Simultaneous diagonalisation of the covariance and ...mandic/research/Quaternion... · the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This

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Signal Processing 148 (2018) 193–204

Contents lists available at ScienceDirect

Signal Processing

journal homepage: www.elsevier.com/locate/sigpro

Simultaneous diagonalisation of the covariance and complementary

covariance matrices in quaternion widely linear signal processing

Min Xiang

a , ∗, Shirin Enshaeifar b , Alexander E. Stott a , Clive Cheong Took

b , Yili Xia

c , Sithan Kanna

a , Danilo P. Mandic

a

a Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2BT, UK b Department of Computer Science, University of Surrey, Surrey GU2 7XH, UK c School of Information Science and Engineering, Southeast University, Nanjing 210018, China

a r t i c l e i n f o

Article history:

Received 31 July 2017

Revised 16 January 2018

Accepted 12 February 2018

Available online 13 February 2018

Keywords:

Quaternion matrix diagonalisation

Complementary covariance

Pseudo-covariance

Widely linear processing

Quaternion non-circularity

Improperness

a b s t r a c t

Recent developments in quaternion-valued widely linear processing have established that the exploita-

tion of complete second-order statistics requires consideration of both the standard covariance and the

three complementary covariance matrices. Although such matrices have a tremendous amount of struc-

ture and their decomposition is a powerful tool in a variety of applications, the non-commutative nature

of the quaternion product has been prohibitive to the development of quaternion uncorrelating trans-

forms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance

and complementary covariance matrices in the quaternion domain, whereby the quaternion version of

the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new

insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quater-

nion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance

matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating

transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.

© 2018 Elsevier B.V. All rights reserved.

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. Introduction

Advances in sensing technology have enabled widespread

ecording from 3-D and 4-D data sources, such as measurements

rom seismometers [1] , ultrasonic anemometers [2] , and inertial

ody sensors [3] . It is convenient to express such measurements

s vectors in the R

3 and R

4 fields of reals, however, the real vec-

or algebra is not a division algebra and is therefore inadequate

hen modelling orientation and rotation [4] . Quaternions have ad-

antages in representing 3-D and 4-D data, owing to their divi-

ion algebra, a generic extension of the real and complex alge-

ras. Quaternions also naturally account for mutual information

etween multiple data channels, provide a compact representation,

nd have proven to offer a physically meaningful interpretation

or real-world applications in the fields of navigation, communi-

ation, and image processing [5,6] . A recent resurgence in research

n quaternion signal processing spans the areas of filtering [7] , in-

ependent component analysis (ICA) [8,9] , neural networks [10,11] ,

nd Fourier transforms [12] .

∗ Corresponding author.

E-mail address: [email protected] (M. Xiang).

i

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ttps://doi.org/10.1016/j.sigpro.2018.02.018

165-1684/© 2018 Elsevier B.V. All rights reserved.

Diagonalisable matrices play a fundamental role in engineering

pplications, whereby the related uncorrelating transforms greatly

implify the analysis of complex problems, both in terms of en-

bling a solution and providing a methodological framework (e.g.

erformance bounds). Computational advantages associated with

atrix diagonalisation include a reduction in the size of the pa-

ameter space from N

2 to N , while the statistical advantages in-

lude the possibility to match source properties (e.g. orthogonal-

ty) in blind source separation. In addition, joint diagonalisation of

atrices in some blind source separation applications has a close

ink with the canonical decomposition problem for tensors [13,14] .

he real linear algebra is a mature area, however, covariance ma-

rices in widely linear signal processing [15] in the complex and

uaternion division algebras and their structures [16] have recently

eceived significant attention [17–21] .

In the context of complex-valued signal processing, for a com-

lex random vector z , both its covariance, C = E{ zz H } , and pseudo-

ovariance, P = E{ zz T } , matrices are necessary to capture complete

econd-order statistical information [22] . To extract the statistical

nformation embedded in the two covariance matrices, simultane-

us diagonalisation of C and P is a prerequisite, while their respec-

ive individual diagonalisations can be performed via the eigende-

omposition and the Takagi factorisation. Such decorrelations exist

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194 M. Xiang et al. / Signal Processing 148 (2018) 193–204

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in their general forms in mathematics [23] . De Lathauwer and De

Moor [15] and Eriksson and Koivunen [16] have given them a prac-

tical value through an important engineering contribution called

the strong uncorrelating transform (SUT), together with its exten-

sion, the generalised uncorrelating transform [24] . Our own recent

contributions include a computationally efficient approximate un-

correlating transform (AUT) [25] , while to preserve the integrity of

the original bivariate sources in the decorrelation process, we have

proposed the correlation preserving transform [26] . These trans-

forms have been used in performance analysis of complex-valued

adaptive filters [27–30] , blind separation of non-circular complex-

valued sources [31,32] , complex-valued subspace tracking [33] , and

separation of signal and noise components in subbands of har-

monic signals [34] .

Real- and complex-valued signal processing have benefited

greatly from making sophisticated use of real- and complex-

valued matrix algebras [35,36] . However, quaternion-valued sig-

nal processing, which is advantageous for 3-D and 4-D data, has

been hindered by the underdevelopment of quaternion-valued

matrix algebra [37] , compared to the well-established real- and

complex-valued matrix algebras [23,36] . This is due to the non-

commutativity nature of quaternion multiplication and the en-

hanced degrees of freedom provided by quaternions. For exam-

ple, a quaternion square matrix has right and left eigenvalues; the

right eigenvalues have been well studied, while the left eigenval-

ues are less known and are not computationally well-posed [37,38] .

The recent theory of quaternion matrix derivatives provides a sys-

tematic framework for the calculation of derivatives of quaternion

matrix functions with respect to quaternion matrix variables [39] .

Advances in structural quaternion matrix decompositions include

tools to diagonalise quaternion matrices [40,41] , while in the con-

text of quaternion widely linear processing, it was shown that

for a quaternion random vector x , the diagonalisation of the Her-

mitian covariance matrix C x = E{ xx H } can be performed straight-

forwardly using the quaternion eigendecomposition, whereas the

diagonalisation of the complementary covariance matrices ( C x ı =E { xx ıH } , C x j = E { xx jH } , C x κ = E { xx κH } ) can be computed via the

quaternion singular value decomposition (SVD) [42] . Furthermore,

the quaternion uncorrelating transform (QUT), a quaternion ana-

logue of the SUT for complex data, was proposed in [43] to jointly

diagonalise the covariance matrix and one of the three comple-

mentary covariance matrices. However, the simultaneous diagonal-

isation of all the four covariance matrices is still an open problem.

