The synchronous (time domain) average revisited

Mechanical Systems and Signal Processing 25 (2011) 1087–1102

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Invited Tutorial Review

The synchronous (time domain) average revisited S. Braun n Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel

a r t i c l e in fo

abstract

Article history: Received 10 May 2010 Accepted 29 July 2010

Synchronous averaging is one of the most powerful techniques for the extraction of periodic signals from a composite signal. It is based on averaging periodic sections, necessitating an a-priori knowledge of the period sought. It is one of the most effective signal processing tools applied to rotating machinery, and has been known and used for decades.It will be shown that synchronous average is actually just one of the many possible ’’synchronous filters’’ which could be used to extract the above periodic components performance. A novel signal analysis, geared to periodic signals will be introduced, with the potential of extracting more complex phenomena typical of some rotating machinery. Examples given are based on periodic oscillating transients, with various additive interferences. The possibility of additional signal processing approaches is also discussed. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Averages Comb filters Time domain average

Contents 1. 2. 3.

4.

5. 6. 7. 8. 9.

n

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A short reminder: averages and averaging filters [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Moving average (MA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Exponential averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Running averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Transfer functions and FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comb filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Formal computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional possibilities—non-equal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block computations—the case of equal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General signal processing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix—The elliptic filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tel: + 972 4 8293156; fax: + 972 4 8295711. E-mail address: [email protected]

0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.07.016

1088 1088 1089 1089 1090 1090 1091 1092 1092 1092 1092 1094 1096 1097 1098 1098 1100 1102

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

k v, w F[] V, W SA SE RA MA FRF

Nomenclature x y n M N f, o fp r

input signal averaged output time index number of elements in period number of periods frequency synchronous frequency period index

index within period derived signals filter signal blocks synchronous average exponential average running average moving average frequency response function

1. Introduction Periodic signal components are one of the most prevalent types encountered generated by mechanical systems, especially those incorporating rotating components. Often the period at hand can be determined from kinematic and geometrical information, or via on-line measurements. In such cases, a powerful method to extract the periodic component is that of synchronous average. While the method has been known and implemented for decades [1–5], its interpretation as a filtering process seems to warrant further investigations. The synchronous average is a special case of comb filters. A more general approach of extracting periodic signal components from a compound signal can actually be presented, enabling the adaptation of specific comb filters to the type of signals and interferences at hand. The paper is organized as follows: Section 2 reviews the basic synchronous average, while Section 3 reviews basic averaging concepts. The principle of comb filters is presented in Section 4, introducing the concept of newly derived signals. Specific examples of extracting periodic signals from interferences are shown in Section 5, comparing synchronous averaging to other comb filters. Section 6 discusses the possibility of applying different comb filters to specific signal sections. Block computations are reviewed in Section 7, further possibilities roughly hinted at in Section 8, and a summary given in Section 9. 2. The classic approach An intuitively simple depiction of the average process is shown by Fig. 1. Data values separated by the exact period are averaged. Any periodic component, synchronous with this period, is thus unchanged, any other will be attenuated and converge asymptotically towards zero. There is some analogy of this process to the computation of power spectral densities (PSD), where spectra of segments are averaged. Hence this process is sometimes called Time Domain Averaging, as contrasted to the Frequency Domain Averaging performed for computation of PSD.

r=1

r=N

r=1

r=N

Synchronous Average Fig. 1. Principle of SA (synchronous average).

S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

1089

N=8 1 0.9 0.8 0.7

|H|

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4 f/fp

5

6

7

8

Fig. 2. FRF of the SA.

