quadratic phase coupling detection in harmonic vibrations ...hera.ugr.es/doi/15000096.pdf ·...

Nonlinear Dynamics19: 273–294, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

Quadratic Phase Coupling Detection in Harmonic Vibrations viaan Order-Recursive AR Bispectrum Estimation

ANTOLINO GALLEGODepartamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

CRISTINA URDIALESDepartamento de Tecnología Electrónica, E.T.S.I. de Telecomunicación, Universidad de Málaga, Campus deTeatinos, 29071 Málaga, Spain

DIEGO P. RUIZDepartamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

(Received: 12 March 1998; accepted: 2 December 1998)

Abstract. The paper proposes a bispectral scheme for quadratic phase-coupling (QPC) detection in harmonicsignals in white and colored noise, possibly generated by non-linear vibrations. This scheme is carried out via anautoregressive bispectrum (AR) using several criteria to fix an optimum order for the AR model used. First, wepropose a recursive-in-order algorithm for AR model-parameter calculation in the bispectrum estimation problem.This algorithm is based on the recursion-in-order minimization of appropriate squared errors with respect tothe reflection coefficients, introduced by the Levinson recursion for Toeplitz and non-Hermitian matrices. Therecursive nature in this method allows us to obtain the bispectrum of several orders up to the desired one, withsignificant computational savings. In computer simulations this method demonstrates its potential both in theestimation of signal bispectra and in QPC bispectral detection problems in noisy environments.

Keywords: Nonlinear signal processing, bispectrum, autoregressive-modelling, colored noise.

1. Introduction

As is known, the final objective in any signal treatment problem consists of processing afinite data set and extracting the greatest amount of information hidden therein. The short-cuts implicit in classic signal processing techniques based on the power spectrum have ledthe scientific community to develop alternative or complementary techniques, named higher-order statistics-based signal analysis, or HOS [1–3]. Generally speaking, HOS or polyspectralanalyses are used in signal processing frameworks for the following reasons:1. Additive Gaussian noise suppresion (white or colored).2. Information about the signal phase.3. Detection and characterization of non-linearities.4. Detection of Gaussian deviations.

Due to its properties, HOS-based analysis has been applied to many diverse scientific areas[4]. The study of mechanical problems in general, and vibrations and oscillations in particular,must make use of this signal processing tool, and for some years much research has provedthe huge potential of bispectral or trispectral analysis for this kind of physical problem [5–8].More precisely, HOS-based treatment has contributed heavily to the detection and character-ization of non-linearities in mechanical systems [9–11]. The dynamics of machining systems

274 A. Gallego et al.

can be represented by linear and non-linear models [9, 12–15]. The information providedby sequences of experiments on mechanical systems can tell us more about the mechanismof generation of the measured signals and the non-linear interactions in the system. This in-formation has proved to be useful for diagnosis [6] (of machine faults, surface roughness, andsystem identification) and vibration analysis [7] (signal pattern recognition, measurement, andknock detection). The presence of non-linearities also plays an important role in the analysisof EEG data [16], oceanography [11], manufacturing [17], laser velocimetry [18], plasmaphysics and geophysics [4, 19].

The present work falls within this general framework, introducing a scheme for bispectraltreatment (i.e., based on third-order statistics) of second-degree non-linear signals. More pre-cisely, the paper is concerned with the use of the bispectrum for the detection of phase-coupledharmonic components occurring, for example, when a second-degree non-linear mechanicalsystem is excited by harmonic input. In this situation the interaction between harmonic com-ponents causes contribution to the power at their sum and/or difference frequencies. Consider,for example, the non-linear processyn = xn + εx2n with xn = Aejw1n+φ1 + B ejw2n+φ2,whereε > 0, |ε| � 1 andφi are independent, uniformly distributed random variables. Then,processyn will contain harmonics with frequencies 2w1, 2w2, w1 + w2 with the correspond-ing phases 2φ1, 2φ2 and φ1 + φ2, apart from those presented inxn. Therefore, the phaserelations among harmonic components are produced due to non-linear interactions betweenthe harmonic components of the input signal.

Basically, three frequencies constitute a quadratic phase coupling (QPC) when one of them(and its phase) is the sum/difference of the other two [1, 2, 20, 22], as in the case shown above.In this type of signals it is necessary, in order to detect and characterize the presence of non-linearities in the physical system, to determine whether peaks at harmonically related positionsin the power spectrum (PS) are phase-coupled or not. Since the PS does not provide anyinformation about phase relations among harmonic components, it cannot provide a solutionto this problem. Thus, HOS-based analysis is required to detect the presence of QPC and dis-criminate among the coupled and uncoupled components. In particular, if the phase couplingis of second-order, the problem is solved in the bispectral domain [20]. This type of analysisalso presents the important advantage of minimizing the negative effects caused by additiveGaussian noise (white or colored) contaminating the signal carrying the information about thephysical system. This occurs since the bispectrum of a Gaussian process is identically zero[1].

