1356 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
Generalized Likelihood Ratios for Testing theProperness of Quaternion Gaussian Vectors
Javier Vía, Member, IEEE, Daniel P. Palomar, Senior Member, IEEE, and Luis Vielva
Abstract—In a recent paper, the second-order statistical anal-ysis of quaternion random vectors has shown that there exist twodifferent kinds of quaternion widely linear processing, which areassociated with the two main types of quaternion properness. Inthis paper, we consider the problem of determining, from a finitenumber of independent vector observations, whether a quaternionGaussian vector is proper or not. Specifically, we derive threegeneralized likelihood ratio tests (GLRTs) for testing the twomain kinds of quaternion properness and show that the GLRTsreduce to the estimation of three previously proposed quaternionimproperness measures. Interestingly, the three GLRT statistics(improperness measures) can be interpreted as an estimate of theentropy loss due to the quaternion improperness. Additionally,we analyze the case in which the orthogonal basis for the repre-sentation of the quaternion vector is unknown, which results inthe problem of estimating the principal -properness direction,i.e., the pure unit quaternion minimizing the -impropernessmeasure. Although this estimation problem is not convex, wepropose a technique based on successive convex approximations,which can be solved in closed form. Finally, some simulationexamples illustrate the performance and practical application ofthe proposed tests.
Index Terms—Generalized likelihood ratio test (GLRT),principal -properness direction, properness, propriety,quaternions, second-order circularity.
I. INTRODUCTION
A PART from its traditional use in aerospace [1], [2]and computer graphics [3] problems, quaternion signal
processing has recently encountered interesting applica-tions in image processing [4]–[8], wind modeling [9]–[12],processing of polarized waves [13], [14], and design (andprocessing) of space–time (and space–time-polarization [15])block codes [16]–[20]. However, the statistical analysis ofquaternion random vectors has received limited attention [9],[14], [21]–[24], and only recently the concept of widely linearprocessing has been extended from complex to quaternion
Manuscript received May 31, 2010; revised October 02, 2010 and December09, 2010; accepted December 11, 2010. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. Jean-YvesTourneret. This work was supported by the Spanish Government, Ministeriode Ciencia e Innovación (MICINN), under projects COMONSENS (CSD2008-00010, CONSOLIDER-INGENIO 2010 Program) and COSIMA (TEC2010-19545-C04-03) and by the Hong Kong RGC 618709 research grant.
J. Vía and L. Vielva are with the Department of CommunicationsEngineering, University of Cantabria, 39005 Santander, Spain (e-mail:[email protected]; [email protected]).
D. P. Palomar is with the Department of Electronic and Computer Engi-neering, Hong Kong University of Science and Technology, Kowloon, HongKong (e-mail: [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.2010.2101067
vectors [25]. A complex random vector is said to be proper if itis uncorrelated with its complex conjugate, which results in theoptimality of the conventional linear processing. However, inthe more general case of (possibly) improper complex vectors,the optimal linear processing is widely linear, i.e., we haveto simultaneously operate on the data vector and its complexconjugate [26]–[34].
Unlike the complex case, we can define two different typesof quaternion widely linear processing [25], which are stronglyrelated to the two main kinds of quaternion properness. In par-ticular, the most general quaternion linear processing (which werefer to as full-widely linear processing) requires the operationon the quaternion vector and its involutions over the three pureunit quaternions in an orthogonal basis . However, for
-proper data, the optimal linear processing reduces to conven-tional linear processing (we do not need to operate on the vectorinvolutions), whereas for -proper vectors, the optimal linearprocessing (referred to as semi-widely linear) only requires theoperation on the quaternion vector and its involution over [25].In other words, taking into account the isomorphism amongquaternion, complex, and real numbers, we can consider threedifferent scenarios. 1) If our quaternion vector is -proper,we can apply conventional quaternion linear processing. 2) If itis -proper, it has to be decomposed into two complex vectors,which will be jointly processed. 3) If is improper (i.e., if it isnot - or -proper), we need to directly operate on the four realvectors composing . Finally, we must note that in [25], the au-thors introduced the definition of -properness, which allowsus to easily relate the two main kinds of quaternion properness.Roughly speaking, we can say that the -properness is all whata -proper quaternion vector needs to become -proper.
Analogously to the complex case [35], algorithms adaptedfor improper signals can fail or suffer from slow convergencewhen they are used for proper signals. This is due to the factthat the number of free parameters in a conventional linearalgorithm is multiplied by four (respectively, by two) in itsfull-widely (respectively, semi-widely) linear counterpart.Therefore, since the complexity of the associated parameter es-timation problem depends on the selected model (conventional,semi-widely, or full-widely), we should follow the principleof parsimony and choose the simplest model exploiting thestatistical properties of the data. As a consequence, it becomescrucial to determine whether a quaternion random vector is
-proper, -proper, or improper. As a practical example,which is illustrated in Section VI, we consider an optical com-munication system based on dual polarization [36]–[38]. Thus,depending on several system parameters, the signals in the fibercan be represented by -proper, -proper, or improper randomquaternions.