Formally [6] yðn DtÞ ¼

X 1 N1 xðn DtrM DtÞ Nr¼0

ð1Þ

where M is the number of elements per period and N the number of section averaged. y(n) is then a sequence of M points, spanning one period of the averaged sections. The frequency response can be computed via the z-transform as YðzÞ ¼

1 1 NX ð1 þ zrM ÞXðzÞ Nr¼0

HðzÞ ¼

YðzÞ 1 1zMN ¼ XðzÞ N 1zM

ð2Þ

The frequency response then being 9Hðf =fp Þ9 ¼ HðzÞ9z ¼ expðjo DtÞ ¼

1 sinðpNf =fp Þ N sinðpf =fp Þ

ð3Þ

with fp ¼

1 M Dt

the frequency of the extracted periodic component. The FRF is shown in Fig. 2. It has a form of a ‘‘comb’’ filter, with main lobes centered around integer multiples of the synchronizing frequency fp. Hence it is ideal for extracting the fundamental as well as all harmonics of the signal, hence the periodic signal itself. Increasing N, the number of section averaged, results in narrower main lobes (improved filtering of the periodic data) and increases the number of secondary lobes. Depicting the process such as a filtering operation has one slightly bothersome aspect. A classical filter generates a continuous output sequence, the result of applying the filter function F to the excitation xn, hence basically yn = F[xn]. The frequency responses (3) however seem to be based on a single period of the synchronous waveform. Thus we now attempt a more tutorial oriented presentation. 3. A short reminder: averages and averaging filters [7] 3.1. Moving average (MA) This is computed as yðnÞ ¼

1 1 NX xðnqÞ Nq¼0

ð4aÞ

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

with N a constant, the number of data values averaged. The data consisting of N samples, within a (moving window), is replaced by its average. Eq. (4a) represents as an MA system. A recursive form is often more convenient, and given below: yðnÞ ¼ yðn1Þ þ

xðnÞxðnNÞ N

ð4bÞ

3.2. Exponential averaging A recursive auto regressive (AR) form is given by yðnÞ ¼ yðn1Þ þ

xðnÞyðn1Þ K

ð5aÞ

where K is a constant. A more familiar form would be yðnÞ ¼ ayðn1Þ þ ð1aÞxðnÞ

ð5bÞ

where a ¼ ð11=KÞ. It is relatively straightforward to show an equivalent MA expression yðnÞ ¼

1  ð0Þ e xðnÞ þeðqÞ xðn1Þ þeð2qÞ xðn2Þ þ    þeðn1Þq xð1Þ K

ð5cÞ

where q ¼ lnð11=KÞ. And the terms to the right behave like a decaying exponential, with less and less weight given to distant past data points. 3.3. Running averaging This is a cumulative average, based on all incoming data . It can be computed via a batch operation, actually an MA form (6a), or often preferably via a recursive AR form (6b): yðnÞ ¼

n1 1X xðnqÞ nq¼0

yðnÞ ¼ yðn1Þ þ

ð6aÞ

xðnÞyðn1Þ n

ð6bÞ

where n is updated continuously with the new incoming data (Fig. 3).

MA 2 1.5 1 0.5 0 −0.5 0 2 1.5 1 0.5 0 −0.5 900

2

4

6

8

10 EX

12

14

16

18

20

950

1000

1050

1100

1150 RA

1200

1250

1300

1350

1400

2

4

6

8

10

12

14

16

18

20

2 1.5 1 0.5 0 −0.5 0

Fig. 3. Types of average.

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3.4. Transfer functions and FRFs All averages perform smoothing operation and can thus be considered as low pass filters—attenuating fast fluctuations. It is thus helpful to compute their frequency response function. For the moving average, applying a z-transform to (4a) results in  1 1 þ z1 þ z2 þ    þ zðN1Þ XðzÞ N YðzÞ 1 1zN ¼ HðzÞ ¼ XðzÞ N 1z1 YðzÞ ¼

ð7aÞ

and the FRF computed via HðzÞ9z ¼ expðjo DtÞ 1 1expðjoN DtÞ N 1expðjo DtÞ 1 1cosðoN DtÞ 2 9HMA ðoÞ9 ¼ N 1cosðo DtÞ HMA ðoÞ ¼

