrix as follows

, (9)

Then, is called the direct-data matrix [29], [30], whose SVD can be expressed by (10)

A. Subspace Decomposition Assume uncorrelated narrowband plane waves with unsimultaneously incident on an known DOAs array of sensors, where . Let be the number of snapshots, the array output vector at snapshot , is defined as and can be written as [1]–[7]

and

In a noise environment, be written as

is perturbed by AGWN and it can (11)

where is the AGWN here means the perturbed value instead matrix. Note that of the estimated value of , and its SVD can be written as (12) It can be seen clearly from (7) and (11) that (13) Therefore, we obtain from (8), (12) and (13) that

(2)

(14)

is mathematical expectation, is Hermitian is the identity matrix and is used throughout the

Hence, the subspace decomposition by the EVD of is the same as that by the SVD of . The reason might not be

YAN et al.: LOW-COMPLEXITY DOA ESTIMATION

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often used in practice is that the dimensions of grow with , while has fix dimensions [29], [30]. In this paper, the EVD of is to be used in Section III to discuss the theory of the proposed C-MUSIC algorithm while the SVD of is to be used in Section V to get a much easier analysis on the proposed method. B. The Standard MUSIC Algorithm The conventional MUSIC estimator [1] involves minimizing its null-spectrum function as follows (15) where is Frobenius norm. According to the orthogonality between the signal- and noisesubspace, the standard MUSIC algorithm attempts to find the source DOAs by searching over with a fine gird such that is minimized. Note that the computational complexity of this spectral search step is substantially higher than that of the subspace decomposition step [14].

Fig. 1. Spatial angular dividing, where the length of the -th angular sector space is and that in space is , in .

where

is a constant given by (19)

Note that and are fixed here, thus and . It is implied by (18) that

only depends on

III. THE PROPOSED C-MUSIC ALGORITHM

(20)

A. Basic Definitions and Assumptions Definition 1: When the number of sensors is no smaller than , we say that the two times that of the sources, i.e., standard MUSIC is compressible. Definition 2: When the number of sensors is larger than that of the sources, but no larger than two times that of the sources, i.e., , we say that the standard MUSIC is incompressible. Throughout this paper, we assume that the standard MUSIC is compressible. Although this is more strict than that of the standard MUSIC, which requires [1], it is to be shown that this assumption can lead to a much lower computation as compared to the standard MUSIC. B. The C-MUSIC Algorithm

where stands for Hadamard product and the vector is defined as . Now, suppose , is the -th source DOA. Then based on the orthogonality between the steering vector and the noise subspace, we have (21) is inner product and where Substituting (20) into (21) gives

is the -th column of

(22) where

In order to reduce the computational complexity of MUSIC, let us divide the total angular field-of-view into small sectors , , which is shown in Fig. 1. Here, the length of each sector in the sine space is given by

is conjugate operation, , and column of a new matrix , which is given by

, ,

, there satisfying

For the sake of convenience, let us define (17)

, Let then we have

satisfies is the -th

(23)

(16) It can be seen from Fig. 1 that , exists a

.

denote the -th element of

,

(24) , is called the NLSC. The intersection Then of NLSC and the orthogonal projection onto this intersection subspace are denoted by (25)

(18)

(26)

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subspace cluster (SLSC) , ilar expression to NLSC as follows

, which has a sim-

(29) Fig. 2. The virtual sources generated by the C-MUSIC spatial spectrum.

where

is the -th column of

, which is defined as (30)

respectively. It can be seen from (22) that is equivalent to , where , , . Since is the intersection subspace, we must have , , and therefore we further obtain (27) and Based on such multiple orthogonality between , we propose a new C-MUSIC estimator for DOA estimation by minimizing its null-spectrum function as follows

The purpose of introducing SLCS here is to get prepare for the performance analysis of C-MUSIC in Section V, where SLCS together with the following theorem revealing the relationship between and is to be used. and are defined as Theorem 1: Assume (23) and (29), respectively. Then is the orthogonal complement subspace of , . In other words, we have (31) Proof: See Appendix A. C. The Noise Subspace of C-MUSIC

