The Higher Order Spectra for Condition Part1

8/6/2019 The Higher Order Spectra for Condition Part1

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The 10 th International Conference of the Slovenian Society for Non-Destructive Testing Application of Contemporary Non-Destructive Testing in Engineering

September 1-3, 2009, Ljubljana, Slovenia, 17-25

THE HIGHER ORDER SPECTRA FOR CONDITIONMONITORING/NDT IN NON-STATIONARY CONDITIONS : PART I.

Keynote paper

L. Gelman, I. Petrunin

Cranfield University, School of Engineering, Department of Process and Systems Engineering,Cranfield, UK

email: [email protected]

1. Introduction

The classical higher order spectra (HOS) have been widely investigated [1-7] for stationary processes. However, for some important practical applications in radar, sonar and mechanicalsystems (e.g. change of the instantaneous frequency of radar and sonar signals, change of the shaftfrequency during machinery operation etc.), it is necessary to process non-stationary signals withfrequency non-stationarities. The classical techniques are not suitable for those signals because theHOS average a non-stationary frequency and thus create estimation errors due to frequency non-stationarity.

Normally, for transient signals from mechanical systems (e.g., turbines, gearboxes, etc.), radar andsonar systems, time variations of the instantaneous frequency could be known a priori or easilyestimated from independent synchronous measurements from tachometers, frequency sensors andsimilar instruments. Therefore, the main problem is processing and recognizing transient signalswith known a priori or independently measured time variation of the instantaneous frequency.HOS have been investigated for non-stationary processes. The Wigner higher order spectra [8] have

been proposed for transient signals with linear variation of the instantaneous frequency in time. Thewavelet bispectrum [9, 10] has been proposed for impulses, quasi-periodic and chaotic signals. Thechirp-Fourier higher order spectral analysis [11] has been proposed for transient signals with linear and nonlinear polynomial variation of the instantaneous frequency in time. Hanssen and Scharf [12]have developed an important theory of higher order spectral analysis for non-stationary stochastic

process belonging to the harmonizable class. Dandawate and Giannakis [13] and Gardner andSpooner [14] have developed an important theory of higher order spectral analysis for higher order cyclostationary processes. The above mentioned works [12-14] could be fairly easily generalized togeneric non-stationary processes.Parameter estimation using nonstationary HOS was investigated [8, 15-16] for the finite sum of theexponentially damped sinusoidal signals with constant frequencies (i.e. signals with amplitude non-stationarity) in additive white or coloured Gaussian noise. Detection by transient HOS of anunknown deterministic transient signal in broad-band additive stationary noise was investigated in[17-18]. A matched filtering and HOS were combined [19] for detection of transient deterministic or


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random non-Gaussian signals on the background of zero-mean Gaussian noise with unknowncovariance sequence.Unfortunately, the problem of using HOS for processing and recognizing transient signals withknown a priori or independently measured time variation of the instantaneous frequency is not addressed in any of the above mentioned works [8-19]. In all these woks investigators used fixed

frequencies, not instantaneous approach .In addition to the HOS theories discussed above, there are various time-frequency techniques thatcould be exploited for higher order spectra analysis of transient signals with time variations of theinstantaneous frequency.Standard matched filters have been widely used [20] for processing of the chirps and higher order chirps (i.e. transient signals with any nonlinear polynomial variation of the instantaneous frequencyin time). A useful approach is to process these signals by the adaptive Kalman filter [21-23]. As analternative, the discrete Gabor expansion was proposed [24] for chirp processing. Another usefulmethod of chirp analysis is the order tracking [25-26]. The order tracking based on the time variantdiscrete Fourier transform (TVDFT) [27] is a special case of the chirp-z transform [28]. A usefulapproach for detecting signals with variable instantaneous frequency is based on synthesizing amulti-channel filterbank [29]. This technique, like the matched filtering approach, differssignificantly from the recently developed advanced techniques: the chirp-Fourier transform, thechirp-Wigner transform and the chirplet transform.The chirplet transform [30] is a generalization of both the short time Fourier transform and thewavelet transform. The chirp-Wigner transform [31-32] has been proposed for amplitude estimationof the higher order chirps. The polynomial Wigner-Ville distribution [33-35] has been proposed andinvestigated for processing the higher order chirps. The higher order local polynomial Wigner distribution has been proposed and investigated by Katkovnik [36-37] and investigated by Stankovic[38] for estimating of the unknown variable instantaneous frequency and its derivatives for thehigher order chirps. A high resolution evolutionary spectral method which uses the linear chirpGabor expansion has been developed and investigated by Akan and Chaparro [39] and Akan [40] for chirps and higher order chirps.All the above mentioned techniques [20-40] could be used in the higher order spectral analysis intransient conditions for nonlinear polynomial variation of the instantaneous frequency. However,use of these transforms for the higher order spectral analysis has not been investigated in literature.Therefore, no one has addressed the problem of using HOS for processing and recognizing transientsignals with known a priori or independently measured time variation of the instantaneousfrequency.The problem considered in this paper is to create time-frequency HOS for processing andrecognizing transient signals with known a priori or independently measured time variation of theinstantaneous frequency. This problem is not addressed in the existing literature.

