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Xia, X-G. Discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Transactions on Signal Processing, 48, 11 (Nov. 2000), 3122—3133. Wang, G. and Bao, Z. ISAR imaging of maneuvering targets based on chirplet decomposition. Optical Engineering, 38, 9 (Sept. 1999), 1534—1541. Lo´ pez-Risuen˜ o, G. and Grajal, J. Multiple signal detection and estimation using atomic decomposition and EM. IEEE Transactions on Aerospace and Electronic Systems, 42, 1 (Jan. 2006), 84—102. Mallat, S. G. and Zhang, Z. Matching pursuits with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41, 12 (Dec. 1993), 3397—3415. Guo, J., Liu, G., and Yang, X. A novel matching pursuit algorithm with adaptive subdictionary. In Proceedings of the 9th International Conference on Signal Processing, Beijing, China, Oct. 2008, 207—210. Hough, P. V. C. Method and means for recognizing complex patterns. U.S. Patent 3069654, 1962. Jost, P., Vandergheynst, P., and Frossard, P. Tree-based pursuit: Algorithm and properties. IEEE Transactions on Signal Processing, 54, 12 (Dec. 2006), 4685—4697. Djuric, P. M. and Kay, S. M. Parameter estimation of chirp signal. IEEE Transactions on Signal Processing, 38, 12 (Dec. 1990), 2118—2126. Dhanoa, J. S., Hughes, E. J., and Ormondroyd, R. F. Simultaneous detection and parameter estimation of multiple linear chirps. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 6, Apr. 2003, VI 129—132. Krstulovic, S. and Gribonval, R. MPTK: Matching pursuit made tractable. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, May 2006, 496—499. Bultan, A. A four-parameter atomic decomposition of chirplets. IEEE Transactions on Signal Processing, 47, 3 (Mar. 1999), 731—745. O’Neill, J. C. and Flandrin, P. Chirp hunting. In Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, Oct. 1998, 425—428. Dachs, A. and Jager, G. D. Filtering in Hough space to enhance or detect linear features. In Proceedings of the 1990 IEEE South African Symposium on Communication and Signal Processing, 164—169.
Coherent Source Localization: Bicomplex Polarimetric Smoothing with Electromagnetic Vector-Sensors
The work presented here considers coherent source localization with bicomplex. A new polarimetric smoothing variant is proposed by using bicomplex modeled subarrays obtained from complete electromagnetic vector-sensor array, and a MUSIC-like algorithm is further developed. The identifiability, computational complexity, and the choice of selection vectors for the proposed method are also addressed. Simulations show that the proposed method can provide better direction-of-arrival estimates than the complex methods in perturbations caused by noise, short data, and model errors.
I. INTRODUCTION Direction-of-arrival (DOA) estimation with electromagnetic (EM) vector-sensors has received growing interest in the past decades. A “complete” EM vector-sensor comprises six EM sensors (for example, orthogonally oriented short dipoles and small loops arranged in a collocated or distributed manner), and is able to measure complete electric and magnetic field components induced by an EM incidence [1, 2]. An “incomplete” EM vector-sensor such as tripole and crossed dipole comprises only a subset of the above-mentioned six EM sensors and is of great interest in some practical applications [3, 4]. In the last two decades, many theoretical issues associated with EM vector-sensors have been investigated and numerous algorithms for DOA and polarization estimation have been developed. For example, Nehorai and Paldi have worked out the Cramer-Rao bound (CRB) on DOA estimation of stochastic sources for both EM vector-sensor array and single EM vector-sensor, as well as a simple cross-product-based DOA estimator using a single EM vector-sensor [1]. The CRB for deterministic pure-tone sources was derived in [5]. Source tracking algorithms for one or multiple sources were Manuscript received February 25, 2010; revised August 20 and December 15, 2010; released for publication January 21, 2011. IEEE Log No. T-AES/47/3/941800. Refereeing of this contribution was handled by J. Lee. This work was supported by the National Natural Science Foundation of China under Contracts 60672084, 60602037, 61072098, and 60736006, and by the Fundamental Research Fund for the Central Universities of China. c 2011 IEEE 0018-9251/11/$26.00 °
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proposed in [6], [7]. The problem of polarimetric modeling with EM vector-sensors was addressed in [8]. The identifiability for one or several complete EM vector-sensors was studied in [9], [10], and was further extended to the fourth-order cumulant domain in [11]. Maximum likelihood strategy was considered for vector-sensors in [12], [13]. The eigenstructure-based algorithms using EM vector-sensors, such as ESPRIT and MUSIC, have been extensively investigated in the literature. For example, Li [2] proposed applying ESPRIT to an array of complete EM vector-sensors, wherein the multiple rotational-invariance property among the dipole and loop outputs was exploited for DOA and polarization estimation. Wong refined Li’s method in [14], and proposed ESPRIT-based methods for single EM vector-sensor [15], or sparse EM vector-sensor array [16]. A MUSIC-like algorithm self-initiated with coarse DOA estimates obtained from ESPRIT was proposed in the spatio-polarizational beamspace [17]. The MUSIC scheme was also revised for sparse EM vector-sensor arrays in [18], and root-MUSIC was extended to EM vector-sensor arrays in [19]. In addition, the subspace fitting technique was addressed in [20], [21], and the virtual rotational-invariance property present in higher order statistics was exploited for vector-sensors [22]. DOA estimation of near-field sources with EM vector-sensors, contrary to the conventional far-field source localization problem, was addressed in [23]. The use of EM vector-sensors in airborne array systems was studied in [24], with particular concerns on the remedy of manifold perturbations. Tensorial methods featuring the use of tensor decomposition techniques (such as parallel factor analysis) were proposed in [25]—[27]. Furthermore, the existence of multipath propagations that are usually encountered in real world (for example, the environment where lots of reflection, refraction, and scattering happen) has been considered in applications of EM vector-sensors [28—32]. In these applications, signals impinging from distinct directions are strongly correlated and thus are usually modeled as coherent signals (fully correlated signals). Moreover, since the rank deficient covariance matrix of coherent sources does not satisfy the full rank requirements of many DOA estimators, a rank restoration scheme such as smoothing is usually carried out as a preprocessing step. In particular, [28] proposed to divide the array of complete EM vector-sensors along the polarimetric dimension into six subarrays with identical spatial configuration, and to perform smoothing among the auto-covariance matrices of these subarray signals as a preprocessing procedure to the following DOA estimation scheme. Reference [29] modified the scheme given in [28] by using nonuniform weights of auto-covariance matrices in smoothing for the purpose of noise cancellation, and [30] proposed to modify the standard complex CORRESPONDENCE
polarimetric smoothing algorithm by using both auto-covariance and cross-covariance matrices in either element domain or subspace domain with nonuniform weights, with some special guiding principles such as noise cancellation or matrix diagonalization. In [31] wideband coherent signals were considered and smoothing was conducted in the polarimetric-time-frequency domain. In [32] the coherent source localization problem was solved by using a specific array geometry. Recently, some efforts have been devoted to formulating the output of vector-sensors within a hypercomplex framework [33—37], wherein the vectorial structure of each vector-sensor is arranged into a hypercomplex scalar with one real part and multiple imaginary parts. Particularly, a quaternion version of the singular value decomposition technique was applied to real-valued polarized wave separation [33] with three-component vector-sensors; MUSIC was extended to the domains of quaternion, biquaternion, and quad-quaternion in [34], [35], and [36], respectively; and ESPRIT was revised within the quaternion framework for a spatially shift-invariant array of crossed dipoles [37]. In these applications, the local vector components of a vector-sensor array are retained and operated in a compact hypercomplex manner, resulting in a more elegant formalism. In addition, due to the stronger constraints that hypercomplex orthogonality imposes on hypercomplex vectors, these methods are shown to offer stronger robustness to array model errors than their complex-based counterparts [34—37]. However, the above-mentioned hypercomplex methods have not taken into consideration the case of coherent signals. In this paper we consider polarimetric smoothing [28—30] within the bicomplex framework. More specifically the array of complete EM vector-sensors is divided into several subarrays with identical spatial configuration. Each element of these subarrays is a two-component sub-vector-sensor selected from the six components of a complete EM vector-sensor, and the output signals of these subarrays could further be represented with bicomplex vectors. Then, after smoothing is carried out among the bicomplex covariance matrices of the subarray outputs, a MUSIC-like algorithm implementing bicomplex manipulations is performed to obtain the DOA estimates. Unlike other existing works based on hypercomplex [33—37], we herein consider a different hypercomplex algebra, namely bicomplex, for the representation and processing of EM vector-sensor array signals. The motivation behind our choice is that bicomplex multiplication is commutative, and this property is crucial for the derivation and analysis of our proposed method, as explained in more detail later on. The rest of the paper is organized as follows. Section II introduces some bicomplex algebra 2269
prerequisites. Section III presents the bicomplex measurement model, the proposed algorithm, and discussions on some related theoretical issues. In Section IV the performance of the new algorithm is demonstrated with simulations. Finally, this paper is concluded in Section V. II. BICOMPLEX ALGEBRA PREREQUISITES
DEFINITION 1 (Bicomplex and Bicomplex Matrix) A bicomplex number b 2 D is defined as ¢ b = b00 +~ib01 + ~j (b10 +~ib11 )
b = c0 + ~j c1 = S(b) + V(b)
DEFINITION 2 (Conjugation and Transposes) The total conjugation of B = C0 + ~j C1 2 DM£N is given by (3)
where C¤n denotes the complex conjugation of Cn 2 CM£N , (n = 0, 1). There also exist some other definitions of conjugation [42, 44]. Moreover, the transpose and total conjugated transpose are defined as
CH 0
(4)
¡ ~jCH 1 (5)
CTn
CH n
where and are the complex transpose and conjugated transpose of Cn , n = 0, 1, respectively. 2270
i=1 j=1
Generally jabj 6= jaj jbj for a, b 2 D, so bicomplex does not form a normed algebra under the above definition of norm. Other definitions of bicomplex norms could be found in [44]. Moreover, vectors a, b 2 DN are mutually orthogonal if jaH bj = 0. We note herein that the bicomplex vector orthogonality imposes stronger constraints on the vector components if we follow a similar analysis as that in [35].