One way to solve the problem therefore consists in performing a bispectral estimationfrom the available noisy data and detecting the existence of QPC in them. The techniquesfor estimation can be divided into two categories: conventional or ‘Fourier-type’ methods andparametric methods [1, 2]. The latter methods can also be grouped into autoregressive (AR),moving-average (MA), and ARMA modeling techniques [1, 2]. In this paper we concentrateon AR-parametric techniques, characterized by the so-called ‘third-order recursion’ (TOR)equations [19], which establish a relationship between the AR parameters and the third mo-ments of the process, and show good efficiency for the detection of QPCs. Moreover, whenonly a finite sample of data is available, estimation algorithms, such as TOR [20], CTOM[21], PREW and POSW [23], must be introduced. In particular, we present a new bispec-trum AR-estimation method for QPC detection which, while maintaining the improvementsassociated with the reduction of windowing effects, is a recursive-in-order algorithm in thesense that it yields higher-order AR-model parameter estimates starting with the lower-orderAR estimates. This situation is very advantageous when the optimum AR-model order is

Quadratic Phase Coupling Detection275

unknown. Therefore, this new method, termed ‘reflection-coefficients method’ (RCM), unlikeexisting methods, does not directly estimate the parameters of a fixed-order AR model, butthe coefficients associated with models of orders ranging from 1 to a preselected maximum.

A problem often associated with the application of AR-based algorithms for QPC detectionis model-order selection, indispensable for a correct solution of the problem. Although severalresearchers have demonstrated the potential of these algorithms for QPC detection [2, 20, 21],none deal with the model-order selection problem. In this work we have therefore introducedcertain criteria to choose the optimum order. These criteria are fundamentally based on usingthe proposed RCM method to obtain the bispectra in a specified order range, and then eval-uating the bispectral energies in the principal region of the bispectral frequency plane or innarrow regions around it termed bispectral masks. The introduction of these criteria is justifiedby the fact that a QPC in the original signal is equivalent to an impulse or peak in the bispectraldomain located exactly in the generating frequency pair.

The paper is organized as follows: in Section 2 we establish the AR bispectral modeling,relating this problem with the theory of linear prediction and thereby obtaining TOR equations[24, 25]. In Section 3, based on the Toeplitz structure of the AR-modeling equations, it isshown how these equations can be solved quickly using the Levinson algorithm [26, 27],thus introducing the reflection coefficient concept, which leads to the proposal in Section 4of the above-mentioned RCM method. In Section 5, the QPC concept is formally defined andits possible detection using the signal bispectrum is analyzed. In Section 6, we show howthe RCM method can be applied to this problem and introduce four model-order selectioncriteria. Finally, in Section 7 we present some simulation results representative of the problemstudied. These results are divided into two groups. In the first group we compare the potentialof the RCM method with respect to other existing methods for bispectrum estimation. In thesecond group we show its application to the QPC detection problem by means of an examplecontaining two couplings. In both cases the results in the presence of colored noise added tothe original signal are presented.

2. AR Bispectrum for Causal and Anticausal Processes

Given a zero-mean real discrete processxn that is third-order stationary, its bispectrum isdefined as [1]

Bx(ω1, ω2) =∞∑

m=−∞

∞∑n=−∞

g(m, n)e−j (ω1m+ω2n), |ω1|, |ω2| ≤ π. (1)

whereg(m, n) is the third-order moments sequence of the process, given by [1]

g(m, n) = E{xkxk+mxk+n}, (2)whereE{·} denotes expectations. The bispectrum so defined is the most commonly usedone and is also called the Third-Order Moment Spectrum. Keeping in mind the definitionof Bx(ω1, ω2) given in Equation (1) and the properties of the momentsg(m, n), it can beshown that the bispectrum presents the following features, among others: (1) it is complexand doubly periodic with period 2π ; (2) it has six symmetry regions in theω1–ω2 plane, so ifit is only known in the triangular regionω2 ≥ 0, ω2 ≥ ω1, ω1 + ω2 ≤ π (principal region),its description can be extended to the entire plane; and (3) ifxn is a stationary zero-meanGaussian process, its bispectrum is identically zero.


Let xn now be a causal, real, stationarypth-order AR process with zero mean, describedby the difference equation

xn +p∑i=1

aixn−i = Wn, (3)

whereWn is a non-Gaussian process withE{Wn} = 0,E{WnWn+k} = Qδ(k) (δ is the impulsefunction) andE{WnWn+kWn+m} = βδ(k,m).

Previous works [2, 20, 24] have shown that the bispectrum ofxn can be obtained by

Bx(ω1, ω2) = βH(ω1)H(ω2)H ∗(ω1+ ω2), (4)whereH(ω) is the transfer function of the process

H(ω) = 11+∑pi=1 ai e−jωi , (5)

andai are the AR-model parameters. Sincexn is stationary, these parameters are related withthe third-moment sequence of the process through the TOR equations, which for the causalcase are [2]

g(−k,−l)+p∑i=1

aig(i − k, i − l) = βδ(k, l), k, l ≥ 0, (6)

wherel andk are integer numbers.The equations given in (6) involve third moments over the entirem–n plane. A restricted

version, with moments only over them = n line, can be obtained by choosingl = k, whichyields the matrix equation

g(0,0) g(1,1) . . . g(p, p)g(−1,−1) g(0,0) . . . g(p − 1, p − 1)

......

. . ....

g(−p,−p) g(−p + 1,−p + 1) . . . g(0,0)

1a1

. . .

ap

=β

0. . .

0

, (7)where the moments matrix,R, is non-symmetric and Toeplitz-type.