The purposes of this paper are (i) to propose and develop new time-frequency techniques: the shorttime instantaneous HOS with adjustment to measured or known a priori time variation of theinstantaneous frequency of non-stationary processes (ii) to compare the instantaneous and the non-instantaneous HOS (i.e., without adjustment to time variation of the instantaneous frequency) for nonlinearity recognition in the non-stationary case.

2. The instantaneous HOS

Let us consider the important general case:


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the transient complex signal ( ) x t is an additive mixture of ( 1)n transientcomponents ( ) j s t , narrowband or sinusoidal on the background of interference; j

is a component number

the instantaneous frequencies of these components vary in time; frequencyvariations are measured from independent measurements or known a priori andcould be different for different componentsthe number of harmonics could be known or unknown a priori or evaluated from asignal; in the last case, the adaptive methods for harmonic detection (e.g. contrastmethods [41], etc.) could be employed

This case is important for many applications, e. g.,:

vibrations from gearboxes during start up or shut down; transient components arenarrowband fundamental and higher mesh harmonics; the instantaneous frequencies

of these harmonics vary in time due to variation of the shaft frequency; frequencyvariations are obviously different for harmonics; a variation of the fundamentalfrequency could be measured from the tachometer signal; variations of the higher harmonics could be estimated as an integer multiple (i.e. harmonic number) of themeasured variation of the fundamental frequencytransient frequency modulated multi-component signals from radar and sonar systems; the instantaneous frequencies of signal components vary in time due to thefrequency modulation; frequency variations are different for components and could

be measured from frequency sensors

New techniques are proposed here for this case: the instantaneous HOS. The main idea of these

techniques is the use of the independently measured or known a priori instantaneous frequencies of transient components for each m -th segment for estimating the appropriate transforms of theclassical (i.e.,. stationary) and any non-stationary HOS . e. g., [8-19]. Therefore, the techniques aresuitable for transient signals with any measured or known a priori variation of the instantaneousfrequency in time.

For the sake of simplicity and without loss of generality, let us consider the proposed approach for the classical HOS based on the Fourier transform. The following steps should be undertaken for estimating the proposed short time instantaneous HOS based on the Fourier transform:

complex transient time domain signal ( ) x t should be divided into overlapping

blocks ( )t by the external time window ( )h t e ; 1,..., K , K defines the totalnumber of overlapping blocks, T is the time centre of the external windoweach block ( )t should be divided into overlapping segments ( )m x t by the internal

time window ( )ih t , 1,...,m M , M defines the total number of overlappingsegments in the time block instantaneous frequency variation should be estimated from independentmeasurements (e.g. from tachometers); this step should be avoided if this variationis known a priorithe instantaneous frequencies should be used for each m -th segment for estimatingthe Fourier transforms of the instantaneous HOS


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The proposed frequency adjustment could be done on-line (i.e., in real time) or off-line.

