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Online tracking of instantaneous frequency and amplitude ofdynamical system response

P. Frank Pai

Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

a r t i c l e i n f o

Article history:

Received 15 February 2009

Received in revised form

29 June 2009

Accepted 8 July 2009Available online 4 September 2009

Keywords:

Instantaneous frequency

Timefrequency analysis

TeagerKaiser algorithm

Sliding-window tracking

HilbertHuang transform

a b s t r a c t

This paper presents a sliding-window tracking (SWT) method for accurate tracking of

the instantaneous frequency and amplitude of arbitrary dynamic response by

processing only three (or more) most recent data points. TeagerKaiser algorithm

(TKA) is a well-known four-point method for online tracking of frequency and

amplitude. Because finite difference is used in TKA, its accuracy is easily destroyed by

measurement and/or signal-processing noise. Moreover, because TKA assumes the

processed signal to be a pure harmonic, any moving average in the signal can destroy

the accuracy of TKA. On the other hand, because SWT uses a constant and a pair of

windowed regular harmonics to fit the data and estimate the instantaneous frequency

and amplitude, the influence of any moving average is eliminated. Moreover, noise

filtering is an implicit capability of SWT when more than three data points are used, and

this capability increases with the number of processed data points. To compare the

accuracy of SWT and TKA, HilbertHuang transform is used to extract accurate time-

varying frequencies and amplitudes by processing the whole data set without assuming

the signal to be harmonic. Frequency and amplitude trackings of different amplitude-

and frequency-modulated signals, vibrato in music, and nonlinear stationary and non-

stationary dynamic signals are studied. Results show that SWT is more accurate, robust,

and versatile than TKA for online tracking of frequency and amplitude.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Every mechanical system (including human ears) has its specific resonant frequency (or frequencies), and its response

to an excitation is primarily determined by the closeness and commensurability (if nonlinear systems with polynomial

types of nonlinearity) between the excitation frequency and the resonant frequency [1]. Moreover, a signal is mainly

described by its amplitude and frequency. Hence,extractionor trackingof a signals frequency and amplitude is one of themajor tasks in signal processing for dynamics-based structural health monitoring, dynamical system identification and

control, recognition of speech and music, and many other scientific applications [26]. For experimental verification and

updating of finite-element models, aircraft flight tests, and many other types of after-fact data analysis, extraction of

frequencies and amplitudes is needed. However, effective control schemes, online structural health monitoring, and many

other real-world applications requireonline casual tracking(i.e., one new output from one new input) of the instantaneous

frequency and amplitude of a signal, which is more challenging because only a limited number of data points can be

processed due to the limited processing speed of hardware and software, possible existence of time-varying nonlinear and/

or non-stationary effects, and availability of data points.

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Mechanical Systems and Signal Processing

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Mechanical Systems and Signal Processing 24 (2010) 10071024
http://-/?-http://www.elsevier.com/locate/jnlabr/ymssphttp://localhost/var/www/apps/conversion/tmp/scratch_6/dx.doi.org/10.1016/j.ymssp.2009.07.014mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_6/dx.doi.org/10.1016/j.ymssp.2009.07.014http://www.elsevier.com/locate/jnlabr/ymssphttp://-/?-

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In traditional data analysis, the first step is to choose a complete, convergent set of orthogonal basis functions, and the

second step is to perform convolution computation between the signal under analysis and the chosen basis functions to

extract components similar to the basis functions [79]. Unfortunately, this approach is not suitable for a nonlinear and/or

non-stationary signal because its frequency and amplitude change with time. Hence, adaptive basis functions need to be

used and they can be derived only from the signal itself.

Fourier transform (FT) is often used to transform time-domain data into frequency-domain data to identify a signals

frequency and amplitude from the obtainedspectrum[7], but a spectrum cannot show the time-varying characteristics of

the processed signal. To overcome this problem, short-time Fourier transform (STFT,spectrogram), wavelet transform (WT),and HilbertHuang transform (HHT) have been developed for timefrequency analysis [1013]. STFT uses windowed

regular harmonics and function orthogonality to simultaneously extract time-localized regular harmonics, and WT uses

scaled and shifted wavelets and function orthogonality to simultaneously extract time-localized components similar to the

wavelets[9]. Unfortunately, STFT and WT cannot accurately extract time-varying frequencies and amplitudes because of

the use of pre-determined basis functions and function orthogonality for simultaneous extraction of components. On the

other hand, HHT uses the apparent time scales revealed by the signals local maxima and minima to sequentially sift

components of different time scales, starting from high-frequency to low-frequency ones [1317]. Because HHT does not

use pre-determined basis functions and function orthogonality for component extraction, it provides more accurate time-

varying amplitudes and frequencies of extracted components. Unfortunately, none of FT, STFT, WT, and HHT is really

appropriate for online tracking of a general signals instantaneous frequency and amplitude because each of these methods

requires a data length that covers at least three periods of the lowest-frequency component of the signal (as explained next

in Section 2.1), and hence the obtained frequency and amplitude are sectionally averaged (not instantaneous) values.

In this paper we derive a sliding-window tracking (SWT) method that can accurately track a signals frequency andamplitude by processing only three (or more) recent data points. Different linear and nonlinear signals are processed to

validate the method.

2. Extraction and tracking of instantaneous frequency and amplitude

2.1. Extraction

To extract the time-varying amplitude a(t) and phase y(t) from an amplitude-modulated and frequency-modulated

(AMFM) signal u(t)=a(t)cosy(t) one can use its conjugate part v(t)=a(t)siny(t) to form

zt ujv aejy; a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 v2

p ; y tan1v=u; 1

where j ffiffiffiffiffiffiffi1p

. Unfortunately, y(t) anda(t) are two unknowns but there is just one known function (i.e., uacosy). One

can use the Hilbert transform (HT) to obtain the other function v(t), butHT(u)=v=a(t)siny(t) only if it satisfies the Bedrosian

theorem[18], which requires the frequency band ofa(t) be lower and non-overlapped with the frequency band of cos y(t) as

shown next.

