performance analysis of quaternion-valued adaptive filters...

1566 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 66, NO. 6, MARCH 15, 2018

Performance Analysis of Quaternion-ValuedAdaptive Filters in Nonstationary Environments

Min Xiang , Member, IEEE, Sithan Kanna , Student Member, IEEE, and Danilo P. Mandic , Fellow, IEEE

Abstract—Quaternion adaptive filters have been widely used forthe processing of three-dimensional (3-D) and 4-D phenomena,but complete analysis of their performance is still lacking, partlydue to the cumbersomeness of multivariate quaternion analysis.This causes difficulties in both understanding their behavior anddesigning optimal filters. Based on a thorough exploration of theaugmented statistics of quaternion random vectors, this paper ex-tends an analysis framework for real-valued adaptive filters to themean and mean square convergence analyses of general quater-nion adaptive filters in nonstationary environments. The extensionis nontrivial, considering the noncommutative quaternion alge-bra, only recently resolved issues with quaternion gradient, andthe multidimensional augmented quaternion statistics. Also, forrigor, in order to model a nonstationary environment, the systemweights are assumed to vary according to a first-order random-walk model. Transient and steady-state performance of a generalclass of quaternion adaptive filters is provided by exploiting theaugmented quaternion statistics. An innovative quaternion decor-relation technique allows us to develop intuitive closed-form ex-pressions for the performance of quaternion least mean square(QLMS) filters with Gaussian inputs, which provide new insightsinto the relationship between the filter behavior and the completesecond-order statistics of the input signal, that is, quaternion non-circularity. The closed-form expressions for the performance ofstrictly linear, semiwidely linear, and widely linear QLMS filtersare investigated in detail, while numerical simulations for the threeclasses of QLMS filters with correlated Gaussian inputs supportthe theoretical analysis.

Index Terms—Quaternion adaptive filters, least mean square,mean square analysis, stability, steady-state performance, im-proper signals, noncircularity.

I. INTRODUCTION

QUATERNIONS have traditionally been used in aerospaceengineering and computer graphics in order to model

three-dimensional rotations and orientations as their algebraavoids numerical problems associated with vector algebras [1].The recently introduced augmented quaternion statistics [2],[3] and HR calculus [4], [5] have triggered a resurgence ofresearch on quaternion-valued signal processing, owing to a

Manuscript received May 2, 2017; revised August 21, 2017 and October 8,2017; accepted November 24, 2017. Date of publication December 25, 2017;date of current version February 7, 2018. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. MasahiroYukawa. (Corresponding author: Min Xiang.)

The authors are with the Department of Electrical and Electronic Engineer-ing, Imperial College London, London SW7 2BT, U.K. (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2017.2787102

compact model of mutual information between data channelsprovided by quaternions, and the inherent physically meaning-ful interpretation for three-dimensional and four-dimensionalproblems. Recent research mainly focuses on adaptive filtering,neural networks, independent component analysis, and spectralestimation [6]–[11]. Quaternions have subsequently found newapplications in communications, motion tracking, and biomed-ical engineering [12]–[15].

Recently, there have been recent extensive works on the-ory and applications of quaternion filters [16]–[20]. Tradi-tional strictly linear quaternion filters based on the strictlylinear model y = hH x with the input vector x ∈ HM ×1 , theweight vector h ∈ HM ×1 , and the output y ∈ H, utilise second-order quaternion statistics based on the standard covarianceand are optimal only for estimating second-order circular(proper) quaternion signals [16]. Advances in quaternion statis-tics have established that widely linear quaternion filters basedon the widely liner model y = hH x + gH xı + uH xj + vH xκ ,where g, u, v ∈ HM ×1 are complementary weight vectors, andxı ,xj ,xκ are involutions of x, exploit the three complemen-tary covariances in addition to the standard covariance, andthus capture complete second-order statistical information inquaternion signals [17], [21]. For quaternion signals with spe-cial second-order statistical properties, the widely linear filtersreduce to semi-widely linear filters based on the semi-widelylinear model yn = hH

n xn + gHn xı

n [22], [23]. Notice that, simi-lar to the duality between complex filters and bivariate real filters[24], [25], quadrivariate real filters are isomorphic to widely lin-ear quaternion filters, but are totally different from strictly andsemi-widely linear quaternion filters.

Contrary to the research on real-valued adaptive filters [26]–[29], the complete performance analysis of quaternion adaptivefilters is still an open problem, causing difficulties in under-standing their behaviour and in the design of optimal filteringstrategies. Since the quadrivariate real filters are not isomor-phic to the whole class of quaternion filters, it is impossible touse the performance of the former to straightforwardly analysethe latter, and thus the performance analysis of the latter mustbe undertaken in the quaternion domain. The mean convergenceanalysis for the quaternion least mean square (QLMS) filterswas proposed in [30], but the non-commutativity of quater-nion products poses a challenge to the application of classi-cal mean square convergence analysis methods for real-valuedand complex-valued adaptive filters to the quaternion domain.For the mean square convergence analysis, the work in [31]discussed a simple case, the univariate strictly linear QLMS

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XIANG et al.: PERFORMANCE ANALYSIS OF QUATERNION-VALUED ADAPTIVE FILTERS IN NONSTATIONARY ENVIRONMENTS 1567

algorithm, and derived the bounds of the step size. However,considering the difficulty in multivariate quaternion analysisdue to the non-commutativity of quaternion products and thelack of quaternion matrix factorisation methods, it is challeng-ing to extend the analysis in [31] to general multivariate QLMSalgorithms.

Based on the analysis framework for real-valued adaptive fil-ters developed in [27], [28], this paper analyses the mean andmean square convergence of general quaternion adaptive filterswith general (proper or improper) quaternion inputs in non-stationary environments, and thus quantifies their transient andsteady-state performance. The proposed analysis not only treatsdifferent adaptation approaches uniformly, but also caters fordifferent linear models for quaternion signal estimation. In or-der to model a non-stationary environment, the system weightsare assumed to vary according to a first-order random-walkmodel. By decorrelating the quaternion regressor vectors, theanalysis of the QLMS filters with Gaussian inputs yields closed-form solutions relating to the second-order statistics of inputdata. The analytical results of the strictly linear QLMS (SL-QLMS), widely linear QLMS (WL-QLMS), and semi-widelylinear QLMS (SWL-QLMS) filters are investigated in detail.The key contributions of our work are as follows: (i) a unifiedperformance analysis for a general class of quaternion adap-tive filters, encompassing various adaptation approaches andquaternion linear estimation models; (ii) the mean square per-formance of QLMS filters with Gaussian inputs is linked to thesecond-order statistics of quaternion inputs by representing afourth-order quaternion moment matrix in terms of covarianceand complementary covariance matrices; (iii) the application ofa new decorrelation technique for random quaternion vectors tothe Gaussian regressors of QLMS filters leads to concise andphysically meaningful closed-form results. This work is there-fore a non-trivial extension of the real-valued analysis [27],[28] to the quaternion domain. Compared to [27], [28], carefulattention is paid to the non-commutative quaternion algebra,and quaternion analysis techniques are exploited to deal withaugmented quaternion statistics. Compared to the mean squareanalysis of univariate SL-QLMS in [31], a general class of mul-tivariate quaternion adaptive filters are considered by using ageneric filter form and quaternion multivariate statistics. Theso established analytical results bring new insights into the be-haviour of the quaternion adaptive filters.

The rest of this paper is organised as follows. Section II pro-vides an overview of quaternions and quaternion adaptive filters.Section III presents a convergence analysis framework for gen-eral quaternion adaptive filters to obtain the bounds of stabilityconditions and steady-state performances, from which the per-formance of QLMS filters with Gaussian regressors is deduced.In Section IV, a decorrelation technique for Gaussian regressorvectors of QLMS filters is exploited to derive concise and intu-itive closed-form analytical results, which are exemplified withthe SL-QLMS, SWL-QLMS and WL-QLMS. Simulations forQLMS filters with correlated improper Gaussian input vectorsare presented in Section V, and Section VI concludes the paper.