(2)
¢
Transpose: BT = CT0 + ~j CT1
(7)
where ªK = [IK , ¡~j IK ], IK is a K £ K identity matrix, (K = M, N), and · ¸ C0 C1 ¢ ÂB = : ¡C1 C0
part of b, and V(b) = b ¡ S(b) is the vector part. We can further define bicomplex matrix B 2 DM£N as the matrix with bicomplex entries. Moreover, addition and multiplication extend naturally to the bicomplex (matrix) case, and thus are not addressed here. We note that bicomplex is multiplicatively commutative, due to the commutativity of ~i and ~j .
B¯ = C¤0 ¡ ~j C¤1
kbk = jbH bj v uM N uX X kBk = t jB(i, j)j2 :
(1)
¢ ¢ where cm = bm0 +~ibm1 , m = 0, 1, S(b) = b00 is the scalar
Total Conjugated Transpose: B = B =
In addition, the norms of vector b 2 DN , and matrix B 2 DM£N could be defined as
DEFINITION 4 (Adjoint Matrix) A bicomplex matrix B = C0 + ~j C1 2 DM£N could be linked to a unique complex adjoint matrix ÂB 2 C2M£2N via the following equation: (8) B = 12 ªM ÂB ªNH
where bmn 2 R (m, n = 0, 1), ~i and ~j are imaginary units such that ~i2 = ~j 2 = ¡1, ~i ¢ ~j = ~j ¢~i. In addition, b can also be expressed as
¯T
(6)
¢
Bicomplex belongs to the family of hypercomplex commutative algebras (or multicomplex algebras). It was first discovered by Segre in 1892 [38—41], and was also denoted as reduced biquaternion in some other related works [42—44]. Different from quaternions, bicomplex is a commutative and associative algebra with zero-divisors. In this section we review the algebra of bicomplex with emphasis on matrix operations that are crucial for our proposed algorithm. Interested readers could refer to [38]—[44] for more details.
H
DEFINITION 3 (Norms and Vector Orthogonality) The norm of b = b00 +~ib01 + ~j (b10 +~ib11 ) 2 D is q ¢ ¯ = b2 + b2 + b2 + b2 : jbj = S(bb) 00 01 10 11
The adjoint matrix could be considered as the complex version of the matrix representation for bicomplex [40, 43, 44]. Also, the properties of adjoint matrix could be obtained similarly from the biquaternion case by taking bicomplex as reduced biquaternion [42—44]. For example, we have the following lemmas: LEMMA 1 ÂAB = ÂA ÂB , where ÂAB , ÂA , ÂB are the adjoint matrices of AB, A, B, respectively. LEMMA 2 A 2 DM£M has a multiplicative inverse A¡1 2 DM£M , such that AA¡1 = A¡1 A = IM , if and only if its adjoint matrix ÂA 2 C2M£2M is invertible. The proofs for these two lemmas could be obtained similarly to the biquaternion case [35]. In addition, from Lemma 2 we note that a non-zero bicomplex scalar b is invertible only when Âb is invertible, or det(Âb ) = c02 + c12 6= 0. In other words, division for fb = c0 + ~j c1 2 D j c02 + c12 = 0g cannot be uniquely defined, and thus these numbers are named zero-divisors. DEFINITION 5 (Rank) The rank of B 2 DM£N , denoted by rank(B), is defined as the largest value of R · min(M, N), such that there exists at least
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one selection of R linearly independent column (row) vectors of B (the bicomplex linear independence is similarly defined as the quaternion case [33, 34]). In addition, denoting ÂB as the adjoint matrix of B, it can be easily proven that rank(B) = 1=2rank(ÂB ). DEFINITION 6 (Eigenvalue Decomposition of Hermitian Bicomplex Matrix) The bicomplex eigenvalue decomposition1 (BC-EVD) of a Hermitian matrix B 2 DN£N (B is Hermitian if B = BH ) is given by 2N ¢X ¸n un uH (9) B= n n=1
where ¸n 2 R and un 2 DN are the nth eigenvalue and eigenvector, respectively, Bun = ¸n un , kun k = 1, and juH n1 un2 j = 0 for n1 6= n2 , where n, n1 , n2 = 1, 2, : : : , 2N. In addition, it could be proven similarly to the biquaternion case [35] that B and its adjoint matrix ÂB share the same eigenvalues, and that un = p ¡1 2 ªM vn , where vn is the nth eigenvector of ÂB . Therefore, the BC-EVD of B could be obtained by the eigenvalue decomposition of ÂB . In addition we have the following remarks on BC-EVD: REMARK 1 (Number of Eigenvalues) A rank R Hermitian matrix B has 2R non-zero eigenvalues. REMARK 2 (Lower Rank Approximation via Truncated Eigenvalue Decomposition) Lower rank approximation of a rank R Hermitian matrix B 2 DN£N is to find a matrix B0 2 DN£N with rank R 0 < R such that kB ¡ B0 k is minimized. This could be realized by truncating the eigenvalue decomposition of B. More exactly, by denoting ¸n as the nth largest eigenvalue of B, and un as the associated eigenvector, n = 1, 2, : : : , 2R, we could approximate B as B0 = P2R0 H n=1 ¸n un un (see Appendix I for the proof). III. PROPOSED ALGORITHM This section presents the proposed bicomplex polarimetric smoothing for coherent source localization. We set up the bicomplex model for complete EM vector-sensor arrays, and then present the details of the proposed algorithm. Remarks and discussions are also given to provide insights into the proposed method. A. Measurement Model Let (μ, Á) and (®, ¯) be the azimuth-elevation 2-dimensional (2D) DOA (see Fig. 1) and polarization of an EM signal, respectively, where 0 < μ · 2¼, jÁj · ¼=2, ¡¼=2 < ® · ¼=2, and j¯j · ¼=4. The complex 1 A more generalized eigenvalue decomposition scheme for an arbitrary square bicomplex matrix was proposed in [43]. Herein we only focus on the Hermitian case for our purpose.