Any function defined over 0rtrTcan be extended by repeating itself to become a periodic function. Then the averaged

amplitudes of different harmonic components ofu(t) over the sampled periodTcan be extracted using the discrete Fourier

transform (DFT) as[7]

utk uk a0 2XN=2i1

ai cosoitk bi sinoitk a0 Re2XN=2i1Uiejoitk ;

ai 1=NXNk1uk cosoitk; bi 1=N

XNk1uk sinoitk;

Ui ai jbi 1=NXNk1ukejoitk ; 2

where oi2pi/T, tk(k1)Dt, k=1,y, N, N is the total number of samples (assumed to be even here), Dt the samplinginterval, 1/Dtthe sampling frequency,T(=NDt) the sampled period, Df(=1/T) is the frequency resolution, and the Nyquist (or

maximum) frequency is 1/(2Dt)(=N/2T). Moreover,Uiis the Fourier spectrum, and aiandbiare amplitudes of the extracted

regular harmonics. The formulas in Eq. (2) for computing aiandbishow that the conformality betweenu(t) and cosoitandsinoit is used to extract regular harmonics from u(t). Apparently, regular harmonics of uniformly spaced frequencies arechosen as basis functions, andu(t) is expressed as the summation ofN/2 harmonics of constant amplitudes and phases.

Because the lowest-frequency harmonic has a frequency o1=2p/T, theTneeds to cover at least one period of the signal inorder to identify the signals lowest-frequency component from the spectrum. Because uN1=u1 is implicitly assumed in

DFT, if sudden change (i.e., discontinuity) exists between uNandu1due to the sampling process and/or transient response,

the o1 harmonic is not a real component. Moreover, in order to account for this discontinuity, many high-frequency

harmonics exist and result in the so-called leakageproblem in frequency domain, which results in the Gibbs phenomenonin the time domain. In order to reduce leakage, the sampled periodTis often chosen to cover at least three cycles of the

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lowest-frequency component to be extracted[19]. Moreover, Shannons sampling theory requires more than two samples

per cycle of a component in order to capture the components frequency and amplitude. Hence, more than six samples

covering three periods are needed in order toextractthe correct (but averaged) frequency and amplitude from a signals

spectrum. More seriously, a small Tresults in a big Df(=1/T), and a smallNresults in a small Nyquist frequency. Hence, this

frequency-domain method is appropriate for extractionof only sectionally averaged frequency and amplitude, but not for

online tracking.

A signals time-varying frequency and amplitude can be extracted as shown next. If the global average of the signal over

0rtrT is zero (i.e., a0=0), the conjugate part v(tk) of the u(tk) in Eq. (2) is given by

vtk vk 2XN=2i1

ai sinoitk bi cosoitk Im2XN=2i1Uiejoitk : 3

Thev(t) can be proved to be the Hilbert transform (HT) ofu(t) [8,13]. Hence, we have

ut jvt 2XN=2i1Uiejoit aejy; a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 v2

p ; y tan1 v=u; 4

_ut j _vt 2XN=2i1joiUie

joit: 5

Eq. (5) shows that the Fourier spectrum Ui needed for computing vk can be used to compute the instantaneous velocityusing the inverse discrete Fourier transform (IDFT), and then to compute the instantaneous frequency without using finite

difference or curve fitting as

o dydt dtan

1 v=udt

u _v _uva2

; 6a

where _u du=dt:If significant noise exists, theoatt=tncan be obtained by averaging over 2pDtto reduce the influence ofnoise as

o Xpip1

ytn iDt ytn i 1Dt=2pDt: 6b

For a lengthy time series without moving averages, this method can accurately extract its time-varying frequency and

amplitude by processing the whole data set. However, online frequency tracking can process only a short length of recentdata, but the discontinuity-induced end effect makes the extracted o(t) anda(t) inaccurate for the current state at one end.Hence, this method is also inappropriate for online tracking.

More seriously, Eq. (6a) does not work for a signal with a moving average. To illustrate this concept, we consider the

following signal u(t) consisting of two harmonics with constant amplitudes u1 and u2:

ut a1 cosf1 a2 cosf2; 7a

wheref1o1t,f2o2t, and o1and o2are constant. It follows from the vector plot on the uvplane and the law of cosinesthat

a12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffia21 a22 2a1a2 cosf1 f2

q ; f12 tan1

a1 sinf1 a2 sinf2a1 cosf1 a2 cosf2

; 7b

o12 _f12 12 o1 o2 12 o1 o2 a2

1 a2

2

a212: 7c

If a1=a2 and o2bo1E0, ut u a2 coso2t with u a1 coso1tnbeing a constant around t=tn. Then, it follows fromEq. (7c) that o12Eo2/2, but the actual instantaneous frequency should be close to o2because the instantaneous trajectoryin theuvplane circles around the tip point O2of the vectora1e

jo1tn with a circular frequency close too2. Becauseo1E0,the period 2p/o1-N and any finite-length sampling will present the signal as u(t)Ea1a2coso2taround t=0. HenceEq. (3) will give v(t)Ea2 sino2t because the a1 would be processed as the global average a0 in Eq. (2). The erroneousprediction of Eq. (7c) results from the assumption thatv(t) a1 sinf1a2 sinf2and use of the originO1in theuvplane tobe the instantaneous trajectory center. Moreover, to obtain v=HT(u)=a1 sinf1a2 sinf2 requires an infinite length ofsampling in order to present the signal as u(t)=a1 cosf1a2 cosf2 and to have a zero global average a0, which is almostimpossible to be numerically realized. Ifo1is not negligibly small, the instantaneous trajectory center will be at a point Othat is different fromO1andO2. All the mistakes and paradoxical theories about instantaneous frequencies in the literature

are mainly caused by the co-existence of these three centers O, O1, andO2and the difficulty in tracking them in the time/frequency domains (see, e.g., [20]).