Throughout the paper, we use boldface capital letters to de-note matrices, A, boldface lowercase letters for vectors, a, and

italic letters for scalar quantities, a. Superscripts (·)T , (·)∗, and(·)H denote the transpose, conjugate, and Hermitian (i.e., trans-pose and conjugate) operators respectively, diag(A) creates acolumn vector containing the diagonal entries of matrix A, andDiag(a) creates a diagonal matrix with the elements of vectora on the diagonal. The symbol IM denotes an M × M identitymatrix, 1M an M × 1 all-ones vector, [A]a,b the element on thea-th row and and b-th column of matrix A, ‖a‖ the Euclideannorm of vector a, E{·} the statistical expectation operator, whileλ(A), λmax(A), and λmin(A) denote the eigenvalue, maximumeigenvalue, and minimum eigenvalue of matrix A.

II. BACKGROUND

A. Quaternion Algebra

The quaternion domain H is a four-dimensional vector spaceover the real field R, spanned by the basis {1, ı, j, κ}. A randomquaternion variable x ∈ H consists of a real part R [·] and aimaginary part I [·] which comprises three imaginary compo-nents, so that

x = R [x] + I [x] = R [x] + Iı [x] ı + Ij [x] j + Iκ [x]κ (1)

where R[x],Iı [x],Ij [x],Iκ [x] are real variables and ı, j, κ areimaginary units with the properties

ı2 = j2 = κ2 = ıjκ = −1, ıj = −jı = κjκ = −κj = ı, κı = −ıκ = j

The conjugate of x is defined as

x∗ = R [x] − I [x] = R [x] − Iı [x] ı − Ij [x] j − Iκ [x] κ (2)

The modulus of x is given by

|x| =√

R2 [x] + I2ı [x] + I2

j [x] + I2κ [x]

and the product of two quaternions x, y ∈ H by

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

+ I[x] × I[y]

where “·” denotes the scalar product and “×” the vector product.The presence of the vector product causes non-commutativity ofthe quaternion product, that is, xy �= yx. The quaternion producthas the following properties:

|xy| = |x||y|, x−1 =x∗

|x|2 , (xy)−1 = y−1x−1 , (xy)∗ = y∗x∗

A quaternion variable x is called a pure quaternion if R [x] = 0.A quaternion variable x is called a unit quaternion if |x| = 1.Similar to complex numbers, a quaternion x can be repre-sented in the polar form as x = |x|eζ , where ζ is a purequaternion, and the exponential can be represented by the sumeζ =

∑∞n=0 ζn/(n!) and satisfies (eζ )∗ = e−ζ [32].

Another important notion is the quaternion involution [33],which defines a self-inverse mapping, analogous to the complexconjugate. The general involution of the quaternion variable xis defined as xα = −αxα, which represents the rotation of thevector part of x by π about a unit pure quaternion α. The quater-nion involutions have the property: (xα )α = x. Accordingly,xα∗ = (xα )∗ = (x∗)α . The three special cases of involutions


about the ı, j and κ imaginary axes are given by

xı = −ıxı = R [x] + Iı [x] ı − Ij [x] j − Iκ [x] κ (3)

xj = −jxj = R [x] − Iı [x] ı + Ij [x] j − Iκ [x] κ (4)

xκ = −κxκ = R [x] − Iı [x] ı − Ij [x] j + Iκ [x]κ (5)

Due to the non-commutativity of quaternion products, quater-nion matrices have different properties from real and complexmatrices. For example, a quaternion square matrix has two typesof eigenvalues [34].

Definition 1: Given A ∈ HM×M , if Ax = xλ for λ ∈ Hand some non-zero x ∈ HM×1 , then λ is called a right eigen-value of A, and x is called a right eigenvector.

Definition 2: Given A ∈ HM×M , if Ax = λx for λ ∈ Hand some non-zero x ∈ HM×1 , then λ is called a left eigenvalueof A, and x is called a left eigenvector.

Right eigenvalues have been well studied in literature, whileleft eigenvalues are less known and are not computationallywell-posed [34]–[37]. Relevant to our analysis, a Hermitianquaternion matrix H ∈ HM×M has exactly M right eigen-values, which are also left eigenvalues, and are real-valued.Note that H can also have non-real left eigenvalues. There ex-ists a unitary quaternion matrix U ∈ HM ×M , which satisfiesUH U = UUH = IM , such that H = UΛUH is the eigen-decomposition of H with the right eigenvalues of H on thediagonal of a diagonal real matrix, Λ. A Hermitian quaternionmatrix is called positive definite, positive semi-definite, negativedefinite, or negative semi-definite if xH Hx is respectively pos-itive, non-negative, negative, or non-positive for any non-zeroquaternion vector x ∈ HM×1 . This is the case if and only if allright eigenvalues of H are positive, non-negative, negative, ornon-positive.

In the subsequent analysis, without loss in generality, wewill apply the eigendecomposition of Hermitian quaternion ma-trices, H = UΛUH , which only involves right eigenvalues.For conciseness, we will use the terminology, eigenvalues andeigenvectors, to denote right eigenvalues and right eigenvectorsof Hermitian quaternion matrices [38].

Notice that the Weyl’s inequality about the eigenvalues ofHermitian complex matrices also applies to Hermitian quater-nion matrices. If A and B are Hermitian quaternion matriceswith eigenvalues λ1(A) ≥ λ2(A) ≥ · · · ≥ λM (A), λ1(B) ≥λ2(B) ≥ · · · ≥ λM (B), then C = A + B is also Hermitianwith eigenvalues λ1(C) ≥ λ2(C) ≥ · · · ≥ λM (C), satisfying

λi (A) + λj (B) ≤ λm (C) ≤ λk (A) + λl (B)if k + l − 1 ≤ m ≤ i + j − 1 (6)

∀i, j, k, l ∈ {1, 2, . . . ,M} [39].The singular value decomposition has also been recently ex-

tended to the quaternion domain [34]. For any A ∈ HM×N , ofrank d, there exist unitary quaternion matrices U ∈ HM ×M andV ∈ HN ×N such that A = U(D

000 )V where D ∈ Rd×d is a

diagonal matrix with d positive singular values of A on thediagonal.

B. Augmented Second-Order Quaternion Statistics

The set of involutions in (3)–(5), together with the origi-nal quaternion random variable x, forms the most frequentlyused basis for augmented quaternion statistics, which is atthe core of the recently proposed widely linear processingmethodology [2], [3]. Benefiting from the involution basis, aug-mented second-order statistics of a zero-mean random quater-nion variable1 x is exploited by ı-, j-, and κ- covariances,Cxxα = E{xxα∗}, η = ı, j, κ, which are referred to as com-plementary covariances, together with the standard covarianceCxx = E{xx∗}. These covariances enable the characterisationof the quaternion impropriety (second-order non-circularity)which arises from the degree of correlation and/or power imbal-ance between imaginary components relative to the real com-ponent. The impropriety coefficients of a quaternion randomvariable x are defined as [31], [40]

rα =∣∣∣∣Cxxα

Cxx

∣∣∣∣ , α = ı, j, κ (7)

which reflect the correlation between x and each of its involu-tions, normalised by the signal power. Note that rα ∈ [0, 1]. Thefour degrees of freedom in the quaternion domain allow for dif-ferent levels of properness: H-properness, Rα -properness andCα -properness.