(28) It can be seen clearly from (27) that for each true DOA , there is to show one spectral peak by the C-MUSIC spatial spectrum in each angular sector simultaneously, where the virtual DOAs are given by , , , respectively. This means that for each true source, the C-MUSIC spatial spectrum generates equivalent virtual sources, which is shown in Fig. 2. Since the -th virtual DOA and the -th virtual DOA are mathematically related by (17), it can be seen from Fig. 2 that the virtual DOAs together with the true DOA are uniformly distributed over the total angular field-of-view in the sine space. Therefore, one of the virtual DOAs can be found by spectral search over only one angular sector instead of the total angular field-of-view. Then the other virtual DOAs can be computed by (17) immediately without spectral search. On the other hand, since the steering vector is orthogonal to the original noise subspace only at the directions of the true incident angle , we can select the true signal DOA among the observed virtual DOAs , by maximizing . Note that the spectral search over the sine domain is physically equivalent to that over the angle domain [28]. Therefore, the limited spectral search by C-MUSIC can also be performed over one sector in the sine space with length , which is as wide as the total field-ofview. Therefore, C-MUSIC realizes a compression for the standard MUSIC, and with this meaning, the new technique is called the compressed MUSIC and is called the compression times for C-MUSIC. Getting the intersection of NLSC is the key to construct the C-MUSIC spatial spectrum. However, before our discussing on the intersection of NLSC, let us first define a new signal-like

In order to compute the noise subspace of C-MUSIC, we must get the intersection of NLSC. The existing methods for calculating the intersection of several subspaces can be classified as the following two different kinds. First, for two subspaces of a Hilbert space, i.e., , their intersection can be obtained by the following lemma. Lemma 1: Using and to define a new matrix (32) . Then we have

whose SVD is expressed by

(33) Proof: See section 12.4.4 of [27]. Second, for more general cases of no less than two closed subspaces of a Hilbert space, i.e., , the APM [24]–[26] technique offers a way for calculating the orthogonal projection onto the intersection of these subspaces in terms of their orthogonal projections. The main result of APM is given by the following lemma. Lemma 2: Let stand for closed subspaces of a Hilbert space, whose intersection is denoted by

and let , tion onto subspace we have

and and

be the orthogonal projec, respectively. Then

(34)


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Proof: This lemma was first proved by Aronszajn in [24], where the special case was addressed. After that, Kayalar and Weinert discussed its error bounds in [25], and Halperin gave a proof for the general case in [26]. It can be seen from (34) that APM has a high computational complexity because it involves an iteration of matrix product. Moreover, APM only gives the orthogonal projection onto the intersection of several closed subspaces instead of the orthogonal basis of their intersection. Although (33) has a lower complexity than APM and also gives the orthogonal basis directly, it is only applicable for the special case . In order to compute the intersection of NLSC without these drawbacks, we propose another method to get the orthogonal basis for the intersection of NLSC directly, whose fundamental is given by the following theorem. Theorem 2: a) With , from (23), we define a new as follows matrix (35) Then that b) Using

is an idempotent Hermite matrix, which means is the orthogonal projection onto subspace . , to define the following matrix (36)

. whose null space is . Then we have Proof: See Appendix B. It is shown by theorem 2 that and are the same space. Therefore, we can use SVD to get the orthogonal basis for , which also gives an orthogonal basis for the intersection of NLSC equivalently. The SVD of can be expressed as

and , , with , , being associated with the non zero- and zero- singular values of , respectively. Using [27], we have (42) Therefore, , satisfy , and are linearly independent. Hence, these vectors offer an , and we known from theorem 2 orthogonal basis for that can be given as (43) must be no bigger than such that the According to (43), is not empty. Therefore, the intersection subspace maximum value for is given by (44) where

stands for round down to integer operation.

D. C-MUSIC for Fast 2-D Source Localizations and Summary of the Proposed Algorithm Although we elaborated on the proposed C-MUSIC algorithm with specific linear arrays, it can be directly extended to the cases of 2-D source localizations. Here, we take the plane array for example. The elevation angle, say , is defined as the angle between the wave direction and the axis while the azimuth angle, say , is defined as the angle between the axis and the projection of waveform onto the plane. Let , stand for the coordinate of the -th sensor, then is given by (45) In order to exploit C-MUSIC for fast 2-D DOA estimations, we perform a transformation as follows

(37) where and are the left- and the right- singular matrix of , respectively. is a diagonal matrix with being the -th singular value of . Since C-MUSIC has equivalarger- and smallerlent sources, there must be diagonal elements in . Hence, , can be sorted as (38) Therefore, (37) can be rewritten as (39)

(46) Then,

can be expressed with

and

as follows (47)

angular plane into Thus, by dividing the 2-D electric sectors, the 2-D transformed steering vector shows a similar property to the linear angular steering vector given by (27). Therefore, for each electric DOA , of the transformed domain, there are candidate electric DOAs , satisfying (48) and