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2.2. Tracking

For casual online frequency tracking, the TeagerKaiser algorithm (TKA) is a popular energy scheme that requires only

four recent data points [21,22]. TKA assumes the signal u(t) to be a regular harmonic and hence

u a cosotf; _u oa sinotf; u o2a cosotf; .u o3a sinotf;

cu

_u2

u u

o2a2; c_u

u2

_u.uo4a2

)o ffiffiffiffiffiffiffiffiffiffi

c _ucus ; a

cuffiffiffiffiffiffiffiffiffiffic _up x: 8

For digital signal processing, one can replace _u with the two-sample backward difference (unun1)/Dt and forwarddifference (un1un)/Dtand u with the three-sample central difference (un12unun1)/Dt2 to obtain

Dt2cun un un1un1 un unun1 2un un1 u2n un1un1 a2 sin2oDt;Dt4c _un un un12 un1 un2un1 un 4a2 sin2oDtsin2oDt=2;

)Dt4c _un

2Dt2cun 2sin2oDt=2 1 cosoDt;

o 1Dt

cos1 1 Dt4c _un

2Dt2cun

!; a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiDt2cun

1 cos2oDt

s ; 9

wheretn(n1)Dt. Because the signal is assumed to be a pure harmonic and it uses finite difference, TKA cannot accuratelytrack frequencies and amplitudes of signals with moving averages and noise-contaminated signals, as shown later in

Section 4 by examples.

To account for a possible moving average C0,to reduce influences of noise, and to use only three or more samples in the

frequency and amplitude tracking, we propose the following sliding-window tracking (SWT) method. The signal is

assumed to have the following form:

ut C0 e1 cosot f1 sinot C0 C1 cosot D1 sinot; 10

where C0, e1, andf1 are unknown constants, t t tnis a localized time coordinate, tn is the observed time instant, andC1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie21 f21

q cosotn f1; D1


q sinotn f1; f1 tan1f1=e1: 11

Theoat the fourth data point is proposed to be estimated using the TKA shown in Eq. (9) by processing the first four data

points, and hence this frequency-tracking method works only after the first 4 data points are available. After the fourthpoint, the previous points frequency will be used as an initial guess of the current points frequency. Then, the three

unknowns C0, C1, and D1 for the current data point at t 0 (i.e., t=tn) can be determined by minimizing the squareerrorEerror Sm1t0 ajijuni uni2, wherem(Z3) is the total number of processed data points,un1denotes the right-handside of Eq. (10) witht ti iDt, anduni denotes the actual u(t) att tn ti. Moreover,ajijis a weighting factor, and theforgetting factora(r1) is chosen by the user. However, a=1 is used for all examples presented in this work. Because thereare only three unknowns, only three recent data points need to be processed. However, more points can be used to reduce

the influence of measurement noise and/or signal-processing errors. After C1 and D1 are determined, the instantaneous

amplitudea, phase angle y, and frequencyo can be obtained as

a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC21 D21

q


q ; y tan1D1=C1 otn f1;o _y: 12

Because the initial guess ofo is updated using Eq. (12), this method is able to adapt and capture the actual variation offrequency. Eq. (12) reveals that D1is the Hilbert transform ofC1. To reduce the influence of noise on calculated frequency,

theo at t=tn can be computed by averaging over p Dtas

o Xp1i0

ytn iDt ytn i 1Dt=pDt: 13

However, this can be calculated only after the (mp)th data point. Eq. (9) shows that TKA requires minimal computation,but SWT requires more computation because the corresponding least-squares fitting requires formation and inversion of at

least a 33 symmetric matrix.

2.3. Dynamic characteristics of nonlinear oscillators

To illustrate that the steady-state response of a nonlinear system to a harmonic excitation is an amplitude- and

frequency-modulated harmonic and the proposed SWT method can track its time-varying frequency and amplitude to

reveal the systems order, type, and magnitude of nonlinearity, we consider the following Duffing oscillator subjected to aharmonic excitation having a frequency O close to its linear natural frequency o, and the corresponding steady-state

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second-order perturbation solution is[1]:

u 2Bo _u o2u au3 FcosOt; 14a

ut a1 cosOtf a3 cos3Ot 3f at cosOtfYt; a3 aa3132O2

5a1; 14b

at ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffia21 a23 2a1a3 cos2Ot 2fq a1 a3 cos2Ot 2f; 14cYt tan1 a3 sin2Ot 2f

a1 a3 cos2Ot 2f a3a1

sin2Ot 2f; 14d

O_

O _Y O 2Oa23 2Oa1a3 cos2Ot 2fa21 a23 2a1a3 cos2Ot 2f

O 2Oa3a1

cos2Ot 2f; 14e

3a8o

2a61

3as4oa41 s2 B2o2a21

F

2o

2 0; s Oo; 14f

f

tan1

Bo

3aa2

1=8o s;

14g

o o 3a8oa21; 14h

where f is the phase difference between the forcing function and the response, and B, a, F, and f are constants. Theamplitudea1is a nonlinear function ofF,B,o, anda, as shown in Eq. (14f) from perturbation analysis [1]. Eq. (14b) showsthatu(t) consists of two synchronous harmonics (i.e., reaching maxima at the same time), but, becausea1ba3, u(t) would

appear as one distorted harmonic (i.e., a harmonic with time-varying amplitude and frequency) with an amplitude a and a

frequency O_

varying at a frequency 2O. This phenomenon can be used to determine the order (cubic or other) of

nonlinearity. Moreover, ifa40, a3/a140;O_

anda are at their maxima when u(t) is at its maxima or minima. Ifao0, a3/a1o0;andO

_

anda are at their minima when u(t) is at its maxima or minima. This phenomenon can be used to determine

the type (hardening or softening) of nonlinearity, and the magnitude of nonlinearity can be estimated using Eq. (14b) and

the time-domain dataa(t) asa

32O2a3=a

3

1

. Moreover, one can use Eq. (14g) to estimate B, use Eq. (14h) to estimate theundamped amplitude-dependent natural frequency o, and use Eq. (14f) to obtain Fand then estimate the mass asm=F0/F,whereF0 is the known excitation amplitude[23].

