Biquaternion Beamspace with Its Application to Vector-Sensor Array Direction Findings and Polarization Estimations Dan Li, Feng Xu, Jing Fei Jiang and Jian Qiu Zhang*

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Li et al. EURASIP Journal on Advances in Signal EURASIP Journal on Advances Processing (2017) 2017:16 DOI 10.1186/s13634-017-0451-6 in Signal Processing

RESEARCH Open Access Biquaternion beamspace with its application to vector-sensor array direction findings and polarization estimations Dan Li, Feng Xu, Jing Fei Jiang and Jian Qiu Zhang*

Abstract In this paper, a biquaternion beamspace, constructed by projecting the original data of an electromagnetic vector-sensor array into a subspace of a lower dimension via a quaternion transformation matrix, is first proposed. To estimate the direction and polarization angles of sources, biquaternion beamspace multiple signal classification (BB-MUSIC) estimators are then formulated. The analytical results show that the biquaternion beamspaces offer us some additional degrees of freedom to simultaneously achieve three goals. One is to save the memory spaces for storing the data covariance matrix and reduce the computation efforts of the eigen-decomposition. Another is to decouple the estimations of the sources’ polarization parameters from those of their direction angles. The other is to blindly whiten the coherent noise of the six constituent antennas in each vector-sensor. It is also shown that the existing biquaternion multiple signal classification (BQ-MUSIC) estimator is a specific case of our BB-MUSIC ones. The simulation results verify the correctness and effectiveness of the analytical ones. Keywords: Array signal processing, Biquaternion beamspace, Biquaternion beamspace music, Direction of arrival estimation, Polarization parameters estimation

1 Introduction six times more than that in a scalar-sensor array with An electromagnetic (EM) vector-sensor, which consists the same array aperture. It means that more computation of six spatially collocated antennas, measures the com- efforts and memory spaces are required for vector-sensor plete electric and magnetic fields induced by EM signals array signal processing. Another is that a computation [1]. Except for spatially and temporally sampling the inci- prohibitive four-dimensional (4D) search over the direc- dent EM source signals as done by a scalar-senor array tion and polarization parameters is required [5], when the [2, 3], an EM vector-sensor array can also record the traditional multiple signal classification (MUSIC) algo- source direction information by the signal responses of rithm for scalar-sensor array processing [2, 3] is directly the six constituent antennas in each vector-sensor (see extended to the vector one. Thus, it is strongly desirable the V(θk, φk) item in (2)). Moreover, an EM vector-sensor to find some fast and computation effective techniques for array-manifold is also sensitive to incident wavefields’ vector-sensor array signal processing. polarization states [4–6], which is useful in applications, To achieve a significant reduction in computation time, for example remote sensing [7]. It has been understood many efforts have been made. For example, the beamspace that an EM vector-sensor array can record more source approaches have been proposed in [8–10] for a scalar- information than the scalar-sensor one. However, its sig- sensor array processing case, in which the original sensor nal processing techniques are also more complex than data are projected into a subspace of a lower dimen- the scalar-sensor ones for at least two reasons. One is sion (i.e., the beamspace) to reduce the data size. In the that the number of antennas in a vector-sensor array is vector-sensor case, Wong and Zoltowski in [5] projected the array data into a spatio-polarizational beamspace to *Correspondence: [email protected] decouple the polarization estimation from the direction Research Center of Smart Networks and Systems, The Department of finding. With their techniques, only a 2D search is needed Electronic Engineering, Fudan University, Handan Road, Shanghai, China

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for estimating the sources’ direction angles before their to 4/9 of the traditional cases, and reduce the multiplica- polarization parameters are estimated. tions of the eigen-decomposition of the data covariance However, their beamspace for each source has to be matrix from O((6M)3) to O((4M)3) for an array of M designed separately to decouple the polarization estima- vector-sensors. tion from direction finding. Another is to blindly whiten the noise coherence of the Then a promising alternative for reducing the compu- selectric or magnetic components of each vector-sensor tation costs of the vector-sensor array processing is the in an array which cannot be done by the methods in the using of the quaternion and biquaternion algebra. Miron literature. et al. in [11] proposed a quaternion MUSIC (Q-MUSIC) After that, two types of the quaternion transformation estimator for two-component vector-sensor array pro- matrices are analytically chosen to conduct two enhanced cessing. Zhang et al. in [12] improved the performance of versions of the BB-MUSIC estimator. Such two estima- the Q-MUSIC estimator for colored noise, but only for a tors are capable of decoupling the estimations of polar- two-component vector-sensor array. Miron’s research was ization parameters from those of the direction angles. later extended to a three-component vector-sensor array One of the proposed BB-MUSIC estimators, termed as via the biquaternion algebra [13, 14], where a biquater- DOA-BB-MUSIC, is able to estimate the direction angles nion version MUSIC (BQ-MUSIC) estimator was intro- of the sources without any polarization information of duced. Gong et al. in [15] improved its robustness to them. However, theoretical analyses also indicate that the colored noise by diagonalizing the biquaternion covari- DOA-BB-MUSIC estimator cannot estimate the polar- ance matrix. Using the calculation rules of the quaternion ization parameters either. Then another version of the and biquaternion algebras, the computation efforts and enhanced BB-MUSIC estimator, nominated as DOA-P- memory spaces of their covariance matrix are significantly BB-MUSIC estimator, is presented to fulfill both the reduced. To handle a six-component EM vector-sensor direction and polarization angle. The DOA-P-BB-MUSIC array, Gong et al. in [16] proposed a quad-quaternion estimator requires a 2D search to find the direction angles MUSIC (QQ-MUSIC) estimator. Although it was shown first. After that the polarization parameters are obtained in [13] by Monte Carlo simulations that the BQ-MUSIC by another 2D search or finding the optimal solutions estimator was robust to the coherent corrupting noise and of the formulated linear equations. However, the above- polarization errors, it was kept unknown what kind of mentioned estimators can only naturally whiten the noise reasons were for the robustness. Furthermore, the BQ- coherence of the electric or magnetic components of each MUSIC was suitable for the direction finding of sources vector-sensor in an array. To construct a biquaternion with known polarization parameters. Unfortunately, both beamspace, which can be used to blindly whiten all of the sources’ directions and polarization angles are nor- the noise coherence of the electric and magnetic com- mally unknown in applications. The QQ-MUSIC estima- ponents of each vector-sensor in an array, an optimal tor in [16] cannot also be used to estimate the polarization problem with a solution is given. Based on such a con- angles of the sources. The geometric algebra model (G- structed biquaternion beamspace, an enhanced DOA-P- MODEL) of an EM vector-sensor array is given in [17]. BB-MUSIC estimator, nominated as DOA-PB-BB-MUSIC It is, however, kept unknown how to use it for array pro- estimator, is then proposed for the direction findings cessing applications. On the other hand, The biquaternion and polarization estimations. The performances of the Capon beamformer was proposed in [18] and [19], but proposed estimators are tested by simulations, which con- they cannot estimate the polarization angles of a source. firm the correctness and effectiveness of our theoretical In this paper, the techniques of projecting the out- analyses. put data of an electromagnetic vector-sensor array into a Besides, the analytical results of the quaternion transfor- biquaternion beamspace via a quaternion beamformer (or mation matrix also show that the BQ-MUSIC in [13] can a quaternion transformation matrix) are first proposed. be considered as a special case of our BB-MUSIC estima- The general form of the biquaternion beamspace multiple tor. Using the biquaternion beamspace theory presented signal classification (BB-MUSIC) estimator is formulated herein, the reasons why the BQ-MUSIC estimator in [13] by using the eigen-decomposition (EVD) of the biquater- is robust to coherent noise and polarization errors are nion matrix [13, 20]. In our biquaternion beamspace, it analytically presented by comparing the covariance of the is found that the quaternion transformation matrix offers complex and biquaternion algorithms. us some additional degrees of freedom to handle the The rest of the paper is organized as follows. In direction finding and polarization estimation. Using these Section 2, the biquaternion beamspace and its measure- degrees of freedom, we achieve the foregoing three goals ment model are given by applying a quaternion trans- as follows. First of all, we report two notable properties of formation matrix to the traditional long vector model. a set of quaternion transformation matrices. One is to save Then in Section 3, the general form of our BB-MUSIC the memory spaces for storing the data covariance matrix estimator is derived, the techniques to de-correlate the Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 3 of 15