It is important to notice that the relationship between the quater-

nion covariance matrices in the widely linear model is governed by

Cheong Took and Mandic [21]

P x =

1

2

(C x ı + C x j + C x κ − C x ) (1)

which demonstrates that the diagonalisation of the pseudo-

covariance matrix, P x = E{ xx T } , requires us to address the follow-

ing issues:

• There is no closed-form solution

1 to perform a simultaneous di-

agonalisation of the covariance matrix and the three comple-

mentary covariance matrices. • Given the dimensionality of the problem and numerous prac-

tical applications, a simple approximate diagonalising trans-

form which would apply to both the covariance and pseudo-

covariance matrices would be beneficial.

In this paper, we therefore set out to propose solutions to these

open problems, and support the analysis with illustrative exam-

ples.

1 Iterative solutions were proposed in the context of quaternion ICA [8] .

s

The rest of this paper is organised as follows. Section 2 provides

n overview of quaternion algebra and statistics. Section 3 pro-

oses novel techniques for the diagonalisation of symmetric

uaternion matrices and a simultaneous diagonalisation of two

uaternion covariance matrices. Section 4 proposes an approx-

mate simultaneous diagonalisation of the four quaternion co-

ariance matrices. Simulation results are given in Section 5 , and

ection 6 concludes the paper. Throughout the paper, we use bold-

ace capital letters to denote matrices, A , boldface lowercase let-

ers for vectors, a , and standard letters for scalar quantities, a . Su-

erscripts ( · ) T , ( · ) ∗ and ( · ) H denote the transpose, conjugate, and

ermitian (i.e. transpose and conjugate), respectively, I the identity

atrix, and E { · } the statistical expectation operator.

. Background

.1. Quaternion algebra

The quaternion domain H is a 4-D vector space over the field

f reals, spanned by the basis {1, ı, j , κ}, where ı = (1 , 0 , 0) , j =(0 , 1 , 0) and κ = (0 , 0 , 1) are three unit vectors in R

3 . A quaternion

ector 2 This does not affect the generality of our results. x ∈ H

L ×1

onsists of a real (scalar) part R [ ·] and an imaginary (vector) part

[ ·] which comprises ı −, j−, and κ− imaginary components, so

hat

= R [ x ] + I [ x ]

= R [ x ] + I ı [ x ] ı + I j [ x ]j + I κ [ x ] κ

= x a + x b ı + x c j + x d κ (2)

here R [ x ] = x a , I ı [ x ] = x b , I j [ x ] = x c , I κ [ x ] = x d are L × 1 real

ectors, and ı, j , κ are orthogonal imaginary units with proper-

ies

j = −j ı = κ jκ = −κj = ı κ ı = −ıκ = j

ı 2 = j 2 = κ2 = −1 (3)

quaternion variable x is called a pure quaternion if it satisfies

[ x ] = 0 . The modulus of a quaternion variable x ∈ H is defined as

x | =

√

x 2 a + x 2 b

+ x 2 c + x 2 d

quaternion variable x is called a unit quaternion if it satisfies

x | = 1 . The product of two quaternions x, y ∈ H are defined as

y = R [ x ] R [ y ] − I [ x ] · I [ y ] + R [ x ] I [ y ] + R [ y ] I [ x ] + I [ x ] × I [ y ]

here “ · ” denotes the scalar product and “ × ” the vector product.

otice that the presence of the vector product in the above expres-

ion causes the non-commutativity of the quaternion product, that

s, xy � = yx .

Another important notion in the quaternion domain is the so-

alled “quaternion involution” [44] , which defines a self-inverse

apping analogous to the complex conjugate. The general involu-

ion of the quaternion vector x is defined as x α = −αx α, and rep-

esents the rotation of the vector part of x by π about a unit pure

uaternion α. The special cases of involutions about the ı, j and κmaginary axes are given by Ell and Sangwine [44]

x

ı = −ı x ı = x a + x b ı − x c j − x d κ

x

j = −jx j = x a − x b ı + x c j − x d κ

κ = −κx κ = x a − x b ı − x c j + x d κ (4)

his set of involutions, together with the original quaternion, x ,

orms the most frequently used basis for augmented quaternion

tatistics [20,21] . While an involution represents a rotation along

2 Throughout this paper, the quaternion vector is assumed to be zero-mean.

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M. Xiang et al. / Signal Processing 148 (2018) 193–204 195

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3 It is called C α-proper in some literature, but we consider the term C α-improper

is more intuitive. 4 They are both right and left eigenvalues, but are simply termed eigenvalues for

conciseness in this paper.

single imaginary axis, the quaternion conjugate operator ( · ) ∗ ro-

ates the quaternion along all three imaginary axes, and is given

y

∗ = R [ x ] − I [ x ] = x a − x b ı − x c j − x d κ (5)

useful algebraic operation which conjugates only one imaginary

omponent is given by

x

ı ∗ = (x

ı ) ∗ = (x

∗) ı = x a − x b ı + x c j + x d κ

x

j∗ = (x

j ) ∗ = (x

∗) j = x a + x b ı − x c j + x d κ

κ∗ = (x

κ ) ∗ = (x

∗) κ = x a + x b ı + x c j − x d κ (6)

ince the only difference between x and x α∗, for α ∈ { ı, j , κ}, is

he sign of the α-imaginary component, a quaternion vector x is

alled ( · ) α∗ invariant if I α[ x ] = 0 . For example, if x has a vanishing

-imaginary part, then x ı ∗ = x . For rigour, the structure and general

roperties of quaternion matrices are given in Appendix A .

.2. Quaternion covariance matrices: Structural insights

The recent success of complex augmented statistics is largely

ue to the simplicity and physical meaningfulness of the statistical

escriptors in the form of the standard covariance, C x = E{ xx H } ,nd the pseudo-covariance, P x = E{ xx T } . In particular, the pseudo-

ovariance enables us to account for the improperness (chan-

el power imbalance or correlation) of complex variables, while

he symmetry of P x implies that its decomposition can be com-

uted using the Takagi factorisation as P x = Q �Q

T , where Q is a

omplex unitary matrix and � a real non-negative diagonal ma-

rix [23] . However, the pseudo-covariance of a quaternion ran-

om vector x = [ x 1 , x 2 , . . . , x L ] T does not exhibit symmetry, ow-

ng to the non-commutativity of the quaternion product, whereby

m

x n � = x n x m

( m � = n ), yield an asymmetric matrix

x = E{ xx

T } =

⎡

⎢ ⎢ ⎣

E{ x 1 x 1 } E{ x 1 x 2 } . . . E{ x 1 x L } E{ x 2 x 1 } E{ x 2 x 2 } . . . E{ x 2 x L } . . .

. . . . . .