VÍA et al.: GLRTs FOR TESTING PROPERNESS OF QUATERNION GAUSSIAN VECTORS 1357
In this paper, we consider the three binary hypotheses testingproblems associated with the evaluation of two out of the threepossible scenarios. In particular, assuming zero-mean quater-nion Gaussian data, we derive three generalized likelihood ratiotests (GLRTs), which are also the key ingredient for solving themore general multiple-hypotheses testing problem. Althoughsuboptimal in the Neyman–Pearson sense, this kind of detectoris easy to obtain, performs well in practice, and in our case re-sults in simple detection rules, which admit straightforward in-terpretations. Specifically, we show that the proposed GLRTsreduce to the estimation of three improperness measures, whichcan be interpreted as the entropy loss due to the different kindsof quaternion improperness.
Two previous works related to our detection problems are[23] and [39]. Specifically, [39] studied the problem of testingwhether “a covariance matrix with complex structure has quater-nion structure,” which can be shown to be equivalent to one ofthe three GLRTs derived in this paper. However, although theformulation in [39] can be useful for obtaining the moments ofthe test statistic, our derivation based on the quaternion com-plementary covariance matrices makes the derivation and inter-pretation of the GLRTs easier, as well as their generalization toarbitrary orthogonal bases. Finally, in [23] the authors consid-ered the detection of -proper quaternion random variables inwhite -proper noise.
Additionally, we consider the case in which the principal-properness direction is unknown, and therefore it has
to be estimated as well. In this case, the estimation problemreduces to the minimization of the -improperness measureor, equivalently, the maximization of the -improperness. Al-though the maximum-likelihood (ML) estimation of resultsin a nonconvex optimization problem in the general vectorcase, we propose an algorithm based on successive convexapproximations [40]–[42] of the nonconvex cost function,which guarantees the convergence to a solution satisfying theKarush–Kuhn–Tucker (KKT) conditions [43]. Finally, after abrief discussion on the general classification problem and thedistribution of the test statistics, several simulation examplesillustrate the accuracy and fast convergence of the proposedalgorithm, as well as the performance and practical applicationof the three derived GLRTs.
A. Notation
In this paper, we use boldfaced uppercase letters to denotematrices, boldfaced lowercase letters for column vectors, andlightfaced lowercase letters for scalar quantities. Superscripts
and denote quaternion (or complex) conjugate,transpose, and Hermitian (i.e., transpose and quaternion conju-gate), respectively. The notation denotes that isan matrix with entries in , where can be , the fieldof real numbers, , the field of complex numbers, or , theskew-field of quaternion numbers. , and de-note the real part, trace, and determinant of matrix . (re-spectively, ) is the Hermitian square root of the Hermitianmatrix (resp. ). The diagonal matrix with vector alongits diagonal is denoted by , and is the column-wise vectorized version of matrix . is the identity matrixof dimension , and is the zero matrix. Finally,
the Kronecker product is denoted by is the expectationoperator, and in general is the cross-correlation matrix forvectors and , i.e., .
II. PRELIMINARIES
A. Quaternion Algebra
Quaternions are four-dimensional hypercomplex numbers in-vented by Hamilton [44]. A quaternion is defined as
(1)
where are four real numbers, and the imaginaryunits ( ) satisfy
(2)
which also implies
(3)
Quaternions form a skew field [45], which means that theysatisfy the axioms of a field except for the commutative law ofthe product, i.e., for in general, although wemust note that . The conjugate of a quaternionis defined as , and the conjugate of theproduct satisfies . The inner product between twoquaternions is defined1 as , and two quaternions areorthogonal if and only if (iff) their scalar product (the real partof the inner product) is zero. The absolute value of a quaternion
is defined as , and it is
easy to check that . The inverse of a quaternionis , and we say that is a pure unit
quaternion iff (i.e., iff and its real part is zero).A particularly important operation is the quaternion involution.
Definition 1 (Quaternion Involution): The involution of aquaternion over a pure unit quaternion is
(4)
and it represents a rotation of angle in the imaginary planeorthogonal to [45].
Some basic properties of the quaternion involution, which canbe easily checked, are [14], [24]
(5)
(6)
(7)
(8)
Here, we must point out that the real representation in (1) canbe easily generalized to other orthogonal bases. Specifically, wewill consider an orthogonal system given by
(9)
1Other definitions of the quaternion inner product are possible; see for in-stance [45].
1358 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
where is a rotation matrix, (i.e., and), which implies
(10)
Thus, any quaternion can be represented as
(11)
where .Finally, a quaternion can also be represented by means of
the Cayley–Dickson construction
(12)
where
can be seen as complex numbers in the planes spanned by, or .
B. Second-Order Statistics of Quaternion Random Vectors
The statistical analysis of a quaternion random vectorcan be based on its real representation
, which will allow us to obtain the mo-ments of the GLRT statistics [39]. However, working with theaugmented quaternion vectorwill make easier the definition of quaternion properness, thederivation of the tests, and the generalization of the results to anarbitrary orthogonal basis . Thus, the second-orderstatistical information of the quaternion vector is given by theaugmented covariance matrix
(13)
which contains the covariance matrix and three
complementary covariance matrices
, and .Finally, we must point out that, given two different orthogonal
bases and related by means of a rotationmatrix as
(14)
we can easily relate the augmented quaternion vectors and co-variance matrices as stated in the following lemmas.
Lemma 1: Given a quaternion random vector andtwo different orthogonal bases and , thecorresponding augmented quaternion vectors are related as
(15)
where is a unitary quaternion operator given by
(16)
, and .Proof: Let us consider the pure unit quaternion
, where is the first row of . Thus,the involution of over is
(17)
Repeating this procedure for and , we obtain the mappingbetween the augmented quaternion vectors in the two differentbases.