ð7bÞ

For the exponential average (EA) YðzÞ 1a ¼ XðzÞ 1az1 1a 2 9HEA ðoÞ9 ¼ 1þ a2 2a cosðo DtÞ

HEA ðzÞ ¼

ð8Þ

For the running average yRA ðnÞ ¼

n1 1X xðnqÞ nq¼0

ð9aÞ

In the next equation (9), contrary to Eq. (7), n is a variable, updated as a new data x(n) is acquired. Hence RA is not a Linear Time Invariant (LTI) system, and the frequency response will vary after every new acquired x(n). One consequence is that the convolution between inputs and impulse responses can no longer be used for the computation of responses in the time domain. This is actually a Linear Time Variable (LTV) system. We use a somewhat simplified approach, where due to the time variability of the system’s parameters, all transforms (and derived Transfer Functions) are a function of the time index n. A more in-depth approach to LTV systems is beyond the scope of this article, and among the many available references for this we cite [8,9]:  1 1 þ z1 þ z2 þ    þ zðn1Þ XðzÞ n Yðz,nÞ 1 1zn ¼ HRA ðz,nÞ ¼ Xðz,nÞ n 1z1

Yðz,nÞ ¼

2

9HRA ðoÞ,n9 ¼

ð9bÞ

1 1cosðon DtÞ n 1cosðo DtÞ

ð9cÞ

FRF of averaging operations 1 Elliptic

0.9 0.8 0.7 0.6

Moving Average

0.5 Exponential

0.4 0.3 0.2 0.1 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Fig. 4. FRF of the 3 average.

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Impulse Responses 0.4 0.35 RA

0.3 0.25

MA

0.2

EA ELL

0.15 0.1 0.05 0 −0.05 −0.1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 5. Impulse response of averages and elliptic filter.

The FRF of the 2 types of averaging systems are shown in Fig. 4. Also shown is the FRF of an elliptic filter (see the Appendix), used later in Section 5 where k= 9 (a =0.8889) for the EA, N = 17 for the MA, and the parameters of the elliptic filter as well, are chosen so as to achieve equal areas under their respective FRF. 3.5. The impulse responses The impulse responses are shown in Fig. 5. The impulse response for the moving and exponential average, as well as the elliptic one, could in principle be computed via the inverse z-transform of the corresponding FRF. This is not the case for the running average, which is not time invariant. All the results shown in Fig. 5 were thus computed numerically, as the result of applying a single impulse at time zero, to the corresponding algorithm. The parameters were the ones used in Fig. 4, and that of the elliptic filter are given in the Appendix. 4. Comb filters 4.1. The principle Eq. (1) dealt with the averaging of data points separated by one period. We now show the general effect of applying a filtering operation to data separated by specific delays. Assume that a normal filtering operation F is applied to the sequence x(n), resulting in y(n)= F[x(n)]. The filters transfer function in the z domain will be H(z), and the filters frequency response computed via HðoÞ ¼ HðzÞ9z ¼ expðjo DtÞ

ð10Þ

We now apply the same operation F[n] to the sequence xM, comprised of data M points apart, see Fig. 6. The transfer HM function will now be related to H via HM ðzÞ ¼ HðzM Þ

ð11Þ

Due to the M point delay the frequency response will then be HM ðoÞ ¼ HM ðzÞ9z ¼ expðjo DtÞ ¼ HðzM Þ9z ¼ expðjo DtÞ ¼ HðzÞ9z ¼ expðjoM DtÞ

ð12Þ

This results in a periodic repetition of a basic H(o) function, see Fig. 7. Hence applying a filter F[n] to the sequence of M separated x(n) values, is equivalent to applying a comb filter, with filter FRF separated by 1/M Dt. It is instructive to note that these comb filter properties are not limited to a specific filter F. The case of synchronous averaging is just one of the many possible filters—a running average one This will be pursued in Section 5. 4.2. Formal computations We denote the sequence consisting of data points separated by M as vðk,rÞ