. . . E{ x L x 1 } E{ x L x 2 } . . . E{ x L x L }

⎤

⎥ ⎥ ⎦

(7)

The quaternion involution basis in (4) is at the core of the re-

ently proposed widely linear processing [20,45] , which provides

heoretical and practical performance gains over traditional strictly

inear processing [46] . For example, the widely linear minimum

ean square error (MMSE) estimation of the quaternion signal y

n terms of the observation x is performed as

ˆ = E{ y | x , x

ı , x

j , x

κ} (8)

hich is analogous to the complex widely linear MMSE estimation

22] , given by

ˆ = E{ y | x , x

∗} hen, the involution basis for the quaternion widely linear process-

ng provides a useful description of quaternion second-order statis-

ics, represented by the standard covariance matrix and the ı −, j−,

nd κ− complementary covariance matrices, which are given by

heong Took and Mandic [21]

x α = E{ xx

αH } =

⎡

⎢ ⎢ ⎣

E{ x 1 x α∗1 } E{ x 1 x α∗

2 } · · · E{ x 1 x α∗L }

E{ x 2 x α∗1 } E{ x 2 x α∗

2 } · · · E{ x 2 x α∗L }

. . . . . .

. . . . . .

E{ x L x α∗1 } E{ x L x α∗

2 } · · · E{ x L x α∗L }

⎤

⎥ ⎥ ⎦

(9)

here α ∈ { ı, j , κ}. The α-complementary covariance matrix is α-

ermitian, that is, C x α = (C x α ) αH , which stems from the fact that

ts diagonal entries are ( · ) α∗ invariant, whereas its off-diagonal

ntries are governed by the relationship C x α [ m, n ] = (C x α [ n, m ]) α∗,here C

α [ m, n ] denotes the entry of C

α in row m , column n . The

x x

elationship between the pseudo-covariance and the three comple-

entary covariances is given in (1) .

emark 1. The knowledge of both the standard covariance matrix

nd the three complementary covariance matrices is necessary to

nsure the exploitation of complete second-order statistical infor-

ation in the quaternion domain.

.3. Quaternion properness

The notion of non-circularity (improperness) is unique to divi-

ion algebras. While non-circularity refers to probability distribu-

ions which are not rotation-invariant, a proper complex random

ector z = z r + z ı ı ( z r , z ı ∈ R

L ×1 ) has a vanishing pseudo-covariance

{ zz T } = 0 . In other words, its real and imaginary parts are uncor-

elated and have equal variance, that is, E{ z r z T ı } = 0 and E{ z r z T r } ={ z ı z T ı } . Similarly, improperness in the quaternion domain is char-

cterised by the degree of correlation and/or power imbalance be-

ween imaginary components relative to the real component. The

dditional degrees of freedom in the quaternion domain allow for

wo types of properness: H -properness and C

α-properness [47] .

efinition 1 ( H -properness ) . A quaternion random vector x is H -

roper if it is uncorrelated with its vector involutions, x ı, x j , and

κ , so that

C x ı = E{ xx

ıH } = 0 C x j = E{ xx

jH } = 0

x κ = E{ xx

κH } = 0 (10)

efinition 2 ( C

α-properness ) . A quaternion random vector x is

α-improper 3 with respect to α = ı, j or κ if it is correlated only

ith the involution x α , so that all the complementary covariances

xcept for C x α vanish.

As shown in Appendix B , a C

α-improper quaternion random

ector can be generated from two proper complex random vectors.

. Diagonalisation of quaternion covariance matrices

Prior to introducing the diagonalisation of the quaternion

seudo-covariance matrix and the simultaneous diagonalisation of

wo quaternion covariance matrices, the following two observa-

ions establish results essential for subsequent analyses.

bservation 1. The eigendecomposition applied to the Herimitian

uaternion covariance matrix, C x , gives C x = Q

H �x Q , where Q is

quaternion unitary matrix and �x a real-valued diagonal matrix

or which the entries are the eigenvalues 4 of C x [48] .

bservation 2. The α-complementary covariance matrix, C x α ,

∈ { ı, j , κ}, can be factorised as C x α = Q

H α�αQ

αα, where Q α is a

uaternion unitary matrix and �α is a real-valued non-negative

iagonal matrix for which the diagonal entries are the singular val-

es of C x α [42] .

Observation 1 provides an algebraic tool to diagonalise the

uaternion covariance matrix. Observation 2 enables the diagonal-

sation of quaternion complementary covariance matrices, and de-

enerates into the Takagi factorisation of complex symmetric ma-

rices if the quaternion vector x has two vanishing imaginary parts

49] .

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196 M. Xiang et al. / Signal Processing 148 (2018) 193–204

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3.1. Diagonalisation of symmetric matrices

The tools for the decomposition of quaternion pseudo-

covariance matrices are still in their infancy, because:

• The tools for the analysis of the symmetric matrices in C cannot

be readily generalised to H . • It is required to simultaneously diagonalise all the three com-

plementary covariance matrices, a task with prohibitively many

degrees of freedom.

We shall start our analysis with the diagonalisation of 2 × 2

symmetric quaternion matrices.

Proposition 1. A 2 × 2 symmetric quaternion matrix A admits the

factorisation A = U �U

T if either its diagonal elements or off-diagonal

elements are real-valued, where U is a quaternion unitary matrix and

� is a real diagonal matrix with the singular values of A on the di-

agonal.

Proof. Based on the SVD of quaternion matrices [37] , A = U �V

H ,

the matrix products AA

∗ and A

∗A can be expressed as

AA

∗ = AA

H = (U �V

H )(U �V

H ) H = U �2 U

H

A

∗A = A

H A = (U �V

H ) H (U �V

H ) = V �2 V

H

The result in Appendix C shows that ( AA

∗) ∗ = A

∗A if either the di-

agonal or off-diagonal elements of A are real-valued. It then fol-

lows that U

∗�2 (U

∗) H = V �2 V

H , where the commutativity of the

product is valid because of the condition that the diagonal or

off-diagonal elements of A are real-valued. Thus, we can obtain

V = U

∗, whereby the Takagi factorisation

5 of quaternion symmet-

ric matrices is identical to that of complex symmetric matrices,

A = U �U

T . �

A consequence of Proposition 1 is that the Takagi factorisation

of a quaternion symmetric matrix A is possible if ( AA

∗) ∗ = A

∗A

holds. However, in general, ( AA

∗) ∗ � = A

∗A , indicating that such a

Takagi factorisation does not exist for general quaternion symmet-

ric matrices.

3.2. Simultaneous diagonalisation of two covariance matrices

We next proceed to introduce a simultaneous diagonalisation

of the covariance and complementary covariance matrices. The

finding that a non-singular transform is sufficient to diagonalise 6

both C x and C x α will form a basis for the quaternion uncorrelat-

ing transform [43] . First, we shall clarify that the unitary trans-

form which can simultaneously diagonalise C x and C x α for a gen-

eral quaternion random vector x does not exist.

Proposition 2. If C x C x α � = (C x C x α ) αH , then there is no quaternion

unitary matrix U such that both U

H C x U and U

H C x α U

α are diagonal.

