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Coherent Source Localization: BicomplexPolarimetric Smoothing with ElectromagneticVector-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.

0018-9251/11/$26.00 c° 2011 IEEE

2268 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 3 JULY 2011

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

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

CORRESPONDENCE 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

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.

DEFINITION 1 (Bicomplex and Bicomplex Matrix) A

bicomplex number b 2 D is defined as

b¢=b00 +~ib01 +~j(b10 +~ib11) (1)

where bmn 2R (m,n= 0,1), ~i and ~j are imaginary

units such that ~i2 =~j2 =¡1, ~i ¢~j =~j ¢~i. In addition,b can also be expressed as

b = c0 +~jc1 = S(b)+V(b) (2)

where cm¢=bm0 +~ibm1, m= 0,1, S(b)

¢=b00 is the scalar

part of b, and V(b)¢=b¡S(b) is the vector part. We

can further define bicomplex matrix B 2 DM£N as thematrix 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.

DEFINITION 2 (Conjugation and Transposes) The

total conjugation of B=C0 +~jC1 2 DM£N is given by

B =C¤0¡~jC¤1 (3)

where C¤n denotes the complex conjugation ofCn 2 CM£N , (n= 0,1). There also exist some otherdefinitions of conjugation [42, 44]. Moreover, the

transpose and total conjugated transpose are defined

as

Transpose: BT =CT0 +~jCT1 (4)

Total Conjugated Transpose: BH = BT =CH0 ¡~jCH1(5)

where CTn and CHn are the complex transpose and

conjugated transpose of Cn, n= 0,1, respectively.

DEFINITION 3 (Norms and Vector Orthogonality)

The norm of b = b00 +~ib01 +~j(b10 +~ib11) 2 D is

jbj ¢=S(bb) =qb200 + b

201 + b

210 +b

211: (6)

In addition, the norms of vector b 2 DN , and matrixB 2 DM£N could be defined as

kbk ¢= jbHbj

kBk=vuut MX

i=1

NXj=1

jB(i,j)j2:(7)

Generally jabj 6= jaj jbj for a,b 2D, so bicomplexdoes 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 jaHbj= 0. Wenote herein that the bicomplex vector orthogonality

imposes stronger constraints on the vector components

if we follow a similar analysis as that in [35].

DEFINITION 4 (Adjoint Matrix) A bicomplex matrix

B=C0 +~jC1 2DM£N could be linked to a uniquecomplex adjoint matrix ÂB 2C2M£2N via the followingequation:

B= 12ªMÂBª

HN (8)

where ªK = [IK ,¡~jIK], IK is a K £K identity matrix,

(K =M,N), and

ÂB¢=

·C0 C1

¡C1 C0

¸:

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 = ÂAÂB , where ÂAB , ÂA, ÂB are theadjoint matrices of AB, A, B, respectively.

LEMMA 2 A 2 DM£M has a multiplicative inverseA¡1 2 DM£M , such that AA¡1 =A¡1A= IM , if and onlyif 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 isinvertible, or det(Âb) = c

20 + c

21 6= 0. In other words,

division for fb = c0 +~jc1 2D j c20 + c21 = 0g cannot beuniquely 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 valueof R ·min(M,N), such that there exists at least


one selection of R linearly independent column

(row) vectors of B (the bicomplex linearindependence is similarly defined as the quaternion

case [33, 34]). In addition, denoting ÂB as the adjointmatrix 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 2DN£N (B is Hermitian if B= BH) is givenby

B¢=

2NXn=1

¸nunuHn (9)

where ¸n 2R and un 2 DN are the nth eigenvalueand eigenvector, respectively, Bun = ¸nun, kunk= 1,and juHn1un2 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 adjointmatrix ÂB share the same eigenvalues, and that un =p2¡1ªMvn, where vn is the nth eigenvector of ÂB .

Therefore, the BC-EVD of B could be obtained by theeigenvalue decomposition of ÂB . In addition we havethe 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 2DN£Nis to find a matrix B0 2 DN£N with rank R0 < R suchthat kB¡B0k is minimized. This could be realizedby 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 =P2R0n=1¸nunu

Hn (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

1A 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.