,

can be computed by

where (49) (40) (41)

where and are the length of each angular sector at the - and - direction, respectively. Therefore, we can similarly exploit the new noise subspace , and the proposed C-MUSIC estimator for fast 2-D DOA source localizations can

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COMPARISON

OF

TABLE I COMPUTATIONAL COMPLEXITY

IV. COMPLEXITY ANALYSIS In this section, we analyze the computational complexity of C-MUSIC and compare it with that of the standard MUSIC as well as with that of the FD-root MUSIC technique [14]. Let stand for the number of spectral points of the total angular field-of-view. For the proposed C-MUSIC algorithm, it for each searched spectral point. has to compute Note that the dimensions of are and C-MUSIC involves a limited search over only one small angular sector with spectral points, flops are required by the spectral search step [8]. For the EVD step on , using the FSD technique [13], its complexity is given flops. As the dimensions of are and by C-MUSIC has a new signal subspace with dimensions , the additional SVD step on requires flops. Therefore, the total computational complexity of the proposed approach is given by

TABLE II DETAILED STEPS FOR IMPLEMENTING THE C-MUSIC ALGORITHM

be performed by minimizing its null-spectrum function in the transformed domain as follows.

(52) , can be obtained by Note that for the special case the SVD of , whose dimensions are . is given by Therefore, the complexity of C-MUSIC for

(50) is Because the spectral search over the physical angles equivalent to that over the electric angles [28], one of the candidate electric DOAs can be found be spectral search over only one angular sector and the other candidate virtual DOAs , can be computed by (49) without spectral search immediately. The inverse transformation of (46) is given by (51) which means that for each given , there are two different pairs of associated , i.e., and . However, since the 2-D steering vector is orthogonal to the original noise only at the true DOAs, virtual DOAs can be simsubspace . ilarly excluded by maximizing It is worth noting that the definitions for and in (46) can be changed with array structures, therefore, in essence C-MUSIC has no dependence on array configurations. The detailed steps for implementing the proposed C-MUSIC algorithm for DOA estimation of multiple narrowband sources are summarized in Table II as follows.

(53) for For the standard MUSIC, it has to compute are each spectral point. Since the dimensions of and all the spectral points have be computed, the spectral search step costs flops [8]. Therefore, the computational complexity of the standard MUSIC is given by (54) Since the complexity of the FD-root MUSIC is not explicitly given in [14], and there is an improved modification for the FD-root MUSIC, i.e., the FD Line-Search MUSIC (FDLSMUSIC) technique, we use FDLS-MUSIC here for complexity comparison. Using FSD, the computational complexity of the FDLS-MUSIC method is given by [8], [13], [14] (55) For spectral search with a rough gird 3 and that with a fine grid 1 , and spectral points have to be computed by the standard MUSIC, respectively. In such cases, we compare the computations of the above three algorithms, which is shown in Table I.


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As can be seen from Table I, C-MUSIC with mild compression times offer a better reduced complexity as compared to the FDLS-MUSIC technique when is not sufficiently larger than . In such cases, we have . On the other hand, when is substantially larger than , the complexity of C-MUSIC is comparable to FDLS-MUSIC. In such cases, we have , which means that the computation of C-MUSIC is even times lower than that of MUSIC. It is worth noting that for fine grid search or the 2-D DOA estimation, is usually very large as compared to , therefore, C-MUSIC has a much lower computational burden as compared to the standard MUSIC. V. PERFORMANCE STUDY The performance study of the proposed C-MUSIC algorithm follows from the theory of subspace perturbation and Taylor’s series expansion [29]–[33]. Unlike many previous asymptotic studies, the analysis presented here makes the assumption of high signal-to-noise ratio (SNR) instead of a sufficiently large amount of snapshots. Under this assumption, the analytic mean square error (MSE) expression for DOA estimation by the proposed C-MUSIC is given, and the relationship between the theoretical MSEs for DOA estimation by the standard MUSIC and those by the new method is also revealed. With perturbation on the direct-data matrix , the noise subspace estimated by the standard MUSIC and that by the proand posed C-MUSIC can be written as , respectively. To establish the distribution of C-MUSIC and compare it with that of MUSIC, we need the following results of the perturbation of and the MSE expression for DOA estimation by the standard MUSIC. Lemma 3: a) The perturbation of estimated by the standard MUSIC algorithm at high SNR can be expressed by a linear function of the AGWN matrix as follows