CORRESPONDENCE
Fig. 1. Coordinates and angle definition.
steering vector of a complete EM vector-sensor then could be written as · ¸· ¸· ¸ pμ+¼=2,0
¢
aμ,Á,°,´ =
|
pμ,Á+¼=2
¡pμ,Á+¼=2
{z
Fμ,Á
pμ+¼=2,0
cos ®
}|
sin ®
cos ¯
¡ sin ® cos ®
~i sin ¯
{z Q®
} | {z } w¯
(10) ¢
T
where pμ,Á =[cos Á cos μ, cos Á sin μ, sin Á] denotes a unit vector with orientation defined by (μ, Á). By definition we have pμ+¼=2,0 = [¡ sin μ, cos μ, 0]T and pμ,Á+¼=2 = [¡ cos μ sin Á, ¡ sin μ sin Á, cos Á]T . Then (10) is actually equivalent to the model given in many existing works (e.g. the one used in [3]) if we substitute the expressions for pμ+¼=2,0 and pμ,Á+¼=2 into (10). Specifically, the polarization state is represented by orientation angle ® and ellipticity angle ¯. Moreover, signals with ¯ = §¼=4 are denoted as circularly polarized where the sign denotes the direction of spin. Moreover, the polarization state could also be represented by polarization amplitude angle 0 · ° · ¼=2 and polarization phase difference angle j´j · ¼, which could be linked to ®, ¯ as follows: tan 2® = tan 2° cos ´ sin ¯ = sin 2° sin ´:
(11)
By denoting q 2 R6 as the selection vector obtained by permuting [1, 2, 3, 4, 5, 6]T arbitrarily, we could further define three bicomplex scalars b1,μ,Á,°,´ , b2,μ,Á,°,´ , and b3,μ,Á,°,´ as follows: ¢ b1,μ,Á,°,´ = aμ,Á,°,´ (q1 ) + ~j aμ,Á,°,´ (q2 ) ¢
b2,μ,Á,°,´ = aμ,Á,°,´ (q3 ) + ~j aμ,Á,°,´ (q4 )
(12)
¢ b3,μ,Á,°,´ = aμ,Á,°,´ (q5 ) + ~j aμ,Á,°,´ (q6 )
where ql denotes the lth entry of q, and aμ,Á,°,´ (ql ) denotes the ql th entry of aμ,Á,°,´ , l = 1, 2, 3. By definition, we know that b1,μ,Á,°,´ , b2,μ,Á,°,´ , and b3,μ,Á,°,´ characterize the responses of three “nonoverlapping” sub-vector-sensors (see Fig. 2) each comprising two 2271
Fig. 2. Sub-vector-sensors.
components selected from the six components of the complete EM vector-sensor in the manner defined by the selection vector q. For an array of N complete EM vector-sensors, the spatial steering vector dμ,Á 2 CN is given by ~
T
~
~
T
T
dμ,Á = [ei¢2¼(k1 pμ,Á =¸) , ei¢2¼(k2 pμ,Á =¸) , : : : , ei¢2¼(kN pμ,Á =¸) ] (13) where kn is the position vector of the nth sensor, ¸ is the wavelength of the impinging signals, and pμ,Á = [cos Á cos μ, cos Á sin μ, sin Á]T . In the scenario that M far-field, narrowband signals are impinging, the bicomplex model for the output signal of the subarray comprising sub-vector-sensors of the same type, as given in (12), is defined as ¢
xl (t) =
M X
m=1
bl,μm ,Ám ,°m ,´m dμm ,Ám sm (t) + nl (t) | {z } | {z }
(14)
dm
bl,m
where (μm , Ám ) and (°m , ´m ) are the 2D DOA and polarization of the mth signal, sm (t) is the complex envelope of the mth signal, nl (t) is the bicomplex additive noise term for the lth subarray, ¢
B. Bicomplex Polarimetric Smoothing DEFINITION 8 (Bicomplex Covariance Matrix) The bicomplex covariance matrix of xl (t) 2 DN with zero-mean is defined as ¢
Rl (t) = E(xl (t)xH l (t))
1, 2, : : : , M. From (14) we note that a complete EM vector-sensor array is divided into three subarrays with three types of sub-vector-sensors given in (12). We herein name these three subarrays as polarimetric subarrays to indicate that the division is conducted along the polarimetric dimension. It is important to note that the above bicomplex model combines two of the six elements within an EM vector-sensor into one bicomplex scalar, and therefore is not the same as the complex model. In addition we note that the bicomplex model and the quaternion model [34] represent two different hypercomplex tools with distinct algebraic properties, both of which can be established by combining two elements of EM vector-sensor into one bicomplex/quaternion. In addition we have the following assumptions. A1) The sources are zero-mean, stationary, mutually coherent, and with identical signal power ¾s2 ; A2) The noises are zero-mean, stationary, spatially white, uncorrelated with the sources, and with identical noise power ¾"2 ;
(15)
where E(¢) denotes the mathematical expectation. Under the assumptions A1 and A2, we know that rm1 ,m2 = ¾s2 , and R",l = ¾"2 IN , where rm1 ,m2 denotes the covariance between the m1 th and m2 th signals, and R",l is the noise covariance matrix of the lth subarray, l = 1, 2, 3, m1 , m2 = 1, 2, : : : , M. Then according to (14) we have Rl = E(xl (t)xH l (t)) = ¾s2
¢
bl,m = bl,μm ,Ám ,°m ,´m , and dm = dμm ,Ám , l = 1, 2, 3, m =
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A3) The sources have distinct DOAs and any arbitrary K (K · N) spatial steering vectors associated with different DOAs are linearly independent. A4) The number of sources M is known and there are more vector-sensors than sources (N > M).
M M X X
2 ¯ bl,m1 dm1 dH m2 bl,m2 + ¾" IN
m1 =1 m2 =1
= ¾s2 [bl,1 d1 , bl,2 d2 , : : : , bl,M dM ] £ 1M£M [bl,1 d1 , bl,2 d2 , : : : , bl,M dM ]H + ¾"2 IN (16) where 1M£M denotes an M £ M matrix with all the elements equal to 1. By using the commutativity of bicomplex multiplications, we could commute bl,m and dm in (16) to further obtain 2 3 bl,1 6 7 .. 7 Rl = ¾s2 [d1 , : : : , dM ] 6 . 4 5 | {z } D
2¯ bl,1 6 £ 1M£M 6 4
..
3
. b¯ l,M
bl,M
7 7 [d , d , : : : , d ]H +¾ 2 I 5 | 1 2 {z M } " N DH
H 2 = D(¾s2 bl bH l )D + ¾" IN
(17)
¢
where bl =[bl,1 , bl,2 , : : : , bl,M ]T . We know from the rank definition of bicomplex matrices (Definition 5)
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that rank(¾s2 bl bH l ) = 1. Therefore, the dimension of the signal subspace identified from Rl via BC-EVD is smaller than the number of sources, and the signal subspace does not span the M-dimensional subspace spanned by d1 , d2 , : : : , dM . As a result the subspace-based methods such as MUSIC could not be implemented directly to Rl for DOA estimation. To solve this problem we consider smoothing over the polarimetric subarrays to restore the rank: 3 ¢X
R=
Rl = D
l=1
=
3 X H 2 (¾s2 bl bH l )D + ¾" IN l=1
D(¾s2 BBH )DH +¾"2 IN |
{z R0
}
(18)
¢
where B = [b1 , b2 , b3 ], and R0 = D(¾s2 BBH )DH is denoted as the signal part of the smoothed covariance matrix R. In addition we assume M · 3 and that B is full rank. Under assumptions A3 and A4 we know that D is full column rank, and we could further obtain rank(R0 ) = M. As a result, the signal subspace identified via truncated BC-EVD of R could be taken as a reasonable estimate of the subspace spanned by d1 , d2 , : : : , dM , and MUSIC-like algorithm could be carried out for DOA estimation by searching within the array manifold for the spatial steering vectors that fall into the estimated signal subspace (usually this is done by finding the steering vectors that are orthogonal to the noise subspace). We note that the bicomplex polarimetric smoothing (BPS) as given in (18) is conducted among three bicomplex sub-vector-sensors of which no common components are shared, and therefore was named nonoverlapping BPS. This scheme uses only three bicomplex subarrays for smoothing and thus the number of identifiable signals could not exceed three. To handle the problem where more than three coherent signals are present, an “overlapping” BPS scheme could be used. The idea is to redivide the complete EM vector-sensor, subject to a different selection vector q0 , to obtain another three bicomplex sub-vector-sensors, denoted as b4,μ,Á,°,´ , b5,μ,Á,°,´ , and b6,μ,Á,°,´ , respectively, which are mutually different from b1,μ,Á,°,´ , b2,μ,Á,°,´ , and b3,μ,Á,°,´ obtained with the selection vector q. Then, based on these six sub-vector-sensors we could obtain six bicomplex subarrays fxl (t) j l = 1, 2, : : : , 6g and the corresponding bicomplex covariance matrices fRl j l = 1, 2, : : : , 6g by (14) and (16), respectively. Therefore, BPS for 4 · P M · 6 could be revised as R = M l=1 Rl . Obviously, the matrix B = [b1 , b2 , : : : , bM ] must not contain collinear columns, so q0 should be selected in the way such that the obtained sub-vector-sensors fbl,μ,Á,°,´ j l = 1, 2, : : : , 6g are mutually different (for example, we may choose q0 = [q2 , q1 , q4 , q3 , q6 , q5 ]T ). However, this CORRESPONDENCE
consideration is not sufficient to guarantee the full rank of B, and the conditions under which B could be guaranteed full rank are addressed in the next subsection. After the above mentioned BPS is conducted, a MUSIC-like scheme could then be performed to R for DOA estimation, as follows. 1) Implement BC-EVD on R as given in (9); we would obtain 2N eigenvalues and eigenvectors. According to Remark 2, Section II, we know that the largest 2M eigenvalues and their associated eigenvectors form an estimation of R0 whose columns span the bicomplex signal subspace. On the other hand, the remaining 2N ¡ 2M eigenvectors u2M+1 , u2M+2 , : : : , u2N form an estimation for the vector bases that span the bicomplex noise subspace. 2) Calculate the noise subspace projector ¢ P2N ˜ Á) ˜ u uH . Then the angular parameters (μ, P= n=2M+1
n n
that minimize kPdμ,Á k are the estimates of the true DOAs, that is ˜ Á) ˜ = arg min kPd k (μ, μ,Á
(19)
μ,Á
where “k ¢ k” denotes the norm of a bicomplex vector, as defined in (7). In practical situations where only finite data are available, the covariance matrix Rl is estimated as follows: K 1X ˜ xl (tk )xH Rl = l (tk ) K
(20)
k=1
where xl (tk ) is the kth sample of xl (t), k = 1, 2, : : : , K, and K is the number of snapshots. C.