Definition 3 (Properness of a random quaternion variable):A random quaternion variable x is H-proper if it is uncorrelatedwith its involutions xı , xj and xκ , so that Cxxı = Cxxj =Cxxκ = 0; x is Rα -proper if it is only uncorrelated with the in-volution xα , so that only Cxxα among the three complementarycovariances vanishes; x is Cα -improper2 if it is only correlatedwith the involution xα , so that all complementary covariancesexcept Cxxα vanish; x is maximally improper (rectilinear) if itsimpropriety coefficients are maximal, so that rı = rj = rκ = 1.

Similarly, augmented second-order statistics of a zero-meanrandom quaternion column vector x is exploited by ı-, j-, and κ-covariance matrices, Cxxα = E{xxαH }, α = ı, j, κ, which arereferred to as complementary covariance matrices, together withthe standard Hermitian covariance matrix, Cxx = E{xxH }.The α-complementary covariance matrix is α-Hermitian, whichmeans Cxxα = (Cxxα )αH . The knowledge on both the co-variance and the three complementary covariance matrices isnecessary to ensure the utilisation of complete second-orderstatistical information. Using such knowledge, the semi-widelylinear processing with the augmented signal vector x �[xT ,xηT ]T , where η ∈ {ı, j, κ}, and the widely linear process-ing with the augmented signal vector x � [xT ,xıT ,xjT ,xκT ]T

have been shown to achieve better performance for improperquaternion signals, compared to the traditional strictly linearprocessing, which is based on only x [42]. The covariance

1Throughout the paper, we assume zero-mean quaternion variables. This doesnot affect the generality of our results.

2It is called Cα -proper in some literature [41], but we consider Cα -improperto be more intuitive.


matrices of x and x can be represented by

Cxx = E{xxH

}=

[Cxx Cxxη

Cηxxη Cη

xx

]

Cxx = E{xxH

}=

⎡⎢⎢⎣

Cxx Cxx ı Cxxj Cxxκ

Cıxx ı Cı

xx Cıxxκ Cı

xxj

Cjxxj Cj

xxκ Cjxx Cj

xx ı

Cκxxκ Cκ

xxj Cκxx ı Cκ

xx

⎤⎥⎥⎦

Likewise, the complementary covariance matrices of x and xcan also be represented by block matrices built from the covari-ance and complementary covariance matrices of x.

Definition 4 (Properness of a random quaternion vector):A random quaternion vector x is H-proper if it is uncorrelatedwith its involutions xı , xj and xκ , so that Cxx ı = Cxxj =Cxxκ = 0; x is Rα -proper if it is only uncorrelated with theinvolution xα , so that only Cxxα among the three comple-mentary covariances vanishes; x is Cα -improper if it is onlycorrelated with the involution xα , so that all complementarycovariances except Cxxα vanish; x is maximally improper(rectilinear) if it is maximally correlated with its involutions,so that the impropriety coefficients of the elements of x are allunits.

C. Quaternion Adaptive Filters

A fundamental problem in quaternion signal processing is toobtain the estimate, yn , of a desired signal, yn ∈ H, from a setof measurements, xn ∈ HL×1 , which carries information aboutyn , at time n. The estimation model yn = f(xn ), which incor-porates the knowledge on the relationship between yn and xn ,is crucial for estimation performance. Three linear estimationmodels for quaternion signals have been proposed and can berepresented in a unified form given by

yn = wHn sn (8)

where sn ∈ HM×1 is the regressor vector and wn ∈ HM×1 theweight vector. These three linear models arise from (8) as fol-lows [2], [42]:

� Strictly linear model:

sn = xn , M = L (9)� Semi-widely linear model:

sn = xn , M = 2L (10)� Widely linear model:

sn = xn , M = 4L (11)

The estimation model in (8) is based on the assumption that thedesired signal arises from the following linear system:

yn = wHn sn + υn (12)

where the system noise υn is i.i.d., zero-mean, independent ofsn , and with variance σ2

υ . For the signal in (12), the optimalweight in minimum MSE estimation based on (8) can be calcu-lated as

E{snsH

n

}−1E {sny∗

n} = E {wn} (13)

To explore the filter behaviour in non-stationary environments,we assume that the system weight vector wn ∈ HM×1 varies

according to a widely used first-order random-walk model givenby [26], [29], [43], [44]

wn = wn−1 + qn (14)

where qn ∈ HM×1 is a random quaternion vector that is i.i.d,zero-mean and independent of sn . The expression in (14) is anapproximation of a Markov model given by [45]

wn = Awn−1 + qn (15)

where the coefficient matrix A is assumed to be close to IM ina number of practical applications. For example, in modellingRayleigh fading channels in a wireless communications envi-ronment, (15) is a first-order approximation of the variation offading coefficients, and A is usually close to IM [29]. In or-der to simplify the derivation during the convergence analysis,the real-valued adaptive filtering literature customarily assumesA = IM to employ the first-order random-walk model in (14),which can be straightforwardly extended to the quaternion do-main. Despite the simplified non-stationary scenarios studiedhere, the essence of the problem that is common to more com-plicated situations is retained.

A number of quaternion adaptive filtering algorithms, basedon (8) and the stochastic gradient decent minimisation of MSE,assume the following form:

yn = wHn sn

en = yn − yn

wn+1 = wn + μf [sn ] sne∗n(16)

where μ is the step size, f [sn ] a real-valued scalar function ofsn , f [sn ] : HM×1 → R. Specific values of f [sn ] correspond tospecific filtering algorithms. For example, f [sn ] = 1 results inthe QLMS, f [sn ] = ‖sn‖−2 results in the normalised QLMS,and f [sn ] = (ε + ‖sn‖2)−1 results in the ε-normalised QLMS[43].

III. PERFORMANCE ANALYSIS OF QUATERNION

ADAPTIVE FILTERS

To analyse the convergence performance of the filters in theform of (16), define the weight-error vector as wn � wn −wn , and express the filter error as en = wH

n sn + υn . Then theweight-error vector recursion is expressed as

wn+1 = wn − μf [sn ] sne∗n + qn (17)

=(IM − μf [sn ] snsH

n

)wn − μf [sn ] snυ∗

n + qn

A. Mean Analysis

Since υn and qn are i.i.d., zero-mean and independent of sn ,the expectation of (17) is given by

E {wn+1} = E{IM − μf [sn ] snsH

n

}E {wn} (18)

A convenient change of coordinates is enabled by the eigen-decomposition, E{f [sn ]snsH

n } = UΛUH , where U is a uni-tary quaternion matrix and Λ is a real diagonal matrixcontaining eigenvalues of E{f [sn ]snsH

n } on the diagonal. De-fine w′

n = UH wn , then (18) becomes

E{w′

n+1}

= E {IM − μΛ}E {w′n} (19)


from which we obtain that limn→∞E{w′

n} = 0 if and only if the

eigenvalues of the matrix E{IM − μΛ} are all within (−1, 1),or equivalently

0 < μ < 2λ−1max

(E

{f [sn ] snsH

n

})(20)

B. Mean Square Analysis

Define the a priori and a posteriori estimation errors as

ea,n � wHn sn − wH

n sn = wHn sn

ep,n � wHn sn − wH

n+1sn = (wn+1 − qn )H sn

and the weighted a priori and a posteriori errors as eΣa,n �

wHn Σsn and eΣ

p,n � (wn+1 − qn )H Σsn , where Σ is a quater-nion Hermitian positive definite weighting matrix. For notationconciseness, denote sH

n Σsn by ‖sn‖2Σ . Because of (17) and

eΣp,n = eΣ

a,n − μf [sn ] ensHn Σsn = eΣ

a,n − μf [sn ] en ‖sn‖2Σ

(21)we obtain

wn+1 + sn

eΣ∗a,n

‖sn‖2Σ

= wn + sn

eΣ∗p,n

‖sn‖2Σ

+ qn (22)