Lemma 2: The augmented covariance matrices in two dif-ferent orthogonal bases are related as
(18)
where the expressions in parentheses make the bases explicit.Proof: This is a direct consequence of Lemma 1 and the
definition of the augmented covariance matrix.
C. Properness of Quaternion Vectors
In the complex case, a vector is proper iff thecomplementary covariance matrix is zero [26],[29], [30], [46]. The quaternion case is a bit more complicated,and we can define different kinds of properness [21], [22], [25],which also have different implications on the optimal linear pro-cessing of a quaternion random vector [25]. In this paper, wefocus on the two main kinds of quaternion properness.
Definition 2 ( -Properness): A quaternion random vectoris -proper iff the three complementary covariance matrices
, and vanish.Definition 3 ( -Properness): A quaternion random
vector is -proper iff the complementary covariance ma-trices and vanish.
These properness definitions satisfy some interesting proper-ties, which include the invariance to linear quaternion transfor-mations and the invariance of the second-order statistics (SOS)to different types of right-Clifford translations [25], [47]—i.e.,right products , with a unit quaternion. Here, we sum-marize the two main properties of the - and -propernessdefinitions [25].
Property 1 ( -Proper Vectors): A quaternion randomvector is -proper iff it is -proper for all pure unitquaternions .
Property 2 ( -Proper Vectors): A quaternion randomvector is -proper iff the vectors in itsCayley–Dickson representation are jointlycomplex-proper, i.e., iff the composite vector isproper.
From a practical point of view, the two previous definitionshave a strong impact on the structure of the optimal linear
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processing. In general, the optimal linear processing of aquaternion random vector takes the form
(19)
with and . That is, wehave to simultaneously operate on the quaternion vector and itsinvolutions, which is referred to as full-widely linear processing.However, in the case of - and -proper vectors, the optimallinear processing simplifies as follows [25].
Property 3 (Linear Processing of -Proper Vectors): The op-timal linear processing of a -proper vector takes the form
(20)
and it is referred to as conventional linear processing.Property 4 (Linear Processing of -Proper Vectors): The
optimal linear processing of a -proper vector takes the form
(21)
and it is referred to as semi-widely linear processing.Finally, in [25] the authors have introduced a third kind of
quaternion properness based on the cancellation of only onecomplementary covariance matrix.
Definition 4 ( -Properness): A quaternion random vectoris -proper iff the complementary covariance matrixvanishes.
Unfortunately, this third kind of quaternion properness doesnot result in a simplification of the optimal linear processing.However, the -properness definition clearly relates the twoprevious kinds of quaternion properness. That is, we can say thatthe - and -properness are complementary and, together,result in -properness. This relationship will become useful inthe derivation of the GLRTs in Section III.
III. GENERALIZED LIKELIHOOD RATIO TESTS
As we have seen, the - and -properness have a strongimpact on the structure of the optimal linear processing ofquaternion random vectors. Therefore, it is crucial to deter-mine whether our quaternion data are -proper,
-proper, or improper. Clearly, in its general formulation, thisis a multiple-hypotheses testing problem. However, here wefocus on the three binary hypotheses testing problems obtainedby considering two out of the three different hypotheses, whichis justified by the two following facts.
• The binary hypotheses testing problems can arise in prac-tical situations when the problem structure yields somea priori information about the properness of the data.For instance, if we consider the problem of detecting thepresence of a zero-mean improper Gaussian signal (withunknown augmented covariance matrix) in zero-mean
-proper Gaussian noise (with unknown covariancematrix), the optimal detector amounts to determiningwhether the observations are -proper or not. That is, inthis situation, the problem structure allows us to discardthe hypothesis of -proper observations (which is stillimplicit in the improper hypothesis).
• The binary hypotheses testing problems result in simpledetection rules, which provide a clear insight about thestructure of the overall testing problem. Moreover, we willshow that the three binary tests can be seen as the core ofa practical multiple-hypotheses test based on the approxi-mation of the a posteriori probabilities of each hypothesis.
In this section, we propose three GLRTs for solving the as-sociated binary hypotheses tests. Although suboptimal in theNeyman–Pearson sense, the GLRT provides satisfactory resultsin practical situations [32], [33], [48]. Furthermore, the deriva-tion of the GLRT is usually simpler than other alternative detec-tors and, in our particular problem, it permits a straightforwardand intuitive interpretation of the detection rules.
A. ML Estimates of the Augmented Covariance Matrix
Let us start by writing the probability density function (pdf)of a quaternion Gaussian vector with zero mean and nonsingularaugmented covariance matrix as [25]
(22)
Thus, given independent realizationsof a quaternion Gaussian vector , we can take the logarithmof the pdf to obtain the log-likelihood function, which (up to ascaling factor and constant terms) is given by
(23)
where
(24)
can be seen as the sample covariance matrix estimatorof . Here, we must note that in the transition from(22) to (23) we have used the relation
. Alternatively, we could
write .Finally, assuming for notational simplicity that is non-
singular2 (which obviously requires ), we are ready toobtain the ML estimates of the augmented covariance matrix
under the three different hypotheses.1) -Proper Vectors (Hypothesis ): In the case of-proper vectors, the ML estimation problem can be written as
(25)
2We must note that, replacing matrix inverses by Moore–Penrose pseudoin-verses, the derived GLRTs can be directly applied in the case of rank-deficientsample covariance matrices �� . See [35] and [49] for the complex counter-part.