Discussion
In this subsection, we present some remarks to provide insights into the proposed DOA estimator. REMARK 3 (On the Identifiability of the Proposed Method) According to Subsection IIIB, the matrix B should be full rank to enable a successful identification of all the M coherent incidences. Herein we give some related results on the identifiability issue, the proofs of which are given in Appendix II. 1) The nonoverlapping BPS-MUSIC could identify up to three signals with distinct DOAs, subject to certain choices of the selection vector q. 2) The overlapping BPS-MUSIC could identify four signals with distinct DOAs, subject to certain choices of the selection vectors q and q0 , if the ellipticity angles of these four signals are not equal. 3) The overlapping BPS-MUSIC could identify five signals with distinct DOAs, subject to certain choices of the selection vectors q and q0 , if exactly two or three sources are circularly polarized with the same spin. 2273
4) The overlapping BPS-MUSIC could identify six signals with distinct DOAs, subject to certain choices of the selection vectors q and q0 , if exactly three signals are circularly polarized with the same spin. The results above show that the proposed method could identify up to six signals with different DOAs, if the signals’ polarization states satisfy certain conditions and the selection vectors are properly chosen. Moreover, it is important to note that the requirements of signal polarization states are sufficient conditions, and therefore signals that do not satisfy the above conditions may also be uniquely identified by our algorithm. We note also that the properly chosen selection vectors are crucial for the proposed algorithm, and this issue is addressed in Remark 4. REMARK 4 (On the Choice of the Selection Vector) From Remark 3 we note that the full rank condition of matrix B relies on the choices of the selection vectors. In addition, it is also desired to find an optimal selection vector under which the noise subspace could be most accurately estimated so as to provide the best DOA estimates. Therefore, we propose the following scheme to determine the selection vector (herein we consider only the nonoverlapping case, the selection vectors for the overlapping scheme could be chosen similarly): μ ¶ ¸2M ¡ ¸2M+1 q˜ = arg max (21) ¸1 q2¡ where ¡ is a set of selection vector candidates, and ¸1 , ¸2 , : : : , ¸2N are eigenvalues of R sorted in descending order among which ¸1 , ¸2M , ¸2M+1 are selected. By definition we could see that, if B is not full rank, both ¸2M and ¸2M+1 are then associated with the noise subspace, and this would result in ¢
a small value of ³ = (¸2M ¡ ¸2M+1 )=¸1 . Therefore
the scheme given in (21) excludes the choices of selection vectors for which B is rank deficient (we assume the DOAs and polarization states of impinging signals satisfy the identifiability requirements given in Remark 3). Moreover, for those selection vectors under which B is full rank, we note by definition that ¸2M is the smallest eigenvalue associated with the signal subspace, and ¸2M+1 is the largest eigenvalue associated with the noise subspace, and ³ then could be considered as a measure of the minimum distance (normalized by the largest eigenvalue ¸1 ) between the estimated signal subspace and noise subspace. Obviously with a larger ³, the noise subspace and the signal subspace are then more distinguishable from each other, and this would result in a more accurate estimation of the noise subspace. In addition, the globally optimal selection vector could be obtained 2274
if ¡ covers all the 60 possible candidates.2 However, this exhaustion procedure may be time consuming and a smaller set of candidates could be used in practice. REMARK 5 (On the Motivation of using Bicomplex) When compared with complex algebra, we note that hypercomplex algebra imposes stronger constraints on vector orthogonality according to [34]—[36], and thus bicomplex algebra-based methods (such as BPS-MUSIC) may be advantageous over complex-based methods, with regards to the robustness to errors caused by noise, short data length, or model errors. When compared with other hypercomplex algebras, we note that bicomplex distinguishes itself from quaternion, biquaternion, and quad-quaternion with the property of multiplicative commutativity, and this property is crucial for the proposed method. More precisely, we note that a key point of the proposed method is that the bicomplex covariance matrix Rl could be reformulated into H the form given in (17) (Rl = D(¾s2 bl bH l )D , for clarity we remove the noise term), wherein the matrix ¾s2 bl bH l in the “center” is multipled by D and DH from the “outside.” This special structure (with D and DH on the outside and ¾s2 bl bH l in the center) enables rank restoration via BPS by noting P that l ¾s2 bl bH l may be full rank, and also enables estimation of the noise subspace orthogonal to the one spanned by columns of D via BC-EVD. Furthermore, we note that the derivation of (17) from (16) requires that bl,m should commute with dm , and thus the multiplicative commutativity is crucial for our algorithm. For clearance, we next show that polarimetric smoothing may fail for coherent signals if quaternion is used instead of bicomplex. More exactly, according to [34] the quaternion covariance matrix is given by PM of the lth subarray P H ¤ q d r d q R0l = M m1 =1 m2 =1 l,m1 m1 m1 ,m2 m2 l,m2 , where ql,m1 is a quaternionic alternative for bl,m1 in (14) by using the quaternion model instead. In addition, in the presence of coherent signals such that rm1 ,m2 = ¾s2 we have P PM R0l = ¾s2 M q d dH q¤ = ¾s2 ¥l 1M£M ¥lH , P 0 m1 =12 Pm2 =1 l,m1 m1 H m2 l,m2 and l Rl = ¾s l ¥l 1M£M ¥l , where 1M£M denotes an M £ M matrix with all the elements equal to 1, and ¢ ¥l =[ql,1 d1 , : : : , ql,M dM ]. Since the multiplication is not commutative, ql,m does not commute with dm so that P 0 P 2 H l Rl = ¾s l ¥l 1M£M ¥l could not be modified into the structure desired by subspace-based methods (a full rank matrix in the center and matrices of vectors that span the signal subspace on the outside). This example shows that the quaternion-MUSIC could only be used for direction finding of incoherent signals. 2 The
order of sub-vector-sensors given in (12) has no influence on the performance of BPS-MUSIC. In addition, it also makes no difference for the proposed method if we reverse the order of the two components within each sub-vector-sensor simultaneously. Therefore, there exist 6!(3!2!) = 60 possible choices of the selection vector.