The short time instantaneous Fourier-based HOS for ( 1)n non-stationary components( ), ( ),..., ( )i j k s t s t s t for m -th time domain segment can be written as follows:

... ( , , ..., , ) ( ) ( )... ( ) ( ... )i j k s s s m mi m mj m mk m mi mj mk H i j k T X f X f X f X f f f , (1)

where mi f , mj f , , mk f are ( 1)n instantaneous frequencies of components ( )i s t , ( ) j s t , , ( )k s t at

m -th segment respectively; 1,..., 1i n ; 1,..., 1 j n ,, 1,..., 1k n ; i , j and k arecomponent numbers;

2

( ) ( )mji f t

m mj m X f x t e dt

,1 22 ( ... )

1 2( ... ) ( ) m m mk i f f f t

m m m mk m X f f f x t e dt

( )m mj X f and 1 2( ... )m m m mk X f f f are the Fourier transforms at the appropriate frequencies of the m -th segment ( )m x t of the external time window ( )eh t , is a symbol of a complex conjugate,n represents an order of the HOS.Frequencies mi f , mj f , , mk f are estimated continuously from the measured instantaneous

frequency variations ( )i f t , ( ) j f t , , ( )k f t for components ( )i s t , ( ) j s t , , ( )k s t respectively.

Measurement time of these frequencies is defined by the time centre of the m -th segment. Values of those frequencies are averaged over the relatively short time interval mt of the internal windowusing the following expressions:

dt t f t

f m

m

t

t l

mml )(

1 2

1

; dt t f t f m

m

t

t j

mmj )(

1 2

1

; dt t f t f m

m

t

t k

mmk )(

1 2

1

(2)

where 1mt and 2mt are time moments of the beginning and the end of the m -th segment.It can be seen that the short time instantaneous HOS are time-frequency techniques; they are n -

dimensional functions (i.e. ( 1)n component numbers and time). If one selects particular signalcomponents (i.e. component numbers), then the short time instantaneous HOS are the onedimensional time domain functions.If amplitude-phase relationships between n Fourier components in equation (1), including higher order phase coupling, are stationary, then the instantaneous HOS (1) could be averaged over M segments:

...1

1( , , ..., , ) ( ) ( )... ( ) ( ... )

i j k

M

s s s m mi m mj m mk m mi mj mk m

H i j k T X f X f X f X f f f M

(3)


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It is impossible to specify the particular frequencies at which the instantaneous HOS (3) areestimated because the instantaneous frequencies of components are time varying. One can state onlythat the instantaneous HOS are estimated for ( 1)n components ( ), ( ), ..., ( )i j k s t s t s t with numbers

, ,....,i j k respectively. So, the instantaneous HOS do not depend on frequencies (like the non-instantaneous HOS describe in literature); they depend on component numbers. The new short timeinstantaneous HOS are generalizations of the appropriate non-instantaneous HOS for the case of transient signals with independently measured or known a priori time variation of the instantaneousfrequency.The rationale of the proposed transforms is explained below for the two main cases of combinationsof instantaneous frequency variation ( ) j f t of component ( ) j s t , frequency resolution f (defined

by the m -th segment length) and length of the external window.Case 1 . The instantaneous frequency change ( ) j f t during duration of the external window is

within the limits 2/;2/ f f f f p p of frequency resolution f at the particular frequency p f .

This case is characterized by relatively

low speed of the frequency variation ( ) j f t

poor frequency resolution (i.e. high value of parameter f ) due to the shortinternal windowshort external window

The considered case could be viewed as a quasi-stationary case. It is obvious that one does not needto use the instantaneous HOS; the single frequency of component ( ) j s t averaged within the length

of the external window could be used.

Unfortunately, this case has limited application because investigators normally use relatively highfrequency resolution and relatively long length of the external window [18] for averaging purposes.For some important cases, investigators are forced to consider relatively high speeds of instantaneous frequency variation.Case 2 . The instantaneous frequency change ( ) j f t during duration of the external window is more

than the frequency resolution f at the particular frequency p f . This case is characterized by

relatively

high speed of the frequency variation ( ) j f t

high frequency resolution (i.e. low value of parameter f ) due to the long internalwindowlong external window

This case is definitely non-stationary and has wide application. Use of the single frequency of component ( ) j s t averaged within the external window creates estimation errors because the

instantaneous frequency ( ) j f t will be in different frequency bins for different segments of the

external window. Therefore, the instantaneous HOS should be used in this case.It is obvious that the instantaneous HOS allow the use of relatively high frequency resolution andrelatively long duration of the external window for HOS estimation at any frequency speed of transient components. It is known that


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high frequency resolution leads to increment of the signal /noise ratio for HOSestimates related to narrowband or sine componentsa long external window leads to better HOS averaging (i. e. to decrement of HOSvariance).