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principle of comb filter 8 7 M=8 6 data M points apart

5 4 3 2 1 0 0

5

10

15

20

25

30

35

40

45

50

Fig. 6. Data separated by specific delays.

prototype Butterworth filter 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 resulting comb filter 1 0.8 0.6 0.4 0.2 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Fig. 7. Filter and resulting comb filter.

where r is the period index and k the index within each period. With r = 1:N, k= 0,1, M 1, there are M such sequences, obtained from the data x(n), n = 0:NM  1. Thus r¼

jnk M

ð13aÞ

where bc is the integer part of n k ¼ nrM

ð13bÞ

Each sequence v(r, k) is now filtered by a filter F, resulting in w(r, k)= F[v(r, k)] (Fig. 8). As an example, for a running average, the filtering operation would be   x½ðr1ÞM þ kw½ðr1Þ,k wðr,kÞ ¼ w ðr1Þ,k þ r

ð14aÞ

The output sequence is now obtained by re-assembling the data points yðnÞ ¼ wðr,kÞ For the full range of r and k.

ð14bÞ

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15

8 6 4 2 0

x 10−3 v(2,k)

10 Filter

v

5

0

0.015 0.01 0.005 0 −0.005 −0.01 −0.015

0.5

1

r=1

0

1.5

r=2

20

30

40 50

r=3

0.5

20

40

8 6 4 2 0

r=4

1

Filter W

0

k

k

k

k

10

8 6 4 2 0

0 0

8 6 4 2 0

1.5

Filter W

0

20

40

0

10 20 8 6 4 2 0 0 8 6 4 2 0 0

30

40 50

20

40

20

40

Fig. 8. (a) Derived sequences and (b) filtering of derived sequences.

comb filters Elliptic

1.05 1 0.95

RA

0.9 EA 0.85 0.95

1

1.05

1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5 2 2.5 3 normalized frequency f/fp

3.5

4

Fig. 9. The 3 types of comb filters.

5. Examples In this section, 3 types of comb filters will be compared—a) running average, b) exponential average and c) elliptic. To facilitate comparisons the exponential averaged and elliptic will have equal noise bandwidth. The running average one is of course non-stationary, with properties evolving as more data is inputted. The elliptic filter used is described in the Appendix. Fig. 9 shows the relevant frequency responses. The periodic signal to be extracted correspond to Fig. 1—a train of exponentially decaying oscillations. Such a signal is often used as a model of a vibration signal generated by a roller bearing with a localized defect [10]. It should be stressed that such a signal model is only an approximate one. In practice, a jittered period is often encountered, as investigated in detail in [11,12]. We choose the approximate signal model, resulting in a periodic impulsive oscillating train, only to exercise the specific signal processing cases described in this paragraph. The number of samples per period is 256, the sampling interval is 1.5 ms, and with 256 samples per period, this basic period is 384 ms, corresponding to approximately 2.6 Hz. The oscillating frequency is 80 Hz. Various noise signals will now be added, and the performance of the comb filters, based on a period of 256 samples will be compared. The total number of periods is N =128.

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Case 1. Additive harmonic signal. The signal to noise ratio (based on maximum amplitude) is approximately 14 db (0.1963), Fig. 10a. The performance of the 3 filters is depicted by Fig. 10b–d. Comparing the attenuation of the harmonic interference for these filters, these are 33.9 db (50), 22.7 db (13.7) and 45.5 db (190) for the RA, EA, and Elliptic, respectively. The elliptic filters gives the highest attenuation for this single harmonic interference, as can also be seen from the FRFs of Fig. 4. Case 2. Additive periodic train of decaying oscillations of 100 Hz, with 300 samples per period (thus a ratio of 300/ 256 =1.1719 between the basic periods), and with half the amplitude of the basic signal train to be extracted (Fig. 11a), The performance of the 3 filters is depicted by Fig. 11b–d. Comparing the attenuation of the interfering pulse train for these filters, these are 32.6 db (42.8), 20.4 db (10.46) and 22.6 db (13.46) for the RA, EA and Elliptic, respectively. For this case, the elliptic filter is not anymore the best one. This follows from the characteristics of the interfering signal, consisting of multiple harmonics of its basic frequency (2.22 Hz). The amplitude of these harmonics is maximum around 100 Hz, and many of these will fall inside the pass band of the lobes corresponding to the elliptic filter. This lobe is much flatter in the pass bands (Fig. 4), and the attenuation is hence less pronounced. The RA is much sharper, hence its performance is better. Case 3. A strong transient is superimposed on the signals of case 2. It occurs in the interval of 3.75–3.8 s, and its magnitude is 50% higher than the signal peak. The three filters have obviously a different memory, related to their impulse response (Fig. 5), and the effect of the transient will persist differently for them—Fig. 12