If xn can said to be an anticausal model, with parametersbi described by

xn +p∑i=1

bixn+i = Vn, (8)

it is also possible to derive the following set of TOR equations:

g(k, l)+p∑i=1

big(k − i, l − i) = βbδ(k, l), k, l ≥ 0, (9)

whereβb = E{V 3n }, Vn being a non-Gaussian zero-mean, white, flat bispectrum process.Sinceg(k, l) is different fromg(−k,−l), the TOR equations for an anticausal model will bedifferent from those obtained for a causal representation, and therefore parametersai andbiwill also be different. In this case, the bispectrum ofxn is

Bx(ω1, ω2) = βbH ∗(ω1)H ∗(ω2)H(ω1+ ω2), (10)


with H(ω) defined by Equation (5) usingbi instead ofai . In addition, by using only thirdmoments over them = n line, the following matrix equation, equivalent to Equation (7), canbe obtained:

g(0,0) g(−1,−1) . . . g(−p,−p)g(1,1) g(0,0) . . . g(1− p,1− p)...

.... . .

...

g(p, p) g(1− p,1− p) . . . g(0,0)

1b1

. . .

bp

=βb

0. . .

0

. (11)The derivation of TOR equations can also be carried out by using a linear prediction theory

[24, 25]. To do so for a causal case, let us consider theqth-order forward-linear-predictionestimation of samplexn defined by [25]

x̂n = −q∑i=1

âixn−i , (12)

where âi (i = 1, . . . , p) are the forward-prediction coefficients. Thus, the forward-linear-prediction error can be expressed as

efn = xn − x̂n = xn +q∑i=1

âixn−i . (13)

Let us now consider the following forward-prediction squared error defined, for each value ofthe integerl, as

Qlf = E{[efn ]2xn−l}. (14)By minimizing this error with respect to the prediction coefficients, i.e. by imposing that

∂Qlf

∂âk= 0, k = 1, . . . , q, (15)

calculating its minimum value,(Qlf )min, using the third-moments definition and keeping inmind thatxn is stationary, the following equations can be obtained [24]:

g(−k,−l)+q∑i=1

âig(i − k, i − l) = (Q0f )minδ(k, l), k, l ≥ 0, (16)

which are structurally identical to the TOR equations for a causal autoregressive process asgiven by Equation (6). The restricted version that uses only third moments over them = n lineof them–n plane can be obtained only by choosing Equation (16) withl = k, which yields

g(−k,−k)+q∑i=1

âig(i − k, i − k) = (Q0f )minδ(k), k = 0, . . . , q, (17)

which, when expressed in matrix form, exhibits a non-symmetric Toeplitz structure analogousto expression (7). Thus, ifxn is strictly an AR process of orderp and the prediction order isq = p, it must be true that the prediction coefficients are similar to the model parameters usedin the bispectrum calculation, i.e.,âi = ai for i = 1, . . . , q, and also(Q0f )min = β.


The connection between the anticausal TOR equations and the backward linear-predictiontheory can be obtained in an analogous manner. In this case, the estimation ofxn and thebackward linear-prediction error are [25]

x̂n = −q∑i=1

b̂ixn+i , (18)

ebn = xn−q − x̂n−q = xn−q +q∑i=1

b̂ixn+i−q . (19)

In the above equations,̂bi are the backward-prediction coefficients. In addition, the set ofbackward-prediction squared errors to be minimized with respect tob̂i is

Qlb = E{[ebn]2xn+l−q }. (20)In this case, the minimization gives the equations

g(k, l)+q∑i=1

b̂ig(k − i, l − i) = (Q0b)minδ(k, l), k, l ≥ 0, (21)

which are similar to the anticausal-TOR expressions given in Equation (9). Thus, ifq = pandxn is strictly an anticausal AR(p) process, it will hold thatb̂i = bi , (Q0b)min = βb. As forthe causal case, the version of these equations with only third moments over them = n linecan be obtained by choosing expression (21) withl = k,

g(k, k)+q∑i=1

b̂ig(k − i, k − i) = (Q0b)minδ(k), k = 0, . . . , q. (22)

3. Solving the TOR Equations with Levinson’s Algorithm

The TOR equations for causal and anticausal AR models obtained by means of the linearprediction, Equations (17) and (22), have a Toeplitz structure, which means that Levinson’srecursive-in-order algorithm [26, 27] can be used for a quick resolution. In addition, themoments matrixR defined in Equation (7) is persymmetric, and consequentlyR−1 is alsopersymmetric [27]. By using this property ofR−1, it can be demonstrated that(Q0f )min and(Q0b)min, appearing in expressions (17) and (22), are equal. We will therefore randomly denotethem by(Qj)min, wherej stands for the prediction order.

