ARTICLE IN PRESS

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 23 (2009) 1447–1457

0888-32

doi:10.1

� Cor

E-m

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

Time domain averaging based on fractional delay filter

Wentao Wu a, Jing Lin b,�, Shaobo Han a, Xianghui Ding a

a Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, PR Chinab School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, Sha’anxi Province 710049, PR China

a r t i c l e i n f o

Article history:

Received 4 October 2008

Received in revised form

19 January 2009

Accepted 23 January 2009Available online 14 February 2009

Keywords:

Time domain averaging

Fractional delay

Phase accumulation error

70/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ymssp.2009.01.017

responding author.

ail address: jing_lin00@sina.com (J. Lin).

a b s t r a c t

For rotary machinery, periodic components in signals are always extracted to investigate

the condition of each rotating part. Time domain averaging technique is a traditional

method used to extract those periodic components. Originally, a phase reference signal

is required to ensure all the averaged segments are with the same initial phase. In some

cases, however, there is no phase reference; we have to establish some efficient

algorithms to synchronize the segments before averaging. There are some algorithms

available explaining how to perform time domain averaging without using phase

reference signal. However, those algorithms cannot eliminate the phase error

completely. Under this background, a new time domain averaging algorithm that has

no phase error theoretically is proposed. The performance is improved by incorporating

the fractional delay filter. The efficiency of the proposed algorithm is validated by

some simulations.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Time domain averaging (TDA) is a classic technique for periodic components extraction, which is especially useful inrotary machinery diagnosis. It improves the signal-to-noise ratio (SNR) by suppressing the asynchronous components insignals [1,2] and has wide applications in machine vibration monitoring and fault diagnosis [3–6]. In this process, signal isfirst truncated into several segments according to the revolution period of the rotating part. Then, all the segments aresummed up together so that noncoherent components and nonsynchronous components are canceled out. To suppressthese interference components, a large number of segments for averaging are required. As a result, this method works wellonly if all the segments are with the same initial phase. Traditionally, this constraint is guaranteed by unitizing an externaltrigger signal provided by an additional hardware such as a tachometer. However, in some circumstances, there is notrigger signal available. Therefore, a further approach is necessary in order to ensure that all the averaging segments havethe same initial phase.

There are two problems to be solved if no trigger signal is available. Firstly, the trigger signal has to be replaced byrotating period estimation. In Ref. [7], TDA is performed by using the estimated revolution period. In this process, therevolution period is confined to a certain range. Then, the exact value is determined by using some rules. This idea alsobenefits the adaptive comb filtering [8,9]. Actually, the revolution period can be estimated exactly by this idea [10]. Thetrigger problem need not be considered. Secondly, even if the revolution period is known exactly, the direct averagingapproach is not sufficiently effective since there is error for the period of the required component, called cutting error,which is caused by fractional sampling interval. It is accumulated by the increment of the segment number, which leads the

ll rights reserved.

ARTICLE IN PRESS

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–14571448

so-called phase accumulation error. Direct TDA algorithm suffers significantly by the cutting error. Braun firstly proposedsome methods less sensitive to the synchronous error and applied them to some specific cases [1,3,4]. McFadden presentedan off-line interpolation technique, which performed TDA for vibration signals from several locations by only using onephase reference [2,5]. Moreover, an improved algorithm was introduced by Liu et al. [11], which limits the cutting error ofevery segment within a half sampling interval. The accumulated error is reduced to a large extent accordingly. However,theoretically, current algorithms cannot eliminate phase accumulation error completely.

Fractional delay filter is a device for band-limited interpolation between samples. It is used in numerous fields, such ascommunications, audio, antenna and transducer array. Only if the sampling rate satisfies the Nyquist criterion [15], thesampled series can represent adequately the original continuous signal. The delay of fractional sampling interval can besolved by viewing a delay as a resampling process. It can be obtained by first reconstructing the continuous band limitedsignal and resampling it after shifting. But there are many design methods to approximate the process, which need notperform reconstruction and resampling explicitly. An excellent survey of the fractional delay filter design has beenpresented in [12,14]. The fractional delay filter can provide an accurate compensation for time delay. In this article, it isincorporated into TDA to improve the performance.

This article is organized as follows. The introduction is given in Section 1. In Section 2, TDA and its performance arereviewed and the error analysis for current algorithms is given. In Section 3, the fractional delay filter is introduced and aTDA algorithm based on fraction delay filter is proposed. Simulations and comparisons of three TDA algorithms are given inSection 4. Finally, conclusions are presented in Section 5.