Thus, the proposed transforms are useful for improving early nonlinearity recognition for relativelyhigh speeds of the instantaneous frequency variation and on the background of interference.It is proposed here to employ the correlation between the external and internal windows from ref.[18], i.e. the internal window length to be the ( 1)n -th root of the external window length. If asignal has fast variations, a shorter external window is required. Then the shorter internal window isrequired [18] which provides poorer frequency resolution and better time resolution. If it is desiredto represent the long term signal evolution, a longer external window is required. Then the longer internal window should be [18] used which provides better frequency resolution and poorer timeresolution.

Normally, 50% overlap of the internal window is used in the HOS to increase the number of averaged segments and avoid redundancy. It is recommended here that the same overlap of theinternal window for the short time instantaneous HOS be used. However, if a shorter externalwindow is required and one wants to maintain the same length of the internal window and number M of averaging, then a higher overlap (e.g. 75-80%) of the internal window is recommended. Ahigher overlap of the internal window increases the accuracy of the instantaneous HOS estimation;however, 90-95% overlap is redundant. The overlap of the external window depends on the requiredtime resolution. Higher overlap leads to increment of the time resolution of the short timeinstantaneous HOS.The proposed techniques can be used with any non-stationary transform (e. g., the Wigner transform, the chirplet transform, etc.). This could be done by substituting a non-stationarytransform for the Fourier transform in equations (1-2). An example of such substitution is discussedin section 3.

3. Comparison of the instantaneous and the non-instantaneous HOS

The proposed transforms were compared with the non-instantaneous HOS (i.e., without adjustmentto time variation of the instantaneous frequency) for recognizing a non-stationary nonlinearity.To compare the instantaneous and the non-instantaneous HOS for recognition of non-stationarynonlinearity, an input transient random cosine signal with constant amplitude, random initial phaseand linearly changed instantaneous frequency in time (i. e., chirp) has been passed via the followingnonlinear (bilinear) system:

2

2

2 cos , 0,

2 cos , 0,

S

C

x hx x A t x

x hx x A t x

(4)

where X

xm

, X is the displacement,mc

h2

, h is damping; ,S C S C k k m m

, m and c

are the mass and damping coefficient respectively, S k and C k are the stiffness for positive

displacement and stiffness for negative displacement respectively, 1 A

Am

, 1 A is the constant


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amplitude of the input signal, dt t t , t is the instantaneous phase, ( ) t is the linearlychanged angular frequency, the initial phase of the each simulated signal has been taken randomlyand is uniformly distributed in the range [0; 2 ].

The bilinear system (4) is widely used in applications for investigation of the sub-harmonicresonances of the following offshore structures: free-hanging risers, tension leg platforms andsuspended loads [42-44], articulated loading towers constrained by a connection to a massive tanker or vessels moored against fenders [45], oscillating parts with clearances and motion limiting stops[46]; the system (4) is also used for investigation of gearboxes [47].Parameters used in simulation of the input signal and system response are as follows: frequencyspeed of the input signal is 4Hz/s; the known linear variation of the signal frequency correspondingto the time range (0-40) s is in the range (1420-1580) Hz for the linear system and (1380-1540) Hz

for the bilinear one; sampling frequency is 20kHz; stiffness ratios c sr c

k k k

k are 0 (linear system)

and 0.1 (bilinear system); the resonance frequencies of the system are 1500 Hz and 1460.5 Hz at thestiffness ratios 0 and 0.1 respectively. So the resonance frequency is centred within the excitationfrequency range for both the linear and the bilinear systems.

a) b)Fig. 1. a) The simulated response of the linear system b) The magnitude of the chirp-Fourier

transform of the simulated response of the linear system.

a) b)Fig. 2. The instantaneous HOS of the simulated responses of the a) linear system b) non-linear

system.