Proof. Define �x = U

H C x U , �α = U

H C x α U

α, where U is a quater-

nion unitary matrix. It is then obvious that �x is a Hermitian ma-

trix, so that if �x is diagonal, then �x is real diagonal. Further-

more, if �α is also diagonal, we can obtain

C x C x α =U �x U

H U �αU

αH =U �αU

αH U

α�x U

αH =C x α C

αx (11)

Because C x is Hermitian and C x α is α-Hermitian, we can readily

obtain

C x α C

αx = C

αH x α C

αH x = (C x C x α )

αH (12)

Combining (11) and (12) yields C x C

α = (C x C

α ) αH . �

x x

5 The Takagi factorisation is a special case of the complex/quaternion SVD when

A = A T . 6 Corollary 4.5.18 (c) in [23] is the corresponding result in the complex domain.

q

t

t

c

Although a unitary transform for the simultaneous diagonalisa-

ion of C x and C x α is inapplicable, a non-unitary transform can be

erived in a similar manner to the SUT in the complex domain.

n the basis of Observation 1 , C x can be factorised as C x = V�x V

H

here V is a quaternion unitary matrix, and �x is a real diago-

al matrix. Define a whitening transform D = V�− 1

2 x V

H , and de-

ote s = Dx , to obtain the covariance matrix of s as

s = DC x D

H = V�− 1

2 x V

H V�x V

H V�− 1

2 x V

H = I

ased on Observation 2 , the α-complementary covariance matrix

f s can be factorised as C s α = W�αW

αH , where W is a quaternion

nitary matrix, and �α is a real diagonal matrix. Now, the non-

ingular uncorrelating transform, Q = W

H D , and its application in

he form y = Qx yield the simultaneous diagonalisation

C y = W

H C s W = W

H IW = I

y α = W

H C s α W

α =W

H W�αW

αH W

α =�α

e refer to the transform Q as the quaternion uncorrelating trans-

orm (QUT), which is summarised in Proposition 3 and Algorithm 1 .

lgorithm 1 Quaternion Uncorrelating Transform (QUT).

1. Compute the eigendecomposition of the covariance matrix

C x = E{ xx H } = V �x V

H .

2. Compute the whitening matrix D = V�− 1

2 x V

H .

3. Calculate the α-complementary covariance of the whitened

data, s = Dx , as C s α = E{ ss αH } . 4. Compute the factorisation of the α-complementary covariance

matrix, C s α = W �αW

αH , where W is a quaternion unitary ma-

trix and �α a real diagonal matrix.

5. The QUT matrix is then Q = W

H D .

roposition 3 (QUT) . For a random quaternion vector x with finite

econd-order statistics, there exists a quaternion non-singular matrix

for which the transformed vector y = Qx has the following co-

ariance and α-complementary covariance matrices: C y = I C y α =α, where α ∈ { ı, j , κ} .

Recall that for a C

α-improper vector x , only the α-

omplementary covariance is non-vanishing while the other two

omplementary covariances vanish. Thus, the QUT of C

α-improper

ignals, for example, for the case α = κ, satisfies

y = I C y ı = �ı C y j = �j C y κ = �κ

here �κ is a diagonal matrix and �ı = �j = 0 . In other words,

or a C

α-improper quaternion vector, the QUT can be regarded

s the quaternion version of SUT [15,16] . For a general improper

uaternion vector, however, the remaining two complementary co-

ariance matrices are still not diagonalised by the QUT. In fact,

o far there is no closed-form solution to joint diagonalisation of

our covariance matrices for general improper quaternion vectors.

o circumvent this problem, we shall next introduce an approxi-

ate way to simultaneously diagonalise all four covariance matri-

es, a practically useful tool in most applications of quaternions,

onsidering the effectiveness of the AUT in complex-valued signal

rocessing [28–30] .

. Simultaneous diagonalisation of all four covariance matrices

Following the approach in Section 3 , we next propose the

uaternion analogue of our proposed approximate uncorrelating

ransform (AUT) for complex-valued data [25] . This will enable

he simultaneous diagonlistion of the covariance matrix and three

omplementary covariance matrices.

Page 5

M. Xiang et al. / Signal Processing 148 (2018) 193–204 197

4

s

�

E

w

c

C

C

T

C

C

∑

w

p

m

∑

h

t

p

n

P

b

v

d

t

t

O

t

o

C

4

t

s

a

c

x

w

i

x

U

E

w

E

a

h

w

T

E

a

E

c

m

t

i

ρ

w

t

t

l

.1. Univariate QAUT

Consider first the univariate quaternion random vector x =( x 1 , . . . , x L )

T . Our aim is to find a unitary transform y = �x which

imultaneously diagonalises the covariance matrix, C y = E {

yy H }

=C x �H , and the three complementary covariance matrices, C y α =

{yy αH

}= �C x α�αH , α ∈ { ı, j , κ}. Notice that if C x C

H x = C x α C

H x α ,

e have C y C

H y = C y α C

H y α .

For N samples of x , N > L , the covariance matrix, and the α-

omplementary covariance matrix can be estimated as

x =

1

N

N ∑

n 1 =1

x ( n 1 ) x

H ( n 1 ) (13)

x α =

1

N

N ∑

n 1 =1

x ( n 1 ) x

αH ( n 1 ) (14)

heir quadratic forms C x C

H x and C x α C

H x α are calculated as

x C

H x =

[

1

N

N ∑

n 1 =1

x ( n 1 ) x

H ( n 1 )

] [

1

N

N ∑

n 2 =1

x ( n 2 ) x

H ( n 2 )

]

=

1

N

2

N ∑

n 1 =1

N ∑

n 2 =1

x ( n 1 )

[

L ∑

l=1

x ∗l ( n 1 ) x l ( n 2 )

]

x

H ( n 2 )

x α C

H x α =

[

1

N

N ∑

n 1 =1

x ( n 1 ) x

αH ( n 1 )

] [

1

N

N ∑

n 2 =1

x

α( n 2 ) x

H ( n 2 )

]

=

1

N

2

N ∑

n 1 =1

N ∑

n 2 =1

x ( n 1 )

[

L ∑

l=1

x ∗l ( n 1 ) x l ( n 2 )

] α

x

H ( n 2 )

Upon applying the approximation

L

l=1

x ∗l ( n 1 ) x l ( n 2 ) ≈[

L ∑

l=1

x ∗l ( n 1 ) x l ( n 2 )

] α

(15)

e obtain C x C

H x ≈ C x α C

H x α , and hence C y C

H y ≈ C y α C

H y α , which im-

lies that if C y is diagonal, so too is C y α , and vice versa . Further-

ore, if

L

l=1

x ∗l (n 1 ) x l (n 2 ) ∈ R (16)

olds approximately, then (15) holds for all α ∈ { ı, j , κ}, and thus

he simultaneous diagonalisation applies to all of the three com-

lementary covariance matrices, producing the following quater-

ion approximate uncorrelating transform (QAUT).

roposition 4 (QAUT) . If any of C x , C x ı , C x j and C x κ is diagonalised

y the unitary matrix �, the structural similarity between the four co-

ariance matrices, displayed in (15) , makes possible the approximate

iagonalisation of the other three covariance matrices by �. From (1) ,

he pseudo-covariance matrix is also approximately diagonalised in

his way.