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TABLE I Computational Efforts for Covariance Matrix Estimation
BPS-MUSIC CPS-MUSIC CPSAC CPDS Q-MUSIC
Memory Requirements (real values)
Real Multiplications
Real Additions
Real Divisions
12N 2
48N 2 T
12N 2
24N 2 T
(48T ¡ 12)N 2
42N 2 12N 2 12N 2
84N 2 T 24N 2 T 48N 2 T
12N 2 8N 2 28N 2 8N 2 12N 2
(16T ¡ 8)N 2
(56T ¡ 28)N 2 (16T ¡ 8)N 2 (48T ¡ 12)N 2
TABLE II The Proposed Algorithm 1) Assume the number of coherent impinging signals is M, the array outputs are collected with N complete EM vector-sensors, taken at K distinct snapshots: fx(tk ), k = 1, 2, : : : , Kg, M · min(6, N). 2) Devise a set of selection vector candidates ¡ = fqg j g = 1, 2, : : : , Gg. For each candidate qg , if M · 3, divide the complete EM vector-sensor array into three arrays of nonoverlapping sub-vector-sensors according to (12) and (14); if 3 < M · 6, divide the ˜ for each subarray by (20), array into M arrays of overlapping sub-vector-sensors. Calculate the sampled covariance matrix R l obtain the smoothed covariance matrix R by (18), perform BC-EVD to R, and choose the selection vector q˜ that maximizes (21). ˜ that corresponds to the optimal selection vector q, ˜ use the eigenvectors associated with its 3) For the smoothed covariance matrix R smallest 2N ¡ 2M eigenvalues to calculate the noise subspace projector P, the DOA estimates are then obtained by (19).
Another important property of the considered bicomplex algebra is the existence of zero-divisors, which is often considered a serious drawback as division could not always be uniquely defined. However, we herein note that this potential drawback has no impact on the proposed algorithm as no “inverse” operations (scalar or matrix inverses) are used. REMARK 6 (Comparisons with Existing Polarimetric Smoothing-Based Methods) The proposed method could be considered as a direct extension of the standard complex polarimetric smoothing algorithm [28] (CPS-MUSIC) to the bicomplex domain, by noting that both methods use auto-covariance only, and that identical weights for all the auto-covariance matrices are used. An extension of the proposed method featuring the use of cross-covariance and nonuniform weights could also be obtained following the similar consideration addressed in [29], [30], with some special guiding principles (such as noise removal or matrix diagonalization). However, these issues are not the main focus herein, and thus are not further addressed. Another issue of interest is the complexity of the proposed algorithm. Herein we only consider the computational complexity involved in the estimation of subarray covariance matrices, as this stage best illustrates the complexity difference between different algorithms [34—36]. We assume that there are N EM vector-sensors, and T snapshots. Table I summarizes the comparison of the proposed BPS-MUSIC, CPS-MUSIC [28], complex polarimetric difference smoothing (CPDS) [29], complex polarimetric smoothing with both auto-covariance and cross-covariances (CPSAC) [30], CORRESPONDENCE
and quaternion-MUSIC (Q-MUSIC) [34], with regards to memory requirements and basic arithmetical operations. It is important to note that these algorithms were originally proposed with regards to different scenarios (e.g. CPSAC may not use all the auto- and cross-covariances for some special cases, Q-MUSIC could not be used for coherent source localization). Therefore, the complexity comparison only considers the general case. The details for the calculation of the subarray covariance matrices are given in Appendix III. From Table I we note that BPS-MUSIC and Q-MUSIC have equal computational complexity, while CPS-MUSIC and CPDS are the most computationally economical. The CPSAC scheme is the most computationally expensive, mainly due to the use of cross-covariance that is not used in other methods. Moreover, we should note that the above-shown computational complexity for BPS-MUSIC is based on the nonoverlapping scheme with one selection vector. If we consider the overlapping BPS and the scheme to optimally choose the selection vectors, the computational complexity of BPS-MUSIC will increase, as a cost of gain in coherent source localization performance. Taking the refinements addressed in Remark 4 into consideration, we summarize the proposed BPS algorithm in Table II. IV. SIMULATION RESULTS In this section we present some simulation results to illustrate the performance of the proposed BPS-MUSIC algorithm. All the statistics shown below are the average of the results obtained from 2275
200 independent trials. We assume the considered array comprises eight complete EM vector-sensors, arranged in “L” shape, each being separated from one another by the wavelength of impinging signals. The impinging signals are far-field, narrowband signals, and the noises are spatially white and uncorrelated with the signals. We use the overall root mean square error (RMSE) of azimuth estimation (assume that the elevation angles are known) to measure the performance of the considered methods as follows: v 3 u 200 X ¢ 1 Xu t 1 Â= [(μm ¡ μ˜n,m )2 ] (22) 3 200 m=1
n=1
where μm is the mth true azimuth, and μ˜n,m is the estimate of μm at the nth trial, m = 1, 2, 3, n = 1, 2, : : : , 200. In addition, the signal-to-noise ratio ¢ (SNR) is defined as SNR = 10 log10 (¾s2 =¾"2 ). In simulation 1 we compare the proposed BPS-MUSIC algorithm with Q-MUSIC [34] in the presence of coherent signals. It is important to note that Q-MUSIC was originally proposed for uncorrelated signals. Hence, to enable a clearer comparison of these two methods we add a smoothing preprocessing step similar to BPS before implementing Q-MUSIC. More exactly, this preprocessing step is carried out by replacing the bicomplex model used in BPS with the quaternion model, as is addressed in Remark 5, Subsection IIIC. We assume that three coherent signals are impinging from (μ1 , Á1 ) = (0± , 50± ), (μ2 , Á2 ) = (22± , 50± ), and (μ3 , Á3 ) = (39± , 50± ). The polarization states of the incident signals are (°1 , ´1 ) = (0± , 0± ), (°2 , ´2 ) = (60± , 90± ), and (°3 , ´3 ) = (90± , 180± ), respectively. In addition, SNR is fixed to 30 dB, the number of snapshots is fixed to 1000, and 20 independent trials are performed, with the selection vector q randomly selected in each run (herein we consider the nonoverlapping BPS-MUSIC). We plot the spectra of BPS-MUSIC and Q-MUSIC in Fig. 3. We can see that BPS-MUSIC can locate all three coherent sources very accurately while Q-MUSIC fails even if a similar smoothing preprocessing step is added. This observation coincides with our analysis in Remark 5, Subsection IIIC, that Q-MUSIC could not be used for coherent source localization due to the noncommutativity of quaternion multiplications. Next we compare the proposed BPS-MUSIC algorithm with complex methods such as CPS [28], CPDS [29], CPSAC [30] in simulations 2—6. It should be noted herein that these complex preprocessing methods could be followed by various direction finding schemes, such as propagator algorithm [29], MUSIC, or the hybrid DOA estimator integrating several existing techniques (for example: root-MUSIC, beamforming, cross-product, etc.) proposed in [30]. Therefore, to enable a clearer comparison we consider 2276
Fig. 3. Spectra of BPS-MUSIC and Q-MUSIC in presence of three coherent signals, SNR is 30dB, number of snapshots is 1000. Solid curves correspond to BPS-MUSIC, dashed curves correspond to Q-MUSIC.
only the MUSIC algorithm in the simulations. In addition the methods proposed in [31], [32] are not included in the comparison as the special requirements imposed in these methods are not the main concerns herein (recall that [31] requires the sources to be broadband, and [32] requires a special array configuration). Note also that CPSAC uses nonuniform weights with different purposes. Herein we only consider the general case wherein all the auto-covariance and cross-covariance matrices are used, and identical weights are assigned to them. We consider the scenario that there are three coherent signals impinging from (μ1 , Á1 ) = (0± , 50± ), (μ2 , Á2 ) = (22± , 50± ), and (μ3 , Á3 ) = (39± , 50± ), respectively. The polarization states of the incident signals are (°1 , ´1 ) = (0± , 0± ), (°2 , ´2 ) = (60± , 90± ), and (°3 , ´3 ) = (90± , 180± ), respectively. Two versions of nonoverlapping BPS-MUSIC are considered: BPS-MUSIC-1 randomly chooses the selection vector in different trials, while BPS-MUSIC-2 optimally chooses the selection vector among ten candidates ¡ = fqg j g = 1, 2, : : : , 10g that are randomly generated in each trial. In simulation 2 we fix the number of snapshots to 1000, and let SNR vary between ¡14 dB and 0 dB. The overall RMSE curves of the above-mentioned algorithms as well as the CRB [1] are plotted in Fig. 4. It was shown that at rather low SNR (¡14 dB—¡10 dB), BPS-MUSIC-2 slightly outperforms both BPS-MUSIC-1 and CPS-MUSIC, while providing considerably more accurate DOA estimates than CPDS-MUSIC and CPSAC-MUSIC. When SNR increases, the overall RMSEs of these methods become almost identical. In addition we note that CPS-MUSIC and BPS-MUSIC-1 provide almost identical accuracy of DOA estimation. The observations above indicate that BPS-MUSIC is able to provide stronger robustness to noise than the
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Fig. 4. Performance of BPS-MUSIC-1, BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, CPSAC-MUSIC and CRB versus SNR in presence of three coherent signals; number of snapshots is 1000.