Levinson’s algorithm means that thej -order prediction coefficientŝajk and b̂j

k (k =1, . . . , j ) can be calculated by knowing the(j−1)-order prediction coefficients,âj−1k andb̂j−1k(k = 1, . . . , j − 1), and coefficientŝajj andb̂jj . Mathematically speaking, their application toEquations (17) and (22) leads us to the following recursive-in-order expressions for thej -ordercoefficients [27]:

âj

k ={ −1j/(Qj−1)min, if k = j,âj−1k + âjj b̂j−1j−k, if k = 1, . . . , (j − 1),

and

b̂j

k ={ −∇j /(Qj−1)min, if k = j,b̂j−1k + b̂jj âj−1j−k , if k = 1, . . . , (j − 1),

(23)


where

1j =j∑

m=1g(m,m)â

j−1j−m; âj−10 = 1,

∇j =j∑

m=1g(−m,−m)b̂j−1j−m; b̂j−10 = 1, (24)

and the recursion for(Qj)min is given by

(Qj)min = (Qj−1)min(1− âjj b̂jj ). (25)Therefore, by using Equations (23), (24), and (25), thej -order coefficients depend only

on the(j − 1)-order coefficients and on the third-moment sequence. The recursion is totallyclosed with the initialization, established forj = 1 as

â11 = −g(1,1)

g(0,0); b̂11 = −

g(−1,−1)g(0,0)

; (Q1)min = (1− â11b̂11). (26)

This method provides a fast way to solve not only Equations (17) and (22) but also, withoutadditional computational cost, all the prediction coefficients with orders from 1 toq. Thecoefficients withk = j are usually referred to as ‘reflection coefficients’. We will refer tothem as0ja (forward-reflection coefficients) and0

j

b (backward-reflection coefficients).

4. Bispectrum Estimation Using an Alternative Minimization

As commented in the introduction, different estimation methods have been proposed for theapproximate solution of TOR equations in the case where only a finite number of samples isavailable for the process. In a manner analogous to that used for relating the TOR equationswith the linear prediction theory, the TOR, CTOM, PREW, and POSW methods can also bederived by minimizing a set of prediction-squared errors similar to Equation (14) with respectto the fixed-order AR-model parametersâqk and b̂

q

k (k = 1, . . . , q), but with summationsinstead of expectations [24]. In this section, an alternative method is used: the ‘reflectioncoefficients method’ (RCM), which, contrary to previous methods, carries out a recursive-in-order minimization with respect to the reflection coefficients procedure, constrained to therecursion condition given by Levinson’s algorithm.

For its development, let us consider the followingj -order forward and backward-squarederrors

Qf l

j = E{[(efn )j ]2xn−l},Qblj = E{[(ebn)j ]2xn−j+l}, (27)

where(efn )j and (ebn)j are the linear-prediction errors with the prediction orderj , respect-ively. By using these expressions and that given by Levinson’s recursion, Equation (23), thesesquared errors can be expressed as

Qf l

j = E[xn + 0jaxn−j +

j−1∑s=1(aj−1s + 0jabj−1j−s )xn−s

]2xn−l

,


Qblj = E[xn−j + 0jbxn +

j−1∑s=1(bj−1s + 0jbaj−1j−s )xn−j+s

]2xn−j+l

, (28)which depend only on thej -order reflection and the(j−1)-order coefficients. By minimizingthese errors with respect to0ja and0

j

b , respectively, i.e., by imposing that

∂Qf l

j

∂0ja

= 0; ∂Qblj

∂0j

b

= 0, (29)

we obtain

E

{[xn + 0jaxn−j +

j−1∑s=1(aj−1s + 0jabj−1j−s )xn−s

][xn−j +

j−1∑s=1

bj−1j−s xn−s

]xn−l

}= 0 (30)

and

E

{[xn−j + 0jbxn +

j−1∑s=1(bj−1s + 0jbaj−1j−s )xn−j+s

][xn +

j−1∑s=1

aj−1j−s xn−j+s

]xn−j+l

}= 0. (31)

Bearing in mind thatxn is stationary, using the third-moments definition and appropriatelyreordering, Equations (30) and (31) lead to thej -order reflection coefficients

(0ja )l = −∑j

s=1Flj,s∑j

s=1Glj,s

, and (0jb )l = −∑j

s=1Hlj,s∑j

s=1Ulj,s

, (32)

where

F lj,s = bj−1j−s g(−s,−l)+j−1∑i=1

aj−1i b

j−1j−s g(i − s, i − l),

Glj,s =j∑i=1

bj−1j−s b

j−1j−i g(i − s, i − l),

H lj,s = aj−1j−s g(s,1)+j−1∑i=1

bj−1i a

j−1j−s g(s − i, l − i),

U lj,s =j∑i=1

aj−1j−s a

j−1j−i g(s − i, l − i). (33)

Note that a subindexl has been added to0ja and0j

b in order to point out that a value forthe reflection coefficients can be obtained for each value ofl. In addition, it can be seen thatthe reflection coefficients for a fixedl depend on the third moments over the entirem–n plane.However, Levinson’s recursion between reflection coefficients and the remaining coefficientsis true if and only if third moments over them = n line are involved, since in the oppositecase, the Toeplitz structure of the TOR equations is lost. That is why, in order for the equationsgiven in (32) to be absolutely true, it is essential to carry out a moments transformation over


the entirem–n plane to them = n line. This transformation can be performed by consideringthat thej -order reflection coefficients0ja and0

j

b are obtained by starting with those calculatedfor each value ofl in expression (32)

0ja = −j∑l=1

∑js=1F

lj,sδls∑j

s=1Glj,sδls

= −∑j

s=1Fsj,s∑j

s=1Gsj,s

,

0j

b = −j∑l=1

∑js=1H

lj,sδls∑j

s=1Ulj,sδls

= −∑j

s=1Hsj,s∑j

s=1Usj,s

, (34)

whereδ is Kronecker’s delta. Thus, keeping expression (33) in mind, Equation (34) leads to