The transformation matrix � can take the form of Q in

bservation 1 or Q α in Observation 2 . For example, the implemen-

ation of the transform Q , obtained from the eigendecomposition

f C x , yields

C x = Q

H �x Q

C

ı x ≈ Q

H �ı Q

ı

C

jx ≈ Q

H �j Q

j

κ H κ

x ≈ Q �κQ (17)

.2. Principle of QAUT

The QAUT is based on the approximation in (16) . To examine

he validity of this approximation for highly improper quaternion

ignals, assume that the four components of the quaternion vari-

ble x l are perfectly correlated and consider the Cayley–Dickson

onstruction [50]

l = x a + x b ı + x c j + x d κ = c 1 + c 2 j

here c 1 = x a + x b ı and c 2 = x c + x d ı are complex variables defined

n the plane spanned by {1, ı}. This gives

∗l ( n 1 ) x l ( n 2 ) = c ∗1 ( n 1 ) c 1 ( n 2 ) + c 2 ( n 1 ) c

∗2 ( n 2 )

+ [ c ∗1 ( n 1 ) c 2 ( n 2 ) − c 2 ( n 1 ) c ∗1 ( n 2 ) ] j (18)

pon applying the statistical expectation operator, we have

{x ∗l ( n 1 ) x l ( n 2 )

}= E { c ∗1 ( n 1 ) c 1 ( n 2 ) } + E { c 2 ( n 1 ) c

∗2 ( n 2 ) }

+ E { c ∗1 ( n 1 ) c 2 ( n 2 ) − c 2 ( n 1 ) c ∗1 ( n 2 ) } j (19)

here

{ c ∗1 ( n 1 ) c 1 ( n 2 ) } = E{ x a ( n 1 ) x a ( n 2 ) + x b ( n 1 ) x b ( n 2 ) } + E{ x a ( n 1 ) x b ( n 2 ) − x a ( n 2 ) x b ( n 1 ) } ı

≈ E{ x a ( n 1 ) x a ( n 2 ) + x b ( n 1 ) x b ( n 2 ) } + E{ x b ( n 1 ) x b ( n 2 ) − x b ( n 2 ) x b ( n 1 ) } ı

= E{ x a ( n 1 ) x a ( n 2 ) + x b ( n 1 ) x b ( n 2 ) } ∈ R (20)

nd likewise for E{ c 2 ( n 1 ) c ∗2 ( n 2 ) } ∈ R . Moreover, if c 1 and c 2 are

ighly correlated, they can be expressed as c 2 (t) ≈ kc 1 (t) + g,

here k and g are complex-valued constants, to give

E { c ∗1 ( n 1 ) c 2 ( n 2 ) − c 2 ( n 1 ) c ∗1 ( n 2 ) }

≈ E { c ∗1 ( n 1 ) [ kc 1 ( n 2 ) + g ] − [ kc 1 ( n 1 ) + g ] c ∗1 ( n 2 ) } = 2 kE { I [ c ∗1 ( n 1 ) c 1 ( n 2 ) ] } + E { g} E { c ∗1 ( n 1 ) } − E { g} E { c ∗1 ( n 2 ) }

≈ 0 (21)

herefore, we obtain approximately

{x ∗l ( n 1 ) x l ( n 2 )

}∈ R (22)

nd the approximation (16) holds given the assumption

{x ∗l ( n 1 ) x l ( n 2 )

}≈ 1

L

L ∑

l=1

x ∗l (n 1 ) x l (n 2 )

From the above analysis, the QAUT holds exactly when the four

omponents of the quaternion variable are perfectly correlated (a

aximally improper x ). For general quaternion signals, however,

hese components are not perfectly correlated, causing the approx-

mation error impacted by the correlation coefficients, given by

ab =

cov [ x a , x b ]

σx a σx b

, ρac =

cov [ x a , x c ]

σx a σx c

, ρad =

cov [ x a , x d ]

σx a σx d

,

ρbc =

cov [ x b , x c ]

σx b σx c

, ρbd =

cov [ x b , x d ]

σx b σx d

, ρcd =

cov [ x c , x d ]

σx c σx d

ith the range [ −1 , 1] , where ‘cov’ denotes the covariance and σhe standard deviation. From the diagonalisation condition in (15) ,

he diagonalisation error of C x α arises from and is positively corre-

ated with

x ∗l ( n 1 ) x l ( n 2 ) −[x ∗l ( n 1 ) x l ( n 2 )

]α

= 2 I β

[x ∗l ( n 1 ) x l ( n 2 )

]β + 2 I γ

[x ∗l ( n 1 ) x l ( n 2 )

]γ

for distinct α, β, γ ∈ { ı, j , κ} (23)

Page 6

198 M. Xiang et al. / Signal Processing 148 (2018) 193–204

ı

j

κ

)

Fig. 1. Scatter plots, indicating the high correlation (improperness) of the three cor-

related C κ -improper quaternion signals, x 1 , x 2 and x 3 .

v

q

s

m

b

t

R

s

n

f

h

t

5

i

q

m

(

5

r

t

t

s

w

i

κ

(

x

x

t

T

a

y1 2 3

Note that the imaginary part of x ∗l ( n 1 ) x l ( n 2 ) is given by

I [x ∗l ( n 1 ) x l ( n 2 )

]= [ x a ( n 1 ) x b ( n 2 ) − x a ( n 2 ) x b ( n 1 ) + x c ( n 2 ) x d ( n 1 ) − x c ( n 1 ) x d ( n 2 ) ]

+ [ x a ( n 1 ) x c ( n 2 ) − x a ( n 2 ) x c ( n 1 ) + x b ( n 1 ) x d ( n 2 ) − x b ( n 2 ) x d ( n 1 ) ]

+ [ x a ( n 1 ) x d ( n 2 ) − x a ( n 2 ) x d ( n 1 ) + x b ( n 2 ) x c ( n 1 ) − x b ( n 1 ) x c ( n 2 ) ]

(24

Similarly to the AUT for complex-valued data [25] , Eq. (24) can be

rewritten as

I [x ∗l ( n 1 ) x l ( n 2 )

]=

(ξab

√

1 − ρ2 ab

+ ξcd

√

1 − ρ2 cd

)ı

+

(ξac

√

1 − ρ2 ac + ξbd

√

1 − ρ2 bd

)j

+

(ξad

√

1 − ρ2 ad

+ ξbc

√

1 − ρ2 bc

)κ (25)

where ξ ab , ξ cd , ξ ac , ξ bd , ξ ad , ξ bc are real-valued coefficients depen-

dent on the statistics of x l .