3. Decomposition of compound signals

Decomposing a dynamic signal into a time-varying amplitude and a time-varying frequency is not always unique

because it is an inverse process and there is only one equation. For example, the following amplitude-modulated signal can

also be expressed as two regular harmonics with constant amplitudes and frequencies, or as a signal with time-varying

amplitude and frequency:

ut 1 e cos2OtcosOt 1 e=2cosOt e=2cos3Ot a cosOtY; 15a

at ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi1 e e2=2 e e2=2cos2Otq ; Yt tan1 e sin2Ot

2 e e cos2Ot: 15bHowever, the third form is also exact, and the three different forms have the same period 2p/O. Since the decomposition isnot unique, the question is which form is better for system identification?

Any signal can always be decomposed into components of a complete, convergent set of basis functions.

Because harmonic functions represent the free undamped motion of any physical system described by a linear second-

order differential equation, harmonic functions are important basis functions for analysis of dynamical systems.

Moreover, perturbation analysis of nonlinear dynamical systems reveals that nonlinear vibrations are characterized as

distorted harmonics [23,24]. An independent regular or distorted harmonic should have a positive amplitude

and a symmetric envelope, and independent harmonics behave differently when loading, initial, and/or boundary

conditions change. A signal consisting of multiple independent harmonics may result in a non-periodic signal, a

negative amplitude, and/or an asymmetric envelope, and it is inappropriate to define its instantaneous frequency

and instantaneous mean. For such a signal, it is better to decompose the signal into independent harmonics before

performing system identification. For example, it is more appropriate to decompose the signal in Eq. (16) into twocomponents because the combined signal may have a negative amplitude and it is aperiodic if O4o and O/o is an

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irrational number:

u a2

cosOot a2

cosOot a cosotcosOt: 16

Hence, for dynamics characterization and system identification, it is physically meaningful to decompose a dynamical

signal into regular or distorted harmonics with positive amplitudes and symmetric envelopes, and such tasks are exactly

what HilbertHuang transform can provide [13,23].

HilbertHuang transform (HHT) combines the empirical mode decomposition (EMD) and Hilbert spectral analysis for

timefrequency analysis of nonlinear and non-stationary signals. HHT is essentially different from Fourier and wavelet

transforms because it does not use pre-determined basis functions and the conformality between the basis functions and

the signal itself to extract components [1317]. The first step of HHT is to use EMD to sequentially decompose a signal u(t)

inton intrinsic mode functions (IMFs) ci(t) and a residual rn as

ut Xni1cit rnt; 17

wherec1has the shortest characteristic time scale and is the first extracted IMF. Once the extrema are identified, compute

the upper envelope by connecting all the local maxima using a natural cubic spline, compute the lower envelope by

connecting all the local minima using another natural cubic spline, subtract the mean of the upper and lower envelopes,

m11, from the signal, and then treat the residuary signal as a new signal. Repeat these steps for Ktimes until the left signal

has a pair of symmetric envelopes (i.e., m1KE0), and then define c1as

c1

u

m11

m1K:

18

This sifting process eliminates low-frequency riding waves, makes the wave profile symmetric, and separates the highest-

frequency IMF from the current residuary signal. During the sifting process for each IMF a deviation Dvis computed from

the two consecutive sifting results as

Dv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi1

c1kti c1k1ti2=XNi1c2

1k1ti

vuut ; 19where, for example, c1kum11?m1k, ti=(i1)Dt, and T(=NDt) is the sampled period. A systematic method to end theiteration is to limitDvto be a small number and/or to limit the maximum number of iterations. Afterc1is obtained, define

the residual r1(uc1), treatr1 as the new data, and repeat the steps shown in Eq. (18) to obtain other ci (i=2,y, n) ascn rn1 mn1 mnK; rn1 ut c1 cn1: 20

The whole sifting process can be stopped when the residual rn(=rn1cn) becomes a monotonic function from which nomore IMF can be extracted.After allci(t) are extracted, one can perform Hilbert transform (see Eq. (3)) to obtaindi(t) from eachci(t), then definezi(t),

and use Eq. (17) with rn being neglected to obtain

zit cit jdit Aiejyi ; Aiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i d2i

q ; yi tan1di=ci;

ut ReXni1

cit jdit Re

Xni1Aitejyit

; 21

whereoi and Ai are the time-varying frequency and amplitude of ci. The EMD is well known to be low in frequencyresolution, as pointed out in[25]. However, there are several signal-conditioning techniques to reduce this problem [13].

Moreover, if a high-frequency wave is added to the signal, the added wave and the signals discontinuities can be extracted

as the first IMF. Then the discontinuities can be accurately extracted by subtracting the added wave from the first IMF, and

this method can be used for noise filtering[23].

4. Numerical results

Signals for different engineering applications cover a wide frequency range. For example, AM radio signals range from

0.535 to 1.705 MHz (apart by 10 kHz between radio stations), and FM radio signals range from 88 to 108MHz (apart by

200 kHz). Because the human audible range is from 20 to 20 kHz, stereo audio is often sampled at 44.1 kHz, considering the

Nyquist theorem and Shannons sampling theory. Moreover, cell phones operate at around 850 MHz. However, after a

mathematical model is derived, one can always normalize the time to make a frequency under consideration to be a

specified low value (e.g., 1 Hz), and then vibration phenomena observed on the normalized signal should also exist in the

original signal. To prove this we consider the following nonlinear oscillator:

m u c_u ku kau3 m u 2Bo _u o2uo2au3 F0 cosOt; 22a

3 u 2Bo _u o2u au3 FcosOt; F F0=m: 22b

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If the timetis normalized into a non-dimensional timet using the linear vibration period 2p/oand the displacementu isnormalized into a non-dimensional displacement u using a characteristic length as

t ot2p

; u u: 23a

Eq. (22b) becomes

d2u

dt2 2B2pdu

dt 2p2

u

a

u3

f0 cosO

t; a

a2

; f04p2F0k ; O

2Opo : 23b

Note that Eq. (23b) is equivalent to Eq. (22b) witho=2pand F=f0. Eq. (23b) has a linear natural frequencyo 2pand a

linear periodT 1 in the normalized time domain. After the answeru is obtained in the t domain, the actual answer can be

obtained by scalingt andu back to the timetand the displacementu using Eq. (23a). In other words, the results obtainedby solving Eq. (22b) witho=2pis valid for any value ofo if an appropriate scaling is used. Next we simulate the trackingand extraction of time-varying frequencies and amplitudes of different signals using different methods and compare their

accuracy.