noise coherence are presented. In Section 4, two extended ( ) = ( ) ( ) ( ) T ( ) = ( ) where el t elx t ely t elz t and hl t [hlx t versions of BB-MUSIC estimator focusing on decoupling T T hly(t) hlz(t) ((·) denoting the vector transpose) are, the polarization estimation from the direction finding are respectively, the three-component electric-field and presented. The construction method of the biquaternion magnetic-field measurement vectors of the lth vector- transformation matrix is introduced in Section 5. Simu- sensor, N l is the additive complex noise, and Sk(t) is the lation and comparison results with other estimators are complex envelope (including amplitude and phase [4]) of reported in Section 6. Section 7 concludes this work. the kth source, i.e., ϕ ( ) = | ( )| I k 2 Biquaternion beamspace and its measurement Sk t Sk t e (4)

model where |·|is the amplitude of a complex number and ϕk is Similar to [13], we will use R, C, H,andHC to denote the the kth source’s random carrier phase. sets of real numbers, complex numbers, quaternions, and Let Y(t), S(t),andN(t) be the column vectors describ- biquaternions, respectively. ing the entire array’s received signals, source incident signals, and noise, respectively, i.e., 2.1 The long vector model T ( ) = T ( ) T ( ) ··· T ( ) Suppose that the completely polarized plane waves from Y t Y 1 t Y 2 t Y M t K narrow-band sources, traveling through a nonconduc- tive homogenous isotropic medium, are impinging on an T S(t) = ST (t) ST (t) ··· ST (t) array of M(M ≥ K) identically oriented electromagnetic 1 2 M vector-sensors locating irregularly in a 3D region. Let the T φ T T T azimuth and elevation angles of the kth source be k N(t) = N (t) N (t) ··· N (t) θ φ ∈ π ) θ ∈ 1 2 M and k,respectively.Thus, k [0, 2 ,and k π π − , . The spatial phase factor for the kth source to The measurements of an electromagnetic vector-sensor 2 2 array can thus be expressed as [4] the lth vector-sensor centered at location (xl, yl, zl) can be written as K 2π Y(t) = q(θ , φ ) ⊗ V(θ , φ )P(γ , η )S (t) + N(t) I λ (xl sin θk cos φk+yl sin θk sin φk+zl sin φk) k k k k k k k ql(θk, φk) = e (1) k=1 = ( ) + ( ) where λ is the length of the electromagnetic wave and I is AS t N t the imaginary unit of a complex number1. (5) The spatial response, in matrix notation, of the sig- where ⊗ is the Kronecker product and A is the 6M × K nal from the kth source to the lth vector-sensor can be matrix, expressed by [4–6] A = [a1, a2, ··· , aK ] (6) (θ φ γ η ) = gl k, k, ⎡k, k ⎤ with a = q (θ , φ ) ⊗ V (θ , φ ) P (γ , η ), − φ φ θ k k k k k k k sin k cos k cos k q (θ , φ ) = q (θ , φ ) , q (θ , φ ) , ··· , q (θ , φ ) T . ⎢ φ φ θ ⎥ k k 1 k k 2 k k M k k ⎢ cos k sin k cos k ⎥ The matrix A is generally assumed to be of full rank. ⎢ − θ ⎥ γ (θ φ ) ⎢ 0 sin k ⎥ cos k ql k, k ⎢ ⎥ Iη It defines a K-dimensional signal subspace in a 6M- ⎢ − cos φk cos θk − sin φk ⎥ sin γke k ⎣ ⎦ dimensional space [6]. − sin φk cos θk cos φk P(γ ,η ) From (5), it can be seen that q(θk, φk) depends on both sin θ 0 k k k the direction angles of the kth source and the locations

V(θk,φk) of the vector-sensors. It has an effect of spatial and tem- (θ φ ) (2) poral sampling of the kth source’s signals [3]. V k, k depends only on the direction angles. It contains the and η ∈ [−π, π) are, respectively, the auxiliary polariza- source’s direction information resulting from the spatial tion and phrase difference angles of the kth source. collocation of the vector-sensor’s six constituent anten- ForthegeneralcaseofK narrow-band sources imping- nas. P(θk, φk) only relies on the polarization angles. It ing on the array, the six-component vector measurements records the polarization information of the kth source. of lth vector-sensor is given by [4] Thus,itisexplicitthatanelectromagneticvector-sensor array records more source information than the scalar- ( ) K sensor one [2], because a scalar-sensor array only samples ( ) = el t = (θ φ γ η ) ( ) + ( ) Y l t gl k, k, k, k Sk t N l t the source signals spatially and temporally [3]. hl(t) k=1 When the source signals S(t) and noise N(t) are two (3) independent stochastic processes, the spectral matrix Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 4 of 15