0.02

0.04 0.03 0.02 0.01 0 −0.01 −0.02

0.01

0

−0.01 −0.02

−4 23

23.05 23.1

0.01 0 −0.01 −0.02 0

5

10

15

−3 4 x 10

25

30

35

40

x 10−3 1.5 1 0.5 0 −0.5 −1 −1.5 21.5 22 22.5 EA

2 0 −2 −4 21.5 −3 x 10 5

20

22

2

−2

signal

0.02

2

−2

23.5

23

4

0

0

22.5

−3 4 x 10

45

21.5 −3 x 10 8 6 4 2 0 −2 −4 −6 −8 0 5

22

22.5

RA

10

15

20

25

30

2

2

1

0

0

−2

−1 23

23.5

−3 5 x 10

0

−4 21.5

22.5

−3 4 x 10

−4 22.5

22.5

22

x 10−3

35

40

45

x 10−4

−2 22.5 elliptic

23

23.5

0

−5

−5 0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

Fig. 10. (a) Signal with harmonic disturbance, (b) RA comb filter, (c) exponential comb filter, and (d) elliptic comb filter.

40

45

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

x 10−3

x 10−3

5

4

5

0

24.6

24.8

25

5

5

0

0

−5

0

0

−2

−2 22

−4 21.5

22.5

22

22.5

RA

−5 0

5

4

10

15

x 10−3

0 −2 −4 21.5 x 10−3

20

25

30

35

40

x 10−3 1.5 1 0.5 0 −0.5 −1 −1.5 22.5 21.5 22 EA

2

5

2

−4 21.5 x 10−3

−5

x 10−4 4

2 0

−5 23 24 25 26 27 28 −3 x 10 signal

x 10−3

22

45

0

5

4

15

20

25

x 10−3

30

2

2

1

0

0

−2

−1

−4 22.5 x 10−3

22.5

10

23

23.5

35

40

45

x 10−4

−2 22.5 elliptic

23

23.5

5

0

0

−5

−5 0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

Fig. 11. Signal periodic pulse train disturbance: (a) signal with pulse train, (b) RA comb filter, (c) exponential comb filter, and (d) elliptic comb filter.

At approximately 25 s, the attenuations of this transient pulse are 13.8 db (4.9), 34.85 db (55.31) and 14.5 db (5.3) for the RA, EA, and elliptic filters, respectively. The actual numbers depend of course on the specific time when the transient occurs, and the time where the attenuation is checked. We also note that for the elliptic filter the attenuation would not decrease monolithically with time, due to the oscillations of the impulse response. The design and implementation of digital filter is now an established technology, made extremely easy by commercially available software. The above examples demonstrate the use of some specific filters. The choice of the filter will be dictated by the desired signal and interference properties. The obtained results of the comb filtering are then relatively easy to understand. 6. Additional possibilities—non-equal filters In the prior examples, the same filter was applied to all derived signals. This is obviously not the only possibility, and in principle, different filters could be applied for different derived signals. As an example, we show a composite signal (Fig. 13). It is composed of 2 periodic oscillating pulse trains, with the same period of 0.3840 s. One signal is delayed by 0.15 s from the other. Both the oscillation frequencies and the decaying factor are unequal. Other components are additive random signals, concentrated (in time) around the periodic impulses. We first apply a synchronous average operation. The signal covers N = 128 periods, and the result is shown in Fig. 14a.The additive random signals are highly attenuated, and basically only the periodic components are shown. The plot (toward the end of the section, around 48 s) shows constant magnitudes for both pulse trains.