Fig. 1. Coordinates and angle definition.

steering vector of a complete EM vector-sensor then

could be written as

aμ,Á,°,´¢=

·pμ+¼=2,0 pμ,Á+¼=2

¡pμ,Á+¼=2 pμ+¼=2,0

¸| {z }

Fμ,Á

·cos® sin®

¡sin® cos®

¸| {z }

Q®

·cos¯

~isin¯

¸| {z }

w¯

(10)

where pμ,Á¢=[cosÁcosμ,cosÁsinμ, sinÁ]T denotes

a unit vector with orientation defined by (μ,Á).

By definition we have pμ+¼=2,0 = [¡sinμ,cosμ,0]Tand pμ,Á+¼=2 = [¡cosμ sinÁ,¡sinμ sinÁ,cosÁ]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 andpμ,Á+¼=2 into (10). Specifically, the polarization stateis represented by orientation angle ® and ellipticity

angle ¯. Moreover, signals with ¯ =§¼=4 are denotedas 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:

tan2®= tan2° cos´

sin¯ = sin2° sin´:(11)

By denoting q 2R6 as the selection vectorobtained 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)+~jaμ,Á,°,´(q2)

b2,μ,Á,°,´¢=aμ,Á,°,´(q3)+~jaμ,Á,°,´(q4)

b3,μ,Á,°,´¢=aμ,Á,°,´(q5)+~jaμ,Á,°,´(q6)

(12)

where ql denotes the lth entry of q, and aμ,Á,°,´(ql)

denotes the qlth entry of aμ,Á,°,´, l = 1,2,3. Bydefinition, 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

CORRESPONDENCE 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 bydμ,Á = [e

~i¢2¼(kT1pμ,Á=¸),e

~i¢2¼(kT2pμ,Á=¸), : : : ,e

~i¢2¼(kTNpμ,Á=¸)]

(13)

where kn is the position vector of the nth sensor,¸ is the wavelength of the impinging signals, and

pμ,Á = [cosÁcosμ,cosÁsinμ, sinÁ]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)¢=

MXm=1

bl,μm ,Ám ,°m ,´m| {z }bl,m

dμm ,Ám| {z }dm

sm(t)+nl(t) (14)

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,

bl,m¢=bl,μm ,Ám,°m,´m , and dm

¢=dμm,Ám , l = 1,2,3, m=

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

¾2s ;

A2) The noises are zero-mean, stationary, spatially

white, uncorrelated with the sources, and with

identical noise power ¾2" ;

A3) The sources have distinct DOAs and any

arbitrary K (K ·N) spatial steering vectors associatedwith different DOAs are linearly independent.

A4) The number of sources M is known and there

are more vector-sensors than sources (N >M).

B. Bicomplex Polarimetric Smoothing

DEFINITION 8 (Bicomplex Covariance Matrix) The

bicomplex covariance matrix of xl(t) 2 DN withzero-mean is defined as

Rl(t)¢=E(xl(t)x

Hl (t)) (15)

where E(¢) denotes the mathematical expectation.Under the assumptions A1 and A2, we know that

rm1,m2 = ¾2s , and R",l = ¾

2" IN , where rm1,m2 denotes the

covariance between the m1th and m2th 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)xHl (t))

= ¾2s

MXm1=1

MXm2=1

bl,m1dm1dHm2bl,m2 +¾

2" IN

= ¾2s [bl,1d1,bl,2d2, : : : ,bl,MdM]

£ 1M£M[bl,1d1,bl,2d2, : : : ,bl,MdM]H +¾2" 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

Rl = ¾2s [d1, : : : ,dM]| {z }

D

2664bl,1

. . .

bl,M

3775

£ 1M£M

2664bl,1

. . .

bl,M

3775[d1,d2, : : : ,dM]H| {z }DH

+¾2" IN

=D(¾2s blbHl )D

H +¾2" IN (17)

where bl¢=[bl,1,bl,2, : : : ,bl,M]

T. We know from the

rank definition of bicomplex matrices (Definition 5)


that rank(¾2s blbHl ) = 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:

R¢=

3Xl=1

Rl =D

3Xl=1

(¾2s blbHl )D

H +¾2" IN

=D(¾2sBBH)DH| {z }

R0

+¾2" IN (18)

where B= [b1,b2,b3], and R0 ¢=D(¾2sBB

H)DH is

denoted as the signal part of the smoothed covariance

matrix R. In addition we assume M · 3 and that B isfull 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 subspaceidentified 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 bicomplexsub-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 correspondingbicomplex covariance matrices fRl j l = 1,2, : : : ,6g by(14) and (16), respectively. Therefore, BPS for 4·M · 6 could be revised as R=PM

l=1Rl. Obviously,

the matrix B= [b1,b2, : : : ,bM] must not contain

collinear columns, so q0 should be selected in the waysuch that the obtained sub-vector-sensors fbl,μ,Á,°,´ jl = 1,2, : : : ,6g are mutually different (for example, wemay choose q0 = [q2,q1,q4,q3,q6,q5]

T). However, this

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 whosecolumns 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

P¢=P2Nn=2M+1unu

Hn . Then the angular parameters (μ, Á)

that minimize kPdμ,Ák are the estimates of the trueDOAs, that is

(μ, Á) = argminμ,Á

kPdμ,Ák (19)

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:

Rl =1

K

KXk=1

xl(tk)xHl (tk) (20)

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 theellipticity angles of these four signals are not equal.


five signals with distinct DOAs, subject to certain

choices of the selection vectors q and q0, if exactlytwo or three sources are circularly polarized with the

same spin.

CORRESPONDENCE 2273


six signals with distinct DOAs, subject to certain

choices of the selection vectors q and q0, if exactlythree 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):

q= argmaxq2¡

μ¸2M ¡¸2M+1

¸1

¶(21)

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

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 notethat hypercomplex algebra imposes strongerconstraints on vector orthogonality accordingto [34]—[36], and thus bicomplex algebra-basedmethods (such as BPS-MUSIC) may be advantageousover complex-based methods, with regards to therobustness to errors caused by noise, short datalength, or model errors. When compared with otherhypercomplex algebras, we note that bicomplexdistinguishes itself from quaternion, biquaternion, andquad-quaternion with the property of multiplicativecommutativity, and this property is crucial for theproposed method. More precisely, we note that a keypoint of the proposed method is that the bicomplexcovariance matrix Rl could be reformulated intothe form given in (17) (Rl =D(¾

2s blb

Hl )D

H , forclarity we remove the noise term), wherein thematrix ¾2s blb

Hl in the “center” is multipled by D

and DH from the “outside.” This special structure(with D and DH on the outside and ¾2s blb

Hl in the

center) enables rank restoration via BPS by notingthat

Pl ¾2s blb

Hl may be full rank, and also enables

estimation of the noise subspace orthogonal tothe 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 withdm, and thus the multiplicative commutativity iscrucial for our algorithm. For clearance, we next showthat polarimetric smoothing may fail for coherentsignals if quaternion is used instead of bicomplex.More exactly, according to [34] the quaternioncovariance matrix of the lth subarray is given byR0l =

PMm1=1

PMm2=1

ql,m1dm1rm1,m2dHm2q¤l,m2 , where ql,m1 is

a quaternionic alternative for bl,m1 in (14) by using thequaternion model instead. In addition, in the presenceof coherent signals such that rm1,m2 = ¾

2s we have

R0l = ¾2s

PMm1=1

PMm2=1

ql,m1dm1dHm2q¤l,m2 = ¾

2s¥l1M£M¥

Hl ,

andPlR

0l = ¾

2s

Pl¥l1M£M¥

Hl , where 1M£M denotes

an M £M matrix with all the elements equal to 1, and

¥l

¢=[ql,1d1, : : : ,ql,MdM]. Since the multiplication is not

commutative, ql,m does not commute with dm so thatPlR

0l = ¾

2s

Pl¥l1M£M¥

Hl could not be modified into

the structure desired by subspace-based methods (afull rank matrix in the center and matrices of vectorsthat span the signal subspace on the outside). Thisexample shows that the quaternion-MUSIC could onlybe used for direction finding of incoherent signals.