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The output signal in the vicinity of resonance consists of the fundamental resonance harmonic (atthe stiffness ratio 0, linear system) and of the fundamental and higher resonance harmonics (at thestiffness ratio 0.1, bilinear system) [44]. The instantaneous frequencies of the harmonics vary intime due to variation of the instantaneous frequency of the excitation of the input signal. Gaussianwhite noise was added to the output signal to hinder the recognition of the nonlinearity and so tomore closely mimic data from the early stage of nonlinearity development; the signal/noise ratio is30dB.An example of the simulated response (acceleration) of the linear system without noise for unityexcitation and the above mentioned parameters is shown in Fig. 1. Using the half-power point

bandwidth [48] with known frequency speed (4 Hz/s), one can find from the curve that the estimateof the damping h of the simulated system is 220 s -1. The estimated maximum amplitude of thelinear system response is 21.4. The instantaneous HOS of the simulated responses of the linear andthe non-linear systems are shown in Fig. 2 a) and b) respectively.The instantaneous HOS based on the chirp-Fourier transform [11] were employed by substitutingthe chirp-Fourier transform for the Fourier transform in equations (1-2). These instantaneous HOSwere compared for nonlinearity recognition with the non-instantaneous HOS (i.e., withoutadjustment to the instantaneous frequency estimation) also based on the chirp-Fourier transform.The chirp-Fourier transform was employed because it is suitable for the linear frequency variationof the output signals.Assuming that the amplitude-phase relations between three chirp-Fourier components are stationaryin the -3dB vicinity around the resonance frequency, system nonlinearity was recognized by theinstantaneous and the non-instantaneous HOS of order 3 at

the fundamental and the second resonance harmonicsthe fundamental and the third resonance harmonics

Parameters and functions used for the non-instantaneous and instantaneous HOS estimation are asfollows: internal window size is 0.225s (i. e., frequency resolution is 4.4Hz); overlap of the internalwindow is 50%; the internal window is the Hamming window; the external window size is 18s(defined as a duration of the resonant oscillations at the -3dB vicinity around the resonancefrequency); the external window is rectangular. The chirp-Fourier transforms used in the bothinstantaneous and non-instantaneous HOS were fully adapted [11] to the known time variation of the instantaneous frequency.In the non-instantaneous HOS only one value of the instantaneous frequency is estimated for eachcomponent at the duration of the external window; an averaged frequency at this duration is used.Both the non-instantaneous and the instantaneous HOS were estimated only for one externalwindow in the vicinity of resonance.Eight hundred simulation tests were performed. The Fisher criterion [49] of the recognition

effectiveness is employed; it is known [49] that higher values of the Fisher criterion provide better recognition effectiveness.

Table: The Fisher criteria for the instantaneous and the non-instantaneous HOS

Transform Harmonics The Fisher criterionThe instantaneous HOS The fundamental and second 398

The non-instantaneous HOS The fundamental and second 6The instantaneous HOS The fundamental and third 554

The non-instantaneous HOS The fundamental and third 22


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The Fisher criteria for the instantaneous and the non-instantaneous HOS and different components,i.e., combinations of the employed harmonics are shown in the table. It can be seen from the tablethat

the instantaneous HOS provide higher estimates of the Fisher criterion incomparison with the non-instantaneous HOSthe gains, the ratios of the Fisher criteria for the instantaneous and the non-instantaneous HOS, are 61.2 and 24.6 for HOS at the fundamental and the secondresonance harmonics and the fundamental and the third resonance harmonicsrespectively.

So the instantaneous HOS provide essential effectiveness gains in comparison with the non-instantaneous HOS and are therefore more effective for nonlinearity recognition.Another important factor for evaluating the instantaneous HOS is the accuracy of instantaneousfrequency estimate. Obviously, the effectiveness of the proposed HOS depends on error infrequency estimation. The above simulation was carried out to investigate this issue. An error infrequency estimation was simulated by adding a random zero mean noise to each point of the timevariation of the instantaneous frequency; error values are as follows: 0.1% and 0.4 % of the currentinstantaneous frequency. The Fisher criteria have decreased to 328.3 and 5.66 and 153.8 and 3.63for the instantaneous and the non-instantaneous HOS at the fundamental and second harmonics anderrors 0.1% and 0.4 % respectively. However, the use of the proposed approach in the considerednoisy environment still provides the following essential effectiveness gains: 58 for 0.1% error and42.3 for 0.4% error respectively. One can note that increment of the frequency estimation error decreases the effectiveness gain.

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