0ja = −∑j

s=1[bj−1j−s g(−s,−s)+

∑j−1i=1 a

j−1i b

j−1j−s g(i − s, i − s)

]∑j

s=1∑j

i=1 bj−1j−s b

j−1j−i g(i − s, i − s)

,

0j

b = −∑j

s=1[aj−1j−s g(s, s)+

∑j−1i=1 b

j−1i a

j−1j−s g(s − i, s − i)

]∑j

s=1∑j

i=1 aj−1j−s a

j−1j−i g(s − i, s − i)

. (35)

Therefore, it can be seen that thej -order reflection coefficients depend only on the(j −1)-order coefficients, and also that their calculation involves only third moments over them = n line. The recursive procedure is initialized with01a = −g(−1,−1)/g(0,0) and01b =−g(1,1)/g(0,0).

However, in a practical situation in which only a finite number of data is available, only anestimation of third moments, and therefore of the reflection coefficients, can be obtained. Theestimation of0ja and0

j

b , which provides a higher bispectral resolution because it diminishesthe windowing effects, is that in which the reflection coefficients are obtained by minimizingthe followingj -order squared errors:

Qf l

j =1

N − qN∑

n=j+1[(efn )j ]2xn−l ,

Qblj =1

N − qN∑

n=j+1[(ebn)j ]2xn−j+l , (36)

instead of those defined in Equation (27), whereN denotes the number of available data.By means of a theoretical development similar to that used to obtain Equation (35), it can

be shown that thej -order reflection coefficients in this case are

0ja = −∑j

s=1[bj−1j−sD

−s,j0,−s +

∑j−1i=1 a

j−1i b

j−1j−sD

−s,j−s,−i

]∑j

s=1∑j

i=1 bj−1j−s b

j−1j−i D

−s,j−s,−i

,

0j

b = −∑j

s=1[aj−1j−s D

−j+s,j−j,−j+s +

∑j−1i=1 b

j−1i a

j−1j−s D

−j+s,j−j+s,−j+i

]∑j

s=1∑j

i=1 aj−1j−s a

j−1j−i D

−j+s,jj+s,−j+i

, (37)


where

Di,j

k,l =N∑

n=j+1xn+kxn+lxn+i , j = 1, . . . , q, i, k, l = −j, . . . ,0. (38)

Finally, the remainingj -order coefficients are recursively obtained by using Equation (23),with the initialization

01a = −D−1,10,−1

D−1,1−1,−1

, 01b = −D

0,1−1,0D

0,10,0

. (39)

In order to diminish the bispectrum-estimation variance, the available data are usuallydivided intoM records withN data in each one. For simplicity of notation, we have onlyconsidered one record in the method presented. In the case ofM records, we may formDi,jk,lof Equation (38) for each record, average theDi,jk,l ’s for all records, and finally use them in thecalculation of the reflection coefficients.

Based on the previous developments, the AR-coefficient estimation algorithm can besummarized as follows:1. CalculateD−1,1−1,−1,D

−1,10,−1,D

0,1−1,0,D

0,10,0 using Equation (38). From these coefficients, calcu-

late the initialization01a and01b .

Fromj = 2 . . . q:2. Compute0ja and0

j

b using Equations (38) and (37);3. Calculate,̂aj−1k andb̂

j−1k (k = 1, . . . , j − 1) using

âj

k = âj−1k + âjj b̂j−1j−k, k = 1, . . . , (j − 1),b̂j

k = b̂j−1k + b̂jj âj−1j−k , k = 1, . . . , (j − 1), (40)with the initializationâj−10 = b̂j−10 = 1; and

4. Use Equation (40) withj = q to obtain, in addition to0qa and0qb , the desired set ofcoefficientsâqk andb̂

q

k (k = 1, . . . , q).

5. Quadratic Phase Coupling: Definition and Bispectral Detection

Let xn now be a superposition oft sinusoids with non-random real amplitudes,r pairs ofwhich are quadratically coupled, and the remainingt − 3r sinusoids uncoupled. The signalis contaminated by a zero-mean, third-order stationary, signal-independent, noisy process,denoted assn. So, the noise-corrupted signal can be written as

xn =(

r∑i=1[x1,in + x2,in + x3,in ] +

t∑i=3r+1

xin

)+ sn, (41)

where

xj,in = Aji ej (ωji n+φji ), 1≤ j ≤ 3; 1≤ i ≤ r,ω3i = ω1i + ω2i; φ3i = φ1i + φ2i , 1≤ i ≤ r, (42)

are the coupled terms, while the uncoupled terms are given by

xin = Ai ej (ωin+φi ), 3r + 1≤ i ≤ t. (43)


Figure 1. (a) Bispectrum of a process with a QPC. (b) Power spectrum of a process with a QPC.

The phases{φji : j = 1,2;1 ≤ i ≤ r} and {φi : 3r + 1 ≤ i ≤ t} are a collectionof i.i.d. random variables, uniformly distributed in the interval[0,2π). So the set of pairs{(ω1i, ω2i) : 1≤ i ≤ r} is called the set of quadratically coupled frequency pairs.