1360 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
where denotes the convex set of -proper augmented co-variance matrices, i.e.,
(26)Lemma 3: Under the hypothesis , the ML estimate of the
augmented covariance matrix is given by
(27)
which results in a log-likelihood function
(28)
Proof: Let us start by noting that, under canbe rewritten as
(29)
where is the Kullback–Leibler diver-gence [25], [50] between two zero-mean quaternion Gaussiandistributions with augmented covariance matrices and
. Therefore, the ML estimation problem reduces to theminimization of the positive term , which van-ishes for .
2) -Proper Vectors (Hypothesis ): In this case, theML estimation problem is
(30)
with the convex set
(31)
Lemma 4: Under the hypothesis , the ML estimate of theaugmented covariance matrix is
(32)
which yields the log-likelihood
(33)
Proof: The proof is identical to that of Lemma 3.3) Possibly Improper Vectors (Hypothesis ): Finally, in
the case of (possibly) improper vectors, we do not need to im-pose any particular structure on the augmented covariance ma-trix, and the ML estimation problem is
(34)
Therefore, the ML estimate of the augmented covariance ma-trix is directly given by the sample covariance matrix estimator,which results in a log-likelihood function
(35)
B. -Properness GLRT
After obtaining the ML estimates of the augmented covari-ance matrix under the three different hypotheses, the derivationof the GLRTs is straightforward. Let us start by considering thefollowing binary hypothesis test:
Null Hypothesis
Alternative Hypothesis
That is, we want to decide whether is -proper or not. Thus,taking the logarithm of the generalized likelihood ratio, we ob-tain the GLRT statistic
(36)
where is defined as the -coher-ence matrix, which is closely related to the multiset extensionsof canonical correlation analysis (CCA) [51]–[53]. Specifically,
appears in the CCA of the quaternion random vectors, and .
Interestingly, the test statistic can be seen as an estimateof the -improperness measure proposed in [25], which isbased on the Kullback–Leibler divergence [50] between twozero-mean quaternion Gaussian distributions and providesthe entropy loss due to the -improperness of the quaternionrandom vector —that is, due to the additional correlation(not contained in ) among the real components of thequaternion vector. Moreover, we must note that satisfiesthe following important properties.
Property 5: is invariant to invertible lineartransformations.
Proof: Consider the linearly transformed data, with an invertible matrix.
Then, it is easy to see that the associated -properness GLRTstatistic is
(37)
where is defined as
(38)
That is, the GLRT statistics for the original and trans-formed data are identical.
Property 6: is independent of the orthogonal basis.
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Proof: Let us write the GLRT statistic as
Now, as a direct consequence of Lemma 2, is indepen-dent of the orthogonal basis. Moreover, it is easy to see that
. Therefore, we can conclude thatdoes not depend on the particular choice of the orthogonal basis
.Summarizing, the -properness GLRT reduces to the com-
parison of with some fixed threshold
(39)
C. -Properness GLRT
Here, we consider the problem of determining whether is-proper or not, i.e., our hypotheses testing problem can be
written as
Null Hypothesis
Alternative Hypothesis
Following the lines in Section III-B, we easily obtain theGLRT statistic
(40)
where now is the -coherencematrix, which appears in the CCA of the random vectors
and .Analogously to the previous case, can be seen as an esti-
mate of the -improperness measure proposed in [25], whichprovides the entropy loss due to the -improperness of , andsatisfies the following invariance property.
Property 7: is invariant to invertible semi-widely lineartransformations.
Proof: Let us define the semi-widely linear transformation, with providing an invert-
ible matrix
(41)
Then, the associated -properness GLRT statistic is
(42)
Moreover, can be rewritten in terms of the vectors in theCayley–Dickson representation as [25]
(43)
where is the coherence matrix for the complex randomvector . That is, defining the augmentedvector , the coherence matrix is obtained as
, where
(44)
is the sample covariance estimator of the complex augmentedcovariance matrix, and
(45)
Interestingly, is also the GLRTstatistic for determining whether is (complex) proper ornot, or equivalently, for determining if and are jointlycomplex-proper [32], [33], [48], [54], [55]. That is, as sug-gested by Property 2, the -properness test reduces to theevaluation, and comparison to a previously fixed threshold, ofthe complex-improperness measure of
(46)
D. -Properness Versus -Properness GLRT
Finally, let us consider the case in which we already knowthat the quaternion random vector is -proper (see the sim-ulations for a practical example). Then, we should determinewhether it is also -proper, and our testing problem is
Null Hypothesis
Alternative Hypothesis
Following the lines in Sections III-A and III-B, we obtain theGLRT statistic
(47)where is defined as the -coher-ence matrix. Analogously to the previous cases, can be seenas an estimate of the -improperness degree in [25], whichis a measure of the entropy loss due to the -improperness ofthe -proper vector , and satisfies the following invarianceproperty.
Property 8: is invariant to invertible lineartransformations.
Proof: This is a direct consequence of the decompositionand the invariances of and .
Therefore, as we have pointed out before, the -proper-ness naturally appears as the difference between the two mainkinds of quaternion properness, and the - versus -proper-ness GLRT reduces to
(48)
where is a previously fixed threshold.
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IV. GLRTS FOR UNKNOWN BASES
In Section III, we have derived the GLRTs for testing theproperness of a quaternion random vector under the a prioriknowledge of the orthogonal basis . That is, we haveassumed that the pure unit quaternion , for the analysis ofthe -properness, is fixed. However, in practice this parametercould be unknown, and we should consider the -propernesstests for all the possible values of .