Fig. 5. Performance of BPS-MUSIC-1, BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, CPSAC-MUSIC and CRB versus number of snapshots in presence of three coherent signals; SNR is ¡10 dB.
complex methods, by using the bicomplex formulation and a properly chosen selection vector.3 However, if the selection vector is randomly chosen, the average DOA estimation accuracy of BPS-MUSIC and CPS-MUSIC is almost equal. In simulation 3 we keep the simulation settings unchanged except that SNR is fixed to ¡10 dB and the number of snapshots varies between 200 and 3 The selection vector used in BPS-MUSIC-2 is not the globally optimal one, since it is only chosen among ten randomly generated candidates.
CORRESPONDENCE
500. The overall RMSE curves of all the considered algorithms and CRB versus the number of snapshots are plotted in Fig. 5. From Fig. 5 we note both BPS-MUSIC-1 and BPS-MUSIC-2 outperform CPS-MUSIC remarkably in short data length (200—350), and these three methods provide almost identical accuracy of DOA estimation when the number of snapshots exceeds 400. Moreover, BPS-MUSIC-1 and BPS-MUSIC-2 outperform CPDS-MUSIC and CPSAC-MUSIC in this scenario. This observation indicates that the proposed BPS-MUSIC is able to provide stronger robustness 2277
Fig. 6. Performance of BPS-MUSIC-1, BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, and CPSAC-MUSIC versus variance of model error in presence of three coherent signals; SNR is 5 dB, number of snapshots is 1000.
to the errors caused by short data length than the complex algorithms. In simulation 4 we compare the above-mentioned algorithms in the presence of model errors which are mainly caused by imprecise knowledge of the array’s spatial configuration. The simulation settings remain the same as simulation 2 except that SNR and the number of snapshots are fixed to 5 dB and 1000, respectively, and that the model error is taken into consideration. More exactly, we introduce the model error as p (23) d˜ m = dm + P"m where d˜ m and dm are the true and ideal spatial steering vectors associated with the mth signal, respectively. "m 2 CN is a stochastic vector drawn from the Gaussian distribution, and P is the variance of model error. We let P vary between 0 and 0.2, and the overall RMSE curves against the variance of model error are plotted in Fig. 6. From the figure we note that all the compared methods offer almost identical performance in the absence of model error (P = 0). Moreover, when mild model errors exist (P = 0:05—0.20), the performance of the complex algorithms and BPS-MUSIC-1 is much worse, while BPS-MUSIC-2 is still able to generate quite accurate DOA estimates. This simulation shows that the proposed BPS-MUSIC is less sensitive to the model errors than the complex methods, subject to a properly chosen selection vector. In simulation 5 we consider the performance of the compared algorithms against the source correlation. We keep the simulation settings the same as simulation 2 except that SNR and the number of snapshots are fixed to ¡5 dB and 1000, respectively, 2278
and that the source correlation varies between 0 and 1. The overall RMSE curves and CRB against the source correlation are plotted in Fig. 7. We note herein that BPS-MUSIC-2 slightly outperforms CPS-MUSIC and BPS-MUSIC-1 at all levels of source correlation. In addition all the considered algorithms provide close DOA estimation accuracy for coherent sources, while at low levels of source correlation CPDS-MUSIC and CPSAC-MUSIC remarkably underform BPS-MUSIC-1, BPS-MUSIC-2, and CPS-MUSIC. The observations further imply that with a properly chosen selection vector, the proposed BPS-MUSIC could provide more accurate DOA estimates than the complex methods at all levels of source correlation. Furthermore, we can also see that for weakly or mildly correlated sources (the incompletely correlated case), the advantages of BPS-MUSIC are even clearer. In simulation 6 we consider the ability of the compared methods to separate closely located signals. The simulation settings are kept the same as simulation 2, except that SNR and the number of snapshots are fixed to 15 dB and 1000, respectively, and the azimuth of the second source varies between 3± and 36± . Recall that the azimuths of the first and third sources are 0± and 39± , respectively; the second source in this scenario actually moves from near the first source to near the third one. Moreover, in addition to the overall RMSE used in simulations 2—5, the probability of detection (PD) is also used to measure the resolution of these methods, defined ¢ as PD = I 0 =I, where I 0 is number of trials that successfully detect all three incident coherent signals, and I is the total number of independent trials. The
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Fig. 7. Performance of BPS-MUSIC-1, BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, CPSAC-MUSIC and CRB versus source correlation in presence of three signals; SNR is ¡5 dB, number of snapshots is 1000.
Fig. 8. Performance of BPS-MUSIC-1, BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, and CPSAC-MUSIC versus azimuth of second source in presence of three coherent signals; SNR is 15 dB, number of snapshots is 1000. (a) Overall RMSE and CRB. (b) PD.
overall RMSE curves and PD curves are plotted in Fig. 8. From Fig. 8 we note that all the compared algorithms fail when the second source is close to the first source (μ2 = 3± ); when the distance between the first and the second signals gradually increases (μ2 = 6± —12± ), BPS-MUSIC-2, CPS-MUSIC, CPDS-MUSIC, and CPSAC-MUSIC are able to locate all the three sources precisely, while BPS-MUSIC-1 underperforms these methods. Moreover, in case that the three sources are distant enough (μ2 = 15± —27± ), all the compared algorithms provide almost equal performance with regards to both overall RMSE and PD. When the second source moves close to the third one (μ2 = 30± —36± ), we note that CORRESPONDENCE
BPS-MUSIC-2 provides almost equal performance as CPDS-MUSIC, and underperforms CPS-MUSIC and CPSAC-MUSIC with regards to both overall RMSE and PD, while BPS-MUSIC-1 underperforms the other algorithms. These observations illustrate that with a properly chosen selection vector, the proposed BPS-MUSIC is able to provide slightly inferior resolution of closely located sources than the complex algorithms. Simulations 2—6 generally illustrate that BPS-MUSIC is able to provide more accurate DOA estimates at all levels of source correlation in perturbations caused by noise, short data length, and model errors than the complex methods, and that BPS-MUSIC slightly underforms the complex 2279
ones with regards to the resolution of closely located sources. The advantages of BPS-MUSIC are mainly due to the stronger constraints that bicomplex vector orthogonality imposes on the vector components. More exactly, this property of bicomplex vector orthogonality may result in a more accurate estimation of the noise subspace spanned by orthogonal eigenvectors via BC-EVD, and thus improves the performance of the MUSIC-like algorithm. Moreover, we note that BPS-MUSIC-2 considered in simulations 2—6 only finds the suboptimal selection vector among ten randomly generated candidates in each trial. Therefore, improved performance of BPS-MUSIC could be expected if the global optimal selection vector is obtained among all the 60 candidates. However, this could be time consuming, and a fast scheme to obtain the global optimal selection vector will be the focus of our future work. In simulation 7 we consider the scenario where six coherent signals are impinging. We assume the elevation angles of all the six signals are all equal to 50± and known, and the azimuth angles of the six sources are μ1 = 10± , μ2 = 30± , μ3 = 60± , μ4 = 80± , μ5 = 120± , and μ6 = 150± , respectively. The orientation angles and ellipticity angles f(®m , ¯m ) j m = 1, 2, : : : , 6g that are given in (10) are used to represent the signal’s polarization states. We fix ®1 = ¡90± , ®2 = ¡45± , ®3 = 0± , ®4 = 45± , ®5 = 90± , ®6 = 60± , ¯2 = 0± , ¯4 = 88± , and ¯1 = ¯3 = ¯5 = 45± . In addition the same array is used as simulation 2, and SNR and the number of snapshots are fixed to 30 dB and 1000, respectively. The overlapping BPS-MUSIC is performed with 20 independent trials, in which the selection vector q is randomly chosen in each trial, and q0 is given by q0 = [q2 , q1 , q4 , q3 , q6 , q5 ]T (recall that q and q0 are two selection vectors for overlapping BPS, as addressed in Subsection IIIB), subject to the following two cases: 1) ¯6 = 25± , 2) ¯6 = 45± . We note that in the first case, there are exactly three sources that are circularly polarized with the same spin, while in the second case there are four circularly polarized sources with the same spin. Simulations of CPS-MUSIC, CPDS-MUSIC, CPSAC-MUSIC with 20 independent trials in the above-mentioned two cases are also taken. The 1D spectra of these methods are plotted in Fig. 9. From Fig. 9(a) it is observed that the proposed BPS-MUSIC could precisely locate all the six coherent sources in the first case. This is in accordance with our analysis in Remark 3, Subsection IIIC, that the overlapping BPS-MUSIC could identify six signals with distinct DOAs, subject to certain choices of the selection vectors q and q0 , if exactly three signals are circularly polarized with the same spin. However, it is also observed that the proposed BPS-MUSIC fails to identify the six coherent sources in the second case. This could be explained as follows: when there are four 2280
circularly polarized signals with the same spin, B = [b1 , b2 , : : : , b6 ] is guaranteed to be rank deficient, where bm , m = 1, 2, : : : , 6, are given in (17), such that the proposed algorithm fails to resolve all the six coherent sources. Moreover, similar behaviors for the complex methods can also be observed from Fig. 9(b)—(d). Thus we conclude that the maximal number of coherent sources that BPS-MUSIC, CPS-MUSIC, CPDS-MUSIC, and CPSAC-MUSIC are capable to resolve is six, conditioned that the polarization states of the sources satisfy certain requirements. V.