. Since is a constant matrix, it can where be seen from (60) that the perturbation of is given by (61) where the second-order term is neglected. Because is obtained by the SVD of , it follows directly at high SNR can be expressed by a from lemma 3 that linear function of . Note that , we have (62) Substituting (56) and (61) into (62) and using (14) gives

(63) where and the fact is used. With the expression given by (63), we find that the estimates , by (28) at high SNR are unbiased and consistent estimations for the true parameters , and the theoretical MSEs of the C-MUSIC estimator can be computed by a close-from expression immediately. These results are shown by the following two theorems. Theorem 3: Assume the AGWNs are Gaussian random variables with zero means, the estimated , obtained by searching the minima of C-MUSIC null-spectrum in (28) at high SNR are unbiased and consistent estimations for the true parameters . Proof: See Appendix C. Theorem 4: Assume the AGWNs are Gaussian random variables with zero means and variance , the MSE for the estimation error of the incident angle , by C-MUSIC at high SNR is given by (64)

(56) where . b) Assume the AGWNs are Gaussian random variables with zero means and variance and let stand for the first-order derivative of . Then, the of incident MSE for the estimation error angle , by the standard MUSIC at high SNR is given by (57) where

,

are defined as follows (58) (59)

Proof: See [29] and [30]. We start our analysis by giving an expression for as follows. Substituting (A.7) into (36), can be rewritten as

where (65) (66) (67) (68) Proof: See Appendix D. Since C-MUSIC is the compressed form of MUSIC, it can be regarded as a generalized version of the standard MUSIC. In other words, the conventional MUSIC must be a special case of C-MUSIC when . As a matter of fact, it can be seen from (36), (39) and (43) that . Therefore, we have . Consequently, the theoretical MSEs for DOA estimation by C-MUSIC with must also agree with those by the standard MUSIC, which is shown by the following equation. (69)

(60)

Proof: See Appendix E for the proof of (69). The purpose of this proof is to reveal the relationship between the standard

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TABLE III ORTHOGONALITY CHECKING FOR EXPERIMENT 1

Fig. 3. MUSIC- and C-MUSIC- spectrum, where , and sources at and are used on a ULA.

MUSIC and the proposed C-MUSIC and also to give a further confirmation on the correctness of (64). VI. SIMULATION RESULTS Computer simulations are conducted to assess the proposed method and to verify the theoretical analysis. To see clearly the performance of the proposed C-MUSIC estimator with various types of array configurations, 500 independent Monte Carlo runs are used on a ULA with half a wavelength apart and on a Minimum-Redundancy Linear Arrays (MRLA) [34] in the firstand second- simulation, respectively. The MSEs for DOA estimation by MUSIC and C-MUSIC are compared in terms of experimental- and theoretical- values, where the theoretical MSEs of the MUSIC- and C-MUSIC- estimator are computed by (57) and (64), respectively. In the first simulation, we conduct four experiments, say experiment 1, 2, 3 and 4, to compare the performance of C-MUSIC with that of MUSIC on a ULA with the scenario where the virtual sources of C-MUSIC are well-separated from each other. Throughout this simulation, the number of sensors and that of the sources are fixed as and , respectively. Thus, the maximum value for is given by . 1) Experiment 1: Comparison of Spatial Spectrums: In the first experiment, we compare the proposed C-MUSIC spatial spectrum with the standard MUSIC spatial spectrum by using well-separated true sources, which is shown in Fig. 3. We can see from Fig. 3 that for each true source, C-MUSIC can generate candidate spectral peaks, which are are uniformly distributed over the space with apart. The candidate DOAs of are given by

Fig. 4. MSEs against the SNR, where and are used on a ULA.

Similarly, the candidate DOAs of

and

sources at

are given by

Thus, one of the candidate DOAs can be found by searching the C-MUSIC spectrum over only one sector instead of the total angular field-of-view, and the other candidate DOAs can be computed by (17) immediately. Then, the true DOAs can be selected among the candidate DOAs by maximizing , as shown in Table III. 2) Experiment 2: Comparison of DOA Estimation MSEs: To see clearly the performance of the proposed C-MUSIC algorithm in such scenario, we compare the MSEs for DOA estimation by C-MUSIC with those by the standard MUSIC in Figs. 4 and 5, where is set as a control parameter for the performance comparison and for different values of , the virtual sources of C-MUSIC are also well-separated from each other. The theoretical predictions for the MSEs of DOA estimation by MUSIC and those by C-MUSIC with are given in Tables IV and V. It is observed from the figures that the MSEs of C-MUSIC are higher as compared to the standard MUSIC when . However, as decreases, the differences between the MSEs of C-MUSIC and MUSIC decrease dramatically and the proposed method shows a very close performance to MUSIC. On the other hand, it can be seen that there is a close match between


Fig. 5. MSEs against the number of snapshots, where sources at and are used on a ULA.