In this section, we generalize the previous results by includingthe estimation of in the GLRTs. As we will see, the ML esti-mation problem amounts to finding the principal -propernessdirection , that is, the pure unit quaternion minimizing the
-improperness measure . Interestingly, this problem canbe seen as that of finding a decomposition of the quaternionrandom vector into two jointly proper complex vectors. Fi-nally, we should also note that after obtaining the ML estimateof , we will be ready to apply the optimal semi-widely linearprocessing .
A. Problem Statement
Following the derivation in Section III, we should start by ob-taining the joint ML estimates of and under the three dif-ferent hypotheses. However, under and , the maximumof the log-likelihood function does not depend on , which canbe seen as a direct consequence of Lemma 2. Therefore, ourproblem reduces to the joint ML estimation of and underthe hypothesis of -proper vectors,3 i.e.,
(49)
where denotes the set of pure unit quaternions. Now, it is clearthat the previous problem can be rewritten as
(50)
or, as a direct consequence of Lemma 4
(51)
Thus, noting that , and sinceis independent of , our ML estimation problem can
be written as
(52)
or equivalently
(53)
That is, as we could expect, we are looking for the pureunit quaternion minimizing (equivalently maximizing) theestimated (equiv. ) improperness measure. Finally,taking into account the ML estimates of and , the overall
-properness test is
(54)
3We say that a vector is -proper iff it is -proper for some �.
and equivalently, the problem of testing -properness versus-properness results in
(55)
B. Formulation of the Optimization Problem
Let us start by rewriting the -coherence matrix as
(56)
where can be seen asan estimate of the complementary covariance matrix of theprewhitened vector . This reduces the -im-properness measure to ,which results in the following ML estimation problem:
(57)
Finally, defining the matrices for all pure unitquaternions , the objective function in (57) can be rewritten as
, and applying Lemma 2 we have
(58)
where are the coordinates of the pure unitquaternion in the arbitrary orthogonal basis , i.e.,
. Thus, (57) can be rewritten as
subject to
(59)
where , and the last constraint, which forces(equivalently ), has been relaxed to an in-
equality because the cost function is monotonically decreasingwith .
C. Proposed Algorithm: Successive Convex Approximations
Unfortunately, the above optimization problem is not convexdue to the cost function in (59), which pre-cludes its solution by means of standard convex optimizationtools [43]. Here, in order to find reliable approximated solu-tions, we propose to apply the successive convex approxima-tions method [40]–[42]. This technique relies on solving a se-ries of convex problems, in which the nonconvex cost function
is replaced by a convex approxima-tion . The following lemmas provide sufficient conditionsfor the convergence of the successive convex approximationsmethod, as well as a particular approximation satisfying the con-vergence conditions.
VÍA et al.: GLRTs FOR TESTING PROPERNESS OF QUATERNION GAUSSIAN VECTORS 1363
Lemma 5: Consider the optimization problem
(60)
where is a convex set and is a nonconvex function. If theconvex approximations of the cost function satisfy:
• for all ;• , where is the optimal solution of the
approximated problem in the previous iteration;• , where is the gradient operator;
then the successive convex approximations method, based onthe solutions of the convex problems
(61)
guarantees the convergence of , and after the convergencesatisfies the KKT conditions of the original problem.
Proof: See [40] and [41] for the proof and some minortechnical details.
Lemma 6: Consider the cost functionand the matrix , where denotes the
value of in the previous iteration. Then, the approximation
(62)
satisfies the convergence conditions in Lemma 5.Proof: Defining , the cost function is
, and is its first-order Taylor’s series approxi-mation (with respect to ) around . Now, it is easy to checkthat the approximation satisfies the second and third conver-gence conditions in Lemma 5. Finally, since the cost functionis concave in , the approximation also satisfies the firstconvergence condition.
Using the proposed approximation, the convex problem to besolved in each iteration is
subject to
(63)
where
(64)
(65)
(66)
Thus, defining the matrix
(67)
the previous problem can be rewritten as
subject to (68)
Algorithm 1: Principal -Properness Direction
Input: Estimates in some basis .Output: Principal -Properness Direction .Initialize: at some arbitrary value.repeat
Obtain .Compute .Obtain , and from (64)–(66).Obtain .Extract as the principal eigenvector of .
until Convergence.
with . Finally, the solution is given by theprincipal eigenvector of the matrix , and the overall algorithmfor the estimation of the pure unit quaternion is summarizedin Algorithm 1.
V. FURTHER DISCUSSION
In this section, we provide some additional details about thedistribution of the test statistics, the particularization of the ob-tained results to the scalar case, and the general multiple-hy-potheses testing problem.
A. Distribution of the GLRT Statistics
As we have seen, the three proposed GLRTs reduce to thecomparison of the estimated improperness measure with athreshold. Typically, the selection of the threshold is basedon some performance criterion, such as a constant false alarmprobability, which in this paper is defined as the probabilityof accepting the alternative (improper) hypothesis when thenull (proper) hypothesis is true. Therefore, the selection ofthe threshold requires the knowledge of the GLRT statisticdistribution under the null hypothesis.