Ry(t) of (5) is given by [2, 5, 6] Compare (7) with (9), one can see that the transformation matrix T in (9) offers us additional degrees of ( ) = E ( ) H ( ) Ry t Y t Y t freedom to handle the spectral matrix R (t).Inthenext (7) z = AE S(t)SH (t) AH + E N(t)N H (t) two sections, we aim at utilizing the additional freedom to achieve the three goals simultaneously, i.e., reducing the E{·} where is the mathematical expectation operator and memory spaces and computation efforts, separating the (·)H is the conjugate transpose of a complex matrix/vector. sources’ direction and polarization parameter searches, Since the six measurements of each vector-sensor are and improving the robustness of the MUSIC-like estima- × concatenated into a 6M 1longvector,(5)isalsocalled tor to coherent corrupting noise. as the long vector model (LV-MODEL) of an electromagnetic vector-sensor array in [11, 13]. As (5) shares 3 The biquaternion beamspace MUSIC estimator the same form as the measurement model of a scalar- Before addressing how to determine the transformation sensor array [2, 3], the traditional MUSIC estimator for matrix T in (9) for our three goals, let us first give the scalar-sensor array processing can directly applied to the biquaternion beamspace MUSIC (BB-MUSIC) estimator. vector-sensor case except for the fact that a 4D search over the parameters (θ , φ , γ , η ) is needed. This vector- k k k k 3.1 BB-MUSIC estimator sensor version of the MUSIC estimator is often called as H 2 Suppose that TE N(t)N(t) T = σ IJ . the long vector MUSIC (LV-MUSIC) estimator [11, 13]. σ 2 is the observation noise power of the lth vector- sensor. 2.2 Biquaternion beamspace and its measurement model In this way, (9) can be rewritten as For large arrays (i.e., a large M), the implementation H H 2 of the signal subspace based algorithms require an Rz(t) = TAE S (t) S (t) A T + σ IJ (10) O((6M)3) eigendecomposition [10]. To achieve a sig- Similar to the scalar-sensor case, we need to constrain nificant reduction in computation time, an efficacious that E S (t) SH (t) is a nonsingular matrix and TA is wayistoprojecttheoriginaldataintoalowerdimen- full rank in order to develop the MUSIC-like estima- sion beamspace via a transformation matrix [8–10]. The tor. Using the biquaternion matrix eigen-decomposition parameter estimations are then carried out on the lower (EVD) techniques as given in [13] or [20]2, the EVD of the dimension beamspace data. The known transformation Hermitian biquaternion matrix R (t) can be obtained as matrices, such as the spatio-polarizational [5] and stan- z ( ) = † + † dard Fourier [10] ones, are the complex ones. Unlike the Rz t EsDsEs EnDnEn (11) traditional beamspace techniques, a quaternion transfor- ∈ H2M×2K ∈ H2M×(4M−2K) mation matrix is here used to project the original array where Es C and En C are, respec- data into our biquaternion beamspace. The measurement tively, eigenvectors corresponding to the signal and noise ∈ R2K×2K ∈ R(4M−2K)×(4M−2K) model in our biquaternion beamspace is defined as subspaces, Ds and Dn are two diagonal matrices whose diagonal elements are, ( ) = ( ) = ( ) + ( ) Z t TY t TAS t TN t (8) respectively, the eigenvalues associated with Es and En. Projecting Ta on the noise subspace, we have3 where T ∈ HJ×6M(K < J < 6M) is a quaternion k = † matrix with orthogonal rows. That is TT IJ ,where (Tak) En = 04M−2K , (k = 1, 2, ···K) (12) denotes the transpose-conjugate of a quaternion matrix where 04M−2K is a 4M − 2K dimensional row vector with as defined in [11] and IJ the J × J identity matrix. Mean- T all zero elements. The BB-MUSIC estimator consists of while, the transformation matrix in (8) plays the role = (θ φ γ η ) as a beamformer to project the original array data Y(t) to finding the set of parameters k k, k, k, k so that our biquaternion beamspace. As Y(t) is a complex vec- the following function is maximized: tor in the LV-MODEL, Z(t) ∈ HJ×1 is a biquaternion c F ( ) = 1 = 1 vector [13]. Correspondingly, the spectral matrix of the BB k ( )† 2 S (Ta )†E E† (Ta ) biquaternion beamspace output is given by Tak En k n n k (13) † Rz(t) = E Z(t)Z (t) · where denotes the norm of a biquaternion vector as given in [13] and S{·} denotes the operator taking the = TAE S(t)SH (t) AH T + TE N(t)N H (t) T scalar part of a biquaternion matrix. Let (θk, φk) = (9) Tq(θk, φk) ⊗ V(θk, φk), (13) can be rewritten as where † is the transpose-conjugate of a biquaternion F ( ) = 1 BB k H (14) matrix/vector [13, 21]. P (γk, ηk) (θk, φk) P (γk, ηk) Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 5 of 15

where will be mapped into a real number equaling to the sum of the three variance items of R, while the covariance (θ , φ ) = S [ (θ , φ )]†E E† [ (θ , φ )] (15) k k k k n n k k items are canceled out by (19). From this point, the map-

P(γk, ηk) can refer to (2). It can be seen that (15) depends ping in (19) can naturally whiten the coherence among a only on the direction angular parameters of the sources. three-component random noise vector. = It is understood that the prerequisite of the traditional Rewrite the noise covariance matrix Rn H 6M×6M subspace estimation algorithms, such as MUSIC [2, 3], E N(t)N (t) ∈ R in (7) as the following and ESPRIT [6, 23], is that the additive noise N(t) ∈ blockwise partition form × ⎡ ⎤ C6M 1 in (7) is a complex Gaussian distributed random Rn(1, 1) ··· Rn(1, 2M) vector with a covariance matrix ⎢ . . ⎥ Rn = ⎣ . . ⎦ (20) H 2 . . E N(t)N (t) = σ I6M (16) Rn(2M,1) ··· Rn(2M,2M) It means that their noise components are required to where each block entry is a 3×3matrix.Rn(2l − 1, 2l − 1) be noncoherent with one another. For the BB-MUSIC and Rn(2l,2l)(l = 1, ··· , M) are, respectively, the estimator, such a prerequisite becomes as follows: noise covariance matrices (symmetrical matrices) of the H 2 TE N(t)N (t) T = σ I2M (17) electric-field antennas (short dipoles [1, 6]) and the magnetic-field antennas (small loops [1, 6]) of the lth which is the basic assumption of our BB-MUSIC esti- vector-sensor. R (m, n)(m = n) is the noise covariance mator as given at the very beginning of this subsection. n matrix of the short dipoles and small loops of a vector- Moreover, if (17) has to be satisfied while (16) holds, one sensor or different vector-sensors. should, obviously, let TT = I . Namely, the rows of T 2M Let T be a 2M × 6M quaternion matrix with the follow- are orthogonal to each other. This is also the reason why ing form our biquaternion beamspace (8) requires TT = I2M.In the next subsection, we will show that, if the transforma- 1 t 0 T = √ IM ⊗ (21) tion matrix T is chosen or constructed properly, (17) may 3 0 t still hold even if E N(t)N H (t) in (16) is not a diagonal From (9), it is known that the noise covariance matrix matrix. It implies that the developed BB-MUSIC may be R of the biquaternion beamspace can be rewritten as applied to some coherent corrupting noise cases. n

Rbn = TRnT 3.2 Robustness to coherent noise components ⎡ ⎤ Rbn(1, 1) ··· Rbn(1, 2M) Before rendering our analyses, let us first introduce a ⎢ ⎥ = . . ∈ H2M×2M mapping. Define t = [ijk],wherei, j and k denote ⎣ . . ⎦ three quaternion imaginary units respectively. Thus t is Rbn(2M,1) ··· Rbn(2M,2M) × a quaternion vector. Let a 3 3 real/complex matrix (22) R given by ⎡ ⎤ where r 1, 1 r 1, 2 r 1, 3 ⎢ ⎥ 1 = ⎣ ⎦ R m, n= tRnm, nt ∈ H (23) R r 2, 1 r 2, 2 r 2, 3 (18) bn 3 r 3, 1 r 3, 2 r 3, 3 is a quaternion number. Based on (19), one can conclude By the operation rules of the biquaternion algebra, we that can define the mapping as follows: 1. The noise variance sums of three short dipoles or Rb = tRt small loops are represented by = + + = 1 ∈ R (r 1, 1 r 2, 2 r 3, 3 ) Rbn m, n 3 tRn m, n t , in which their (19) + ( − ) + ( − ) covariance items are canceled out. r 2, 3 r 3, 2 i r 3, 1 r 1, 3 j 2. R m, n= 1 tR m, nt ∈ HC(m = n) describes + ( − ) bn 3 n r 1, 2 r 2, 1 k the noise coherence among the short dipoles and The mapping of (19) transforms a 3 × 3realorcomplex small loops of the same vector-sensor or that among matrix R into a quaternion or biquaternion number Rb. the different vector-sensors. The three quaternion T When R is a symmetric matrix, i.e., R = R , Rb is a real imaginary parts of (19) carry out three subtraction number equaling to (r 1, 1 + r 2, 2 + r 3, 3).WhenR operations for the covariance items (positive real is a general complex or real matrix, Rb is a biquaternion numbers) of Rn m, n . These operations can weaken or quaternion number. Moreover, if R is the covariance the noise coherence because the difference of two matrix of a three dimensional random noise vector, R positive numbers is smaller than either of them. Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 6 of 15