2. Time domain averaging technique and error analysis

2.1. Review of time domain averaging

Usually, signals from rotating machinery are a combination of periodic signals with random noise. Consider a signal x(t)composed of a periodic signal x0(t) with a fundamental period T0 (the fundamental frequency is f0 accordingly) and a noisycomponent n(t):

xðtÞ ¼ x0ðtÞ þ nðtÞ (1)

The periodical components x0(t) can be extracted by using the time domain averaging. The method can be expressed as

yðtÞ ¼1

M

XM�1

r¼0

xðt þ rT0Þ (2)

where M is the number of the averaged segments, y(t) is the averaged result. Suppose signal is sampled by Dt. Then

xðiDtÞ ¼ x0ðiDtÞ þ nðiDtÞ (3)

yðiDtÞ ¼1

M

XM�1

r¼0

xðiDt þ rN �DtÞ; i ¼ 0;1; . . . L� 1 (4)

where N ¼ round (T0/Dt). The transfer function of time domain averaging can be expressed as

HðjoÞ�� �� ¼ 1

M

sinðpMo=o0Þ

sinðpo=o0Þ

��������; o0 ¼

2pT0

(5)

Fig. 1 is the frequency response of the TDA. It can be easily found that all the synchronous components – harmonics arereserved, however others are suppressed. The signal-to-noise ratio is improved accordingly. The number of segments is 16in this example. The SNR will increase as the number of segment increases.

It is convenient to deal only with one of the periodic signal of the transfer function. Firstly, a new variable F ¼

f=f 0 � K ; K ¼ 0;1;2;3 is defined. This causes all the harmonics to overlap and be centered around the origin. Thus, weonly need to investigate the transfer function within the fundamental frequency bin. Take T equal to the fundamentalfrequency period, the comb filter will center on the fundamental and harmonic frequencies. As hinted in Ref. [1], theperiodic waveforms are extracted and the noises are rejected. To evaluate the performance of rejecting the white noises, theequivalent noise bandwidth (ENB) is given by [1]

ENB ¼

Z 0:5

�0:5

1

N

sin pNF

sin pF

� �2

dF ¼p

N2

Z p=2

�p=2

sin Nx

sin x

� �2

dx ¼ p2=N (6)

As expected, the averaged white noise decreases by 1/N. For the interfering periodic signals, the exact attenuationdepends on the specific wave shape. To achieve the attenuation by a, the smallest necessary order is N4a=p2F, N must beestimated somewhat conservatively.

The triggering error effect is very important in TDA, because it may cause synchronous error. In some circumstances,there is even no phase reference available. It is also necessary to synchronize all the segments for averaging. Usually, the

ARTICLE IN PRESS

M=161

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.3

0

0.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5The normalized frequency

Am

plitu

de (d

B)

Fig. 1. Frequency response of the TDA.

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–1457 1449

revolution period is roughly a priori. Some methods have already been proposed on how to determine the exact period [10].In this paper, the revolution period is supposed known exactly.

2.2. Error analysis

Suppose that the Fourier series of signal x(t) are given by

xðtÞ ¼Xþ1

m¼�1

jAmj expðj2pmf 0t þ jfmÞ (60)

where f0 is the fundamental frequency, jAmj is the amplitude of the Fourier coefficients, fm is the phase of the Fourier coefficients.Assume tr is the starting time of sample and Dt is the sampling interval, the (r+1)th segment for averaging is given by

xrðiDtÞ ¼ xðiDt � trÞ

¼Xþ1�1

jAmj exp½j2pmf 0iDt � j2pmf 0tr þ jjm� (7)

The phase of the mth harmonic in the (r+1)th segment is

fr;m ¼ �2pmf 0tr þjm (8)

The phase of the mth harmonic in the first segment is

f0;m ¼ �2pmf 0t0 þjm (9)

Therefore, the phase difference cr,m between jr,m and j0,m is

cr;m ¼ fr;m � f0;m ¼ 2pmf 0ðt0 � trÞ (10)

In practice, time difference between the (r+1)th segment and the first segment is always approximated by using severalsampling intervals for simplicity. Suppose lr is the number of the intervals. Then we have

xrðiDtÞ ¼ xðiDt � lrDtÞ ¼Xþ1�1

jAmj exp½j2pmf 0iDt � j2pmf 0lrDt þ jjm� (11)