Although the theoretical derivation of the distributions is ingeneral a very difficult problem, the properties of the test statis-tics allow us to draw some interesting conclusions. First, the in-variance of the statistics (resp. ) under linear (resp.semi-widely linear) transformations can be easily exploited tonumerically determine the thresholds for a fixed false alarmprobability in the case of known . That is, following the linesin [33] for the case of complex vectors, the distributions of thestatistics under the null hypothesis can be obtained by simula-tion using . Thus, as illustrated in Section VI-B,we only need to tabulate the simulation results for the differentvalues of (vector dimensionality) and (number of availableobservations). Second, we must note that any complementarycovariance matrix satisfies , whichimplies that the diagonal elements are orthogonal to . There-fore, a complementary covariance matrix iscompletely specified by real numbers ( in the di-agonal and above the diagonal). Thus, as a direct con-sequence of the Wilks’ theorem [33], [56], we have that underthe null hypothesis
(69)
1364 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
where indicates convergence (for ) in distribution,denotes a chi-square random variable with degrees of freedom,and
(70)
The case of estimated is more complicated. Obviously,is independent of the orthogonal basis, and the previous resultsapply. However, the -properness test statistic is not invariantunder semi-widely linear transformations, which means that itsdistribution under the null hypothesis cannot be obtained fromsimulations using . In this case, if is not -proper,the direct application of the Wilks’ theorem provides an asymp-totic distribution
Finally, the distribution of under the null hypothesis canbe obtained by means of simulations with , but theWilks’ theorem does not apply due to the inconsistency4 of the
estimates [56].The derivation of the small sample distributions is much
more complicated. However, focusing on the case of known ,and taking into account the equivalence between the -proper-ness test and the GLRT for testing the improperness of complexrandom vectors [32], [48], [54], [55], we can directly applythe results for the complex case [33], [39]. Furthermore, it canbe proved that the -properness versus -properness GLRTis equivalent to the problem of “testing the hypothesis thata covariance matrix with complex structure has quaternionstructure,” which was studied by Anderson et al. in [39]. Thus,using the results in [39] for the moments (of order ) of the teststatistics under the null hypothesis, we have
(71)for , and
(72)
for , where is the gamma function. Now, followingthe lines in [33] for the application of the Box’s approximationmethod [57], we conclude that under the null hypothesis
(73)where denotes approximated distribution. Moreover, takinginto account that and are independent under the nullhypothesis [39], it is clear that and are also inde-pendent, and therefore the th moments of are given (for
) by the product of (71) and (72), which can be exploitedto approximate the null distribution of . Here, we must notethat, for , these approximations become equivalent tothe Wilks’ approach in (69).
4Note that under the null ( -proper) hypothesis, we cannot define a true valueof the principal -properness direction �.
Finally, the distributions of the test statistics under the alter-native (improper) hypothesis are not easy to obtain. There aresome available results in the complex case [35], [49], but ingeneral we can only say that, for and true improper-ness measures close to zero, the test statisticsare approximately distributed as [58]–[60]
(74)
(75)
(76)
where now denotes the noncentral chi-square distribu-tion with degrees of freedom and noncentral parameter , and
depend on the actual distribution [58]–[60].
B. Particularization to the Scalar Case
The GLRT statistics provide additional insights in the scalarcase . Specifically, defining thereal vector and the unitary matrix
(77)
we can write , and
(78)
which is the GLRT statistic for the well-known sphericity test ofthe real vector [61]. Furthermore, using the Cayley–Dicksonrepresentation , and defining the vector
, we obtain [25]
(79)
That is, as previously pointed out, the -properness GLRT re-duces to testing the complex properness of , and the -proper-ness GLRT is the sphericity test for the complex vector .
Finally, we must note that in the scalar case, the cost func-tion in (59) reduces to a quadratic function, and the optimiza-tion problem can be solved in closed form. Thus, the principal
-properness direction is obtained after the first iteration of theproposed successive convex approximations algorithm.
C. Multiple-Hypotheses Testing Problem
As we have previously shown, the -properness is all whata -proper vector needs to become -proper, which is con-firmed by the relationship . Interestingly,this fact can also be used to easily relate the three proposedGLRTs. In particular, consider the classification problem withhypotheses , and , assume as true the ML estimatesof the augmented covariance matrices, and assign some a prioriprobabilities satisfying
(80)
VÍA et al.: GLRTs FOR TESTING PROPERNESS OF QUATERNION GAUSSIAN VECTORS 1365
Fig. 1. Space of values for the two improperness measures (GLRT statistics)�� and �� in the case � � � �� , which divides the space into threeregions corresponding to the different properness hypotheses. (a) Scenario witha priori knowledge of the -properness direction �. (b) Scenario with unknown�. The solid lines represent the achievable pairs of values for different valuesof �.
Then, it is easy to prove that the maximum a posteriori5 (MAP)classification rule would be based on the thresholds
(81)
and more importantly
(82)
As an example, Fig. 1(a) shows the space of values of the pair, as well as the regions associated with a particular
choice of the thresholds , and . Addi-
5Note that this is not a true MAP classification technique because we aredirectly plugging the ML estimates of the augmented covariance matrices.
TABLE ISECOND-ORDER STATISTICS FOR THE SIMULATION EXPERIMENTS
Fig. 2. Convergence example. Evolution of the test statistic �� for five inde-pendent experiments, each one with 100 different initialization points. (a) �and� � ��. (b)� and� � ��. (c)� and� � ��. (d)� and� � ��.
tionally, Fig. 1(b) represents the case without previous knowl-edge of the principal -properness direction. As can be seen, theoptimization in moves the points toward the upper left corner,and in particular, point moves from the improperness regionto the -properness region (for some pure unit quaternion ).