Thus, when the transformation matrix is chosen as The point is not new because Miron et al. in [13] (21), the noise coherence among the electric-field or have given such results by the calculation rules of magnetic-field components of a vector-sensor can be biquaternions. Furthermore, it should be noticed that our totally whitened, while the noise coherence among other biquaternion beamspace also helps to reduce the com- components is weaken (or partially decorrelated). As a putation complexity of the EVD for the MUSIC like result, our BB-MUSIC estimator can be extended to the estimators. It has been understood that the EVD of a coherent noise cases satisfying the following assumptions: M × M complexmatrix requires O(M3) multiplications [10]. It implies that the multiplications for the EVD of A1. the noise components of the different vector-sensors 3 Ry(t) are O((6M) ). By contrast, the multiplications for in an array are noncoherent one another. 3 the EVD of Rz(t) are O((4M) ).ThisisbecausetheEVD A2. a vector-sensor’s electric-field noise is noncoherent of the 2M × 2M biquaternion matrix are obtained via with its magnetic-field noise. the EVD of its complex representation matrix with a size A3. the electric-field noise of a vector-sensor is allowed 4M × 4M [20]. be coherent or noncoherent, and so is the magnetic-field noise. 4 Direction finding and polarization estimation in With the above analytical results, it can be seen that the biquaternion beamspace BB-MUSIC estimator will outperform the traditional long In this Section, two distinct versions of the BB-MUSIC vector type estimators under the condition of coherent estimator based on the different transformation matrix in corrupting noise. (24) are reported. One is developed to estimate the source It should be mentioned that there are many different direction angles without any prior polarization informa- transformation matrices yielding the equivalent results as tion. Another is built to use a 2D search to find the source (19). In order to explain such a point, let us change the direction angles first and then polarization ones. transformation matrix in (21) into the following forms as 4.1 Decoupling polarization angles from direction finding 1 ta 0 Unlike the self-initiating MUSIC estimator in [5] decou- T = √ IM ⊗ (24) 3 0 tb pling the polarization estimation from direction finding via the spatio-polarizational beamforming, we approach in which t and t are 1 × 3 quaternion vectors, whose a b to that by the transformation matrix as given in (24). three entries are unit quaternions√ orthogonal√ to each = = = / = / If the transformation matrix in (24) satisfies ta tb other, such√ as ta 1 3[ijk√]andt√b 1 √3[kji] = / = / ( + ) ( − [e1 e2 e3], we have or ta 1 3[ijk]andtb 2 6[ 3 i√ j 2 i j + k)(√i − j + k)]. If one takes ta = 1/ 3[ijk]and t = 1/ 3[0 0 0], it can be seen that the BQ-MUSIC is a 1 e1 e2 e3 000 b T = √ IM ⊗ (25) special case of our biquaternion beamspace model by tak- 3 000e1 e2 e3 ing the transformation matrix as (24). In a general way, let ta =[e1 e2 e3],tb =[e4 e5 e6], where e1, e2, e3 and e4, e5, e6 Thus, we have the following theorem for decoupling the are any two unit quaternion orthogonal vectors, respec- polarization estimation from the direction finding: tively. Thus, ta and tb can be changed into more general T γ = π/ η =±π/ ( )† = forms as t =[e e e ] = U ijk and t = [e e e ] = Theorem 1 When 4 and 2, Tak En a 1 2 3 a b 4 5 6 T 0 − is equivalent to Ub ijk ,whereUa and Ub are any 3 × 3realorthog- 4M 2K onal matrix. By directly calculation, one can easily verify that (19) is still hold for these chosen t and t . The differ- † a b 1 ences are that the last three terms in (19) will be changed q (θ , φ ) ⊗ E = 0 − , (k = 1, 2, ···K) k k u n 4M 2K with the different T in (24). Even so, our BB-MUSIC esti- k mator can still work correctly under the assumptions A1, (26) A2 and A3. As a side note, it should be pointed out the fact that the where uk = cos φk cos θki + sin φk cos θkj + mapping of (19) can also reduce the memory spaces of R sin θkk, (k = 1, 2, ..., K) in (18). Obviously, R needs 9 or 18 memory units to store a3× 3 real or complex matrix, whereas 4 or 8 memory Proof See Appendix 1. units are enough for storing Rb. As a result, when taking the entire array into consideration, (9) indicates that From Theorem 1, it can be seen that (26) is nothing to 2 2 32M memory units are required to restore , while 72M do with the polarization parameters (γk, ηk) because both ( ) memory units are required to restore Ry t . q(θk, φk) and uk only depend on the direction angular Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 7 of 15

parameters (θk, φk) of the sources. Thus, the expression, 4.2 Direction finding and polarization estimation To maintain the polarization information in the BB- F (θ φ ) = 1 DOA−BB k, k MUSIC spectrum, let us consider the transformation † 2 = 1 matrices in (24) satisfying ta tb and introduce a new q (θ , φ ) ⊗ En k k u quaternion beamspace matrix Tp : k (27) 1 e e e 000 T = √ I ⊗ 1 2 3 (30) p M 000e e e can first be used to find the directions angles of the 3 4 5 6 sources with polarization parameters γk = π/4andηk = One can very easily verify that Tp is a quaternion matrix ±π/2. After that, one substitutes γk = π/4andηk = with orthogonal rows and can make (19) hold. Using (30), ±π/2 into (14) and then use it to search the directions the BB-MUSIC spectrum in (14) is changed into angles of the sources with the two polarization states. In F ( )= /( γ 2 + γ 2 + γ η this way, the traditional 4D search in the MUSIC methods k 1 a 1 cos k a 2 sin k a 12 sin 2 k cos k is reduced to three 2D searches for our estimators (27) and − b 12 sin 2γk sin ηk) (14). As (27) can only find the direction of arrival (DOA) (31) of the sources, we call it as the DOA-BB-MUSIC estimator for the latter discussion convenience. Proof Reduction details: Although the transformation matrix of (25) enables us to decouple the polarization parameters from the source See Appendix 3. direction finding, the following Theorem 2 tells us that One can easy find from (31) that the polarization param- the estimates of the polarization parameters cannot be eters cannot be decoupled. However, a 4D search can help handled via the transform matrix as (25). for the direction finding and polarization angle estimations. In other words, the polarization parameters estima- Theorem 2 If the transformation matrix of the tion is achievable via introducing the new biquaternion biquaternion beamspace is taken as (25), (θ , φ ) in (15) k k beamspace matrix Tp. As the 4D search is time consum- will be changed into ing and computation prohibitive, we next discuss how a −bI to decouple the polarization estimations of (31) from its (θ , φ ) = (28) k k bI a direction finding. Equation (12) can be rewritten as where a and b are two real numbers depending only on † = ( = ··· ) (θk, φk). Moreover, the larger 4M − 2K, the smaller |b/a|. Tpak En 04M−2K , k 1, 2, K (32) Inserting (28) into (14), the BB-MUSIC spectrum can be and then (32) can be changed into rewritten as 1 1 F ( k) = (33) FBB ( k) = (29) H (γ η ) (θ φ ) (γ η ) + b γ η P k, k p k, k P k, k a 1 a sin 2 k sin k (θ φ ) = S (θ φ ) † † (θ φ ) where p k, k [ k, k ] EnEn [ k, k ] . Proof See Appendix 2. To maximize (33) over k,wecanfirstfixthedirec- tion (θk, φk) and maximize (33) over polarization angles It can be seen that, although the shape of the spectrum (γk, ηk). In this way, the 4D search over k is reduced to a (θ φ ) defined by FBB( k) changes with a, b,andsin2γk sin ηk, 2D search over k, k . This idea was first used by Ferrara the extrema of (29) for polarization parameters are only and Parks in [24] for direction finding via diversely polar- taken at fix locations with | sin 2γk sin ηk|=1, i.e., ized antennas. Thus, the polarization decoupled spectrum (γ = π/4, η =±π/2). It indicates that (29) cannot be of (33) is used to find the polarization parameters when the transformation matrix is taken as (25). F(θ φ ) = 1 k, k maxγ η H As a matter of fact, the key point why (25) is incapable k, k P (γk, ηk) p (θk, φk) P (γk, ηk) of estimating the polarization parameters is the form of (34) (θk, φk) in (28) making the extrema of (29) fall away H them. To develop an estimator capable of doing the direc- Since P (γk, ηk)P(γk, ηk) = 1and(θk, φk) is a com- tion finding and polarization estimation both, the trans- plex Hermitian matrix, according to the Rayleigh’s princi- formation matrix different from (25) should be chosen to ple [25], we obtain (θ φ ) break the form of k, k in (28). We will investigate this 1 = 1 max H (γ η ) (θ φ ) (γ η ) λ { (θ φ )} (35) topic in the next subsection. γk,ηk P k, k p k, k P k, k min p k, k Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 8 of 15