2The 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.


TABLE I

Computational Efforts for Covariance Matrix Estimation

Memory Requirements

(real values) Real Multiplications Real Additions Real Divisions

BPS-MUSIC 12N2 48N2T (48T¡ 12)N2 12N2

CPS-MUSIC 12N2 24N2T (16T¡ 8)N2 8N2

CPSAC 42N2 84N2T (56T¡ 28)N2 28N2

CPDS 12N2 24N2T (16T¡ 8)N2 8N2

Q-MUSIC 12N2 48N2T (48T¡ 12)N2 12N2

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 thearray into M arrays of overlapping sub-vector-sensors. Calculate the sampled covariance matrix Rl for each subarray by (20),

obtain the smoothed covariance matrix R by (18), perform BC-EVD to R, and choose the selection vector q that maximizes (21).

3) For the smoothed covariance matrix R that corresponds to the optimal selection vector q, use the eigenvectors associated with its

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],

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

CORRESPONDENCE 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:

Â¢=1

3

3Xm=1

vuut 1

200

200Xn=1

[(μm¡ μn,m)2] (22)

where μm is the mth true azimuth, and μn,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¢=10log10(¾

2s =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 theincident 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

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 thatare 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 slightlyoutperforms 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


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 dBand the number of snapshots varies between 200 and

3The selection vector used in BPS-MUSIC-2 is not the globally

optimal one, since it is only chosen among ten randomly generated

candidates.

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

CORRESPONDENCE 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

dm = dm+pP"m (23)

where dm and dm are the true and ideal spatial steering

vectors associated with the mth signal, respectively.

"m 2CN 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,

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 andthird sources are 0± and 39±, respectively; the secondsource 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 I0 is number of trials that

successfully detect all three incident coherent signals,

and I is the total number of independent trials. The


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 almostequal 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

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

CORRESPONDENCE 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 sixsources are μ1 = 10

±, μ2 = 30±, μ3 = 60

±, μ4 = 80±,

μ5 = 120±, and μ6 = 150

±, respectively. The orientationangles and ellipticity angles f(®m,¯m) jm= 1,2, : : : ,6gthat 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 overlappingBPS, 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 andq0, if exactly three signals are circularly polarizedwith 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

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 bicomplexmatrix with rank R, and B0 2DN£N is a rank R0 matrix


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

B0 =2R0Xn=1

¸nunuHn (24)

where ¸n is the nth largest eigenvalue of B, and unis 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

un as un =p2¡1ªMvn. Therefore, we further have

Â0B =P2R0n=1¸nvnv

Hn which indicates that Â

0B could be

obtained from ÂB via truncated EVD. Obviously, Â0B

is the best rank 2R0 approximation of ÂB in the sensethat kÂB¡Â0BkF is minimized, where k ¢ kF denotesFrobenius norm in the complex domain. Moreover,

denoting B=C0 +~jC1, and B0 =C00 +~jC

01 we have

the following equation according to (6), (7), and

Definition 4:

kB¡B0k2 = kC0¡C00k2F + kC1¡C01k2F= 2¡1kÂB¡ÂB0 k2F: (25)

As a result kB¡B0k is also minimized. This showsthat truncating the BC-EVD of B yields its best rankR0 approximation.

APPENDIX II. PROOF OF THE RESULTS GIVEN INREMARK 3

Assume B=C0 +~jC1, and then according toDefinition 5 we have rank(B) = rank(ÂB)=2, whereÂB is the adjoint matrix of B:

ÂB =

·C0 C1

¡C1 C0

¸: (26)

First we consider the nonoverlapping BPS where

B= [b1,b2,b3], bl = [bl,1,bl,2, : : : ,bl,M]T, and bl,m =

bl,μmÁm°m´m is given in (12), l = 1,2,3, m= 1,2, : : : ,M.