If xn is a zero-mean process, its third-order moments are given by

gx(n,m) =r∑i=1

λi[ej (ω1in+ω2im) + ej (ω1im+ω2i n)] + γsδ(n)δ(m), (44)

whereγs is the skewness of the noise, andλi = A1i A2i A3i is the skewness associated withthe ith QPC pair(ω1i, ω2i). From this result two conclusions should be noted: (1) if sn isGaussian, thenγs = 0, and so its contribution togx(n,m) is null. More generally, if the noisehas a symmetric distribution, its bispectrum is null, independently of its spectral content. Thus,any symmetrically distributed white or colored process added to the noiseless signal does notcontribute, theoretically, to the QPC detection problem in the bispectral domain; (2) thet−3runcoupled vibrationsxin, 3r + 1 ≤ i ≤ t , do not contribute to the third-order momentssequencegx(n,m). Thus, third-order statistics, instead of second-order, is the valid domain todetect QPC harmonic components and discriminate them from the uncoupled pairs.

Without any loss of generality, let us consider the particular caset = 6 andr = 1, withω3 = ω1 + ω2; ω4 = ω5 + ω6; φ4 = φ5 + φ6, whereφi (i = 1, . . . ,6) are independentrandom variables, uniformly distributed in[0,2π). Thus, the pair(ω5, ω6) is coupled, whereas(ω1, ω2) is uncoupled, i.e. only the frequenciesω4 andω5 appear in Equation (43). This factleads to a bispectrum ofxn, shown in Figure 1a. As can be seen in this figure, this bispectrumconsists of a peak in the frequency pair(ω5, ω6), whereas for the rest of theω1–ω2 plane,including the frequency pair(ω1, ω2), the bispectrum is theoretically zero. This result makesthe bispectrum a useful tool for QPC detection. On the other hand, as shown in Figure 1b,discrimination between coupled components and uncoupled components is not possible usingthe power spectrum (PS). In this example, the PS consists of six peaks, with the frequencypairs(ω1, ω2) and(ω5, ω6) being indistinguisable.

Consequently, the problem stated above is reduced to the calculation of the bispectrumof a random process. As mentioned in the introduction, in this work we have focused ourattention on the determination from the bispectrum for QPC detection using AR modeling.To accomplish this, we can use Equation (4) with parametersak (k = 1, . . . , p), estimatedusing any of the methods existing in the literature. Specifically, in the simulations shown inSection 7 we have used the AR parameter estimation method proposed in the previous section.

6. Optimum-Order Selection Criteria

A problem associated with this type of techniques lies in the selection of the appropriateorder of the model. In practice, this problem is not evident when using methods based solely


on the visualization of the bispectrum. It is therefore necessary to obtain the bispectrum fordifferent orders of the AR model and to introduce optimum-order selection criteria. In thispaper we therefore propose to use the RCM method introduced in Section 4 to estimate theak parameters for obtaining the bispectrum. The advantage of this method resides in the factthat the calculation of the bispectrum for a new order is based on the updating of resultsfor previous orders, thus assuring a drastic reduction in computational cost since each newbispectrum parts from the calculation associated with the previous order. So, if we keep inmind the comments from the previous section on the characteristics of the presence of theQPC, then, at the same time as we calculate the bispectrum for each prediction order, theselection of an optimum order,jop, and therefore ofBop(ω1, ω2), can be made as follows:

CRITERIUM 1. Let us define parameter(C1)j as the mean of the module of bispectrumBj(ω1, ω2) that is obtained for orderj and calculated over the main triangular region of spaceω1–ω2, that is,

(C1)j = 1L

∑m

∑r

|Bj(ωm, ωr)|, (45)

where(ωm, ωr) are the equidistant frequency pairs in the main region of the bispectrum andL is the total of frequency pairs in this region. Keeping in mind this definition, Criterium 1consists of determining the maximum ofC1 versusj . Let us justify this question. Since thenoise existing on the estimated bispectrum (generated by estimation errors) remains meanconstant, a sharp increase in the value ofC1 with respect to the order implies the existence ofcouplings near the order for whichC1 is maximum. This occurs because the mean value of thebispectrum in the presence of couplings is dominated, theoretically, by a delta function (seeFigure 1a). Therefore, when the model is over-parameterized, the value ofC1 should decrease.

CRITERIUM 2. Let us now define parameter(C2)j as the difference between the maximumof bispectrumBj(ω1, ω2) (the highest peak) and its mean value for orderj , that is,

(C2)j = Cj − (C1)j , Cj = max{|Bj(ωm, ωr)|; (ωm, ωr) ∈ main region}. (46)Likewise, Criterium 2 consists in finding a maximum forC2, since in this case the differencebetween the coupling peak present forjop and the noise level is much greater than thatbetween this peak and any background value.

CRITERIUM 3. Parameter(C3)j is defined as the pondered mean of the normalized valuesof the parameters used in the above criteria, that is,

(C3)j = λ (C1)jmax{C1} + (1− λ)(C2)j

max{C2} , (47)where λ is a sensitivity factor withλ ∈ [0,1] that allows an optimal adaptation to theexperiment. Therefore, in very noisy environments, the influence of parameterC2 must befavored since the difference in peaks that characterizes this parameter provides much greatersensitivity thanC1, which is subject to the considerable influence of the noise present. Incontrast, in environments with high SNR, the existence of peaks is better characterized byhigh values ofC1. Therefore, if SNR is low,λ < 0.5, while if SNR is high, the best choice isλ > 0.5.