Evaluating energies of both sides of (22) yields

‖wn+1‖2Σ +

∣∣eΣa,n

∣∣2‖sn‖2

Σ

= ‖wn‖2Σ +

∣∣eΣp,n

∣∣2‖sn‖2

Σ

+ ‖qn‖2Σ

+ 2R

[qH

n Σ

(wn + sn

eΣ∗p,n

‖sn‖2Σ

)]

which can be rewritten by combining (21) as

‖wn+1‖2Σ = ‖wn‖2

Σ1+ μ2 |υn |2 f 2 [sn ] ‖sn‖2

Σ + ‖qn‖2Σ

+ 2R

[qH

n Σ

(wn + sn

eΣ∗p,n

‖sn‖2Σ

)]

− 2μR[f [sn ]

(eΣa,n − μf [sn ] ‖sn‖2

Σ ea,n

)υn

]

Σ1 � Σ − μ(f [sn ] snsH

n Σ + f [sn ]ΣsnsHn

)

+ μ2f 2 [sn ] ‖sn‖2Σ snsH

n

The application of the statistical expectation operator to theabove equation yields

E{‖wn+1‖2

Σ

}= E

{‖wn‖2

Σ1

}+ E

{‖qn‖2

Σ

}

+μ2σ2υE

{f 2 [sn ] ‖sn‖2

Σ

} (23)

For simplicity, we assume that the sequence of vectors sn isi.i.d.. Thus (23) becomes

E{‖wn+1‖2

Σ

}= E

{‖wn‖2

Σ ′

}+ E

{‖qn‖2

Σ

}

+μ2σ2υE

{f 2 [sn ] ‖sn‖2

Σ

} (24)

Σ′ � E {Σ1} = Σ + μ2E{

f 2 [sn ] ‖sn‖2Σ snsH

n

}

− μ(E

{f [sn ] snsH

n

}Σ + ΣE

{f [sn ] snsH

n

})

1) Transient Analysis: Consider now the transient behaviourof E{‖wn‖2} by letting Σ = IM in (24). Following the analysisframework proposed in [27], we vectorise matrix Σ′ into thevector vec(Σ′) by stacking the columns of Σ on top of oneanother, and refer to ‖ · ‖2

Σ ′ as ‖ · ‖2vec(Σ ′) . In this way, (24) can

be rewritten as

E{‖wn+1‖2

}= E

{‖wn‖2

Fvec(IM )

}+ E

{‖qn‖2

}(25)

+ μ2σ2υE

{f 2 [sn ] ‖sn‖2

}

F � IM 2 − μA + μ2B

A � IM ⊗ E{f [sn ] snsH

n

}+

(E

{f [sn ] snsH

n

})T ⊗ IM

B � E{

f 2 [sn ](snsH

n

)T ⊗R(snsH

n

)}

where the symbol “⊗” denotes the left Kronecker product

A ⊗ B �

⎡⎢⎣

[A]1,1 B . . . [A]1,M B...

. . ....

[A]M,1 B . . . [A]M,M B

⎤⎥⎦

and “⊗R” the right Kronecker product [46]

A ⊗R B �

⎡⎢⎣

B [A]1,1 . . . B [A]1,M...

. . ....

B [A]M,1 . . . B [A]M,M

⎤⎥⎦

Notice that the eigenvalues of F are all real-valued as F is aHermitian quaternion matrix [38].

Based on the Cayley-Hamilton theorem, [27] has proved that(25) is stable for real variables if and only if all eigenvalues ofF are within (−1, 1), which is equivalent to

0 < μ < min{

λ−1max

(A−1B

),

max{

λ

([A/2 −B/2IM 2 0

])∈ R+

}−1 }(26)

This conclusion also holds for quaternion variables since theCayley-Hamilton theorem applies to the quaternion domain[34], implying that (26) is also the mean square stability condi-tion of quaternion adaptive filters. In contrast to real and complexanalyses [25], [29], the steady-state performance of quaternionadaptive filters is difficult to quantify in a closed form via theabove analysis approach, since the required vectorisation ofquaternion matrices is hindered by the non-commutativity ofquaternion products.

2) Performance Bounds: The sufficient and necessary stabil-ity condition shown in (26) is difficult to interpret owing to thecomplicated structure of matricesA andB. Next, we shall derivesimple sufficient and necessary conditions for the mean squarestability. Set Σ = IM , then the evolution of E{‖wn+1‖2} canbe deduced from (24) as

E{‖wn+1‖2

}= E

{‖wn‖2

Σ ′

}+ E

{‖qn‖2

}(27)

+ μ2σ2υE

{f 2 [sn ] ‖sn‖2

}

Σ′ = IM − 2μE{f [sn ] snsH

n

}+ μ2E

{f 2 [sn ] ‖sn‖2 snsH

n

}


Notice that the Cauchy–Schwarz inequality can be applied tothe Hermitian quaternion matrix E{f 2 [sn ]‖sn‖2snsH

n } [47],yielding

Σ′ IM − 2μE{f [sn ] snsH

n

}+ μ2 (

E{f [sn ] snsH

n

})2

=(IM − μE

{f [sn ] snsH

n

})2(28)

Observe that Σ′ is positive semi-definite, and its eigendecom-position is given by Σ′ = TΛΣ ′TH , where T is a quaternionunitary matrix, and ΛΣ ′ a real diagonal matrix with non-negativeeigenvalues of Σ′ on the diagonal. Thus, we have

E{‖wn‖2

Σ ′

}= E

{wH

n TΛΣ ′TH wn

}(29)

From (29), we next obtain

E{‖wn‖2

Σ ′

}≤ λmax (Σ′) E

{‖wn‖2

}(30)

where λmax (Σ′) is bounded by

λmax (Σ′) ≤ ξ1 � 1 − 2μλmin(E

{f [sn ] snsH

n

})

+μ2λmax

(E


n

})

which is based on (6). This inequality, together with (27) and(30), yields

E{‖wn+1‖2

}≤ ξ1E

{‖wn‖2

}+ E

{‖qn‖2

}

+μ2σ2υE

{f 2 [sn ] ‖sn‖2

} (31)

which is stable if ξ1 < 1, thus implying the following sufficientcondition for the mean square stability

0 < μ <2λmin

(E

{f [sn ] snsH

n

})

λmax

(E


n

}) (32)

From (29), we also have

E{‖wn‖2

Σ ′

}≥ λmin (Σ′) E

{‖wn‖2

}(33)

where λmin(Σ′) is bounded by

λmin (Σ′) ≥ ξ2 � 1 − 2μλmax(E

{f [sn ] snsH

n

})

+μ2λmin

(E


n

})

This inequality, together with (27) and (33), yields

E{‖wn+1‖2

}≥ ξ2E

{‖wn‖2

}+ E

{‖qn‖2

}

+μ2σ2υE

{f 2 [sn ] ‖sn‖2

} (34)

Thus, a necessary condition for the convergence of E{‖wn‖2}is ξ2 < 1, or equivalently

0 < μ <2λmax

(E

{f [sn ] snsH

n

})

λmin

(E


n

}) (35)

From (28), another necessary condition for the conver-gence of E{‖wn‖2} is that the eigenvalues of the matrix(IM − μE{f [sn ]snsH

n })2 are all less than unit, which yieldsthe same step size bound as in (20).

Remark 1: Any value of μ that satisfies the mean square sta-bility condition given in (26) also guarantees the mean stability.