where λmin p(θk, φk) denotes the operator taking the subspace as equal as possible, all of the coherent noise smallest eigenvalue of the matrix p (θk, φk). From (35), it of vector-sensor reduced may be expected. Mathemati- can be seen that our the BB-MUSIC spectrum for DOA cally, it can be formulated as the following optimization estimation is problem

(λ λ ···λ ) F (θ φ ) = 1 min var 2K+1, 2K+2, 4M DOA k, k (36) = (37) λmin p (θk, φk) s.t. TpTp I2M

where var(·) means to get the variance of the Once the direction angular (θk, φk) is found by the 2D λ2K+1, λ2K+2, ···λ4M. The idea of (37) is that we try to search of (36), one can further substitute (θk, φk) into (33), perform a 2D search to find the polarization parame- find a biquaternion transformation matrix Tp which can ters belonging to the sources with direction parameters make the noise eigenvalue changes of the transformed covariance matrix be as small as possible. It must be (θk, φk). Of course, when the DOA has been obtained, one can emphasized that the derivation of matrix Tp does not do polarization estimation with the LV-MUSIC. However, use any prior knowledge of noise. Nevertheless, extra 2 computation cost must be paid. while 72M memory units are required to restore Ry(t) in (7), it also increase the computation efforts of the eigen- Using the fmincon function, taking the active-set algo- decomposition of the data covariance matrix. rithm, in Matlab optimization toolbox, (37) can be solved. In summary, when choosing the transformation matrix Once the biquaternion beamspace matrix Tp is got, the DOA parameters and polarization estimations can in (24) with ta = tb, we can first perform a 2D search of (36) to estimate the direction angles of the sources, be obtained by using the same method as DOA-P- then estimate the corresponding polarization parame- BB-MUSIC algorithm. These procedures are named as ters. To avoid confusion, we nominate this method as DOA-PB-BB-MUSIC algorithm. Refer to (3.4) in [4], the DOA and polarization estimation BB-MUSIC (DOA-P- Cramer-Rao bound (CRB) of the DOA-PB-BB-MUSIC BB-MUSIC) estimator. Correspondingly, the techniques algorithm in the presence of the coherent corrupting noise used in (36) can also be applied to the traditional LV- is given as MUSIC estimator. We will terms this new version LV- ! 2 −1 MUSIC estimator as DOA-LV-MUSIC because it can be σ bT CRB( )= Re btr 1 ⊗ U D † D used to estimate the DOA of the sources without any 2L bq bq cbq bq polarization information. (38)

5 Construction of biquaternion beamspace Proof See Appendix 4. matrix In Section 3, we have known that there are an infinite 6 Simulation results number of biquaternion beamspace matrices making (19) The performances of the three newly proposed estima- satisfied. When choosing the transformation matrix T as tors are evaluated in this section. Since both DOA-BB- (25), the noise coherence among the electric field com- MUSIC and DOA-P-BB-MUSIC estimators are capable ponents or magnetic field ones can be eliminated. How- of decoupling the polarization estimation from the direc- ever, the noise coherence between the electric-field and tion finding, their direction finding performances will be magnetic-field components of each vector-sensor cannot compared with two other polarization decoupled esti- be whitened. Thus, it is desirable to find an appropriate mators, namely the DOA-LV-MUSIC and ESPRIT [23]. transformation matrix Tp as (30) which can whiten all of Besides, the polarization estimation performances of the coherent noise in each vector-sensor. DOA-P-BB-MUSIC will also be compared with the Assuming that the eigenvalues of the signal and ESPRIT estimator in [23]. noise in (11) are, respectively, λ1, λ2, ···λ2K and In all the following simulations, a linear uniform λ2K+1, λ2K+2, ···λ4M with λ1 λ2 ··· λ4M,it array with all vector-sensors along the x-axis with 0.5- is well known that, if the noise in a vector array is a wavelength inter-element spacing is chosen. additive white noise, the eigenvalues of the noise sub- To evaluate the performances of direction find- space of an ideal data covariance matrix are equal, i.e. ings and polarization estimations, we define the λ2K+1 = λ2K+2 = ··· = λ4M. However, an ideal data composite root-mean-square (RMS) errors of the covariance matrix with the infinite data for an applica- direction and polarization angles for the K sources as " $ " $ tion is unpractical. If the additional degrees of freedom 1 1 #K 2 #K 2 offered to us by the quaternion transformation matrix θ 2 + φ2 /( ) γ 2 + η2 /( ) k k 2K and k k 2K , canbeemployedtomaketheeigenvaluesofthenoise k=1 k=1 Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 9 of 15

respectively, where θk, φk, γk and ηk are, that both the DOA-LV-MUSIC and DOA-P-BB-MUSIC respectively, the estimation errors of θk, φk, γk and ηk. successfully find the fourth sources via 2D searches over First, we test the direction finding performances of the direction angles. For the DOA-BB-MUSIC estimator, the proposed estimators. We consider an array of 15 we first employ a 2D search with (27) to find the direc- vector-sensors and 4 uncorrelated narrow-band incident tion angles of the first and second source because their sources4 with the following simulation parameters: polarization angles satisfy the requirements of Theorem 1. Its estimate results are shown in Fig. 1d. Then, as φ = θ =− γ = η = 1 4.5rad, 1 0.8rad, 1 0.7rad, 1 0.3rad depicted in Fig. 1e and f, respectively, we use another φ2 = 3rad, θ2 =−0.1rad, γ2 = 0.1rad, η2 =−1.5rad two 2D searches to estimate the direction angles of the γ = π/ φ3 = 3.8rad, θ3 = 0.4rad, γ3 = π/4rad, η3 = π/2rad third and fourth sources by inserting 4and η =±π/2 into (14). As it can be seen in Fig. 1e, except φ4 = 1.5rad, θ4 = 0.7rad, γ4 = π/4rad, η4 =−π/2rad for the highest peak generated by the third source, two Noncoherent noise is injected with an SNR = 10 dB, lower peaks of the first and second sources also emerge. and 1000 snapshots are used. Among the four sources, Similar phenomenon also accompanies with the fourth the polarization angles of the first and second ones sat- source as given in Fig. 1f. This is due to the fact that (29) isfy the requirements of Theorem 1, while those of the depends slightly on (γk, ηk) when |b/a| is small. More- third and fourth sources do not satisfy the ones. Since the over, the smaller |b/a|, the higher possibility as we can LV-MUSIC estimator requires to perform a 4D search, we see the first and second sources in Fig. 1e and f. If all just fix its polarization parameters to γ4 and η4 to per- sources’ directions and polarization angles are different form a 2D search, while the other estimators perform 2D from each other5, this phenomenon will not induce any searches directly because they are capable of decoupling ambiguities by simply adding the spectrums of Fig. 1d–f the polarization estimations from the direction finding. together, as shown in Fig. 1g. Using the QQ-MUSIC esti- From Fig. 1a, it can be seen that the LV-MUSIC estima- mator, the peak of the fourth source is lower than the tor can only correctly find the fourth sources, because others, as shown in Fig. 1h. The above results confirm we fix the polarization parameters to the fourth sources the analytical correctness of our DOA-BB-MUSIC and and perform a 2D search. In Fig. 1b and c, one can find DOA-P-BB-MUSIC estimators.