CRITERIUM 4 (bispectral mask criterium). As demonstrated in the next section, and partic-ularly in environments with a high level of noise, the above criteria, although providing anadequate environment of possible order, does not provide the precise position of the QPC.Therefore, a fine adjustment of the optimum order can be carried out by using, in combinationwith the above criteria, a new measure that reflects the similarity of the possible peaks detectedto the delta-type theoretical functions. The greater the similarity, the more localized the QPCwill be and therefore the more precise the modeling order will be. Let us examine how tomathematically establish this similarity by the following steps:

Step 1: Let us take once more the maximum value of the bispectrum for the orderj , Cj ,and its bispectral coordinates(ωm, ωr), as well as the coordinates of all the bispectral pointsfor which the amplitude of the bispectrum is greater than or equal to 80% of such value. Thesepoints are noted byPi = (ω′1i, ω′2i).

Step 2: We construct a bispectral mask in the main region of the bispectrum such that allits points are zero except in the points where the peaks calculated in step 1 are located. Therewe center parallelepipeds of base 3× 3 and height unity. Mathematically, this is expressed asfollows:

Mj(ω1, ω2) ={

1 if (ω1, ω2) ∈ E[(ω′1i , ω′2i),2ωs]0 elsewhere

}, (48)

whereE[·] denotes the interval centered in(ω′1i , ω′2i) and 2ωs wide, beingωs the samplingfrequency.

Step 3: The mask is applied to the bispectrum, obtaining a masked bispectrumBmaj (ω1, ω2),that is,

Bmaj (ω1, ω2) = Bj(ω1, ω2)Mj(ω1, ω2). (49)Step 4: We calculate the percentage of masked bispectral energy with respect to the total

energy, obtaining a new parameter(C4)j , which is

(C4)j =1L

∑m

∑r |Bmaj (ωm, ωr)|(C1)j

. (50)

Likewise, a maximum value in(C4)j indicates that most of the energy is concentrated ina narrow area around the detected maxima, thus measuring their similarity to delta-typefunctions.

According to these four defined criteria, we propose the following multilevel hierarch-ical algorithm to determine the optimum order in a practical case, summarized in the blockdiagram in Figures 2a (level 1) and 2b (level 2):

Level 1: It consists of calculating the maximum of parameter(C3)j in an area near 1,[j − 1, j, j + 1]. The value ofj for which we obtain this maximum is the candidate for theoptimum order in level 1. This order is termedjLEVEL1.

Level 2: It consists of calculating the maximum of parameter(C4)j in an area near 2,[jLEVEL1−2, jLEVEL1−1, jLEVEL1, jLEVEL1+1, jLEVEL1+2]. The final position of this maximumis jop. It is immediately evident that level 2 encompasses and improves level 1, but given itshigh computational cost it is better to limit its field of application to a reduced area arounda more imprecise value given by level 1. Thus, we reduce the numerical complexity of theglobal algorithm.


Figure 2. Block diagram of the hierarchical multilevel optimum-order selection algorithm: (a) level 1.

7. Simulation Results

The results presented in this section are divided into two parts. First, we show the potentialof the RCM method proposed for AR bispectrum estimation. Once the RCM has been tested,we apply it to the QPC detection problem, in conjunction with the order-selection criteriaintroduced in Section 6.


Figure 2. Continued. (b) level 2.

7.1. BISPECTRUM ESTIMATION OF AR MODELS

To test the performance of the RCM method, we have compared it with the TOR and CTOMmethods for AR causal processes generated by means of Equation (3). In particular, weshow in this subsection the results obtained for an AR process with order 3 and parameters[−0.7,0.2,0.15]. The behavior of the three methodsversusthe number of data is presentedin Figures 3a and 3b by means of the following error:

ER= 100∑p

k=1(âEk − aTk )2∑p

k=1(aTk )

2, (51)

whereâEk are the estimated parameters andaTk are the theoretical ones. In these figures, ER

is plottedversusN for M = 16 andM = 30, respectively. It can be seen that the RCMmethod improves on both the TOR and CTOM methods, the differences being more evidentas the number of data decreases. In contrast, asN increases, the three methods asymptoticallytend to yield similar results. Likewise, it can be noted that the differences between the threemethods are smaller forM = 30 than forM = 16.

In order to compare the TOR, CTOM, and RCM methods in the presence of whiteand colored Gaussian noise, numerous simulations have been carried out. The results forthis example are presented in this section. The added colored noise has been synthes-ized by filtering white noise through a 15-order FIR filter whose parameters are given by[0.5,0.6,0.7,0.8,0.7,0.6,0.5,0.0,0.0,0.5, 0.6, 0.7,0.8,0.7,0.6,0.5]. The definition of theSNR used in this paper is given by


Figure 3. Error ERversusN for an AR(3) process: (a)M = 16; (b)M = 30.