VI. SIMULATION RESULTS
In this section, the performance of the proposed GLRTs is il-lustrated by means of some simulation results, which have beenobtained using the MATLAB quaternion Toolbox [62]. Unlessotherwise stated, the experiments are based on i.i.d realiza-tions of a four-dimensional quaternion Gaussian vector (i.e.,
) with zero mean and second-order statistics as illustratedin Table I, where the diagonal matrices and are
Thus, with the specified SOS, the principal -properness direc-tion (under and ) is .
A. Convergence of the Successive Convex ApproximationsMethod
The first set of examples illustrate the convergence of thesuccessive convex approximations method. Specifically, Fig. 2
1366 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
Fig. 3. Mean square error in the estimates of the principal -properness direc-tion � after four iterations of the proposed algorithm.
Fig. 4. Cumulative distribution function of the test statistic �� under the nullhypothesis� � � � � � � for � � �.
shows the evolution of the -improperness measure in fourdifferent scenarios, considering five independent examples foreach scenario. In all the cases, the proposed algorithm has beeninitialized in 100 randomly generated values of and, after afew iterations, we can see that the algorithm converges to thesame solution. Based on these and other similar results, we havelimited the proposed algorithm to four iterations. Finally, Fig. 3shows the mean square error (MSE) in the estimate of the prin-cipal -properness direction, where we can see that the pro-posed algorithm provides reliable estimates both in the case of
-proper and improper random vectors.
B. Cumulative Distribution Function of the GLRT StatisticsUnder the Null-Hypothesis
As we have previously pointed out, the invariances of the teststatistics , and can be exploited for obtaining their
Fig. 5. Cumulative distribution function of the test statistic �� under the nullhypothesis � � � � � for � � �.
Fig. 6. Cumulative distribution function of the test statistic �� under the nullhypothesis � � � � � � � for � � �.
distribution under the null (proper) hypothesis by means of sim-ulations. Figs. 4–6 show the numerically obtained cumulativedistribution functions (CDFs) of the three test statistics, both inthe case of known and unknown -properness direction. Addi-tionally, Tables II–IV show the critical values of the thresholdsfor three different probabilities of false alarm and severalvalues of and . Finally, we must remember that the distribu-tion of for estimated depends on the actual second-orderstatistics. Table III has been obtained using underthe null distribution.
C. Receiver Operating Characteristic Curves
In this set of examples, we show the receiver operating char-acteristic (ROC) curves for the three proposed GLRTs. Specif-ically, Figs. 7–9 show the probability of miss (probability ofaccepting the null hypothesis when the alternative is true) as
VÍA et al.: GLRTs FOR TESTING PROPERNESS OF QUATERNION GAUSSIAN VECTORS 1367
TABLE IICRITICAL VALUES � FOR THE -PROPERNESS TEST
TABLE IIICRITICAL VALUES � FOR THE -PROPERNESS TEST
a function of the false alarm probability. Interestingly, the fig-ures show that the third GLRT ( -properness versus -proper-ness) is more affected by errors in the estimate of the principal
-properness direction. As previously pointed out, this is dueto the inconsistency of the estimates under the -proper hy-pothesis.
D. Practical Example
In the final example, we show a practical application of thederived GLRTs. In particular, we consider an optical communi-cation system with dual polarization that, considering a singlefrequency, can be modeled as [36]–[38]
(83)
where (resp. ) is a complexvector in the plane representing the transmitted (resp.received) signals in the two orthogonal principal states of
TABLE IVCRITICAL VALUES � FOR THE - VERSUS -PROPERNESS TEST
Fig. 9. Receiver operating characteristic. -properness versus -propernessGLRT.
20 dB and random matrices and . Furthermore, we considera possible IQ imbalance in the transmitted signals , wherethe IQ imbalance factor is defined as ,with and representing the power in the in-phase andquadrature components of the transmitted signals. Thus,defining the quaternions and ,our goal consists in applying the linear, semi-widely linear, orfull-widely linear quaternion LMS algorithm [9]–[11] for re-covering from . In particular, the quaternion LMS updatingrule can be written as
(86)
where is the estimate of is the error, isthe system equalizer, and is the input for the LMS algorithm.Here, we consider four different scenarios.
• Balanced system : In this case, the sourcequaternion is -proper, and the channel preserves the
Fig. 10. Convergence of the quaternion LMS algorithm in the four differentscenarios. The curves show the conventional linear LMS (Linear), the semi-widely linear LMSs (SWL � and SWL � ), the full-widely linear LMS (FWL),and the proposed technique (Classifier). (a) Balanced system. (b) System withIQ imbalance. (c) System with PDL. (d) System with PDL and IQ imbalance.
quaternion structure. Therefore, is also -proper, andthe most appropriate processing is the conventional linearmodel. That is, we should select for the quaternionLMS.
• System with IQ imbalance : In thiscase, the source and observation are -proper. Theoptimal linear processing is semi-widely linear in , whichmeans that we should select .
• System with PDL : Although thesource quaternion is -proper, the channel introducesa power imbalance, and the received quaternion is
-proper. Therefore, the most appropriate processing issemi-widely linear in .