Next, the steady-state performance of quaternion adaptive fil-ters will be bounded through a similar approach. At the steadystate, we have limn→∞E {‖wn+1‖2} = limn→∞E {‖wn‖2},

and thus the bounds of MSD = limn→∞E {‖wn‖2} can be de-rived from (31) and (34) as

MSD ≤ μ2 σ 2υ Tr(Css )+Tr(Cqq )

2μλm in (E {f [sn ]sn sHn })−μ2 λm a x (E{f 2 [sn ]‖sn ‖2 sn sH

n })MSD ≥ μ2 σ 2

υ Tr(Css )+Tr(Cqq )2μλm a x (E {f [sn ]sn sH

n })−μ2 λm in (E{f 2 [sn ]‖sn ‖2 sn sHn })

(36)Since

E{‖wn‖2

Css

}= E

{wH

n KΛssKH wn

}

where KΛssKH is the eigendecomposition of Css , we obtain

λmin (Css) E{‖wn‖2

}

≤ E{‖wn‖2

Css

}≤ λmax (Css) E

{‖wn‖2

}

so that EMSE = limn→∞E{‖wn‖2Css

} is bounded by

λmin (Css) · MSD ≤ EMSE ≤ λmax (Css) · MSD (37)

C. Performance Analysis of QLMS

When f [sn ] = 1, the general quaternion adaptive filtering al-gorithm in the form of (16) reduces to the QLMS. Therefore,the performance of QLMS filters can be obtained by substitutingf [sn ] = 1 into the above analysis results for general quaternionadaptive filters. Specifically, the mean weight-error vector re-cursion in (18) becomes

E {wn+1} = (IM − μCss) E {wn} (38)

while the mean stability condition in (20) reduces to

0 < μ < 2λ−1max (Css) (39)

which is the same as the mean stability condition obtained in[30].

The mean square stability condition in (26), which is also theoverall stability condition, becomes

0 < μ < min{

λ−1max

(A−1B

),

max{

λ

([A/2 −B/2IM 2 0

])∈ R+

}−1 }

A � IM ⊗ Css + CTss ⊗ IM

B � E{(

snsHn

)T ⊗R(snsH

n

)}(40)

Furthermore, if the regressor vector sn arises from a Gaus-sian distribution, the analysis results shown in (32), (35)–(37)can be simplified by expressing the fourth-order moment ma-trix E{‖sn‖2snsH

n } in terms of covariance and complementarycovariance matrices of sn as

E{‖sn‖2 snsH

n

}

= Tr (Css)Css +12

(C2

ss +∑

α= ı,j,κ

Cssα CHssα

)(41)


which is proved in the Appendix. Then, we have

λmax

(E

{‖sn‖2 snsH

n

})≤ Tr (Css) λmax (Css)

+12

[λ2

max (Css) +∑

α= ı,j,κ

p2max (Cssα )

]

λmin

(E

{‖sn‖2 snsH

n

})≥ Tr (Css) λmin (Css)

+12

[λ2

min (Css) +∑

α= ı,j,κ

p2min (Cssα )

]

where pmax(Cssα ) and pmin(Cssα ) are maximum and mini-mum singular values of Cssα respectively. A sufficient stabilitycondition is deduced from (32) as

0 < μ <

4λmin (Css)2Tr (Css) λmax (Css) + λ2

max (Css) +∑

α= ı,j,κ p2max (Cssα )

(42)

while a necessary stability condition is deduced from (35) as

0 < μ <

4λmax (Css)2Tr (Css) λmin (Css) + λ2

min (Css) +∑

α= ı,j,κ p2min (Cssα )

(43)

The upper and lower bounds of MSD are then deduced from(36), as in (44) shown at the bottom of this page. The upper andlower bounds of EMSE can be calculated based on the MSDbounds in (44), as in (37).

IV. MEAN SQUARE ANALYSIS OF QLMS WITH GAUSSIAN

REGRESSORS USING QUATERNION DATA DECORRELATION

As shown in Section III-B and Section III-C, unlike the anal-ysis for real and complex adaptive filters [25], [29], the meansquare performance of quaternion adaptive filters is difficult toevaluate exactly in a closed form, owing to the non-commutativequaternion algebra and the lack of structural quaternion matrixdecomposition methods. In order to obtain a concise closed-form solution to the mean square convergence equation (24) forQLMS with Gaussian regressors, we next use a novel quaterniondata decorrelation technique to simplify the involved quater-nion covariance matrices. The simplification is based on thefollowing assumption on the augmented second-order statisticsof quaternion Gaussian regressor vector sn .

Assumption 1: There exists a unitary transform sn = GH sn

such that

Css = Λss , Cssα = Λssα , α = ı, j, κ (45)

where Λss is a diagonal real matrix containing the eigenvalues ofCss on the diagonal, and Λssα are diagonal quaternion matrices.

It can be verified that Assumption 1 holds exactly when oneof the following conditions on sn is met:

1) The covariance and complementary covariance matricesof sn are related by scalar multiplication. This condi-tion holds for regressor vectors with specific second-orderstatistics, such as an H-proper regressor vector (Cssα =0Css) and a rectilinear regressor vector (Cssα = ραCss ,where ρα is a unit quaternion).

2) sn can be expressed by applying a unitary transform toa white Gaussian random quaternion vector vn , that is,sn = Gvn , where G is a unitary quaternion matrix.

For quaternion inputs that do not meet the above two con-ditions, Assumption 1 holds approximately as the result of therecently proposed approximate uncorrelating transform [48]–[50]. Under Assumption 1, we shall define wn = GH wn ,qn = GH qn , Σ = GH ΣG and convert (24) into

E{‖wn+1‖2

Σ

}= E

{‖wn‖2

Σ ′}

+ μ2σ2υE

{‖sn‖2

Σ

}

+ E{‖qn‖2

Σ

}

Σ′ = Σ − μ(E

{sn sH

n

}Σ + ΣE

{sn sH

n

})

+ μ2E{‖sn‖2

Σ sn sHn

}(46)

where

E{‖sn‖2

Σ sn sHn

}= Tr

(ΣΛss

)Λss

+12Σ

(Λ2

ss +∑

α= ı,j,κ

Λssα Λ∗ssα

)(47)

is derived similarly to the derivation of (41). Let Σ be a diagonalreal matrix, then Σ′ is also diagonal and real, and hence these twomatrices can be completely characterised by their diagonal en-tries. Defining σ � diag(Σ), σ′ � diag(Σ′), λs � diag(Λss),λq � diag(E{qn qH

n }), and referring to ‖ · ‖2Σ and ‖ · ‖2

Σ as‖ · ‖2

σ and ‖ · ‖2σ′ , we obtain the compact forms of (46) and

(47) as

E{‖wn+1‖2

σ

}= E

{‖wn‖2

Fσ

}+

(μ2σ2

υλs + λq)T

σ

F = IM − μA + μ2B

A = 2Λss

B =12

(Λ2

ss +∑

α= ı,j,κ

Λssα Λ∗ssα

)+ λsλ

Ts (48)

Similar to the stability condition for (26), (48) is stable ifand only if all eigenvalues of F are within (−1, 1). Since these

MSD ≤ 2μ2σ2υ Tr (Css) + 2Tr (Cqq)

4μλmin (Css) − μ2[2Tr (Css) λmax (Css) + λ2

max (Css) +∑

α= ı,j,κ p2max (Cssα )

]

MSD ≥ 2μ2σ2υ Tr (Css) + 2Tr (Cqq)

4μλmax (Css) − μ2[2Tr (Css) λmin (Css) + λ2

min (Css) +∑

α= ı,j,κ p2min (Cssα )

] (44)


eigenvalues are easily seen to be greater than −1, we only needto investigate how to guarantee that they are less than unit. Itcan be proved that they are upper bounded by unit if and only if0 < μ < λ−1

max(A−1B), which is verified to be equivalent to

0 <

M∑m=1

2μλs,m

4 − μλs,m

(1 +

∑α= ı,j,κ r2

sα,m

) < 1 (49)

where λs,m is the m-th element of λs , and rsα,m is the α-impropriety coefficient of the m-th element of sn . Accordingto Remark 1, the condition in (49) is sufficient and necessaryfor the algorithm stability both in the mean and mean squaresense.