S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

4

x 10−3

2 0 −2 −4 24.5 5

x 10−4

1.5 1 0.5 0 −0.5 −1 −1.5 25

−3 4 x 10

0 −2 24.9

25

−4 24.5

25.1

EA

x 10−3

1.5 1 0.5 0 −0.5 −1 −1.5

2

25.5

5

1097

25

x 10−4

24.9

25.5

25

25.1

RA

x 10−3

0

0

−5

−5 0

5

10

15

20

25

30

35

40

−3 4 x 10

0 −2

5

x

10−3

0

5

5

10

15

−4 1.5 x 10 1 0.5 0 −0.5 −1 −1.5 24.9 25 25.5 elliptic

2

−4 24.5

0

45

25

20

25

30

35

40

45

25.1

0

−5 10

15

20

25

30

35

40

45

Fig. 12. Signal with transient disturbance: (a) exponential comb filter, (b) RA comb filter, and (c) elliptic comb filter.

Next we apply a high pass Butterworth filter (order 8, critical frequency equals 0.1 ) to the derived signals spanning the second pulse train, and synchronous average filters to all other. The results are shown in Fig. 14b. The periodic components of the first pulse train are extracted (as in Fig. 14a). However the periodic component of the second pulse train is now eliminated, and only the additive random components around it are now visible. We note that these have a significant magnitude. The importance of these examples lies mainly in their implications. The theory/design and application of filters are well established, and deciding to use any of these for synchronous computations (i.e. as comb filters) would almost be trivial. 7. Block computations—the case of equal filters While Section 4.2 develops a general approach to comb filters (and Sections 5 and 6 apply this approach), it is cumbersome and not necessary for the special case of equal filters applied to all v(r, k) sequences. As computing with signal blocks, or vectors, is relatively easy with signal processing software and hardware modules, a filtering approach base on such vectors can be more intuitive and practical. We now define signal blocks, based on the notations used in 4.1 For the input signal x ¼ ½ðVÞ1 ðVÞ2    ðVÞr    ðVÞN 

ð15aÞ

ðVÞr ¼ ½vðr,1Þ vðr,2Þ    vðr,kÞ    vðr,MÞ

ð15bÞ

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

0.02

Fc=80 Hz

0.01 0 −0.01

Fc=110 Hz

−0.02 0

0.5

1

1.5

0.02

0.02

0.01

0.01

0

0

−0.01

−0.01

−0.02

−0.02 0

0.5

1

1.5

0

0.5

1

1.5

Filter W2

Filter W1 Fig. 13. Composite signal.

And output signal y ¼ ½ðWÞ1 ðWÞ2    ðWÞr    ðWÞN 

ð16aÞ

ðWÞr ¼ ½wðr,1Þ wðr,2Þ    wðr,kÞ    wðr,MÞ

ð16bÞ

And basically the filtering would be ðWÞr ¼ F½ðVÞr 

ð17Þ

For example, the synchronous (running) average would be computed via ðWÞr ¼ ðWÞr1 þ