• System with PDL and IQ imbalance :The source is -proper, but is improper. Furthermore,the principal -properness direction and the -improper-ness measure depend on the particular value of and . Ingeneral, the optimal linear processing is full-widely linear
.The proposed GLRTs, including the principal -properness
direction algorithm, are applied to a set of observa-tions for solving the multiple-hypotheses testing problem andselecting the most convenient kind of processing. In case of se-lecting the semi-widely linear processing, we use the estimatedprincipal -properness direction , i.e., we useas input for the LMS algorithm. In order to achieve the samesteady-state error in the -proper case, the learning rate hasbeen selected as in the case of conventional linearprocessing, for semi-widely linear processing, and
for full-widely linear processing. The results, aver-aged for 1000 independent simulations, are shown in Fig. 10.As can be seen, the performance of the proposed technique isalways very close to that of the most convenient kind of pro-cessing, and the small deviations in the -proper and improper
VÍA et al.: GLRTs FOR TESTING PROPERNESS OF QUATERNION GAUSSIAN VECTORS 1369
scenarios are due to classification mistakes and residual errorsin the estimate of the principal -properness direction.
VII. CONCLUSION
In this paper, we have presented three generalized likelihoodratio tests (GLRTs) for testing the properness of a quaternionrandom vector. This is an important problem because the typeof quaternion properness will determine the required kindof linear processing (full-widely linear, semi-widely linear,or conventional linear processing). The proposed tests havebeen derived under the Gaussian assumption, and they re-duce to the estimation, and comparison to a fixed threshold,of three previously proposed improperness measures. Addi-tionally, we have presented an algorithm for the estimationof the principal -properness direction, or equivalently, todecompose the quaternion vector into two complex vectorswith the lowest improperness degree. The proposed techniqueis based on the successive convex approximations method,which guarantees the convergence to a solution satisfying theKarush–Kuhn–Tucker conditions. Finally, the performanceand practical application of the proposed techniques have beenillustrated by means of several simulation examples.
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Javier Vía (S’04–M’08) received the Telecommuni-cation Engineer Degree and Ph.D. degree in electricalengineering from the University of Cantabria, San-tander, Spain, in 2002 and 2007, respectively.
In 2002, he joined the Department of Communi-cations Engineering, University of Cantabria, wherehe is currently an Assistant Professor. He has spentvisiting periods with the Smart Antennas ResearchGroup, Stanford University, Stanford, CA, andwith the Department of Electronics and ComputerEngineering, Hong Kong University of Science and
Technology, Hong Kong. He has actively participated in several European andSpanish research projects. His current research interests include blind channelestimation and equalization in wireless communication systems, multivariatestatistical analysis, quaternion signal processing, and kernel methods.
Daniel P. Palomar (S’99–M’03–SM’08) receivedthe Electrical Engineering and Ph.D. degrees (bothwith honors) from the Technical University of Cat-alonia (UPC), Barcelona, Spain, in 1998 and 2003,respectively.
He is an Associate Professor with the Depart-ment of Electronic and Computer Engineering,Hong Kong University of Science and Technology(HKUST), Hong Kong, which he joined in 2006.He had previously held several research appoint-ments, namely at King’s College London (KCL),
London, U.K.; UPC; Stanford University, Stanford, CA; the Telecommuni-cations Technological Center of Catalonia (CTTC), Barcelona, Spain; theRoyal Institute of Technology (KTH), Stockholm, Sweden; the Universityof Rome “La Sapienza,” Rome, Italy; and Princeton University, Princeton,NJ. His current research interests include applications of convex optimizationtheory, game theory, and variational inequality theory to signal processing andcommunications.
Dr. Palomar is an Associate Editor of the IEEE TRANSACTIONS ON
INFORMATION THEORY and has been an Associate Editor of the IEEETRANSACTIONS ON SIGNAL PROCESSING, a Guest Editor of the IEEE SignalProcessing Magazine 2010 Special Issue on “Convex Optimization for SignalProcessing,” a Guest Editor of the IEEE JOURNAL ON SELECTED AREAS IN
COMMUNICATIONS 2008 Special Issue on “Game Theory in CommunicationSystems,” and the Lead Guest Editor of the IEEE JOURNAL ON SELECTED
AREAS IN COMMUNICATIONS 2007 Special Issue on “Optimization of MIMOTransceivers for Realistic Communication Networks.” He serves on the IEEESignal Processing Society Technical Committee on Signal Processing forCommunications (SPCOM). He was the General Co-Chair of the 2009 IEEEWorkshop on Computational Advances in Multi-Sensor Adaptive Processing(CAMSAP). He is a recipient of a 2004–2006 Fulbright Research Fellowship;the 2004 Young Author Best Paper Award by the IEEE Signal ProcessingSociety; the 2002–2003 best Ph.D. prize in Information Technologies andCommunications by the UPC; the 2002–2003 Rosina Ribalta first prize forthe Best Doctoral Thesis in Information Technologies and Communicationsby the Epson Foundation, and the 2004 prize for the Best Doctoral Thesis inAdvanced Mobile Communications by the Vodafone Foundation and COIT.
Luis Vielva was born in Santander, Spain, in 1966.He received the Licenciado and Ph.D. degrees inphysics from the University of Cantabria, Santander,Spain, in 1989 and 1997, respectively.
In 1989, he joined the Department of Communi-cations Engineering, University of Cantabria, wherehe is currently an Associate Professor. His current re-search interests include systems biology, convex op-timization, and geometric algebra.