By further denoting h(0)m¢=[am(q1),am(q3),am(q5)]

T, and

CORRESPONDENCE 2281

h(1)m¢=[am(q2),am(q4),am(q6)]

T where am = aμm,Ám ,°m ,´mand qk is the kth element of the selection vector

q, we have C0 = [h(0)1 ,h

(0)2 , : : : ,h

(0)M ]

T, and C1 =

[h(1)1 ,h(1)2 , : : : ,h

(1)M ]

T. 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 aresult 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. Bydenoting h(0)m

¢=[am(q1),am(q3),am(q5),am(q

01), : : : ,

am(q02M¡7)]

T, and h(1)m¢=[am(q2),am(q4),am(q6),am(q

02)

, : : : ,am(q02M¡6)]

T, we have C0 = [h(0)1 ,h

(0)2 , : : : ,h

(0)M ]

T 2CM£M and C1 = [h

(1)1 ,h

(1)2 , : : : ,h

(1)M ]

T 2 CM£M . Inaddition we note that rank[C0,C1] = rank[A,A

0] =rank(A), wherein A

¢=[a1,a2, : : : ,aM]

T 2 CM£6, and A0 2CM£(2M¡6) contains the q0lth column of A as its lthcolumn, l = 1,2, : : : ,2M ¡6. Moreover, it is possibleto choose the selection vectors q and q0 such that C0contains rank(A) linearly independent columns, andrank(C0) = rank[C0,C1] = rank(A). Therefore, for thespecial 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) = 5in 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

that they are linearly dependent. Then there exist

c1,c2, : : : ,c6 not all zero, such that:

c1a1 + c2a2 + ¢ ¢ ¢+ c6a6 = o6£1: (28)

We can further deduce that c1,c2, : : : ,c6 6= 0. To seethis, 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 berearranged as follows:264c1Q1w1c2Q2w2

c3Q3w3

375=¡(c4§¡1F4Q4 + c5§

¡1F5Q5 + c6§¡1F6Q6)w§¼=4

(29)

where real-valued matrix §¢=[F1 F2 F3] is invertible

according to [10, Theorem 1 and Theorem 2]. Writing

cl = jclje~i!l , !l 2 (¡¼,¼], l = 4,5,6, and using [10,Lemma 5], we could obtain264c1Q1w1c2Q2w2

c3Q3w3

375=¡(jc4j§¡1F4Q®4§!4 + jc5j§¡1F5Q®5§!5+ jc6j§¡1F6Q®6§!6 )w§¼=4: (30)

Now by using the strategy adopted in [10,

Appendix D], (30) could be rewritten as

264c1Q1w1c2Q2w2

c3Q3w3

375=¡

26666666664

x1 y1

¡y1 x1

x2 y2

¡y2 x2

x3 y3

¡y3 x3

37777777775w§¼=4

=

266664c1

qx21 + y

21Q®1

c2

qx22 + y

22Q®2

c3

qx23 + y

23Q®3

377775w§¼=4 (31)

where ·xk yk

¡yk xk

¸2R2£2

is non-zero, ®k = arctan(yk=xk) 2 (¡¼=2,¼=2], andck 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=4which is a contradiction. As a result, a1,a2, : : : ,a6 are

linearly independent.


APPENDIX III. CALCULATION OF THECOMPUTATIONAL 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 +~jXl,1,

where Xl,0,Xl,1 2CN£T, and the sampled covariancematrix is thus given by Rl = T

¡1XlXHl . By definition

we know that Rl has N2 bicomplex entries, and

thus occupies the memory of 4N2 real scalars.

In addition, each element of Rl is obtained via

T bicomplex multiplications, T¡ 1 bicomplexadditions 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 16N2T(M) + 12N2T(A) + 4N

2(T¡ 1)(A)+4N2(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 12N2 real scalars, and the total

number of basic arithmetical operations is 48N2T(M) +

(48T¡ 12)N2(A) + 12N2(D). In addition it is easy to verifythat 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 12N2

and 24N2T(M) + (16T¡ 8)N2(A) + 8N2(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

matrices are 42N2 and 84N2T(M) + (56T¡ 28)N2(A) +28N2(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

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Comment on “One-Step Solution for the MultistepOut-of-Sequence-Measurement Problem inTracking”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.

1Bar-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.

0018-9251/11/$26.00 c° 2011 IEEE

CORRESPONDENCE 2285

coherent source localization: bicomplex polarimetric smoothing...

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