ðVÞr ðWÞr1 r

ð18Þ

8. General signal processing method All the examples shown up to now (Sections 5–7) showed the application of filtering operations to the derived signals. We may however generalize, and apply any existing signal processing to these, as shown schematically in Fig. 15 As an example we show a composite signal, Fig. 16. It is obviously difficult to see any clear pattern. Fig. 17 shows the four components used to generate this signal. The first one (17a) is a periodic pulse train oscillations at 80 Hz. The second one (17b) shows a repetitive random narrow band pulse train (around 80 Hz), actually a cyclo-stationary signal. The 3rd one (17c) is similar to the 2nd one, but the random narrow band pulse train is now around 110 Hz, with an amplitude modulation of 2 Hz, and with pulses located in between those of the 2nd component). The 4th one is a wideband random signal, orders of magnitude larger than the first 3 ones. The composite made of the sum of the first 4 signals, is shown in (17d), the signal appearing before in Fig. 16. We now apply specific signal processing methods, intended to see the fluctuations in the sum of energies of the derived sequences around indices corresponding to time intervals 0.1950–0.2220 s and their integer multiples, corresponding to the location of the pulses of the 3rd components. Computing a spectral density of the derived energy sequence (Fig. 18), we note a peak around .2 Hz, showing a harmonic energy fluctuation due to the modulation. This information is not evident via the PSD computation of the total signal, even if the same resolution is used for both. It should be noted that the derived signals in this case, are of square units, hence comparing the ordinates of the 2 spectra should be meaningless. The example is only intended to show the recognition of a 0.2 Hz fluctuation (in this case of the energy in a repetitive time window). The above example is mainly intended to hint at possibilities once general processing methods for derived sequences are considered. Obviously some ad hoc knowledge is needed, to choose potential methods geared to show specific patterns. 9. Summary It has long been known that the synchronous average can be considered, in the frequency domain, as a comb filter. In this paper we have attempted to generalize the approach, by deriving a set of signals, consisting of signal elements

S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

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0.06 0.04

2nd Pulse Train

0.02 0 −0.02 −0.04 1rst Pulse Train

−0.06 47

47.5 48

30

35

48.5 49

0.06 0.04 0.02 0 −0.02 −0.04 −0.06

0

5

10

15

20

25

6

40

45

x 10−3

4

2nd Pulse Train

2 0 −2 −4 −6

1rst Pulse Train

6

48.6 48.65 48.7 48.75 48.8

x 10−3

4 2 0 −2 −4 −6

0

5

10

15

20

25

30

35

40

45

Fig. 14. (a) SA operation and (b) high pass comb filtering.

separated by one (synchronous) period. Synchronous Average is shown to be a regular averaging operation applied to these derived signals. This generalized approach enables one to envision the application of a multitude of other processing methods. One possibility is to apply any filter to the derived signals, with the whole catalogue of existing filters at our disposition. Whether to consider such an option will of course depend on the signals and the interferences at hand. A further question considered is whether to apply different filters to different derived signals, according to the section in the synchronous period of interest. Applying the same filters to all, results in the possibility of using block computations.

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

r=1

r=N

r=1

r=N

General SP Fig. 15. General SP possibilities.

20 15 10 5 0 −5 −10 −15 0

2

4

6

8

10

12

14

16

Fig. 16. A composite signal.

Some examples are given in this paper, but mainly as a demonstration. Whether to use specific type of filters, and/or choose regions in the synchronous period where to apply them, these are basically an additional option, to be judged by the user. A further contemplated generalization is to consider any signal processing method to any of the derived signals. Only a simple example has been given, as a more general presentation would be beyond the intended scope of this presentation. As a final remark, it seems somewhat surprising that anything ’’new’’ could have been presented, for a signal processing technique well known and used for decades. This might suggest the desirability to revisit additional ’’well known’’ techniques, see Fig. 19 Appendix—The elliptic filter Such filters have ripples in both the pass and stop band (often equi-ripple).We also decided to choose a filter with the same area under its FRF as the ones used in all examples (Sections 3 and 5). The filter chosen (Section 5) was chosen with the following requirements: a) A maximum of 0.1 db ripple in a pass band, up to 5% of the Nyquist frequency. b) 60 db attenuation (from the pass band) in the stop band, from 20% of the Nyquist frequency Wp = 0.05.