4.1. AM and FM signals

Eq. (24) represents an AMFM signal contaminated by a moving average C0and noise, where the number of data points

N=498 and Dt=0.1 are used, and noise is aN1 vector of normally distributed zero-mean random numbers with a standarddeviation of 0.004:

ut C0 1 0:3sinoatsinot 3cosopt noise;

C0 0:31 e0:1t; o 2p; oa 0:08o; op 0:05o: 24

IfC0=0 and there is no noise,Figs. 1a, b show the signal and its narrow-band Fourier spectrum,Figs. 1c, d and e,f show the

instantaneous frequency and amplitude tracked using the TeagerKaiser algorithm (TKA) and the sliding-window tracking

(SWT), respectively, andFigs. 1g, h show the frequency and amplitude extracted by HHT. BecauseC0=0, one can also use the

time-domain method shown by Eqs. (3)(6b) without using HHT. Eq. (9) shows that TKA always uses four consecutive

samples (un2, un1, un, un1) and predicts the frequency attn=(n1)Dt(instead oftn1, i.e., delay by one Dt). The presentedresults from SWT use 4 consecutive samples samples (un3, un2, un1, un) and predict the frequency at tn(no delay). SWT

can use just three or more points for tracking. Four consecutive local frequencies are used in Eq. (13) (or Eq. (6b) for HHT)

for these three methods to improve the tracked/extracted local frequency. Although TKA and SWT are derived by assumingthe processed signal to be harmonic, they adapt during tracking and are able to track time-varying frequencies and

amplitudes, as shown in Figs. 1cf. Both TKA and SWT have about the same level but different types of errors. Figs. 1g, h

show that the time-varying frequency and amplitude extracted by HHT are accurate everywhere except at the two ends,

which is due to Gibbs phenomenon in the time domain and leakage in the frequency domain caused by the discontinuity

uN1au1. There are methods for alleviation of the end effect in HHT [13,23], but this is not the main concern of this work

and is not presented here.

IfC0=0 but the noise is added,Figs. 2ad and e, f show the instantaneous frequencies and amplitudes tracked by TKA and

SWT, and those extracted by HHT, respectively. Because TKA uses a finite-difference scheme (see Eq. (9)) that does not

preserve the harmonic wave form to fit four recent samples, its accuracy is seriously reduced by noise, as revealed in Figs.

2a, b. On the other hand,Figs. 2c, d show that, because SWT insists on the use of a harmonic wave form (see Eq. (10)) to fit

four recent samples, it can filter out some noise. However, because of the insistence of a harmonic wave form, the step-

shape errors happen twice per cycle in the tracked frequency and amplitude. Moreover, because the first frequency in SWT

is estimated using TKA, it affects the accuracy of SWT at the beginning. Figs. 2e, f show that the frequency and amplitudeextracted by HHT are accurate because noise is significantly averaged out by the convolution computation (see Eq. (2)) over

the long sampled period T(E50S). However, the end effect caused by uN1au1 still exists.

If both C0 and noise are added,Fig. 3a shows the signal,Figs. 3b, e and f show the moving average C0, frequency, and

amplitude tracked by SWT, respectively andFigs. 3c, d show the frequency and amplitude tracked by TKA. Because a four-

point window length is not able to enforce orthogonality among C1,D1, andC0in Eq. (10), the C0tracked by SWT is not very

accurate but it is good enough for SWT to accurately track the frequency and amplitude, as shown inFigs. 3e, f. On the other

hand,Figs. 3c, d show that the small moving averageC0 totally destroys the tracking accuracy of TKA. Figs. 3g, h show the

time-varying frequency and amplitude of the first intrinsic mode function (IMF) extracted from HHT analysis, respectively.

During the sifting process for the first IMF, the signals upper and lower envelopes are not accurately determined because

the sampling interval Dtis not small enough to accurately show the signals peaks and/or because the moving average

changes the locations of peaks, and hence errors exist also at points away from two data ends. This type of error can be

reduced by using a smaller Dt, and the end effect can be reduced by extending the left end to begin from zero and the right

end to end with zero or other methods [23]. For example, Fig. 4 shows the time-varying frequency and amplitude areaccurately extracted by HHT when Dt=0.05S is used and the two ends are extended to begin and end with zero.

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Fig. 1. Tracking/extraction of the frequency and amplitude of Eq. (24) withoutC0and noise: (a)u, (b) spectrum, (c, d) from TKA, (e, f) from SWT, and (g, h)

from HHT. The thin curves represent the exact time-varying frequency and amplitude.


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Frequency tracking using TKA or SWT can start after four data points are available. However, TKA works only for a signal

with no moving average and noise. On the other hand, SWT works for a signal with any non-zero moving average and noise

and is more robust, but it depends on TKA for the first frequency estimation. After the fourth data point, SWT can use only

three recent data points for frequency tracking, but using more data points can eliminate more noise. Because HHT suffers

from the end effect, it is more appropriate for frequency extraction (instead of tracking) because it requires a long window

length in order to reduce the edge effect. Hence, if the sampling frequency and the number of data points used for

frequency tracking are fixed, TKA and SWT are good for tracking high-frequency signals, and HHT is good for extracting

low-frequency signals.