The difference between the phase difference cr,m of the mth harmonic within the (r+1)th segment and that within thefirst segment is

cr;m ¼ fr;m � f0;m ¼ 2pmf 0lrDt (12)

There are two ways to determine lr in current algorithms. In conventional algorithm, lr is given by

lr ¼ r � roundFs

f 0

� �(13)

cr;m ¼ fr;m � f0;m ¼ 2pmr roundFs

f 0

� �f 0

Fs(14)

ARTICLE IN PRESS

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–14571450

In the algorithm proposed in [4], lr is given by

lr ¼ roundr Fs

f 0

� �(15)

cr;m ¼ fr;m �f0;m ¼ 2pm roundr Fs

f 0

� �f 0

Fs(16)

When Fs/f0 is not an integer, there are phase errors in both algorithms mentioned above. In conventional algorithm,

lr ¼ r � roundFs

f 0

� �¼ r �

Fs

f 0þDM

� �(17)

where DM is the rounded error, �0.5pDMo0.5.The phase error

cr;m ¼ fr;m �f0;m ¼ 2pmrFs

f 0

þ rDM

� �f 0

Fs¼ 2pmr 1þ DM

f 0

Fs

� �(18)

m and r are integers; therefore, the phase error is given by

cr;m ¼ fr;m �f0;m ¼ 2pmrDMf 0

Fs(19)

The phase error solely depends on r since DM and f0/Fs are constants. As the phase error increases with r, the error can beconsidered as accumulation error.

In the algorithm mentioned in [11],

lr ¼ roundrFs

f 0

� �¼ r

Fs

f 0þ DL (20)

DL is the rounded error, �0.5pDLo0.5. The phase error is

cr;m ¼ fr;m �f0;m ¼ 2pmrFs

f 0

þDL

� �f 0

Fs¼ 2pm r þ DL

f 0

Fs

� �(21)

It can be seen in formula (20) that the equivalent phase error is 2pmDLðf 0=FsÞ and the equivalent time delay is DL�Dt,which is less than the half sampling interval. It eliminates the accumulation error. However, it fails to reduce the phaseerror.

We may use the so-called Phase Error Accumulation Factor (PEAF) [13] to quantify the accumulation error. Inconventional algorithm,

PEAF ¼f 0

FsDM (22)

Then the result of conventional TDA is given by

yrðiDtÞ ¼XN�1

r¼0

xrðiDtÞ ¼Xþ1�1

GmjAmj exp½j2pmf 0iDt � j2pmf 0lrDt þ jjm� (23)

Gm is the magnitude of the TDA, as

Gm ¼sinðpmN � PEAFÞ

N sinðpm� PEAFÞ

�������� (24)

where m is the harmonic order and the N is the number of segments. If the PEAF is small, Gm can be approximated by Taylorseries:

Gm ¼ 1�p2m2 � PEAF2

6ðN2� 1Þ þ OðPEAF4

Þ (25)

In the main lobe, the attenuation of the amplitude increases with quadratic of the PEAF, harmonic order m and thenumber of segments. Fig. 2 gives the relationship between the amplitude of gain and the number of segments.

3. The TDA algorithm based on fractional delay filter

3.1. Fractional delay filter

Delaying of a continuous time signal xc(t) by td is conceptually simple. The problem can be described by a linearoperator, Lc, which yields its output yc(t) as

ycðtÞ ¼ LcfxcðtÞg ¼ xcðt � tdÞ (26)

ARTICLE IN PRESS

m PEAF=0.0005m PEAF=0.001m PEAF=0.01m PEAF=0.1

1

0.9

0.8

0.7

0.5

0.4

0.3

0.2

0.1

00 100 200 300 400 500 600 700 800 900 1000

0.6

The

gain

of t

he a

mpl

itude

The number of the averaging

Fig. 2. Magnitude of gain vs. the number of segments.

1

0.5

0

1

0.5

0

-0.5-4 -2 0 2 4 6

-0.5-4 -2 0

Time2 4 6

Time

Fig. 3. Ideal fractional delay filter: (a) D ¼ 1 samples and (b) D ¼ 1.3 samples.

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–1457 1451

However, delaying of a uniform sampled band-limited digital signal is not clearly defined. Simply converting continuoustime signal to digit signal by sampling at t ¼ nT, where n is an integer and T is the sampling interval, we obtain

yðnÞ ¼ LfxðnÞg ¼ xðn� DÞ (27)

where D is a positive real number that can be split into the integer and fractional part as

D ¼ intðDÞ þ d (28)

However, formula (27) is meaningful only for integral values of D. For non-integral values of D, the output value will liesomewhere between two adjacent samples. We may employ a so-called fractional delay filter to solve this problem.