S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

Peridic

5

x 10−3

5

0

−5 0

2

4

6

8

10 0.02

0

0

CS1

0.05

−0.05

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

−0.02 0

5

10

15

20 0.02

0.02 CS2

x 10−3

0

−5

0

0

−0.02

−0.02 0

Sum

1101

5

10

15

20

10

10

0

0

−10

−10 0

5

10

15

20

0

0.2

0.4

0.6

0.8

Fig. 17. (a) Periodic component, (b) first cyclo-stationary component, period= 1/2.6 s, (c) 2nd cyclo-stationary component, period= 1/2.6 s modulated 0.2 Hz, and (d) noise corrupted sum.

Combined signal

Derived signal

15

15

10 10 5 0

5

−5 0 −10 −15

−5 0

2

50

100 150 200 sec spectrum of combined signal

0

50

100 150 sec spectrum of derived signal

200

60 50

1.5 40 1

30 20

0.5 10 0

0 0

20

40

60 Hz

80

100

0

0.5

1

1.5

Hz

Fig. 18. Signal and derived signal (time—upper, spectra—lower).

The design can be based on any of the many commercial signal processing software packages. The result obtained and used was   B ¼ 0:0013 0:0033 0:0045 0:0033 0:0013   A ¼ 1:0000 3:6950 5:1520 3:2112 0:7547

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S. Braun / Mechanical Systems and Signal Processing 25 (2011) 1087–1102

The Synchronous Average

revisited

The ??

revisited

Fig. 19. The importance of re-visiting.

where A and B are the polynomial coefficients for the Z transfer function H= B(z)/A(z). The frequency response is that shown in Fig. 4. References [1] P.D. McFadden, M. Toozhy, Application of synchronous averaging to vibration monitoring of rolling element bearing, Mechanical Systems and Signals Processing 14 (2000) 891–906. [2] P.D. McFadden, Technique for calculating the time domain averages of the vibration of the individual planet gears and the sun gear in an epicyclic gearbox, Journal of Sound and Vibration 144 (1991) 163–172. [3] P.D. McFadden, A revised model for the extraction of periodic waveforms by time domain averaging, Mechanical Systems and Signal Processing 1 (1) (1987) 83–95. [4] P.D. McFadden, A revised model for the extraction of periodic waveforms by time-domain averaging, Mechanical Systems and Signal Processing 1 (1) (1987). [5] F. Bonnardot, M. El Badaoui, R.B. Randall, J. Danie re, F. Guillet, Time domain averaging across all scales: a novel method for detection of gearbox faults, Mechanical Systems and Signal Processing 22 (2) (2008) 261–278. [6] S. Braun, The extraction of periodic waveforms by time domain averaging, Acustica 32 (1975) 69–77. [7] J.F. Kenney, E.S. Keeping, ’’Moving Averages.’’ y14.2 in Mathematics of Statistics, Part 1, 3rd ed., Van Nostrand, Princeton, NJ, 1962, pp. 221–223. [8] L. Tzong_Yeu, J.K. Aggarwal, Recursive implementation of LTV filters-frozen-time transfer function versus generalized transfer function, Proceedings of the IEEE 72 (7), 1984, pp. 980–981. [9] G. Matz, F. Hlawatsch, Extending the transfer function calculus of time varying linear systems: a generalized underspread theory, in: Proceedings of the IEEE ICASSP, Seattle, WA, May 1998, pp. 2189–2192. [10] D. Ho, R.B. Randall, Optimisation of bearing diagnostic techniques using simulated and actual bearing fault signals, Mechanical Systems and Signal Processing 14 (5) (2000) 763–788. [11] J. Antoni, R.B. Randall, A stochastic model for simulation and diagnostics of rolling element bearings with localised faults, ASME Journal of Vibration and Acoustics 125 (2003) 282–289. [12] R.B. Randall, J. Antoni, Rolling element diagnostics—a tutorial, Mechanical Systems and Signal Processing 25(2), in press, doi:10.1016/j.ymssp.2010. 07.017.