Measurement noise may come from different sources. If the measurement noise does not change with Dt, numerical

simulations show that the tracking accuracy of SWT and TKA increases when the window length is equal to or greater than

one quarter of the signals period. (To cover one quarter of a signals period with fixed three or four data points, one can

downsample the recent available data.) On the other hand, if the window length is too long, the computed frequency is anaverage over the window length, not the instantaneous frequency. Under these constraints, SWT and TKA are not good at

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Fig. 2. Tracking/extraction of the frequency and amplitude of Eq. (24) without C0: (a, b) from TKA, (c, d) from SWT, and (e, f) from HHT. The thin curves

represent the exact time-varying frequency and amplitude.


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Fig. 3. Tracking/extraction of the frequency and amplitude of Eq. (24): (a) u, (b)C0from SWT, (c, d) from TKA, (e, f) from SWT, and (g, h) from HHT. The

thin curves represent the exact time-varying frequency and amplitude.


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frequency tracking of a signal modulating at a frequency close to or higher than its fundamental frequency, as shown later

in Section 4.3. Fortunately, AM and FM signals for engineering applications often have a carrier frequency ten or more times

of the frequency of the signal to be transferred.

4.2. Vibrato in music

For music composition, synthesis, and performance, vibrato (i.e., frequency modulation) is very important. Because

string music comes from nonlinear vibration of strings and is based on sympathetic resonance, vibrato is even more

important for string music[19]. In order to model and analyze vibrato during notes and in note transition, accurate vibrato

rate (frequency) and vibrato extent (amplitude) are needed. To compare the three methods for tracking/extraction of

frequency and amplitude, we consider the transition of a harmonic modulated around o1to another harmonic modulatedaround o2 as

u acosZ t

0odt noise a cos o2 o1

2 to2 o1

2 trln

cosht ta=trcoshta=tr

avov

sinovtf

noise;

a

a2 a1

2

a2 a1

2

tanht ta

tr

1

0:2cos

ovt

;

o o2 o12

o2 o12

tanht tatr

av cosovtf; 25

where the number of data points N=5000, Dt=1/5000 s,noise is aN1 vector of normally distributed zero-mean randomnumbers with a standard deviation of 0.002, the exact time-varying amplitude and frequency areaand o, the fundamentalfrequency of the first note iso1=4402p, the frequency of the second note is o2=4932p, transition moment ta=0.52 s,transition parameter tr=0.003 s, magnitude of the vibrato av=302p, frequency of the vibrato ov=52p, phase on thevibrato f=p/2,a1=1, anda2=0.6. Note that tanh[(tta)/tr] represents a function that changes from 1 to 1 aroundt=ta witha transition period determined by tr.

Fig. 5shows the tracking/extraction of frequency and amplitude from TKA, SWT, and HHT analyses. The noise causes

errors inFigs. 5a, b from TKA, the discontinuity (uN1au1) causes the end effect inFigs. 5e, f from HHT, and the magnitude

of error inFigs. 5c, d from SWT is about the same as that in Figs. 5e, f from HHT. The exact time-varying frequency and

amplitude are also plotted inFigs. 5af, but they are covered by the tracked/extracted results. However, zoom-in viewsshow that, around the transition momentta=0.52 s, the HHT results closely follow the exact ones, but the SWT results show

minor step-shape errors (seeFigs. 3e,f).

4.3. Nonlinear stationary and non-stationary signals

First we consider the nonlinear oscillator shown in Eq. (14a) with

o 2p; B 0:03; a o2=6 6:5797; m 2; F0 20 Fm; O o: 26

Because sin uEuu3/6, this system mimics the nonlinear angular vibration of a vertical pendulum subject to a harmonictangential force. Direct numerical integrations are performed using the RungeKutta method with Dt=0.05 s. The power

spectral density (PSD) inFig. 6b indicates that the signal inFig. 6a mainly consists ofO and 3Oharmonics (and negligibly

small 5O and 7O harmonics). Figs. 6g, h show that the frequency and amplitude modulate at 2O, indicating a cubic

nonlinearity (see Eqs. (14c, e)). Moreover, because the frequency and amplitude are at their minima whenuis at its maximaor minima, ao0. The average amplitude a1 and the fluctuation amplitude a3 can be calculated from Fig. 6h to be

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Fig. 4. Analysis of Eq. (24) using HHT with Dt=0.05s: (a) frequency and (b) amplitude.


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a1=1.2495 and a3=0.0101, and then Eq. (14b) can be used to estimate a 32O2a3=a31 6:5407. Furthermore,we obtain f=0.3079 from Fig. 6a and Eqs. (14a,b) and B 3aa21=8o stanf=o 0:0308 from Eq. (14g)

with s(=Oo). Then Eq. (14f) with s=0 is used to obtain F=10.0405(=F0/m) and hence m=F0/F=1.9919 is estimated. Thekey point here is that all a, B and m can be estimated from just one steady-state response to a harmonicexcitation with OEo. The estimations can be refined if more sets of responses under different F0 and/orO are available. Simulations show that the method is very sensitive and reliable because, even if F0=1,

the small variations of amplitude and frequency at 2O are still clearly revealed. On the other hand, Figs. 6cf show that

the frequency and amplitude tracked by TKA and SWT are not accurate, especially those from TKA. This is expected

because SWT and TKA are not able to track a signal modulating at a frequency close to or higher than the

fundamental frequency, as discussed in Section 4.1. However, SWT and TKA are able to capture the modulation at 2O,

indicating cubic nonlinearity. Numerical simulations also show that, for an oscillator with quadratic nonlinearity,

SWT and TKA can capture the frequency and amplitude modulation at O (as the corresponding perturbation

solution predicts [23]) although the tracked frequency and amplitude are not accurate. However, the

errors of SWT inFigs. 6e, f can be significantly reduced by using more data points (e.g., six or eight points) in Eq. (13).

Next we consider the nonlinear oscillator shown in Eq. (14a) with

o 2p; B 0:03; a o2=6 6:5797; m 2; F0 0; u0 2; u_ 0 0: 27

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Fig. 5. Tracking/extraction of the frequency and amplitude of Eq. (25): (a, b) from TKA, (c, d) from SWT, and (e, f) from HHT.