Remark 2: The stability condition in (49) indicates that theupper bound of μ decreases with the increase in the length of theregressor vector, M , the increase in the power of the regressorreflected by λs,m , and the increase in the impropriety degree ofsn reflected by rsı,m , rsj,m , rsκ,m .

MSD and EMSE analysis. Iterating (48) gives

E{‖wn‖2

σ

}= E

{‖w0‖2

Fn σ

}+

(μ2σ2

υλs + λq)T

n−1∑l=0

Flσ

(50)The MSD and EMSE at iteration n can now be derived from(50) by letting σ = 1M and σ = λs , respectively. Equation (50)also yields the iteration

E{‖wn+1‖2

σ

}= E

{‖wn‖2

σ

}+ E

{‖w0‖2

Fn (F−IM )σ

}

+(μ2σ2

υλs + λq)T Fn σ (51)

which characterises the evolution of MSD when σ = 1M , andcharacterises the evolution of EMSE when σ = λs .

At the steady state, limn→∞ E{‖wn+1‖2σ} = limn→∞

E{‖wn‖2σ}, so that (48) yields

limn→∞E

{‖wn‖2

(μA−μ2 B)σ

}=

(μ2σ2

υλs + λq)T

σ

Finally, the steady-state MSD and EMSE become

MSD = limn→∞E

{‖wn‖2

1M

}

=(μ2σ2

υλs + λq)T (

μA − μ2B)−1 1M

EMSE = limn→∞E

{‖wn‖2

λs

}

=(μ2σ2

υλs + λq)T (

μA − μ2B)−1

λs

which can be rewritten as

MSD =2∑M

m=1μ2 σ 2

υ +λq , m λ−1s , m

4−μλs , m (1+∑

α = ı , j , κ r 2sα , m )

μ − 2μ2∑M

m=1λs , m

4−μλs , m (1+∑

α = ı , j , κ r 2sα, m )

(52)

EMSE =2∑M

m=1μ2 σ 2

υ λs , m +λq , m

4−μλs , m (1+∑

α = ı , j , κ r 2sα, m )

μ − 2μ2∑M

m=1λs , m

4−μλs , m (1+∑

α = ı , j , κ r 2sα, m )

(53)

Remark 3: Observe from (52) and (53) that the steady-stateEMSE and MSD increase with the length of the regressor vector,

TABLE ISUMMARY OF ANALYSIS RESULTS

S: Sufficient; N: Necessary.

M , the power of the regressor reflected by λs,m , the improprietydegree of sn reflected by rsı,m , rsj,m , rsκ,m , and the varianceof the system weight variation reflected by λq,m .

Table I summarises the sufficient stability conditions, neces-sary stability conditions, necessary and sufficient stability condi-tions, MSDs, and EMSEs for general quaternion adaptive filtersand QLMS filters with Gaussian inputs. The closed-form repre-sentations of the performance of SL-QLMS, SWL-QLMS, andWL-QLMS with Gaussian regressors satisfying Assumption 1can be obtained by replacing s with xn , xn , and xn respectivelyin (49), (52) and (53). Exploring the second-order statistics ofinput signal, the following three subsections further address theperformance of the three classes of QLMS algorithms.

A. SL-QLMS

1) Scalar Input: The stability condition of the SL-QLMSwith a scalar regressor xn is obtained from (49) as

0 < μ <4

σ2x

(3 + r2

xı + r2xj + r2

xκ

) (54)

where σ2x is the variance of xn . This condition is the same as the

analysis result obtained in [31]. The steady-state performancecan be derived from (52) and (53) as

MSD =2(μ2σ2

υσ2x + σ2

q

)

μσ2x

[4 − μσ2

x

(3 + r2

xı + r2xj + r2

xκ

)] (55)

EMSE =2(μ2σ2

υσ2x + σ2

q

)

μ[4 − μσ2

x

(3 + r2

xı + r2xj + r2

xκ

)] (56)

where σ2q is the variance of the weight variation qn .

2) H-Proper Input Vector: The diagonalisation ofCxx com-pletely decorrelates an H-proper quaternion input vector xn . Forsuch inputs, rxα,m = 0,∀m,α, so that the stability condition in(49) and the steady-state performance in (52) and (53) become

0 <

M∑m=1

2μλx,m

4 − μλx,m< 1 (57)


MSD =2∑M

m=1μ2 σ 2

υ +λq , m λ−1x , m

4−μλx , m

μ(1 − μ

∑Mm=1

2λx , m

4−μλx , m

) (58)

EMSE =2∑M

m=1μ2 σ 2

υ λx , m +λq , m

4−μλx , m

μ(1 − μ

∑Mm=1

2λx , m

4−μλx , m

) (59)

3) Rectilinear Input Vector: If the Gaussian input vector xn

is rectilinear, it can be expressed as xn = eζzn , where zn is arandom Gaussian real vector, and ζ a constant pure quaternionscalar [51]. For such inputs, the covariance and complemen-tary covariance matrices can be represented by Cxx = Czz ,Cxxα = eζ (e−ζ )α Czz , α = ı, j, κ. Therefore, Cxx ,Cxxα canbe simultaneously diagonalised via the diagonalisation of thereal symmetric matrix Czz . Note rxα,m = 1,∀m,α, so that thestability condition in (49) and the steady-state performance in(52) and (53) become

0 <

M∑m=1

μλx,m

2 − 2μλx,m< 1 (60)

MSD =

∑Mm=1

μ2 σ 2υ +λq , m λ−1

x , m

2−2μλx , m

μ(1 − μ

∑Mm=1

λx , m

2−2μλx , m

) (61)

EMSE =

∑Mm=1

μ2 σ 2υ λx , m +λq , m

2−2μλx , m

μ(1 − μ

∑Mm=1

λx , m

2−2μλx , m

) (62)

B. SWL-QLMS

1) H-Proper Input Vector: If the random quaternion vectorxn is H-proper, Cxx reduces to a block diagonal matrix. Denotethe eigendecomposition of Cxx by Cxx = QΛxxQH , whereΛxx is a diagonal real matrix containing all eigenvalues of Cxx

on the diagonal, and Q is a unitary quaternion matrix containingeigenvectors of Cxx . We can find a unitary quaternion matrix

G =1√2

[Q −QQη Qη

]

so that sn = GH xn is a Cη -improper vector and satisfies

Css = I2 ⊗ Λxx , Cssη = Diag(1,−1) ⊗ Λxx

This reduces the stability condition in (49) and the steady-stateperformance in (52) and (53) to

0 <

M/2∑m=1

2μλx,m

2 − μλx,m< 1 (63)

MSD =

∑M/2m=1

2μ2 σ 2υ λx , m +λq , m +λq , m + M / 2

λx , m (2−μλx , m )

μ(1 − 2μ

∑M/2m=1

λx , m

2−μλx , m

) (64)

EMSE =

∑M/2m=1

2μ2 σ 2υ λx , m +λq , m +λq , m + M / 2

2−μλx , m

μ(1 − 2μ

∑M/2m=1

λx , m

2−μλx , m

) (65)

2) Rectilinear Input Vector: As shown in Section IV-A3, thecovariance matrix of a rectilinear vector xn is real-valued, so itseigendecomposition can be represented by Cxx = QΛxxQT ,

where Q is a real orthogonal matrix. We can find a quaternionunitary matrix

G =1√2

[Q −eζ

(e−ζ

)η Q(eζ

)ηe−ζQ Q

]

so that sn = GH xn satisfies

Css = Diag(2, 0) ⊗ Λxx

Cssη = Diag(2eζ(e−ζ

)α, 0) ⊗ Λxx , α = ı, j, κ

which reduces the stability condition in (49) and the steady-stateEMSE in (53) to

0 <

M/2∑m=1

μλx,m

1 − 2μλx,m< 1 (66)

EMSE =

∑M/2m=1

2μ2 σ 2υ λx , m +λq , m

1−2μλx , m+

∑Mm=1+M/2λq,m

2μ(1 − μ

∑M/2m=1

λx , m

1−2μλx , m

) (67)

In this case, the steady-state MSD is uncertain as the weightsolution is not unique.