A fractional delay filter is a device for fractional delay of digital signal [12]. Generally, the problem can be solved byviewing a delay as a resampling process. The process consists of reconstructing the continuous signal and resampling itafter shifting.

As shown in Fig. 3, an ideal fractional delay element is a digital version of a continuous-time delay line. The delaysystem must use an ideal lowpass filter so that the delay merely shifts the impulse response in the time domain. Theoperation relates to interpolation and approximation technique. There are two methods that can be used forapproximation: FIR approximation and all-pass filter approximation. There are some problems in IIR, such as limit cyclesand potential instability in coefficient quantization. Therefore, FIR approximation is used in this article.

ARTICLE IN PRESS

MAGNITUDE RESPONSES

D=0

D=0.1

D=0.2

D=0.3

D=0.4

D=0.5

D=0.5

D=0.4

D=0.3

D=0.2

D=0.1

D=0

1.2

1

0.8

0.6

0.4

0.2

5.6

5.5

5.4

5.3

5.2

5.1

5

4.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NORMALIZED FREQUENCY

NORMALIZED FREQUENCY

PHASE DELAY RESPONSES

MA

GN

ITU

DE

PH

AS

E D

ELA

Y

Fig. 4. Response of the Lagrange filter, L ¼ 11.

Fig. 5. Diagram of the TDA based on fractional delay filter.

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–14571452

In the FIR approximation, there are many classic algorithms, such as window function, maximally-flat FIRapproximation and max–min approximation. Lagrange interpolation algorithm is a popular way to design an FIR filterfor a given fractional delay approximation [12]. In this algorithm, the delay z�D is approximated by

z�D �XN

n¼0

hðnÞz�n (29)

where filter coefficients h(n) have the explicit form

hðnÞ ¼YN

k¼0;kan

D� k

n� k; n ¼ 0;1; . . . ;N. (30)

Illustrations are given in Fig. 4 to show the attributes of Lagrange interpolation. The length of the filter L ¼ N+1 ¼11. Thevalue of D is assigned 0, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively. The magnitude response and the phase delay response areboth shown in Fig. 4, which shows that Lagrange interpolation is able to take this task.

ARTICLE IN PRESS

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–1457 1453

3.2. The TDA algorithm based on fractional delay filter

The TDA algorithm based on fractional delay filter is given in Fig. 5. There are three steps in the algorithm, as indicatedin the blocks. In block (1), the signal is divided into M segments, where xrðnÞ ¼ x n� rFs=f 0

� �� �. These M segments are

synchronized in block (2). Time delay of the (r+1)th segment is tr ¼ Dlr=Fs, where Dlr ¼ ðrFs=f 0Þ � rFs=f 0

� �. In this process,

fractional delay filter is employed for compensation. In block (3), all segments are summed up to suppress theasynchronous components.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5

-4

-3

-2

-1

0

1

2

3

4

5the waveform of Xp(t)

Time

Am

plitu

de

Fig. 6. Time plot of XP(t).

The waveform of g (t)

The waveform of Xr (t)

5

0

-5

5

0

10

5

0

-5

-100

-5

0 0.01 0.02 0.03 0.04 0.05 0.06Time

The waveform of Xn (t)Time

Time

0.07 0.090.08 0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.090.08 0.1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.090.08 0.1

Am

plitu

deA

mpl

itude

Am

plitu

de

Fig. 7. Time plots of simulated signal.

ARTICLE IN PRESS

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–14571454

4. Simulations and discussions

Some simulations are given in this section to test the performance of the proposed TDA algorithm. Two of current TDAalgorithms are used to process the same simulated data for comparisons, which are direct TDA algorithm and the TDAalgorithm proposed by Liu [4]. Direct TDA algorithm is defined as Algorithm 1. The algorithm proposed in [4] is defined asAlgorithm 2 and the proposed algorithm based on fractional delay filter is defined as Algorithm 3.