CONCLUSION
In this paper we have proposed a new algorithm for coherent source localization based on EM vector-sensor arrays within a bicomplex framework. This method divides a complete EM vector-sensor array along the polarimetric dimension into several subarrays, each being represented with the bicomplex formulism, and performs smoothing with all these subarrays for decorrelation. In addition, a MUSIC algorithm using bicomplex matrix operations is followed for direction finding. Discussion and simulation results have shown the following. 1) Only commutative algebra can be adopted for hypercomplex based polarimetric smoothing. This is the reason for our choice of bicomplex (commutative) instead of quaternion (noncommutative) in this work. 2) The proposed BPS-MUSIC could identify at least three coheret signals with distinct DOAs, and at most six coherent signals with distinct DOAs, subject to certain choices of the selection vectors, if certain requirements on the polarization states of the sources are satisfied. Particularly, we have provided for the first time the exact condition for which six coherent signals can be decorrelated by polarimetric smoothing. 3) BPS-MUSIC with properly chosen selection vectors could provide more accurate DOA estimates than the complex ones such as complex polarimetric smoothing, CPDS, and CPSAC, complex polarimetric in the presence of noise, short data length, model errors, and incompletely correlated signals. In this paper, we have presented a novel scheme for the optimum determination of the element selection vectors. 4) The potential drawbacks of BPS-MUSIC are mainly in the following two aspects: 1) the computational complexity of BPS-MUSIC is heavier than the complex methods; 2) BPS-MUSIC has no advantages over complex methods with regards to the ability to resolve close sources. APPENDIX I. PROOF OF REMARK 2 Assume B 2 DN£N is a Hermitian bicomplex matrix with rank R, and B0 2 DN£N is a rank R 0 matrix
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Fig. 9. Spectra of BPS-MUSIC, CPS-MUSIC, CPDS-MUSIC, and CPSAC-MUSIC in presence of six sources, SNR is 30 dB, number of snapshots is 1000. Solid curves correspond to case of three circularly polarized signals with the same spin are present, dashed curves correspond to case of four circularly polarized signals with same spin are present. (a) BPS-MUSIC. (b) CPS-MUSIC. (c) CPDS-MUSIC. (d) CPSAC-MUSIC.
obtained by truncating the BC-EVD of B as
Definition 4:
0
0
B =
2R X
¸n un uH n
(24)
n=1
where ¸n is the nth largest eigenvalue of B, and un is the associated eigenvector. Moreover, we denote the adjoint matrices of B and B0 by ÂB and Â0B , respectively. Then according to the properties of BC-EVD addressed in Definition 6 we know that ¸n is also the nth largest eigenvalue of ÂB , and the nth eigenvector vn of ÂB could be linked to p ¡1 un as un = 2 ªM vn . Therefore, we further have P 0 H 0 Â0B = 2R n=1 ¸n vn vn which indicates that ÂB could be obtained from ÂB via truncated EVD. Obviously, Â0B is the best rank 2R 0 approximation of ÂB in the sense that kÂB ¡ Â0B kF is minimized, where k ¢ kF denotes Frobenius norm in the complex domain. Moreover, denoting B = C0 + ~j C1 , and B0 = C00 + ~j C01 we have the following equation according to (6), (7), and CORRESPONDENCE
kB ¡ B0 k2 = kC0 ¡ C00 k2F + kC1 ¡ C01 k2F = 2¡1 kÂB ¡ ÂB0 k2F :
(25)
As a result kB ¡ B0 k is also minimized. This shows that truncating the BC-EVD of B yields its best rank R 0 approximation. APPENDIX II. PROOF OF THE RESULTS GIVEN IN REMARK 3 Assume B = C0 + ~j C1 , and then according to Definition 5 we have rank(B) = rank(ÂB )=2, where ÂB is the adjoint matrix of B: · ¸ C0 C1 : (26) ÂB = ¡C1 C0
First we consider the nonoverlapping BPS where B = [b1 , b2 , b3 ], bl = [bl,1 , bl,2 , : : : , bl,M ]T , and bl,m = bl,μm Ám °m ´m is given in (12), l = 1, 2, 3, m = 1, 2, : : : , M. ¢
T By further denoting h(0) m =[am (q1 ), am (q3 ), am (q5 )] , and
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¢
T h(1) m =[am (q2 ), am (q4 ), am (q6 )] where am = aμm ,Ám ,°m ,´m and qk is the kth element of the selection vector (0) (0) T q, we have C0 = [h(0) 1 , h2 , : : : , hM ] , and C1 = (1) (1) (1) T [h1 , h2 , : : : , hM ] . According to [10, Theorem 2], every three complex steering vectors of a complete EM vector-sensor with distinct DOAs are linearly independent, and therefore rank[C0 , C1 ] = M if M · 3. Moreover, it is possible to design a selection vector q such that C0 contains M linearly independent columns, and rank(C0 ) = rank[C0 , C1 ] = M. As a result for this selection vector and assuming M · 3, we have rank(B) = rank(ÂB )=2 = rank(C0 ) = M which completes the proof of the first result in Remark 3. Next we consider the overlapping BPS scheme, where B = [b1 , b2 , : : : , bM ], bl = [bl,1 , bl,2 , : : : , bl,M ]T , b1,m , b2,m , b3,m are obtained by (12) with selection vector q, and b4,m , : : : , bM,m are associated with selection vector q0 , 4 · M · 6, l = 1, 2, : : : , M. By ¢ 0 denoting h(0) m =[am (q1 ), am (q3 ), am (q5 ), am (q1 ), : : : , ¢ 0 am (q02M¡7 )]T , and h(1) m =[am (q2 ), am (q4 ), am (q6 ), am (q2 ) (0) (0) T , : : : , am (q02M¡6 )]T , we have C0 = [h(0) 1 , h2 , : : : , hM ] 2 (1) (1) (1) CM£M and C1 = [h1 , h2 , : : : , hM ]T 2 CM£M . In addition we note that rank[C0 , C1 ] = rank[A, A0 ] = ¢ rank(A), wherein A =[a1 , a2 , : : : , aM ]T 2 CM£6 , and A0 CM£(2M¡6) contains the q0l th column of A as its lth
c1 a1 + c2 a2 + ¢ ¢ ¢ + c6 a6 = o6£1 :
(28)
We can further deduce that c1 , c2 , : : : , c6 6= 0. To see this, we assume one of them, say c1 , is zero. Then implies that a2 , a3 , : : : , a6 are linearly dependent, which contradicts with the result given in [10, Theorem 5]. ¢ ¢ ¢ We denote Fl = Fμl ,Ál , Ql = Q®l , and wl = w¯l , l = 1, 2, : : : , 6, where Fμl ,Ál , Q®l , and w¯l are defined in (10), and assume without loss of generality that the last three sources are circularly polarized with the same spin: ¯4 = ¯5 = ¯6 = §¼=4, then (28) could be rearranged as follows: 3 2 c1 Q1 w1
7 6 4 c2 Q2 w2 5 c3 Q3 w3
= ¡(c4 § ¡1 F4 Q4 + c5 § ¡1 F5 Q5 + c6 § ¡1 F6 Q6 )w§¼=4
(29) ¢
2