C. WL-QLMS

1) H-proper Input Vector: If xn is H-proper, Cxx reduces toa block diagonal matrix. Considering the eigendecomposition ofCxx , Cxx = QΛxxQH , we can find such a quaternion unitarymatrix given by

G =12

⎡⎢⎢⎢⎣

Q −Q −Q −QQı −Qı Qı Qı

Qj Qj −Qj Qj

Qκ Qκ Qκ −Qκ

⎤⎥⎥⎥⎦

that sn = GH xn satisfies

Css = I4 ⊗ Λxx

Cssı = Diag(1, 1,−1,−1) ⊗ Λxx

Cssj = Diag(1,−1, 1,−1) ⊗ Λxx

Cssκ = Diag(1,−1,−1, 1) ⊗ Λxx

The stability condition in (49) and the steady-state performancein (52) and (53) become

0 <

M/4∑m=1

2μλx,m

1 − μλx,m< 1 (68)

MSD =∑M/4

m=14μ2 σ 2

υ λx , m +λq , m +λq , m + M / 4 +λq , m + M / 2 +λq , m + 3 M / 4

2λx , m (1−μλx , m )

μ(1 − 2μ

∑M/4m=1

λx , m

1−μλx , m

) (69)

EMSE =∑M/4

m=14μ2 σ 2

υ λx , m +λq , m +λq , m + M / 4 +λq , m + M / 2 +λq , m + 3 M / 4

2(1−μλx , m )

μ(1 − 2μ

∑M/4m=1

λx , m

1−μλx , m

) (70)


2) Rectilinear input vector: If xn is rectilinear, the quater-nion unitary matrix

G =

12

⎡⎢⎢⎢⎣

Q −Q −Q −Q(eζ

)ıe−ζQ − (

eζ)ı

e−ζQ(eζ

)ıe−ζQ

(eζ

)ıe−ζQ(

eζ)j

e−ζQ(eζ

)je−ζQ − (

eζ)j

e−ζQ(eζ

)je−ζQ(

eζ)κ

e−ζQ(eζ

)κe−ζQ

(eζ

)κe−ζQ − (

eζ)κ

e−ζQ

⎤⎥⎥⎥⎦

allows the transformed vector sn = GH xn to satisfy

Css = Diag(4, 0, 0, 0) ⊗ Λxx

Cssα = Diag(4eζ(e−ζ

)α, 0, 0, 0) ⊗ Λxx , α = ı, j, κ

which reduces the stability condition in (49) and the steady-stateEMSE in (53) to

0 <

M/4∑m=1

2μλx,m

1 − 4μλx,m< 1 (71)

EMSE =

∑M/4m=1

4μ2 σ 2υ λx , m +λq , m

1−4μλx , m+

∑Mm=1+M/4λq,m

2μ(1 − 2μ

∑M/4m=1

λx , m

1−4μλx , m

) (72)

In this case, the steady-state MSD is uncertain as the weightsolution is not unique.

V. SIMULATIONS

The performances of the three considered classes of QLMSalgorithms with Gaussian inputs were evaluated by averag-ing results over 1000 independent numerical simulation tri-als. Throughout the simulations, the length of the syntheticcorrelated input vector xn was L = 4, and the eigenvaluesof Cxx were kept the same. For simplicity, the elements ofxn were set to have an equal impropriety coefficient, that is,rm,ı = rm,j = rm,k = r.

A. SL-QLMS

In the simulations for the SL-QLMS, the desired signal wasgenerated from the strictly linear model in (9) and (12). Thevariance of the additive Gaussian noise υn was σ2

υ = 0.01, andthe Euclidean norm of the Gaussian weight variation qn was0.0016. Fig. 1 depicts the theoretical and simulated EMSE andMSD learning curves of the SL-QLMS for an improper inputwith r = 0.5 and μ = 0.02. These curves exhibit a good matchbetween the theory and simulation results. Figs. 2 and 3 illustratethe theoretical and simulated steady-state EMSEs for differentvalues of the step size μ when xn is H-proper and when xn

is improper with r = 0.5, respectively. The theoretical boundsof μ are also indicated. Both figures show that there was agood match between theory and simulations and their differenceincreased with μ, which is similar to the behaviour of real-valued LMS filters [27]. Observe that the decrease in the stabilitybound for the improper input conforms with Remark 2. Fig. 8illustrates a match between the theoretical and simulated steady-state EMSEs for inputs with various impropriety coefficientsr ∈ [0, 1] and μ = 0.2. The increase in the steady-state EMSEswith r conforms with Remark 3. As Remark 2, Figs. 2 and 3show, the upper bound of μ decreases with the increase in r.

Fig. 1. Theoretical and simulated learning curves of SL-QLMS for an im-proper input with r = 0.5 and μ = 0.02.

Fig. 2. Theoretical and simulated EMSEs of SL-QLMS as a function of thestep size, for an H-proper input.

Fig. 3. Theoretical and simulated EMSEs of SL-QLMS as a function of thestep size, for an improper input with r = 0.5.

So, for a fixed μ, the theoretical EMSE for highly improperinputs is less accurate, as μ is closer to the upper bound. Thisexplains the difference between the theoretical and simulatedsteady-state EMSEs for a large r in Fig. 4 as well as in Figs. 8and 12.


Fig. 4. Theoretical and simulated EMSEs of SL-QLMS as a function of theimpropriety coefficient, for μ = 0.2.

Fig. 5. Theoretical and simulated learning curves of SWL-QLMS for animproper input with r = 0.5 and μ = 0.02.

B. SWL-QLMS

In the simulations for the SWL-QLMS, the desired signal wasgenerated from the semi-widely linear model in (10) and (12).The variance of the additive Gaussian noise υn was σ2

υ = 0.01,and the Euclidean norm of the Gaussian weight variation qn

was 0.0008. Fig. 5 depicts theoretical and simulated EMSE andMSD learning curves of the SL-QLMS for an improper inputwith r = 0.5 and μ = 0.02. These curves exhibit a good matchbetween the theory and simulation results. Figs. 6 and 7 showtheoretical and simulated steady-state EMSEs for different val-ues of the step size μ when xn is H-proper and when xn isimproper with r = 0.5. The theoretical bounds of μ are alsoindicated. Both figures illustrate a good match between theoryand simulations and an increase in their difference with μ. Thedecrease in the stability bound for the improper input conformswith Remark 2. Fig. 8 illustrates a match between theoretical

Fig. 6. Theoretical and simulated EMSEs of SWL-QLMS as a function of thestep size, for an H-proper input.

Fig. 7. Theoretical and simulated EMSEs of SWL-QLMS as a function of thestep size, for an improper input with r = 0.5.

Fig. 8. Theoretical and simulated EMSEs of SWL-QLMS as a function of theimpropriety coefficient, for μ = 0.1.

and simulated steady-state EMSEs for inputs with various im-propriety coefficients r ∈ [0, 1] and μ = 0.1. The increase in thesteady-state EMSEs with r conforms with Remark 3.