Fig. 1 Direction finding with four sources in the presence of noncoherent noise: a direction finding using the LV-MUSIC estimator when γ = π/4, η =−π/2, b direction finding using the DOA-LV-MUSIC estimator, c direction finding using the DOA-P-BB-MUSIC estimator, d direction finding using the DOA-BB-MUSIC estimator when γ = π/4, η =−π/2, e direction finding using the DOA-BB-MUSIC estimator when γ = π/4, η =−π/2, f direction finding using the DOA- BB-MUSIC estimator when γ = π/4, η =−π/2, g the sum of the spectrums in d–f,and h direction finding using the QQ-MUSIC estimator Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 10 of 15

The second part of the simulation will verify the validity one can find that the performances of the DOA-PB-BB- of the DOA-PB-BB-MUSIC algorithm, and then com- MUSIC yield almost same performances in both coherent pares the results with the DOA-LV-MUSIC and DOA- and noncoherent corrupting noise situations. As a con- BB-MUSIC respectively. We consider an array of seven trast, the composite RMS errors of the DOA-LV-MUSIC vector-sensors and two uncorrelated narrow-band inci- in the coherent case is almost 10 times larger than those dent sources with the following simulation parameters: in the noncoherent case with low SNRs, while the RMS errors of the ESPRIT in the coherent case is five times

φ1 = 2.8rad, θ1 = 0.1rad, γ1 = 0.7rad, η1 = 0.3rad larger than those in the noncoherent case with low SNRs. φ2 = 2.4rad, θ2 = 0.6rad, γ2 = 0.1rad, η2 =−1.5rad As a result, the performances of the DOA-LV-MUSIC and ESPRIT are strongly degraded by the coherent noise. Assume that the noises of the vector-sensors are To evaluate the performances of the DOA-LV-MUSIC uncorrelated, while the noises among the three electric and ESPRIT after the correlation of noise is whiten, we and magnetic antennas in a vector sensor are correlated. assume that the noise covariance matrix can be estimated With an SNR = −15 dB and 2000 snapshots is used, the and then is used to whiten the received signal covariance simulation results are shown in Fig. 2. matrix as done in [26]. It is understood by means of the It can easy find from Fig. 2 that the correlated noise method in [26] that it is impossible for one to completely in each vector sensor has relatively greater effect on the whitened the correlated noise due to the estimation errors performances of DOA-LV-MUSIC and DOA-BB-MUSIC of the noise covariance matrix. Thus, it is assumed that algorithms when the SNR is lower. The performance of the noise correlation has dropped to one tenth of the the DOA-PB-BB-MUSIC algorithm is greatly improved original after whitening. Figure 3c illustrates the perfor- since the constructed beamspace has whitened the mances of the ESPRIT and standard MUSIC estimators coherent noise. after whitening the coherent noise and the DOA-PB-BB- The third simulation tests robustness of the direction MUSIC estimator in the presence of coherent noise. Even finding performances of the estimators to corrupting so, it can be seen that the performances of the DOA-PB- BB-MUSIC estimator is still better than the ESPRIT and noise. Two sources with {φ1 = 2.8rad, θ1 = 0.1rad, γ1 = standard MUSIC estimators. It also verifies that our esti- 0.7rad, η1 = 0.3rad} and {φ2 = 2.4rad, θ2 = 0.6rad, γ2 = mators are more robust to coherent corrupting noise than 0.1rad, η2 =−1.5rad} are taken. The number of vector- sensors in the array is 7. For each point on Fig. 3, the sim- the known ones. ulations with 300 independent Monte Carlo experiments Using the DOA-PB-BB-MUSIC estimator, we will next are run. Figure 3a plots the composite RMS errors of the test its robustness to the polarization parameters. The direction finding versus the different SNRs in the presence same array and source configuration as the second simu- of noncoherent noise. Figure 3b illustrates the behaviors lation is taken into consideration. of the different estimators in the presence of coherent Since the DOA-BB-MUSIC and QQ-MUSIC estimators noise. When the corrupting noise is noncoherent, the cannot estimate the polarization angles, we only calculate performances of the DOA-LV-MUSIC are the best while the composite RMS errors of the polarization estimations the DOA-BB-MUSIC and ESPRIT estimators are better of the DOA-PB-BB-MUSIC and ESPRIT estimators with than the DOA-PB-BB-MUSIC one as shown in Fig. 3a. different SNRs under both the noncoherent and coherent However, in the coherent noise case, the performances of noise conditions. the DOA-BB-MUSIC and DOA-PB-BB-MUSIC are much Based on the direction finding results in the second sim- better than those of the DOA-LV-MUSIC and ESPRIT ulation in each Monte Carlo run, the algorithm via (36) is ones as depicted in Fig. 3b. Seeing from the y-axes of Fig. 3, used to estimate the polarization parameters. From Fig. 4a

Fig. 2 Direction finding with two sources in the presence of coherent noise: a DOA-LV-MUSIC, b DOA-BB-MUSIC, and c DOA-PB-BB-MUSIC Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 11 of 15

Fig. 3 The composite RMS estimation errors of direction finding versus SNR: a The composite RMS estimation errors of direction finding in presence of noncoherent noise, and b The composite RMS estimation errors of direction finding in presence of coherent noise, and c The composite RMS estimation errors of direction finding after whitening the coherent noise and b, we find that the performances of the ESPRIT esti- noise case and is more robust to general coherent noise mators in the coherent case are obviously degraded. It can cases. Three enhanced versions of BB-MUSIC estima- also be seen that its composite RMS errors in Fig. 4b, are tor are also presented. The three estimators are capa- almost 4 times larger than those in Fig. 4a. Contrastively, ble of decoupling polarization estimation from direction the composite RMS errors of our DOA-PB-BB-MUSIC finding. In addition, we did not solve the optimization estimator are almost the same for both cases, illustrating problem (37) in theory, instead we use the numerical the robustness of our estimator to corrupting coherent solution. The optimal solution will be considered in our noise again. future work. As an alternative way for vector-sensor array pro- 7Conclusions cessing, by choosing proper transformation matrix, the In this paper, we have illustrated the methods of elec- biquaternion beamspace theory presented here may be tromagnetic vector-sensor array signal processing in our applied to other cases, such as enhancing the signals from biquaternion beamspace. In our biquaternion beamspace, favored directions and block interferences from other the memory spaces for storing the data covariance matrix directions. These topics will also be the focus of our and the computation efforts for performing its EVD future work. are reduced. Techniques for choosing the transforma- Endnotes tion matrix matrices to blindly whiten the coherence 1 of the electric and magnetic noise in a vector-sensor In order not to confuse the imaginary unit of a complex have been given. These techniques ensure that our BB- number with the three imaginary units of a quaternion, MUSIC estimator can be extended to a certain coherent we use the notations in [13] that I is the imaginary unit