It is now obvious that when the components of quaternion data

become more correlated, (25) decreases and the diagonalisation er-

ror hence decreases. Owing to the cumulative nature of the error

in (16) , the approximation error also increases with the data seg-

ment length L .

4.3. Multivariate QAUT

We next extend the univariate QAUT to the multivariate case

when an M -variate quaternion signal is represented by a data ma-

trix X =

[x 1 , . . . , x M

]T , where the m -th column random vector

x m

=

[x m

(1) , . . . , x m

(N) ]T

( N > M ≥ m ) represents N samples of

the m -th variate. The covariance and α-covariance matrices of the

M variates are then given by C X =

1 N E{ XX

H } and C

αX

=

1 N E{ XX

αH } .Our aim is to find a unitary transform

Y = �X (26)

which simultaneously diagonalises the covariance matrix, C Y =E{ YY

H } = �C X �H , and the three complementary covariance matri-

ces, C Y α = E{ YY

αH } = �C X α�αH , α ∈ { ı, j , κ}. Notice that if C X C

H X

=C X α C

H X α

, then C Y C

H Y

= C Y α C

H Y α

.

The covariance and α-complementary covariance matrices can

be estimated from X as

C X =

1

N

XX

H (27)

C

αX =

1

N

XX

αH (28)

Their quadratic forms C X C

H X

and C X α C

H X α

are calculated as

C X C

H X =

1

N

2 XX

H XX

H =

1

N

2 X (

M ∑

m =1

x

∗m

x

T m

) X

H

C X α C

H X α =

1

N

2 XX

αH X

αX

H =

1

N

2 X (

M ∑

m =1

x

∗m

x

T m

) αX

H

Following the analysis in (18) –(21) , we can prove that the [ n 1 , n 2 ]

element of the matrix E{ x ∗m

x T m

} , E{ x ∗m

(n 1 ) x m

(n 2 ) } , is real-valued,

for all n 1 , n 2 ∈ { 1 , . . . , N} , whereby E{ x ∗m

x T m

} ∈ R

N×N and

M ∑

m =1

x

∗m

x

T m

∈ R

N×N

Under this condition, we obtain C x C

H x ≈ C x α C

H x α , so that C Y C

H Y

≈C Y α C

H α , which implies if C Y is diagonal, so too is C Y α , and vice

Y

ersa . This means that the four covariance matrices of multivariate

uaternion data can be simultaneously diagonalised via the QAUT,

uch as in (17) .

Similarly to the univariate case, the diagonalisation error of the

ultivariate QAUT decreases with an increase in the correlation

etween the components of quaternion data, and increases with

he number of variates, M .

emark 2. The QAUT becomes exact for quaternion signals with

pecial second-order statistical properties, such as H -proper sig-

als, the signals with fully correlated components, and the signals

or which the covariance and complementary covariance matrices

ave the same singular vectors. This can be verified by exploring

he underlying matrix structures.

. Simulation studies

The effectiveness of the proposed QUT and QAUT techniques

s illustrated via simulations on both synthetic and real-world

uaternion signals. In the experiments, the covariance and comple-

entary covariance matrices were estimated based on (13), (14),

27) and (28) .

.1. Performance of QUT for C

η-improper signals

In the first experiment, the QUT was applied to the decor-

elation of multivariate C

κ -improper quaternion signals. First,

hree uncorrelated C

κ -improper quaternion signals were generated

hrough the approach introduced in Appendix B and were sub-

equently mixed through a 3 × 3 matrix for which the elements

ere drawn from a standard normal distribution. The mixed C

κ -

mproper signals, x 1 , x 2 and x 3 , were correlated in terms of the

-complementary covariance, with the correlation coefficients 0.69

between x 1 and x 2 ), 0.34 (between x 1 and x 3 ), and 0.91 (between

2 and x 3 ). Fig. 1 shows 3-D scatter plots of the mixed signals, x 1 ,

2 and x 3 , illustrating their high correlation, as indicated by ellip-

ical shapes of the scatter plots at an angle to the coordinate axes.

he QUT was used to decorrelate x 1 , x 2 and x 3 into y 1 , y 2 and y 3 ,

s illustrated in Fig. 2 , which shows much more circular nature of

, y and y , as desired.

Page 7

M. Xiang et al. / Signal Processing 148 (2018) 193–204 199

Fig. 2. Scatter plots, indicating the low correlation (properness) of the three C κ -

improper quaternion signals, y 1 , y 2 and y 3 , decorrelated via the QUT.

5

o

a

e

ε

t

s

c

w

t

s

s

i

m

t

m

g

d

d

s

m

d

m

o

n

t

c

b

l

t

Fig. 3. Squared diagonal error [in %] against the correlation degree and the data

segment length for synthetic univariate data.

5

g

v

n

t

r

Q

c

l

a

c

d

.2. Performance of QAUT for synthetic improper signals

To assess the performance of QAUT, the squared diagonal error

f the complementary covariance matrix C x α was measured by an

verage power ratio of the off-diagonal, λα,ij , versus diagonal, λα,ii ,

lements of the approximately diagonal matrix �α , given by

2 α =

L ∑

i, j=1

∣∣λα,i j

∣∣2

(N − 1) L ∑

i =1

λ2 α,ii

× 100% , i � = j

Synthetic quaternion signals with a varying degree of correla-

ion between the real, ı, j and κ components were considered. For

implicity, we assumed the six correlation coefficients of the four

omponents to be equally distributed in (0, 1] and denoted by ρ ,

here ρ = 1 indicates full correlation and ρ = 0 no correlation.

In the univariate simulations, a synthetic quaternion data vec-

or with a varying ρ was generated from a quaternion white Gaus-

ian signal. Then, the performance of the univariate QAUT was as-

essed comprehensively against the degree of correlation present

n the data components, ρ ∈ [0.5, 1], and the length of data seg-

ent, L ∈ [5, 20]. Conforming with the analysis, Fig. 3 shows that

he diagonalisation error increased as ρ or L increased. The perfor-

ance was excellent, for example, for a moderate correlation de-

ree, ρ = 0 . 5 , and a long data segment, L = 20 , where the squared

iagonalisation error was less than 1%, while for a high correlation

egree, ρ is close to unit, where the error was negligible.