column, l = 1, 2, : : : , 2M ¡ 6. Moreover, it is possible to choose the selection vectors q and q0 such that C0 contains rank(A) linearly independent columns, and rank(C0 ) = rank[C0 , C1 ] = rank(A). Therefore, for the special choices of q and q0 , and assuming 4 · M · 6, we have rank(B) = rank(ÂB )=2 = rank(C0 ) = rank(A): (27) According to [10, Theorem 4], rank(B) = rank(A) = 4 if the four sources have distinct DOAs and nonidentical ellipticity angles. In addition, according to [10, Theorem 5], rank(B) = rank(A) = 5 in the presence of five sources with distinct DOAs of which exactly two or three sources are circularly polarized with the same spin. Hence the second and third results in Remark 3 are proven. As for the case that six signals with distinct DOAs are impinging, we have rank(B) = rank(A) = 6 when exactly three are circularly polarized with the same spin, by introducing the following theorem: THEOREM 1 Steering vectors of a complete EM vector-sensor corresponding to six sources with distinct DOAs are linearly independent if exactly three sources are circularly polarized with the same spin. PROOF Consider six steering vectors a1 , a2 , : : : , a6 of a complete EM vector-sensor associated with distinct DOAs (μl , Ál ), l = 1, 2, : : : , 6, among which exactly three of them correspond to circularly polarized signals with the same spin. We assume on the contrary 2282
that they are linearly dependent. Then there exist c1 , c2 , : : : , c6 not all zero, such that:
where real-valued matrix § =[F1 F2 F3 ] is invertible according to [10, Theorem 1 and Theorem 2]. Writing cl = jcl je~i!l , !l 2 (¡¼, ¼], l = 4, 5, 6, and using [10, Lemma 5], we could obtain 3 2 c1 Q1 w1 7 6 4 c2 Q2 w2 5 = ¡(jc4 j§ ¡1 F4 Q®4 §!4 + jc5 j§ ¡1 F5 Q®5 §!5 c3 Q3 w3
+ jc6 j§ ¡1 F6 Q®6 §!6 )w§¼=4 :
(30)
Now by using the strategy adopted in [10, Appendix D], (30) could be rewritten as 3 2 x1 y1 7 6 6 ¡y1 x1 7 3 2 7 6 c1 Q1 w1 7 6 x 7 6 2 y2 7 6 7w 4 c2 Q2 w2 5 = ¡ 6 6 ¡y2 x2 7 §¼=4 7 6 c3 Q3 w3 7 6 4 x3 y3 5
¡y3 x3 3 2 q cˆ 1 x12 + y12 Q®ˆ 1 7 6 q 7 6 2 2 6 = 6 cˆ 2 x2 + y2 Q®ˆ 2 7 7 w§¼=4 5 4 q cˆ 3 x32 + y32 Q®ˆ 3
where
·
xk
yk
¡yk
xk
¸
(31)
2 R2£2
is non-zero, ®ˆ k = arctan(yk =xk ) 2 (¡¼=2, ¼=2], and cˆ k 2 f¡1, 1g, k = 1, 2, 3. By [10, Lemma 2] and its subsequent remark, we could finally obtain that ¯1 = ¯2 = ¢ ¢ ¢ = ¯6 = §¼=4 which is a contradiction. As a result, a1 , a2 , : : : , a6 are linearly independent.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3
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APPENDIX III. CALCULATION OF THE COMPUTATIONAL COMPLEXITY We assume that there are N EM vector-sensors, and T snapshots, then the bicomplex output for one subarray could be given by Xl = Xl,0 + ~j Xl,1 , where Xl,0 , Xl,1 2 CN£T , and the sampled covariance ˜ = T¡1 X XH . By definition matrix is thus given by R l l l 2 ˜ we know that Rl has N bicomplex entries, and thus occupies the memory of 4N 2 real scalars. ˜ is obtained via In addition, each element of R l T bicomplex multiplications, T ¡ 1 bicomplex additions and a division by a real scalar. Note that one bicomplex multiplication implies 16 real multiplications and 12 real additions, one bicomplex addition implies 4 real additions, and the division by a real scalar implies 4 real divisions. Therefore, the number of basic arithmetical operations needed for calculating the bicomplex auto-covariance matrix of one subarray is 16N 2 T(M) + 12N 2 T(A) + 4N 2 (T ¡ 1)(A) 2 +4N(D) , where subscripts (M), (A), and (D) denote real multiplication, real addition, and real division, respectively. The nonoverlapping BPS scheme involves calculation of three bicomplex auto-covariance matrices, and thus the total memory is equal to that of 12N 2 real scalars, and the total number of basic arithmetical operations is 48N 2 T(M) + 2 2 + 12N(D) . In addition it is easy to verify (48T ¡ 12)N(A) that the complexity of Q-MUSIC in calculating all the three quaternion subarray auto-covariance matrices is equal to that of BPS-MUSIC. Similarly, we obtain that the total memory and number of basic arithmetical operations for CPS-MUSIC in calculating all the six complex subarray covariance matrices are 12N 2 2 2 and 24N 2 T(M) + (16T ¡ 8)N(A) + 8N(D) , respectively. Moreover, the complexity of CPDS algorithm is equal to that of CPS-MUSIC by noting that the only difference of these two algorithms is the weights of auto-covariance matrices. For CPSAC, we note that all the auto/cross-covariances are used, so that the total memory and number of basic arithmetical operations for CPSAC in calculating all the 6 complex auto-covariance matrices and 15 cross-covariance 2 matrices are 42N 2 and 84N 2 T(M) + (56T ¡ 28)N(A) + 2 28N(D) , respectively. XIAO-FENG GONG Faculty of Electronic Information and Electrical Engineering Dalian University of Technology Dalian, 116024, China E-mail: (
[email protected]) ZHI-WEN LIU YOU-GEN XU School of Information and Electronics Beijing Institute of Technology Beijing 100081, China CORRESPONDENCE
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Comment on “One-Step Solution for the Multistep Out-of-Sequence-Measurement Problem in Tracking”1
To incorporate out-of-sequence measurements (OOSMs) in a Kalman filter, the cited paper proposed algorithms that reduced the multistep problem into a single-step one, by defining one “equivalent measurement” that has the same dimension as the state vector. However, the authors did not prescribe formulas for the case when such a “full-dimension” equivalent measurement cannot be defined. This correspondence fills the gap by deriving formulas that are always applicable, regardless of the dimension of the equivalent measurement.
I. INTRODUCTION AND PROBLEM STATEMENT The out-of-sequence measurement (OOSM) processing problem arises in many applications with (and sometimes without) communication delays. After a filter has processed in-sequence measurements and obtained an estimate for the current time instant t, a measurement may arrive with a time stamp td such that td < t, and the problem becomes how to update the estimate using such a measurement. Algorithms that have been proposed in the literature fall essentially into two categories. One is anticipatory in the sense that relevant computations are started right after time td when an OOSM is believed to have occurred and the measurement is expected to be received some time in the future. Such algorithms include those proposed in [1] and [2]. The other category is reactive in the sense that relevant computations are started at time t when the OOSM with time stamp td has been received. Such algorithms include those proposed in [3], [4], and [5]. Reactive algorithms are very attractive for situations in which OOSMs occur frequently and the complexity of bookkeeping in anticipatory algorithms becomes hard to deal with. A decentralized formation flying application for NASA’s TPF-I mission (see [6] and [7]) is one such scenario that motivated this study. 1 Bar-Shalom, Y., Chen, H., and Mallick, M., IEEE Transactions on Aerospace and Electronic Systems, 40, 1 (Jan. 2004), 27—37.
Manuscript received August 28, 2009; revised October 3, 2010; released for publication November 7, 2010. IEEE Log No. T-AES/47/3/941801. Refereeing of this contribution was handled by P. Willett. This work was supported in part by NASA/JPL SBIR Contract NNC08CA34C. c 2011 IEEE 0018-9251/11/$26.00 ° CORRESPONDENCE
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