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and

Fig. 6. Probability of Resolution against the SNR, where sources at and are used on a ULA.

and

TABLE IV THEORETICAL PREDICTIONS FOR THE MSES(/ ) BY THE STANDARD MUSIC AND BY THE PROPOSED C-MUSIC WITH

TABLE V THEORETICAL PREDICTIONS FOR THE MSES(/ ) BY THE STANDARD MUSIC AND BY THE PROPOSED C-MUSIC WITH

Fig. 7. Probability of Resolution against the number of snapshots, where and sources at and are used on a ULA.

the experimental results and their theoretical predictions for the C-MUSIC estimator, which verifies the performance analysis of Section V. It is also seen from Tables IV and V that the MSE predictions by C-MUSIC with are the same as those by the standard MUSIC, which verifies the performance analysis of Section V as well. 3) Experiment 3: Comparison of Probabilities of Resolution: To further compare the performance of the new method with that of the standard MUSIC in such scenario, we compare the probabilities of resolution (PR) by using closely-spaced true sources in Figs. 6 and 7. The two sources are said to be successfully resolved if and only if (70)

where stands for the spectral value. Here, the sub-spectrum of the proposed C-MUSIC in the sector that contains the true DOAs is used, and is also set as a control parameter for the performance comparison. It can be concluded from Figs. 6 and 7 that the proposed method has an improved resolution probability for two closelyspaced true sources as compared to the standard MUSIC. On the other hand, it is also seen that the resolution probability by C-MUSIC gets higher as increases. 4) Experiment 4: Comparison of Simulation Time: To verify the efficiency of C-MUSIC and the computational complexity analysis in Section IV, we use a MALAB 7.0 to compare the simulation time costed by different algorithms in Table VI. The simulations presented here are performed on a personal computer whose CPU configurations and RAM are given by Intel(R) Core(TM) Duo T5870 2.0 GHz and 1 GB, respectively.

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TABLE VI , COMPARISON OF SIMULATION TIME (/SECOND), WHERE AND SOURCES AT 30 AND 40 ARE USED ON A ULA

TABLE VII MRLA CONFIGURATIONS

It can be seen from Table VI that the traditional root-MUSIC is the most efficient method among the above four algorithms although it is only applicable for ULAs. It is also seen that the simulation time costed by C-MUSIC is comparable to those costed by the FDLS-MUSIC technique and is about of those costed by the standard MUSIC. Therefore, our method is much more efficient than the standard MUSIC. In the second simulation, we conduct three experiments, say experiment 5, 6 and 7, to compare the performance of C-MUSIC with that of the standard MUSIC on a MRLA [34], whose configuration is shown in Table VII. In this simulation, the worst scenario for the proposed method is considered, where the virtual sources of C-MUSIC are closely-spaced. We fix and in experiment 5, 6 and 7 and fix in experiment 5 and 6, respectively. 5) Experiment 5: Comparison of Spatial Spectrums With Overlapped Candidate DOAs: First, we compare the proposed C-MUSIC spectrum with the standard MUSIC spectrum by using well-separated sources, whose DOAs and , respectively. Since are set as in such case, one of the virtual sources of overlaps the true source of , and vise verse. It can be seen clearly from Fig. 8 that there are only virtual sources for C-MUSIC in such scenario and the other virtual sources are lost. However, since the proposed method involves a further process of originality checking, the true DOAs in this scenario can also be estimated by maximizing , which is shown in Table VIII. 6) Experiment 6: Comparison of DOA Estimation MSEs With Two Closely-Spaced Candidate Sources: To see more clearly the performance of the proposed C-MUSIC algorithm in the worst scenario, where the candidate sources of C-MUSIC are closely-spaced. We set the DOAs of the two true source as and , respectively and set as a control parameter for the performance comparison. Thus, the virtual source of lies close to the true source of and vise verse. and plot the MSEs against the SNR in Fig. 9 We fix and then fix the and plot the MSEs against the number of snapshots in Fig. 10. It can be seen from the figures C-MUSIC shows a comparable performance to MUSIC as increases. For small , the proposed method shows a worse performance as compared to other cases. However, as it is seen from the figures that

Fig. 8. MUSIC- and C-MUSIC- spectrum, where , and sources at and are used on a MRLA.