SNR= 10 logExEn, (52)

whereEx andEn are the energy of the noise-free signal and the noise, respectively. Figure 4ashows the error, ER,versusthe SNR obtained with the three methods by usingM = 16records withN = 17, when white noise has been added to thexn process. Moreover, theresults are a mean value over 50 Monte Carlo runs, using a different seed in each one in orderto generate the white noise. Figure 4b shows the ER when colored noise is present. In all cases,the estimation is poorer with the TOR method and the best results are obtained by using theRCM method. It is also important to note the following interesting results: (1) the differencesamong the three methods, and specially for the CTOM and RCM methods, becomes moreevident as the number of data is reduced, the RCM method yielding the best performance;(2) as the noise level decreases, the error obtained with each method tends to be constantand equal to the noise-free value. In addition, this value is identical in the white and coloredcases. Likewise, it can be observed that for the entire range the errors obtained, specially forthe TOR and RCM methods, are very low. This demonstrates that the presence of Gaussian


Figure 4. Error ERversusSNR for an AR(3) process withN = 17: (a) white noise; (b) colored noise.

noise does not greatly affect the estimation and that therefore the errors are rather due toparticular estimations associated with each method. All these results are logical since theadded noise is statistically Gaussian and thus has a zero bispectrum, so its contribution to thenoisy bispectrum, independent of its spectral content, is theoretically null.

7.2. QPC DETECTION BY THE RCM METHOD

As an application of the RCM method to the problem of QPC detection, we have chosena signalxn with t = 6 and r = 2, i.e. two coupled pairs in the normalized frequencies(0.10,0.18) and(0.15,0.25) with amplitudesAji = 1 for 1 ≤ j ≤ 3 and 1≤ i ≤ 2. Allthe results presented here used 2096 signal samples, divided into 64 records containing 64samples each. The noise level has been set to 5 dB and the results are a mean value over 50Monte Carlo runs. This noise has been obtained by passing white noise through an ARMAfilter with parameters AR[1,−2.2,1.77,−0.52] and MA[1,−1.25].


Figure 5. Optimum-order selection parametersversusthe AR model order.

Figure 6. Bispectral mask forj = 9.

The four parameters defined in the section above have been calculated for this example (seeFigure 5). As may be seen in this figure, Criterium 1 alone is not enough to select an optimumorder, since the presence of a high noise level surrounding the orders where the peaks arelocated, masks the presence of an absolute maximum. This problem gets worse when theorder is increased as the system models the noise better. Criterium 2 shows a better behavior,although the presence of maxima at high orders could lead to the detection of false alarms. Thetwo criteria (i.e., Criterium 3) clearly indicate order 8 as the best choice, since Criterium 2 hasmore weight due to the low SNR value. Finally, Criterium 4 corrects the chosen order throughlevel 2 revealing a choice ofjop = 9. The maks used in this case and forjop = 9 are given inFigure 6.

To confirm these results, Figures 7 and 8 represent, by way of example, the bispectrumfor j ∈ [5,12]. Both figures show that whenj > 6, two peaks arise corresponding to thetwo QPC existing in the original signal. Furthermore, it is observed that the highest peak isobtained forj = 8, clearly identifiable from the rest of the bispectrum. This fact explains why


Figure 7. Bispectrum of numerical example signals.


Figure 8. Bispectrum contours of the signal.


Criteria 2 and 3 reach their maximum values forj = 8. However, forj = 9, the two peaksshow a very similar height, since the bispectral energy is divided among them, the secondpeak being sharper than in thej = 8 case and the first one being lower than in thej = 8 case.This feature means that the fourth criterium gives its maximum value forj = 9, since thebispectral mask used contains the maximum amount of bispectral energy as it includes bothpeaks in the mask (see Figure 6).

Moreover, the location of the bispectral energy around the position of the two couplingfor different orders can be observed in Figure 8. Although bispectal peaks begin to appear forlow orders (Figure 7;j = 5, 6, and 7), the energy is quite dispersed (the contours are moreseparated). Likewise, upon overparametizing the order (j = 10, 11, and 12) there is onceagain a dispersion of the energy in the frequency planeω1–ω2.

8. Conclusions

In this work we have presented a scheme for QPC detection in harmonic vibrations using ananalysis based on signal analysis using third-order statistics. More concretely, we have pro-posed a new algorithm with the following steps: (1) first, the signal bispectrum is determinedusing AR modeling. To do so we have generated a new method whose main property is therecursivity in the model order. Thus, this step provides, without additional computational cost,the AR model parameters with orders lower than or equal to a previously assumed maximumvalue; (2) then, in the set of calculated bispectra, four optimum-order selection criteria areproposed for QPC detection, assuming that a QPC is shown, theoretically, in the bispectrum asan impulse situated in the frequency pair that characterizes it. The first three criteria are basedon evaluating the mean value of the bispectrum and its maximum value in the principal regionof the bispectral frequency plane. The fourth criterium is based on computing the bispectralenergy in narrow bands, named bispectral masks, that attempt to isolate possible peaks orimpulses in the bispectral domain. In order to choose the optimum order we have proposed amultilevel hierarchical algorithm based on the use of these four criteria.

Different simulation examples have been used to show that the RCM method is an improve-ment over previous methods, both for noise-free processes and for processes contaminated byadded white and colored Gaussian noise. The RCM method is especially useful for situationsin which only few data are available for the process. In addition, the potential of this methodfor QPC detection in colored noise environments has been demonstrated. Furthermore, theapplication of the four order-estimation criteria as a whole allows us to fit the best bispectrumfor the set of established orders.

Acknowledgement

This work was partially supported by ‘Comisión Interministerial de Ciencia y Tecnología(CICYT)’ of Spain under project number TIC95-0465.

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