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The two peaks in Fig. 7b indicate the existence of cubic nonlinearity because they are at 0.95 and 3 0.95Hz. Figs. 7e, freveal that the nonlinearity is a softening type because the frequency increases when the vibration amplitude decreases.

ForFigs. 7e,f, p=6 is used in Eq. (13) in order to have better accuracy. Although the first frequency provided by TKA is not

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Fig. 6. Tracking/extraction of the frequency and amplitude of Eqs. (14a) and (26): (a) u, (b) PSD, (c, d) from TKA, (e, f) from SWT, and (g, h) from HHT.


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Fig. 7. Tracking/extraction of the frequency and amplitude of Eqs. (14a) and (27): (a) u, (b) PSD, (c, d) from TKA, (e, f) from SWT, and (g, h) from HHT.


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accurate,Fig. 7e shows that SWT adapts quickly to the correct frequency.Figs. 7d, f, h show that each method accurately

captures the amplitudes at the peak locations of the signal (thin light curves), but they have different types of errors at

other locations. Moreover, the thin light curves inFigs. 7c, e and g represent the amplitude-dependent natural frequency opredicted by Eq. (14h). Note that the modulation of frequency at 2 o in Fig. 7e indicates cubic nonlinearity (see Eqs.(14c, e)), but the modulation in Fig. 7g from HHT is not clear. Moreover, the time-varying frequency extracted by HHT

suffers from the serious end effect caused by the big discontinuity uN1au1. (But this end effect can be reduced by

extending the signal to begin and end with zero as demonstrated inFig. 4.)Figs. 7c, e show that SWT is more accurate than

TKA for frequency tracking. Moreover, if noise is added to the signal, the accuracy of TKA is totally destroyed, but SWT isstill able to track the frequency and amplitude.

Lastly we consider the following non-stationary system:

m u c_u rku au3 m u 2Bo _u ro2u au3 F0 cosOt;

r0:975 0:025tanh t 7

trfor0otr9

0:925 0:025tanh t 11tr

for9ot;

8>>>>>:

o 2p; B 0:03; a 0:1; m 2; F0 2 Fm; O o; tr 0:05: 28

After integration for 40 s usingDt=0.05 s, the systems stiffness is reduced by 5% att=7 s and then another reduction of 5% at

t=11 s. A normally distributed zero-mean random noise with a standard deviation of 0.0005 is added to the numerically

integrated solution. The PSD inFig. 8b does not reveal nonlinearity and transient events.Figs. 8c, d show that the accuraciesof frequency and amplitude tracked by TKA are seriously destroyed by the added small noise. However,Figs. 8g, h show that

the frequency extracted by HHT is able to reveal the two time instants of sudden stiffness reduction but the amplitude is

not really able to. Although the two ends ofu have been extended to begin and end with zero in order to reduce the end

effect, the end effect still reduces the extraction accuracy of HHT. On the other hand,Figs. 8e, f show that the frequency and

amplitude tracked by SWT clearly reveal the two time instants of stiffness reduction, wherep=6 is used in Eq. (13) to reduce

the fluctuation of the tracked frequency.

4.4. Discussions

As exampled by Eqs. (14c, e), the frequency and amplitude of a nonlinear dynamical systems response may modulate at

a frequency higher than the fundamental frequency, and SWT is not accurate for such signals. However, if the order of

nonlinearity is known, the accuracy of SWT for frequency tracking can be significantly improved. For example, if Eq. (10) is

replaced with

ut e1 cosot f1 sinot e2 cos3ot f2 sin3ot C0 C1 cosot D1 sinot C2 cos3ot D2 sin3ot C0: 29

Fig. 9 shows the frequency and amplitudes of the signal in Fig. 6a tracked by SWT. Because the starting data points

frequency is estimated using TKA and is inaccurate, the tracked frequency and amplitudes fluctuate at the beginning but

quickly converge to the correct values, i.e., o=2p, a1=1.2495, and a3=0.01 (see Eq. (14c)). For nonlinear systems withquadratic nonlinearity, 2oneeds to be used in Eq. (29) for the second pair of harmonics. Because there are five unknownsin Eq. (29), the number of processed data points, m, needs to be Z5. Unfortunately, when both frequency and amplitude

change with time (e.g., Fig. 7a), this approach cannot accurately track frequencies and amplitudes because the

orthogonality among the five functions in Eq. (29) cannot be strictly enforced.

To show the convergence of SWT using only three recent data points, we consider the signalu=sin(2pl)noise, where the

number of data points N=4000,D

t=0.0025 s,T=10 s, and noise is aN1 vector of normally distributed zero-mean randomnumbers with a standard deviation of 0.05. By downsampling, the time step can be increased to Dt pDt and hence the

period covered by three data points becomes 2Dt 2pDt. Fig. 10 shows that, when the covered period increases, the

convergence of the mean value is faster than the standard deviation, and the standard deviation is less than 3% (i.e.,

0.03Hz) when the covered period is larger than 0.25s (i.e., 25%).

For a second-order dynamical system, the autoregressive moving average (ARMA) modeling is a popular method for

online system identification. Considering the state at t=ti1 and using _u i1 ui1 ui2=Dt and u i1 ui 2ui1 ui2=Dt2 yieldsm u i1 c_ui1 kui1 fi13ui a1ui1 a2ui2 b1fi1; 30a

where

a1 2 cDt=mx kDt2=m 1 a2 kDt2=m; a2 cDt=m 1; b1 Dt2=m; 30b

ti(i1)Dt, and i=1,y, N. This is the so-called ARMA(2, 1) model. Each of Eq. (30a) requires the use of three consecutivedata points, and it needs at least three such equations (hence at least five data points). Moreover, the forcing function needs

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Fig. 8. Tracking/extraction of the frequency and amplitude of Eq. (28): (a) u, (b) PSD, (c, d) from TKA, (e, f) from SWT, and (g, h) from HHT.