In the multivariate simulations, multiple channels of quaternion

ignals with a varying ρ were generated and then mixed using a

atrix the elements of which were drawn from a standard normal

istribution, to generate multivariate quaternion data. The perfor-

ance of the multivariate QAUT was assessed against the degree

f correlation present in the data components, ρ ∈ [0.5, 1], and the

umber of variates, M ∈ [5, 20]. Fig. 4 shows that the diagonalisa-

ion error increased as ρ or M increased. Similarly to the univariate

ase, for a moderate correlation degree, ρ = 0 . 5 , and a large num-

er of data variates, M = 20 , the squared diagonalisation error was

ess than 1%, while for a high correlation degree, ρ is close to unit,

he error was negligible.

.3. Performance of QAUT for real-world signals

The QAUT was also applied to 4-D real-world electroencephalo-

ram (EEG) signals by combining four adjacent channels of real-

alued EEG signals into a quaternion-valued signal. The four chan-

els measuring adjacent brain regions exhibited strong correla-

ion with each other while the channels measuring distant brain

egions were relatively weakly correlated [51,52] . We tested the

AUT on such quaternion EEG signals with low, medium and high

orrelations between the four dimensions, with the data segment

ength L varying between 5 and 20. Fig. 5 illustrates that the di-

gonalisation error increased with either an increase in L or a de-

rease in the degree of correlation present in the data. The squared

iagonal error was less than 0.2%.

Page 8

200 M. Xiang et al. / Signal Processing 148 (2018) 193–204

Fig. 4. Squared diagonal error [in %] against the correlation degree and the number

of variates for synthetic multivariate data.

Fig. 5. Squared diagonal error [in %] against the correlation degree and the data

segment length for EEG data.

r

X

w

t

i

a

i

5.4. QAUT for rank reduction

In many practical applications the acquired data are highly cor-

related or even collinear, with redundant elements that do not

contribute to the accuracy of processing but require excessive com-

putational resources and may affect stability of algorithms through

the associated singular correlation matrices. It is therefore desir-

able to use the minimum number of variables to describe the in-

formation within a data set; this is typically achieved through a

low-rank matrix approximation. For data requiring rank reduction,

the assumption that the components of variables are highly cor-

related is sensible, that is, the approximation conditions for the

QAUT hold. The multivariate QAUT in (26) can be therefore rear-

ranged to decompose the data matrix X into a sum of uncorrelated

ank one vector outer products

= �H Y =

M ∑

m =1

φH m

y m

here φm

and y m

are rows in the matrices � and Y , respec-

ively, and the transformed variables, y m

, are monotonically non-

ncreasing in the variance. The use of variables for which the vari-

nce is above a defined threshold and the disposal of the remain-

ng variables, which account for noise, provides the reduced rank

Page 9

M. Xiang et al. / Signal Processing 148 (2018) 193–204 201

Fig. 6. Rank reduction using the QAUT.

Fig. 7. Scatter plots showing high correlation between the correlated x 1 and x 2 vari-

ables.

a

X

w

m

c

i

q

w

p

Q

v

n

r

m

a

F

c

o

s

q

b

n

q

t

t

t

t

l

t

Fig. 8. Scatter plots showing no correlation between the transformed y 1 and y 2 variables.

Fig. 9. The logarithm of the ratio of the variance explained by each variable to the

total variance. Insert diagram : The cumulative ratio of variance against the number

of variables for the original and decorrelated corn data.

d

t

5

c

m

c

e

pproximation of X in the form

ˆ =

P ∑

m =1

φH m

y m

, where P < M

hich is illustrated in Fig. 6 . This is analogous to the real do-

ain where the SVD provides a change of coordinates to yield un-

orrelated variables for which the variance is monotonically non-

ncreasing.

To illustrate the potential of the QAUT in rank reduction of

uaternion data, a 2-D quaternion process, x 1 , x 2 , was generated

here x 1 and x 2 were highly correlated as shown in Fig. 7 , which

lots the component relationship in the real, ı, j and κ planes. The

AUT decorrelated x 1 , x 2 into y 1 and y 2 . Fig. 8 shows significant

ariation only in the y 2 axis while the y 1 variation corresponds to

oise, indicating that the true dimensionality of the process is one

ather than two.

Rank reduction is commonly used in chemometrics where

any sets of measurements are required to examine a process

nd the important information may not be directly observable.

or example, Near Infrared Spectra (NIR) are used to examine the

hemical content of a data sample and are often highly correlated

r collinear. We applied the QAUT to the NIR obtained from 80

amples of corn using three different spectrometers [53] . A pure

uaternion variable was formed for each corn sample by com-

ining its spectra from the three spectrometers. The QAUT diago-

alised the data covariance matrices successfully to show that one

uaternion variable was sufficient to express most of the informa-

ion, as the largest diagonal element was 39 compared to 0.01 for

he next largest, as shown in Fig. 9 , which displays the logarithm of

he ratio of variance explained by each variable, compared to the

otal variance in the data. The insert plot in Fig. 9 shows the cumu-

ative ratio of variance explained by including further variables for

he original and transformed data. Observe that the transformed

ata with much fewer variables explain the same information as

he original, thus verifying the QAUT.

.5. QAUT for imbalance detection in three-phase power systems

As shown in Section 4 , when the components of data are highly

orrelated, the QAUT diagonalises all complementary covariance

atrices at the expense of a small error, however, this error in-

reases for lower degrees of correlation. We now show a way to

xploit this data dependence of the QAUT and the structure of the

Page 10

202 M. Xiang et al. / Signal Processing 148 (2018) 193–204

Fig. 10. Three-phase power signal and imbalance detection.

v

m

b

p

o

o

Q

e

a

l

D

v

d

C

e

T

t

a

i

o

6

v

w

o

d

c

s

t

a

o

c

l

d

a

g

d

a

i

q

t

p

A

p

A

l

x

x

complementary covariance matrices to detect imbalance in a three-

phase power system.

In general, the instantaneous voltages of a three-phase power

system are given by Talebi and Mandic [54]

v 1 (n ) = A 1 sin (2 π f�T n + ϕ 1 )

v 2 (n ) = A 2 sin (2 π f�T n + ϕ 2 )

v 3 (n ) = A 3 sin (2 π f�T n + ϕ 3 ) (29)

where A 1 , A 2 , A 3 are the instantaneous amplitudes, ϕ1 , ϕ2 , ϕ3 the

instantaneous phases, f the system frequency, and �T the sampling

interval. In a balanced three-phase system, A 1 = A 2 = A 3 , ϕ 2 − ϕ 1 =2 π3 , ϕ 3 − ϕ 2 =

2 π3 , and the correlation between each component is

equivalent [55] . The power grid is usually designed to operate op-

timally under balanced conditions; however, faults in the power

system can cause imbalanced operating conditions that propagate

through the network, threatening its stability. Therefore, it is im-

portant in fault detection and mitigation applications to identify

the incidences when the power grid is operating in an unbalanced

fashion [56] .