TABLE VIII ORTHOGONALITY CHECKING FOR EXPERIMENT 4

Fig. 9. MSEs against the SNR, where and are used on a MRLA.

and

sources at

both the experimental and the theoretical MSEs of C-MUSIC in such worst scenario are very small (smaller than rad as shown in Figs. 9 and 10), therefore, C-MUSIC makes an efficient trade-off between complexity and MSEs. 7) Experiment 7: Comparison of Complexities When C-MUSIC and MUSIC Show the Equal MSE Performance: In this experiment, we show the difference in the computational complexity between the proposed method and conventional MUSIC when their estimation performances are equal. We fix for C-MUSIC, and set as a control parameter for


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Fig. 10. MSEs against the number of snapshots, where sources at and are used on a MRLA.

and

Fig. 12. MSEs against the SNR, where and are used on a MRLA.

and

sources at

of higher MSEs. Therefore, we can see that the relationship between the proposed C-MUSIC and the standard MUSIC is similar to the pros and cons between the Mini-Norm and MUSIC methods [5], [29]. For Mini-Norm [5], only one transformed vector of the noise subspace is used while the C-MUSIC algorithm uses the intersection noise subspace of NLSC, whose dimensions decrease as increases. In our view, this is the reason why the accuracy of C-MUSIC gets worse while the resolution of C-MUSIC for closely-spaced true sources gets higher as increases. VII. CONCLUSIONS

Fig. 11. MSEs against the SNR, where is used on a MRLA.

and

source at

MUSIC such that the accuracy of the latter is comparable to the former. As it is shown clearly in Figs. 11 and 12 that C-MUSIC with sensors and sectors obtains the same performance as MUSIC with sensors. Assuming there are spectral points, we can compute by using (52) and (54) that in Fig. 11, and in Fig. 12. Hence, C-MUSIC is more efficient than MUSIC without degrading the accuracy. Comparing the first simulation to the second one, we see that for small numbers of snapshots and low SNRs, the MSEs of C-MUSIC in the second simulation are a little higher than those in the first simulation. However, as the number of snapshots or the SNRs increase, the new method with the worst scenario shows a similar performance to the first simulation. On the other hand, it is worth noting that the improved resolution probability of C-MUSIC for closely-spaced true sources comes at the price

A novel computationally efficient method for DOA estimation with no dependence on array configurations is presented based on a newly developed C-MUSIC spatial spectrum, which can generate spectral peaks at the true DOAs and several virtual DOAs simultaneously. Interestingly, the standard MUSIC is a special case of C-MUSIC and consequently, it is proved that the MSE expression for DOA estimation by the standard MUSIC given in [29] is also a special case of that by C-MUSIC given in the present work. The essence of the new technique is to substitute the intersection of NLSC for the original noise subspace of the standard MUSIC. However, since the existing algorithms for getting the intersection of several subspaces are either of huge complexity or only applicable for several specified cases, a new method is also proposed in the present work to exploit the noise subspace for C-MUSIC. It is shown by theoretical and experimental results that the new method has a much lower complexity as well as an improved resolution probability for closely-spaced sources as compared to MUSIC, and hence it trades-off accuracy efficiently by computational complexity and resolution. The future research includes developing C-MUSIC with fewer sensors. APPENDIX A PROOF OF THEOREM 1 Since columns of

and ,

and are spanned by the , respectively. and , and can be expressed as a

1926


linear combination of the columns of and , respectively. Therefore, we obtain by using (23) and (29) that

Now, let us prove the second part of theorem 2. Suppose , then we have , . is the orthogonal projection onto , we must Since have , which implies that . Therefore, we obtain

(A.1) (A.2) where

and are unknown coefficients. Using the facts and , we have

Thus, we have , and therefore, is a arbitrary vector of , we have

(B.1) . Since (B.2)

On the other hand, assume

, then we obtain

(A.3) (B.3) which means that (A.4) Because

Hence, we have (B.4)

, we have (A.5)

On the other hand, it follows from (23) that the is given by of

-th element

Note that have

is the projection of onto . Therefore, we further obtain

, we must

(B.5) (A.6) and are the -th and -th row of where Combining (A.5) and (A.6) leads to

, respectively.