Fig. 4 The composite RMS estimation errors of polarization estimation versus SNR: a the composite RMS estimation errors of polarization estimation in presence of noncoherent noise and b the composite RMS estimation errors of polarization estimation in presence of coherent noise Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 12 of 15

of a complex number while i, j and k denote the three where imaginary units of a quaternion. ta 0 2 Ta = I ⊗ (q(θ , φ ) ⊗ V (θ , φ ) P(γ , η )) In [13], the EVD of a biquaternion matrix is obtained k M 0 t k k k k k k b from the EVD of its quaternion adjoint matrix. Alter- v −v = q (θ , φ ) ⊗ k1 k2 P (γ , η ) natively, the EVD of a biquaternion matrix can also k k v v k k k2 k1 be achieved by the EVD of its complex representation 1 Iηk = q (θk, φk) ⊗ vk1 cos γk − vk2 sin γke matrix [20]. 3uk 3 ThiscanbegetdirectlyfromtheEVDofthequater- ( ) 1 † nionadjointmatrixofRz t [22] and the Q-MUSIC Let W = q (θ , φ ) ⊗ , = W E , ρ = k k 3u k n k estimator in [11], or the EVD of the complex repre- k Iη † vk1 cos γk − vk2 sin γke k , and then insert them into sentation matrix of Rz(t) and the traditional MUSIC (12), we have ρkk = 04M−2K . estimator [2, 3]. Since 4 The third and fourth sources are well known as the left- η η † ρ †ρ = γ − γ I k γ − γ I k circularly polarization and right-circularly polarization k k vk1cos k vk2sin ke vk1cos k vk2sin ke sources respectively. = 1/3 − Iuk sin 2γk sin ηk, 5 If there are some sources happen to have same we have direction angles but different polarization angles, the 1 polarization decoupled MUSIC like estimators may fail to (1/3 + Iu sin 2γ sin η ) ρ †ρ = (1−(sin 2γ sin η )2). k k k k k 9 k k discriminate these sources because only a 2D search over so (θk, φk) is performed. † 1 (1/3 + Iuk sin 2γk sin ηk) ρk ρkk = Appendix 1: Proof of Theorem 1 9 2 Since (1 − (sin 2γk sin ηk) )k = 04M−2N . = =− φ + φ vk1 tav1 sin ke1 cos ke2 When γk = π/4, ηk =±π/2, we get = 04M−sK . That is: vk2 = tav2 =−cos φk sin θke1 − sin φk sin θke2 + cos θke3, † 1 we have q (θk, φk) ⊗ En = 04M−2K , (k = 1, 2, ···K) 3uk uk =vk1vk2 =sin θke1e2 −sinφkcos θke1e3 +cosφkcosθke2e3 Appendix 2: Proof of Theorem 2 and Let = φ 2 + φ 2 − φ φ ( + ) ta 0 vk1vk1 sin k e1e1 cos k e2e2 sin kcos k e1e2 e2e1 , (θk, φk) = IM ⊗ (q (θk, φk) ⊗ V (θk, φk)) . 0 tb where e1, e2, e3 are unit quaternions orthogonal to one + = = = = We have another. Since e1e2 e2e1 0ande1e1 e2e2 e3e3 −1/3, v v =−1/3. Similarly we can also get v v = t 0 k1 k1 k2 k2 (θ , φ ) = I ⊗ a (q (θ , φ ) ⊗ V (θ , φ )) −1/3. k k M 0 t k k k k b In the same manner, we have ta 0 = IMq (θk, φk) ⊗ V (θk, φk) 0 tb vk2vk1 =−vk1vk2. vk1 −vk2 = (q (θk, φk) ⊗ I2) . As a result, the following equations can be deduced: vk2 vk1 (θ φ ) = (θ φ ) ⊗ (θ φ ) ukuk = vk1vk2vk1vk2 = vk1 (−vk1vk2) vk2 =−1/9 If Q k, k q k, k I2,then k, k can be = = (− ) = / rewritten as ukvk1 vk1vk2vk1 vk2vk1 vk1 1 3vk2 − ukvk2 = vk1vk2vk2 =−1/3vk1 vk1 vk2 (θk, φk) = Q (θk, φk) vk2 vk1 From (12) we know From (15), it has known that, if k = ( )† = ( = ··· ) (θ φ ) † † (θ φ ) (θ φ ) = Tak En 04M−2K , k 1, 2, K [ k, k ] EnEn [ k, k ],onehas k, k Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 13 of 15