We focus on how powerful the proposed method is on interference and noise suppression for mechanical signatures.Signals are simulated by incorporating the characteristics of rotating machinery signals. In most cases, the repetitivesignatures are hidden in the signals produced by rotating machines or some alternating mechanisms. The signals fromthese machines can be modeled as repetitive components plus residual components. As pointed in [3], the repetitive signalis composed of a truly periodic component and some random periodic components. The signal can be described by thefollowing equation:

XðtÞ ¼ XPðtÞ þ XnðtÞ þ nðtÞ (31)

where XP(t) is the periodic term, Xn(t) represents the random repetitive term and n(t) is wide-band noise. XP(t) would begenerated by a truly periodic mechanism. XP(t) represents a pure periodic character which is given in Fig. 6. But in practice,some process involving friction would contribute a so-called random repetitive component Xn(t). Xn(t) represents arepetitive non-periodic process. Xn(t) can be modeled as

XnðtÞ ¼ gðtÞXrðtÞ (32)

where g(t) is a deterministic periodic function with period T and Xr(t) is band-limited noise. g(t), Xr(t) and Xn(t) are shown inFigs. 7(a), (b) and (c), respectively.

In the simulated digital signal, the sampling frequency is 4000 Hz. The period of the Xp(t) and g(t) are both 79 Hz. Theamplitude of Xp(t) is about 5. The band-limited noise Xr(t) is in the frequency bin of 0–50 Hz. The random periodiccomponent Xn(t) is determined according to formula (32).

By using formula (31), we can get the simulated signal X(t), shown in Fig. 8. The frequency spectrum of X(t) is also givenfor more information. In both figures, we cannot observe any periodic signature because of low SNR. We will employ threeTDA algorithms to suppress the interference and noise for comparisons in the following parts.

The waveform of x (t)20

15

1.5

1

0.5

00 200 400 600 800 1000 1200 1800 200016001400

10

5

Am

plitu

deA

mpl

itude

0

0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01

-5

-10

-15

-20

Time

The frequency spectrum of X (t)

Frequency (Hz)

Fig. 8. Time plot and frequency plot of simulate signal.

ARTICLE IN PRESS

Result processed by algorithm 1

Result processed by algorithm 2

Result processed by algorithm 3

5

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5

5

0

-5

Time

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Time

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Time

Am

plitu

deA

mpl

itude

5

0

-5

Am

plitu

de

Fig. 9. Results of three algorithms.

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–1457 1455

Fig. 9 gives the results obtained by the three algorithms. The number of segments for averaging is 100. In Fig 9(a), wecan hardly observe any periodic components. However, the periodic components are very clear in Figs. 9(b) and (c). Thisphenomenon shows that the direct TDA algorithm is inferior to Algorithm 2 and Algorithm 3.

Generally, the results obtained as Algorithm 2 and Algorithm 3 are similar. However, there are still some differencesbetween them. For example, the amplitude of each impulse in Fig. 9(c) is larger than that in Fig. 9(b). In Fig. 9(c), theamplitude is above 4, while the amplitude in Fig. 9(b) is just 3. It seems that the SNR of the result obtained by Algorithm 3is better. To verify this point, we will use the square of residual signal (SRS) as the indicator to evaluate the performance.

The residual signal Xres(t) is obtained by the averaged result minus the original periodic signal XP(t). A high SRS meanssevere interference and noise contained in the output result. Consequently as a result, we may expect the SRS takes theminimal value. In Fig. 10, the SRSs with respect to Algorithm 2 and Algorithm 3 are given. It can be easily found that the SRSobtained by Algorithm 3 is much smaller than that obtained by Algorithm 2, which manifests that Algorithm 3 is betterthan Algorithm 2.

To investigate how the performance of each algorithm varies with different input signals, we use the following two tests.In both tests, the energy of Xp(t) is 2.5, say

PtX

2pðtÞ ¼ 2:5. The SNR used in the following is defined by

SNR ¼

PX2

PðtÞPX2

resðtÞ(33)

In the first test, we will fixP

X2nðtÞ to 2.4 and vary

Pn2ðtÞ from 1 to 50, then calculate SNR of the averaged result for each

input signal by using formula (33). The relationship between SNR andP

n2ðtÞ is given in Fig. 11. Generally, Algorithm 3 iswith the best performance of three algorithms. However, when the content of

Pn2ðtÞ is increasing, the SNR of the averaged

result is decreasing dramatically.In the second test, we fix

Pn2ðtÞ to 2 and vary

PX2

nðtÞ from 2 to 128, then calculate SNR of the averaged result for eachinput signal by using formula (33). The relationship between SNR and

PX2

nðtÞ is given in Fig. 12. Similarly, Algorithm 3 iswith the best performance of three algorithms. At the same time, when the count of

PX2

nðtÞ is increasing, the SNR of theaveraged result is decreasing.