Therefore, we have . This implies that , , and we finally have . As is a arbitrary vector of , we have (B.6)

(A.7) Similarly, we can prove that (A.8) Thus, it follows from (A.7) and (A.8) that

(A.9) Combining (A.4) and (A.9) gives which completes the proof.

,

It is implied by (B.2) and (B.6) that and the proof is completed.

,

APPENDIX C PROOF OF THEOREM 3 , First, let us prove the consistency of the estimates by C-MUSIC. We start by examining the performance of the estimated orthogonal projection onto at high SNR. Using , can be expressed by

(C.1) APPENDIX B PROOF OF THEOREM 2 In this appendix, we prove theorem 2. By using , , , it can be easily proved that . On the other hand, we obviously have . is an idempotent Hermite matrix, which Therefore, means that is the orthogonal projection onto , , and the first part of theorem 2 is proved.

is neglected. where the second order term Since the elements of are Gaussian random variables with zero mean, we have and therefore

(C.2)


1927

where denotes the real part of the embraced matrix. Substituting (C.2) into (C.1), we have

and are the -th column and the -th row of , where is the -th element of . respectively, and , we have Proof: It is easily seen that , (D.3)

(C.3) Therefore, the C-MUSIC null-spectrum of (28) converges to the true one with probability one, and the minima of is achieved if and only at high SNR. Thus, the estimates approach the if true parameters with probability one at high SNR. at high SNR are also unbiased esSecond, let us prove timates for the true parameters . As is obtained by min, and it is a consistent estimate for at imizing high SNR, we can obtain the second-order approximation of the about the true value as follows derivative of (see [29]–[33], and references therein) (C.4) where the second- and higher- order terms are neglected. The with refirst- and second-order derivatives of and spect to are defined as

is the -th column of . Expanding as the where , expanding as the weighted weighted columns of and using (2), we have rows of

(D.4) and are the -th element of where ment of , respectively. In a similar way, we obtain

and -th ele-

, respectively. By performing a forward derivation, we can obtain that (C.5) (C.6) where the second order term is neglected in (C.6). From (C.4)–(C.6), we can obtain by performing a straightforward derivation which leads to the first-order exas pression for the estimation error

(C.7) where in the denominator of (C.7) can be replaced by the without affecting the performance of estimate true one (see [29]–[33], and references therein). Substituting (C.1) into , we have (C.7) and using the fact

(D.5) which completes the proof of (D.1) and (D.2). Now, let us derive the MSE expression for DOA estimation by C-MUSIC. Here, we also need the following equation on the product of any given three matrices , and .

(C.8) It follows from (C.2) that . Therefore, unbiased at high SNR, and the proof is completed.

are

(D.6) where is the -th column of , is the -th row of and is the -th elment of . Substituting (63) into (C.8) as well as using the notations of (65)–(68), we have

APPENDIX D PROOF OF THEOREM 4 In this appendix, we derive the MSE expression for DOA estimation by C-MUSIC with the assumption of high SNR. First, let us prove the following two equations. (D.1) (D.2)

(D.7)

1928

where


and

In a similar way, it can be proved that

are defined as

(D.13) (D.8) Therefore, it follows from theorem 3 and (D.7) that

(D.14) where

and

are defined as follows (D.15) (D.16)

Now, let us prove the following two equations (D.17) (D.9)

Proof: We begin the proof by defining (D.18)

Expanding as the weighted sum of the columns of , expanding as the weighted sum of the and using (D.1), we have rows of

(D.19) Thus, by using

,

can be rewritten as

(D.20) (D.10) and are the -th and -th element of where respectively. In a similar way, we can prove that

and

Using the fact

, we have

, (D.21)

(D.11) It follows form (D.10) and (D.11) that the first term of (D.9) is zero. Now, consider the second term of (D.9). Using (D.2) and is a scalar variable, we have (D.6) and noting that

where is the eigenvalue of , which is associated with On the other hand, it is easily to prove that

.

(D.22) and is the -th column of and , respectively, where as is shown in (29). Substituting (D.21) into (D.20) and using (D.22), we have

(D.23)

(D.12)

Since must have

is the intersection subspace of NLSC, we , . On the


1929

other hand, it is shown by (31) that , and we obtain Therefore,

.