S {k},where From (42), we have

= (θ φ ) † † (θ φ ) H k [ k, k ] EnEn [ k, k ] = x v v + Iv x v − 1/3x + 1/3x k12 10 k1 k2 k1 1q k2 30 30 † v −v v −v + − † + − = k1 k2 (θ φ )† † (θ φ ) k1 k2 vk2x3qvk2 vk1x3qvk1 x20vk1vk2 Ivk2x2qvk1. Q k, k EnEnQ k, k . vk2 vk1 vk2 vk1 S ( ) = / ( − H ) + (39) Thus, we can know k12 1 3 x30 x30 × S Iv x v + S Iv x v . Apparently is a pure Let X = QH (θ , φ ) E E†Q (θ , φ ).BasedonX ∈ H2 2 k1 1q k2 k2 2q k1 k12 k k n n k k C imaginary number. In this way, we can make =−bI, and X = X†, X can be expressed as follows: k12 " $ where b is a real number. † S ( ) = / − / H + + H + Similarly let k12 1 3x30 1 3x = x10 Ix1q x30 x3q 30 X (40) S Iv x v + S Iv x v = bI,sowecanget x30 + x3q x20 + Ix2q k2 1q k1 k1 2q k2 S(k11) S(k12) a −bI (θk, φk) = S{k}= = . where x10 and x20 are real numbers, x30 is a complex S(k21) S(k22) bI a number, x1q and x2q arepurequaternions,x3q is a pure biquaternion. Insert (40) into (39), we have Appendix 3 † According to (40), k can be changed into vk1 −vk2 vk1 −vk2 k = X vk2 vk1 vk2 vk1 − † − † + H + † = tav1 tav2 tav1 tav2 = − † 1 x10 Ix1q x30 x3q 1 − X [vk1 vk2] [vk1 vk2] . k tbv2 tbv1 tbv2 tbv1 3uk x30 + x3q x20 + Ix2q 3uk − † + H + † − (41) = tav1 tav2 x10 Ix1q x x tav1 tav2 t v t v 30 3q t v t v b 2 b 1 x30 + x3q x20 + Ix2q b 2 b 1 Since vk1, vk2 and uk are pure quaternions, we can get † † † (43) vk1 = vk1 =−vk1 , vk2 = vk2 =−vk2 and uk = uk =−uk. In this way, (39) can be rewritten as: Suppose that vk1 = tav1, vk2 =−tav2, vk3 = tav2, vk4 = −v tav1,allofvk1, vk2, vk3, vk4 are pure biquaternions. One can = k1 x + x + Ix − 3x u + 3xH u k v 10 20 1q 30 k 30 k get k2 − + † − − =−( + + 3ukx3q 3x3quk I9ukx2quk [vk1 vk2] . k11 vk1x10vk1 Ivk1x1qvk1 vk3x30vk1 H † (42) + vk3x3qvk1 + vk1x vk3 + vk1x vk3 30 3q + + ) = k11 k12 (θ φ ) = S{ }= vk3x20vk3 Ivk3x2qvk3 Let k ,then k, k k k21 k22 S( ) S( ) k11 k12 . Notice that as far as any two arbi- S(k21) S(k22) and trary pure quaternions q1 and q2 are concerned, we know S q q q = S q q q = 0, where S(·) stands for tak- S ( ) = / + / − + H S ( ) 1 2 1 2 1 2 k11 1 3x10 1 3x20 x30 x30 vk1vk3 ing the scalar part of the biquaternion. In other words, − S + S † q1q2q1 and q2q1q2 arepurequaternions.Sinceallof vk3x3qvk1 vk1x3qvk3 vk1, vk2, uk, x1q and x2q are pure quaternions, ukx2quk is = = also a pure quaternion. When qb1 vk2x3qvk1 and qb2 S +S † S ( ) ∈ R †v Since vk3x3qvk1 vk1x3qvk3 is real, k11 . vk1x3q k2 ,wecanget Thus, we can let S (k11) = a 1 ∈ R. Similarly, S ( ) = / + / − S ( ) − S ( ) k11 1 3x10 1 3x20 qb1 qb2 . † † H † = † † = = S (k22) = 1/3x10 + 1/3x20 − x30 + x S (vk2vk4) It can be seen that qb1 vk1 x3qvk2 vk1x3qvk2 30 S( ) = S( )∗ S( )+ qb2. As a result, one has qb1 qb2 .Thatis qb1 † − S vk4x3qvk2 + S vk2x vk4 ∈ R. S(qb2) ∈ R.Sincebothx10 and x20 are real numbers, we 3q have S (k11) = 1/3x10 +1/3x20 −(S (qb1) + S (qb2)) ∈ R S( ) = Additionally, S ( ) can be written as and then let k11 a. k12 Similarly, since S (q ) + S (q ) =−2S v x v ,it b3 b4 k2 3q1 k1 S ( ) =− S − S can be deduced that k12 I vk1x1qvk2 I vk3x2qvk4 − S ( ) − H S ( ) S ( ) + S ( ) =−(S ( ) + S ( )) x30 vk3vk2 x30 vk1vk4 qb3 qb4 qb1 qb2 † − S vk3x3qvk2 − S vk1x vk4 . Therefore S (k22) = S (k11) = a. 3q Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 14 of 15

Since S(k12) is a complex number, one can let T, the CRB of the DOA-PB-BB-MUSIC algorithm in the S(k12) = a 12 + Ib 12.Itiseasytofindoutthat presence of the coherent corrupting noise can be given by S ( ) =− S − S − k21 I vk2x1qvk1 I vk4x2qvk3 σ 2 bT 1 CRB( )= Re btr 1⊗U D † D − S ( ) − H S ( ) 2L bq bq cbq bq x30 vk4vk1 x30 vk2vk3 (52) − S − S † vk4x3qvk1 vk2x3qvk3 where (S ( ))∗ = S ( ) S( ) = Due to k12 k21 and taking k21 − U = P (A †A P + σ 2I) 1A †A P (53) a 12 − Ib 12,wehave bq s bq bq s bq bq s a 1 a 12 + Ib 12 (θk, φk) = S {k} = † a 12 − Ib 12 a 2 cbq = EnEn (54) (44) it can be seen that the a 1, a 2, a 12 and b 12 in (44) are Dbq = TD (55) only related to DOA parameters. Insert (44) into (14), and we have Inserting (50), and (51) into (53), (54), and (55) respectively, one has F ( ) = /( γ 2 + γ 2 k 1 a 1 cos k a 2 sin k − H 2 1 H U = Ps A T TAPs + σ I A T TAPs (56) + a 12 sin 2γk cos ηk − b 12 sin 2γk sin ηk) bq

Appendix 4 † = H H Refer to (3.4) in [4], the CRB on the covariance matrix of Dbq cbqDbq D T TEnlvEnlv T TD (57) any (locally) unbiased estimator of the vector sensor array = ( ) = ( ) If the matrix T satisfies T T I,CRB CRB lv. in the presence of the noncoherent corrupting noise is Funding σ 2 −1 This work was supported in part by the NSF of China under Grant 61571131, H bT CRB( ) = Re btr (1 ⊗ U) D cD 11604055. lv 2L (45) Authors’ contributions DL and JQZ conceived and designed the study. DL, FX, and JFJ performed the where experiments. DL and JQZ wrote the paper. DL, FX, JFJ, and JQZ reviewed and edited the manuscript. All authors read and approved the manuscript. − H 2 1 H U = Ps A APs + σ I A APs (46) Authors’ information Dan Li received the the M.S. and Ph.D. degrees in electronic engineering from Fudan University, Shanghai, China, in 2006 and 2013, respectively. He is H c = EnlvEnlv (47) currently a lecturer at the Department of Electronic Engineering, Fudan University. His research interests include array signal processing and biquaternion with applications in signal processing and image processing. = (1) ··· (1) ··· (n) ··· (n) Feng Xu received the Ph.D. degree in electronic engineering from Fudan D D1 Dq1 D1 Dqn (48) University, Shanghai, China, in 2010. He is currently an Electronic Engineer at the Department of Electronic Engineering, Fudan University. His research interests include array signal processing and biquaternion with applications in ( ) ∂a D k = k (49) signal processing and image processing. l ∂ (k) Jing Fei Jiang received the B.Sc. degree and M.Sc. degrees in electronic l engineering from Fudan University, Shanghai, China, in 2008 and 2011, respectively. His research interests include array signal processing and where Ps is a matrix of the signal power, L denotes geometric algebra with applications in signal processing and image snapshots, Enlv are the eigenvectors corresponding to the processing. noise subspaces in the LV-MODEL. We use a orthogo- Jian Qiu Zhang received the B.Sc. degree from East of China Institute of nal quaternion transformation matrix T to project the Engineering, Nanjing, in 1982, and the M.S. and Ph.D. degrees from Harbin Institute of Technology (HIT), Harbin, China, in 1992 and 1996, respectively. He original array data into the biquaternion beamspace, So is currently a Professor with the Department of Electronic Engineering, Fudan one has University, Shanghai, China. From 1999 to 2002, he was a Senior Research Fellow at the School of Engineering, University of Greenwich, Chatham Abq = TA (50) Maritime, UK In 1998, he was a Visiting Research Scientist at the Institute of Intelligent Power Electronics, Helsinki University of Technology, Espoo, Finland. He was an Associate Professor from 1995 to 1997 and a Lecturer from = 1989 to 1994 with the Department of Electrical Engineering, HIT. During 1982 En TEnlv (51) to 1987, he was an Assistant Electronic Engineer at the 544th Factory, Hunan, China. His main research interests are signal processing and its application for Since the coherent noise of the six constituent anten- advanced sensors, intelligent instrumentation systems and control, and nas in each vector-sensor has been whitened by the matrix communications. Li et al. EURASIP Journal on Advances in Signal Processing (2017) 2017:16 Page 15 of 15

Competing interests The authors declare that they have no competing interests.

Received: 24 August 2016 Accepted: 6 February 2017

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