C. WL-QLMS

In the simulations for the WL-QLMS, the desired signal wasgenerated from the widely linear model in (11) and (12). Thevariance of the additive Gaussian noise υn was σ2

υ = 0.01, andthe Euclidean norm of the Gaussian weight variation qn was0.0004. Fig. 9 depicts theoretical and simulated EMSE andMSD learning curves of the SL-QLMS for an improper in-put with r = 0.5 and μ = 0.02. These curves exhibit a goodmatch between the theory and simulation results. Figs. 10 and11 show theoretical and simulated steady-state EMSEs for dif-ferent values of the step size μ when xn is H-proper and whenxn is improper with r = 0.5. The theoretical bounds of μ arealso indicated. From both figures, we can observe a good match


Fig. 9. Theoretical and simulated learning curves of WL-QLMS for an im-proper input with r = 0.5 and μ = 0.02.

Fig. 10. Theoretical and simulated EMSEs of WL-QLMS as a function of thestep size, for an H-proper input.

Fig. 11. Theoretical and simulated EMSEs of WL-QLMS as a function of thestep size, for an improper input with r = 0.5.

Fig. 12. Theoretical and simulated EMSEs of WL-QLMS as a function of theimpropriety coefficient, for μ = 0.05.

between theory and simulations and an increase in their differ-ence with μ. The decrease in the stability bound for the improperinput conforms with Remark 2. Fig. 12 illustrates a match be-tween theoretical and simulated steady-state EMSEs for inputswith various impropriety coefficients r ∈ [0, 1] and μ = 0.05.The increase in the steady-state EMSEs with r conforms withRemark 3.

VI. CONCLUSION

Convergence of quaternion adaptive filtering algorithms innon-stationary environments has been analysed by extending ananalysis framework for real-valued adaptive filters to the quater-nion domain. For rigour, the first-order random-walk modelhas been employed to model the variation of system weightsin a non-stationary environment. The bounds on transient andsteady-state performance of general quaternion adaptive filtershave been derived. For QLMS filters with Gaussian inputs, aninnovative quaternion decorrelation technique has simplified theanalysis and enabled the step size bounds, MSDs and EMSEsto be expressed in closed forms. The analytical results, whichare shown to be influenced by the second-order statistics of in-put signals, provide new insights into the statistical behaviourof the considered filtering algorithms. Effectiveness of the pro-posed analysis has been verified by numerical simulations forSL-QLMS, SWL-QLMS and WL-QLMS filters with correlatedGaussian inputs. The proposed analysis framework may be ex-tended to other division algebras, such as the analysis of octo-nion adaptive filters, a topic of future research.

APPENDIX

PROOF OF (41)

Based on Isserlis’ theorem [52], the fourth-order moments ofreal and complex variables can be represented by their second-order moments. However, Isserlis’ theorem does not apply toquaternion variables because the quaternion products are non-commutative. To circumvent this difficulty, we represent quater-nion variables using the Cayley-Dickson construction, that is,the m-th element of sn is written as sm = cm + dm j, wherecm and dm are complex variables defined on the basis {1, ı}[18]. By applying Isserlis’ theorem to the constitutive complexvariables, the element on the a-th row and b-th column of the


fourth-order moment matrix E{‖sn‖2snsHn } can be represented

by the second-order statistics of sn as[E

{‖sn‖2 snsH

n

}]a,b

= E

{M∑

m=1

sm s∗m sas∗b

}

= E

{M∑

m=1

(cm + dm j) (c∗m − dm j) (ca + daj) (c∗b − dbj)

}

=M∑

m=1

(E{cm c∗m cac∗b + cm c∗m dad∗b + dm d∗m cac∗b

+ dm d∗m dad∗b} + E{cm c∗m dacb + dm d∗m dacb − cm c∗m cadb

− dm d∗m cadb}j)

=M∑

m=1

E{cm c∗m + dm d∗m}E{cac∗b + dad∗b + (dacb − cadb) j}

+M∑

m=1

(E {cm ca}E {c∗m c∗b} + E {cm c∗b}E {c∗m ca}

+ E {cm da}E {c∗m d∗b} + E {cm d∗b}E {c∗m da}+ E {dm ca}E {d∗m c∗b} + E {dm c∗b}E {d∗m ca}+ E {dm da}E {d∗m d∗b} + E {dm d∗b}E {d∗m da} )

+M∑

m=1

(E {cm da}E {c∗m cb} + E {cm cb}E {c∗m da}

+ E {dm da}E {d∗m cb} + E {dm cb}E {d∗m da}− E {cm ca}E {c∗m db} − E {cm db}E {c∗m da}− E {dm ca}E {d∗m db} − E {dm db}E {d∗m ca} ) j

We also observe

[Css ]a,b = E {sas∗b} = E {(ca + daj) (c∗b − dbj)}= E {cac∗b + dad∗b} + E {dacb − cadb} j

[Cssı ]a,b = E {sası∗b } = E {(ca + daj) (c∗b + dbj)}

= E {cac∗b − dad∗b} + E {dacb + cadb} j

[Cssj ]a,b = E{sasj∗

b

}= E {(ca + daj) (cb − d∗bj)}

= E {cacb + dadb} + E {dac∗b − cad∗b} j

[Cssκ ]a,b = E {sasκ∗b } = E {(ca + daj) (cb + d∗bj)}

= E {cacb − dadb} + E {dac∗b + cad∗b} j

From the above equations, we can verify[E

{‖sn‖2 snsH

n

}]a,b

= Tr (Css) [Css ]a,b +12[C2

ss

]a,b

+12

∑α= ı,j,κ

[Cssα Cαssα ]a,b

Therefore, (41) holds.

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Min Xiang (M’17) received the B.Eng. and M.Eng.degrees from Beihang University, Beijing, China.He is currently working toward the Ph.D. degree inelectrical engineering at the Imperial College Lon-don, London, U.K. His research interests includequaternion-valued statistical signal processing andadaptive filtering.

Sithan Kanna (S’13) received the M.Eng. de-gree in electrical and electronic engineering withmanagement from the Imperial College London,London, U.K., in 2012. His research interests includecomplex-valued statistical signal processing and fre-quency estimation in the smart grid. He was the re-cipient of full scholarship by The Rector’s Fund at theImperial College London to pursue the Ph.D. degreein adaptive signal processing.

Danilo P. Mandic (M’99–SM’03–F’12) received thePh.D. degree in nonlinear adaptive signal processingfrom the Imperial College London, London, U.K., in1999. He is a Professor of Signal Processing with theImperial College London, where he has been involvedin nonlinear adaptive signal processing and nonlineardynamics. He is a Deputy Director of the FinancialSignal Processing Laboratory, Imperial College Lon-don. He has been a Guest Professor with KatholiekeUniversiteit Leuven, Leuven, Belgium, the TokyoUniversity of Agriculture and Technology, Tokyo,

Japan, and Westminster University, London, U.K., and a Frontier Researcherwith RIKEN, Wako, Japan. He has authored/coauthored two research mono-graphs titled Recurrent Neural Networks for Prediction: Learning Algorithms,Architectures and Stability (Wiley, 2001) and Complex Valued Nonlinear Adap-tive Filters: Noncircularity, Widely Linear and Neural Models (Wiley, 2009), anedited book titled Signal Processing Techniques for Knowledge Extraction andInformation Fusion (Springer, 2008), and more than 200 publications on signaland image processing. He has been a member of the IEEE Technical Commit-tee on Signal Processing Theory and Methods, and an Associate Editor for theIEEE SIGNAL PROCESSING MAGAZINE, the IEEE TRANSACTIONS ON CIRCUITS

AND SYSTEMS II, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and theIEEE TRANSACTIONS ON NEURAL NETWORKS. He has produced award winningpapers and products resulting from his collaboration with the industry.

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