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to be known in order to estimatea1, a2, andb1. After that, one can use the results to estimate the system parameter using

o2 km 1 a1 a2

Dt2 ; c

m a2 1

Dt ; o2d o2

a2 12Dt

2; m Dt

2

b1: 30c

If it is a free vibration (i.e., fi1), m cannot be estimated. It is obvious that, if it is a nonlinear system and the form of

nonlinearity (e.g., cubic nonlinearity) is known, the algebraic equations in Eq. (30a) will be nonlinear and are difficult to

solve. Moreover, because finite difference is used, noise would affect the estimation accuracy, as happens to TKA.

5. Concluding remarks

Here we derive and numerically validate a sliding-window tracking (SWT) method that uses only three (or more) recent

data points to accurately track the instantaneous frequency and amplitude of a general noise-contaminated signal with a

moving average. On the other hand, the popular TeagerKaiser algorithm (TKA) requires four recent data points for online

frequency tracking, and it cannot track noise-contaminated signals and signals with moving averages. However, SWT

depends on TKA to provide the first estimation of frequency in order to start online tracking. For comparison of TKA and

SWT, HilbertHuang transform is used to extract accurate time-varying frequencies and amplitudes. Results show that

SWT is more accurate, versatile, and robust than TKA. However, for a signal with frequency and/or amplitude varying at a

frequency close to or higher than the signals fundamental frequency, the tracking accuracy of SWT decreases. This SWT

method can be used for many applications in online adaptive system control, structural health monitoring and damagedetection, and system characterization.

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Fig. 9. Frequency and amplitude of the signal in Fig. 6a tracked by SWT using Eq. (29): (a) the main frequencyo and (b) amplitudes a1(thick line) and1.2a3(thin line).

1.5

1

0.5

0

0 0.2 0.4 0.6 0.8

Covered period (s)

Standarddeviation,

Mean(Hz)

Fig. 10. Convergence of the tracked frequency of a noise-contaminated 1-Hz harmonic using the three-point SWT method, where the broken line is the

mean value and the solid line is the standard deviation.


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References

[1] A.H. Nayfeh, D.T. Mook, in: Nonlinear Oscillations, Wiley-Interscience, New York, 1979.[2] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage identification and health monitoring of structural and mechanical systems from changes

in their vibration characteristics: a literature review, Report No. LA-13070-MS, Los Alamos National Laboratory, 1996.[3] H. Sohn, C.R. Farrar, F.M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, A review of structural health monitoring literature: 19962001, Report No.

LA-13976-MS, Los Alamos National Laboratory, 2003.[4] F.K. Chang (Ed.), Structural hea lth monitoringadvances and challenges in design and implementation, in: Proceedings of the Fifth International

Workshop on Structural Health Monitoring, Stanford, CA, September 1417, 2005. Destech Publishing.

[5] K. Worden, G.R. Tomlinson, in: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Physics Publishing, Bristoland Philadelphia, 2001.

[6] R. Pintelon, J. Schoukens, in: System Identification: A Frequency Domain Approach, IEEE Press, New York, 2001.[7] E.O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.[8] S.L. Hahn, Hilbert Transforms in Signal Processing, Artech House, Boston, 1996.[9] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Lecture Notes 61, SIAM, Philadelphia, 1992.

[10] P. Flandrin, in: TimeFrequency/Time-scale Analysis, Academic Press, 1999.[11] S.A. Neild, P.D. McFadden, M.S. Williams, A review of timefrequency methods for structural vibration analysis, Engineering Structures 25 (2003)

713728.[12] D.F. Shi, L.S. Qu, N.N. Gindy, General interpolated fast Fourier transform: a new tool for diagnosing large rotating machinery, Journal of Vibration and

Acoustics 127 (2005) 351361.[13] N.E. Huang, N.O. Attoh-Okine (Eds.), The HilbertHuang Transform in Engineering, CRC Press, Boca Raton, FL, 2005.[14] N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the Hilbert

spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society of London Series AMathematical Physical andEngineering Sciences 454 (1998) 903995.

[15] N.E. Huang, Z. Shen, S.R. Long, A new view of nonlinear water waves: the Hilbert spectrum, Annual Reviews of Fluid Mechanics 31 (1999) 417457.[16] N.E. Huang, M.C. Wu, S.R. Long, S.S.P. Shen, W. Qu, P. Gloersen, K.L. Fan, A confidence limit f or the empirical mode decomposition and Hilbert spectral

analysis, Proceedings of the Royal Society of London Series A

Mathematical Physical and Engineering Sciences 459 (2003) 23172345.[17] Z. Wu, N.E. Huang, A study of the characteristics of white noise using the empirical mode decomposition method, Proceedings of the Royal Society of

London Series AMathematical Physical and Engineering Sciences 460 (2004) 15971611.[18] E. Bedrosian, On the quadrature approximation to the Hilbert transform of modulated signals, Proceedings of the IEEE 51 (1963) 868869.[19] P. Rossignol, P. Desain, H. Honing, State-of-the-art in fundamental frequency tracking, in: Proceedings of the Workshop on Current Research

Directions in Computer Music, Barcelona, Spain, November 1517, 2001, pp. 244254.[20] P.J. Loughlin, B. Tacer, Comments on the interpretation of instantaneous frequency, IEEE Signal Processing Letters 4 (1997) 123125.[21] P. Maragos, J.F. Kaiser, T.F. Quatieri, Energy separation in signal modulations with application to speech analysis, IEEE Transactions on Signal

Processing 41 (1993) 30243051.[22] D. Vakman, On the analytic signal, the TeagerKaiser energy algorithm, and other methods for defining amplitude and frequency, IEEE Transactions

on Signal Processing 44 (1996) 791797.[23] P.F. Pai, A.N. Palazotto, Detection and identification of nonlinearities by amplitude and frequency modulation analysis, Journal of Mechanical Systems

and Signal Processing 22 (2008) 11071132.[24] P.F. Pai, Highly Flexible Structures: Modeling, Computation, and Experimentation, AIAA, Reston, VA, 2007.[25] G. Rilling, P. Flandrin, One or two frequencies? The empirical mode decomposition answers, IEEE Transactions on Signal Processing 56 (2008) 8595.

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