Substituting (E.4) and (E.5) into (57), we have (E.6)

(D.24) Substituting (D.23) and (68) into (D.13) and using (D.24) gives

On the other hand, using the fact substituting (E.1)–(E.3) into (67) and (68) leads to

and

(E.7) where

. Therefore, we have (E.8)

(D.25) Similarly, we can prove that

can be rewritten as follows

Substituting (E.8) into (64) and using (E.6), we have

(D.26) (E.9) Therefore, by using (D.24), we have

Using theorem 2, (D.6) as well as (E.3), we have

(D.27) Thus, the proof of (D.17) is completed. Substituting (D.10)–(D.12) and (D.17) into (D.9) gives (64), which completes the proof of theorem 4. APPENDIX E PROOF OF (69) In this appendix, we derive (69). Throughout this appendix, . It is easily the compression times for C-MUSIC is set as , is given by seen that when .. . For any given matrix

..

.

.. .

(E.1)

, we obtain form (D.6) that

(E.2) is the sum of all the elements of the embraced where matrix. It follows from (E.1) that and consequently, , , . Thus, we further have (E.3) Using the facts (58) and (59) that

as well as

, we obtain from (E.4) (E.5)

(E.10) Substituting (E.10) into (E.9) gives (69), which completes the proof. REFERENCES [1] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276–280, Mar. 1986. [2] A. G. Jaffer, “Maximum likelihood direction finding of stochastic sources: A separable solution,” Proc. ICASSP, vol. 5, pp. 2893–2896, May 1988. [3] A. Swindlehurst and M. Viberg, “Subspace fitting with diversely polarized antenna arrays,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1687–1694, Dec. 1993. [4] A. Paulraj, R. Roy, and T. Kailath, “A subspace rotation approach to signal parameter estimation,” IEEE Trans. Signal Process, vol. 74, pp. 1044–1046, Jul. 1986. [5] R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 1, pp. 134–139, Jan. 1983. [6] H. Lee and M. Wengrovitz, “Resolution threshold of beamspace MUSIC for two closely spaced emitters,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 9, pp. 1545–1559, Sep. 1990. [7] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp. 720–741, May 1989. [8] Y. Zhang and B. P. Ng, “MUSIC-Like DOA estimation without estimating the number of sources,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1668–1676, Mar. 2010.

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[31] P. Stoica and K. C. Sharman, “Maximum likelihood methods for direction-of-arrival estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 7, pp. 1132–1143, 1990. [32] P. Stoica and T. Soderstrom, “Statistical analysis of a subspace method for bearing estimation without eigendecomposition,” Proc. Inst. Electr. Eng. F, vol. 139, no. 4, pp. 301–305, 1992. [33] J. Xin and A. Sano, “Computationally efficient subspace-based method for direction-of-arrival estimation without eigendecomposition,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 876–893, 2004. [34] C. Chambers, T. C. Tozer, K. C. Sharman, and T. S. Durrani, “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. Fenggang Yan (S’12) was born in Shaanxi province, China, in February 1982. He received the B.E. and M.S. degrees in information and communication engineering from Xi’an Jiaotong University in 2005, Xi’an, China, and The Graduate School of Chinese Science of Academic in 2008, Beijing, China, respectively. Since July 2008, he became a member 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. In March 2011, he began working toward the Ph.D. degree with the Department of Electronics information Engineering, Harbin Institute of Technology, Harbin, China. His current research interests include array signal processing and statistical performance analysis. Ming Jin was born in Liaoning Province, China, in 1968. He received the B.E., M.S., and Ph.D. degrees in information and communication engineering from Harbin Institute of Technology, China, in 1990, 1998 and 2004, respectively. From 1998 to 2004, he was with the Department of Electronics Information Engineering, Harbin Institute of Technology. Since 2006, he has been a Professor with The School of Information and Electricity Engineering, Harbin Institute of Technology, Weihai. His current interests are in the areas of array signal processing, parallel signal processing, and radar polarimetry. Xiaolin Qiao was born in the Inner Mongolia Autonomous Region, China, in June, 1968. He graduated from the Department of Electronics Information Engineering, Harbin Institute of Technology, Harbin, China, and received the B.E., M.S., and Ph.D. degrees in information and communication engineering in 1976, 1983, and 1991, respectively. He was with the Department of Electronics information Engineering, Harbin Institute of Technology, from 1983 to 1993. Since 1994, he has been a Professor with The School of Information and Electricity Engineering, Harbin Institute of Technology. During 1994–2011, he was the president of the Harbin Institute of Technology at Weihai, Weihai, China. During the past 15 years, he has authored or coauthored nearly 70 publications. His research interests are in the areas of signal processing, wireless communication, special radar, parallel signal processing, and radar polarimetry.