ARTICLE IN PRESS

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7Algorithm 2(SRS )

Time

Am

plitu

de

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.101

2

345

67

8Algorithm 3(SRS )

Time

Am

plitu

de

Fig. 10. Residual signal of two algorithms.

Algorithm 1Algorithm 2Algorithm 3

120

100

80

60

SN

R

40

20

00 5 10 15

power of noise ∑ n2 (t)

20 25 30 35 40 5045

Fig. 11. Relationship between SNR of the averaged result andP

n2ðtÞ (P

X2nðtÞ is fixed to 2.4).

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–14571456

From Figs. 11 and 12 we can find that Algorithm 3 is superior to the other two whether for non-harmonic interference ornoise suppression.

5. Conclusions

A new TDA algorithm, which can synchronize all the averaged segments accurately without using phase referencesignal, is proposed in this article. The performance is improved by incorporating fractional delay filter in the

ARTICLE IN PRESS

Algorithm 1Algorithm 2Algorithm 3

25

20

15

10

5

00 20 40 60 80 100 120 140

power of noharmonic ∑ Xx2 (t)

SN

R

Fig. 12. Relationship between SNR of the averaged result andP

X2nðtÞ (

Pn2ðtÞ is fixed to 2).

W. Wu et al. / Mechanical Systems and Signal Processing 23 (2009) 1447–1457 1457

synchronization. Unlike other current algorithms, in the proposed algorithm, time delay can be compensated at anyprecision theoretically as long as the fundamental frequency is known exactly. There are many existing methods todetermine the exact period. Simulation shows the proposed algorithm is superior to other existing algorithms whether fornon-harmonic interference or noise suppression.

Acknowledgement

The authors would like to thank the reviewer and the editor for very helpful suggestions that led to great improvementin content and presentation of the paper. The financial sponsorship from National Natural Science Foundation of China (No.10674149) is gratefully acknowledged.

References

[1] S. Braun, The extraction of periodic waveforms by time domain averaging, Acustica 32 (2) (1975) 69–77.[2] P.D. McFadden, A revised model for the extraction of periodic waveforms by time domain averaging, Mechanical Systems and Signal Processing 1

(1987) 83–95.[3] S. Braun, B. Seth, Analysis of repetitive mechanism signatures, Journal of Sound and Vibration 70 (4) (1980) 513–526.[4] S. Braun, B. Seth, On the extraction and filtering of signals acquired from rotating machines, Journal of Sound and Vibration 65 (1) (1979) 37–50.[5] P.D. McFadden, Interpolation techniques for time domain averaging of gear vibration, Mechanical Systems and Signal Processing 3 (1989).[6] P.D. McFadden, M.M. Toozhy, Application of synchronous averaging to vibration monitoring of rolling element bearings, Mechanical Systems and

Signal Processing 14 (6) (2000) 891–906.[7] M. Feder, Parameter estimation and extraction of helicopter signals observed with a wide-band interference, IEEE Transactions on Signal Processing

41 (1) (1993) 232–244.[8] A. Nehorai, B. Porat, Adaptive comb filtering for harmonic signal enhancement, IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-34

(5) (1986) 1124–1138.[9] D. Cyrill, J. McNames, M. Aboy, Adaptive comb filter for quasiperiodic physiologic signal, in: Proceedings of the Annual International Conference on

IEEE Eng. Med. Biol., Cancun, Mexico, 2003, pp. 2439–2442.[10] L. Zhu, H. Ding, X. Zhu, Extraction of Periodic Signal Without External Reference by Time-Domain Average Scanning, IEEE Transactions on Industrial

Electronics 55 (2) (2008).[11] H. Liu, H. Zuo, C. Jiang, L. Qu, An improved algorithm for direct time-domain averaging, in: Mechanical Systems and Signal Processing, vol. 14,

Academic, London, UK, 2000, pp. 279–285.[12] T.I. Laakso, V. Valimaki, M. Karjalainen, U.K. Laine, Splitting the unit delay: Tools for fractional delay filter design, IEEE Signal Processing Magazine

(1996) 30–60.[13] G. Sheng, L. Tao, Y. Xu, The research of phase accumulation error of time domain averaging, Journal of Vibration Engineering 20 (4) (2007)

(in Chinese).[14] V. Valimaki, T.I. Laakso, Principles of Fractional delay filters, in: Proceedings of IEEE ICASSP’2000, January 1996, pp. 30–60.[15] C.W. Farrow, A continuously variable digital delay element, in: Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS-88),

vol. 3, Espoo, Finland, June 6–9, 1988, pp. 2641–2645.