Pure quaternions have been used to deal with data record-

ings from all three phases simultaneously as x (n ) = v 1 (n ) ı +v 2 (n )j + v 3 (n ) κ [54] . Here we organise the recorded data

into a pure quaternion vector, x = v 1 ı + v 2 j + v 3 κ, where v m

=[ v m

(1) , v m

(2) , . . . , v m

(n )] T , m = 1 , 2 , 3 . We can then apply the uni-

variate QAUT to x and employ the diagonalisation errors of the

complementary covariance matrices as an indicator of power im-

balance. This can be explained by the overall diagonalisation error

given by

I [x ∗l ( n 1 ) x l ( n 2 )

]= [ v 2 ( n 2 ) v 3 ( n 1 ) −v 2 ( n 1 ) v 3 ( n 2 ) ] ı

+ [ v 1 ( n 1 ) v 3 ( n 2 ) −v 1 ( n 2 ) v 3 ( n 1 ) ] j

+ [ v 1 ( n 2 ) v 2 ( n 1 ) −v 1 ( n 1 ) v 2 ( n 2 ) ] κ (30)

which is a reduced form of (24) and is positively correlated with

the diagonalisation error. When the system is balanced, the covari-

ances E{ v p v T q } are equivalent for all phases p, q ∈ {1, 2, 3}, so the

three imaginary parts of x ∗l ( n 1 ) x l ( n 2 ) in (30) are equal in absolute

alue, leading to equal diagonalisation errors of the three comple-

entary covariance matrices. On the other hand, when the system

ecomes unbalanced, the correlation degrees between the three

hase voltages are different, and hence the three imaginary parts

f x ∗l ( n 1 ) x l ( n 2 ) differ, resulting in different diagonalisation errors

f the three complementary covariance matrices. This allows the

AUT to be used in an imbalance detection application. To this

nd, we applied the QAUT to a sliding window of 0.04 seconds of

three-phase power system for which a fault occurred at 0.25 s

asting for 0.25 s before the system returned to a balanced state.

uring this fault, v 1 decreased significantly in amplitude whereas

2 and v 3 increased in amplitude ( v 2 > v 3 ). Fig. 10 shows that the

iagonalisation error for C x j and C x κ increased while the error for

x ı decreased and therefore analysing the difference between the

rrors is an appropriate detection method for the power imbalance.

his can be explained via (30) and (23) . The decrease in the ampli-

ude of v 1 and the increase in the amplitudes of v 2 and v 3 induced

n increased ı-imaginary part and decreased j -imaginary and κ-

maginary parts of x ∗l ( n 1 ) x l ( n 2 ) , so that the diagonalisation error

f C x ı declined and the diagonalisation errors of C x j and C x κ rose.

. Conclusion

Novel simultaneous matrix factorisation techniques for the co-

ariance and complementary covariance matrices, which arise in

idely linear quaternion algebra, have been introduced. The diag-

nalisation of quaternion symmetric matrices has been first ad-

ressed, as the Takagi factorisation of complex symmetric matri-

es cannot be readily extended to the quaternion domain. We have

hown that the symmetry is insufficient for a quaternion matrix A

o be diagonalisable, and instead the condition ( AA

∗) ∗ = A

∗A must

lso hold, which in general is not the case. The simultaneous diag-

nalisation of two quaternion covariance matrices, quaternion un-

orrelating transform (QUT), has then been introduced as an ana-

ogue to the strong uncorrelating transform (SUT) in the complex

omain. It has also been shown that for improper quaternion data,

typical case in quaternion signal processing applications, a sin-

le eigendecomposition of the covariance matrix is sufficient to

iagonalise approximately the three complementary matrices. The

nalysis of the so introduced quaternion approximate uncorrelat-

ng transform (QAUT) has demonstrated its usefulness for improper

uaternion signals, a typical case in practical applications. Simula-

ion studies on synthetic and real-world signals support the pro-

osed QUT and QAUT techniques.

ppendix A. Properties of quaternion matrices

For two general quaternion matrices, A and B , the following

roperties hold [42] :

P1. A

T ∗ = (A

∗) T = (A

T ) ∗ and A

α∗ = (A

∗) α = (A

α) ∗, P2. A

αT = (A

α) T = (A

T ) α and A

αH = (A

α) H = (A

H ) α,

P3. (A

α) β = (A

β ) α = A

γ for distinct α, β, γ ∈ { ı, j , κ} , P4. ( AB ) ∗ � = A

∗B

∗,

P5. ( AB ) T � = B

T A

T ,

P6. ( AB ) H = B

H A

H ,

P7. ( AB ) α = A

αB

α,

P8. ( AB ) αH = B

αH A

αH .

ppendix B. C

α-improper quaternion random vectors

For distinct α, β ∈ { ı, j , κ}, the Cayley–Dickson construction al-

ows for the complex representation of a quaternion random vector

, as

= z 1 + z 2 β

Page 11

M. Xiang et al. / Signal Processing 148 (2018) 193–204 203

w

s

i

A

A

A

B

(

N

t

o

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

here z 1 and z 2 are complex random vectors defined in the plane

panned by {1, α}. The quaternion random vector x is C

α-improper

f and only if both of the following two conditions are fulfilled [47] :

1) z 1 and z 2 are proper complex vectors, which is achieved when

their real and imaginary parts are uncorrelated and with the

same variance,

2) z 1 and z 2 have different variances.

ppendix C. Symmetric quaternion matrices

Consider a 2 × 2 symmetric quaternion matrix A , whereby

A

∗ =

[a 11 a 12

a 21 a 22

][a ∗11 a ∗12

a ∗21 a ∗22

]

=

[a 11 a

∗11 + a 12 a

∗21 a 11 a

∗12 + a 12 a

∗22

a 21 a ∗11 + a 22 a

∗21 a 21 a

∗12 + a 22 a

∗22

]

∗A =

[a ∗11 a ∗12

a ∗21 a ∗22

][a 11 a 12

a 21 a 22

]

=

[a ∗11 a 11 + a ∗12 a 21 a ∗11 a 12 + a ∗12 a 22

a ∗21 a 11 + a ∗22 a 21 a ∗21 a 12 + a ∗22 a 22

]y considering the symmetry A = A

T ( a 12 = a 21 ), the matrices

AA

∗) ∗ and A

∗A can be structured as

(AA

∗) ∗ =

[| a 11 | 2 + | a 12 | 2 a 12 a ∗11 + a 22 a

∗12

a 11 a ∗12 + a 12 a

∗22 | a 12 | 2 + | a 22 | 2

]

A

∗A =

[| a 11 | 2 + | a 12 | 2 a ∗11 a 12 + a ∗12 a 22

a ∗12 a 11 + a ∗22 a 12 | a 12 | 2 + | a 22 | 2 ]

ote that (AA

∗) ∗ = A

∗A if their off-diagonal elements are commu-

ative, implying that either the diagonal or off-diagonal elements

f A are real-valued.

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