Unsupervised noise cancellation for vibration signals

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part II—a novel frequency-domain algorithm. J. Antoni*, R.B. Randall. School of Mechanical and Manufacturing Engineering, The University of New South Wales ...
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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 103–117 www.elsevier.com/locate/jnlabr/ymssp

Unsupervised noise cancellation for vibration signals: part II—a novel frequency-domain algorithm J. Antoni*, R.B. Randall School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia

Abstract Vibration analysis of machines gains greatly in efficiency if periodic vibrations can be separated out from non-deterministic ones. Part II of this paper introduces a solution in the frequency domain, which is faster and simpler to use than adapative algorithms (part I). The performance of this new algorithm is thoroughly investigated and compared to those of the self-adaptive noise cancellation algorithm. Finally, convincing examples are given of the application to the separation of actual bearing signals from gear signals in gearboxes. r 2003 Published by Elsevier Science Ltd.

1. Introduction The first part of this paper [1] addressed the issue of separating periodic signals from broadband signals by means of only one sensor. The issue was conveniently phrased in terms of prediction theory, and it was shown how the difference in correlation times of the two types of contribution allows for their separation. The optimal filter that achieves the separation was shown to be the solution of the so-called Wiener–Hopf equations. From these equations, a well-known class of recursive algorithms can be implemented, which ultimately converge to the optimal filter [2]. One of them, the self-adaptive noise cancellation algorithm (SANC), was discussed in detail. Therefore, as long as we are concerned with stationary signals, the SANC algorithm is nothing other than a convenient tool for solving the Wiener–Hopf large system of equations. Indeed, if the signals are known to be stationary, there may be other approaches for obtaining the same solution without the need for going through an adaptation process. One such approach was inspired by the estimation of the so-called H1 filter as done in experimental modal analysis [3]. It is based on *Corresponding author. Universitie de Technologie de Compiegne, GI-Heudiasyc BP 20529, 60205 Compi"egne C!edex, France. E-mail address: [email protected] (J. Antoni). 0888-3270/04/$ - see front matter r 2003 Published by Elsevier Science Ltd. doi:10.1016/S0888-3270(03)00013-X

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the use of a series of Fourier transforms of blocks of data which are averaged to estimate the optimal Wiener frequency response. The benefit of this approach comes from the computational efficiency of the FFT and the fact that it gives the optimal filter in one run. Indeed, SANC was often found to have convergence problems, which required processing the same signal many times before reaching the supposed optimal solution.

2. Principle of the algorithm The Wiener–Hopf equations describe a convolution of the finite-length noise cancellation filter with the autocorrelation of the signal. A natural idea is to solve for the convolution in the frequency domain where it simplifies into a product. The Fourier transform of the autocorrelation sequence is then classically estimated as an average of products of short-time Fourier transforms taken over the signal. For this procedure to be efficient, the short-time Fourier transforms should be computed on short sequences that at least span the filter length. However, this introduces unavoidable distortions due to windowing effects, which must be analysed in detail. Before going further, let us rephrase the problem in terms of short-time sequences. Let Xk ðnÞ be a properly windowed sequence of length N taken at time kT; i.e. Xk ðnÞ ¼ X ðn þ kTÞwN ðnÞ; n ¼ 0; y; N  1 with wN ðnÞ a weighting window of length N: In the same way, define Xkd ðnÞ as a sequence taken at time kT  N  D; or kT on the delayed signal, i.e. Xkd ðnÞ ¼ X ðn þ kT  N  DÞwN ðnÞ; n ¼ 0; y; N  1: The construction of sequences Xk ðnÞ and Xkd ðnÞ is shown schematically in Fig. 1. The objective is to find the filter which best predicts Xk ðnÞ from Xkd ðnÞ: Note that in spite of the windowing, the deterministic components in the signal remain deterministic and are still perfectly predictable. On the other hand, the broadband noise will get rejected from the predicted Xkd ðnÞ if it is uncorrelated with it, which places as a constraint that D is strictly positive (see Fig. 1). Under these conditions, the optimal filter achieving the minimum mean squared prediction error has a frequency response given by Hðf Þ ¼

Sxd xk ðf Þ k

Sxd xd ðf Þ k k

¼

Spd pk ðf Þ k

ð1Þ

Spd pd ðf Þ þ Srd rd ðf Þ k k

k k

where SUV ðf Þ is the cross-power spectrum between two arbitrary signals U and V ; and Xk ðnÞ ¼ pk ðnÞ þ rk ðnÞ with pk ðnÞ and rk ðnÞ the deterministic and non-deterministic parts. Labelling X* k;M ðf Þ N



N

x (n )

x kd (n )

xk (n )

Fig. 1. Short-time sequences used in the frequency-domain algorithm.

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and X* dk;M ðf Þ as the M-long ðMXNÞ discrete Fourier transforms of Xk ðnÞ and Xkd ðnÞ; a simple estimator of Hðf Þ on K sequences is then given by PK  * *d k¼1 Xk;M ðf ÞXk;M ðf Þ # : Hðf Þ ¼ PK X* d ðf ÞX* d ðf Þ k¼1

k;M

ð2Þ

k;M

This quantity may now be transformed back into the time domain to form an M-long noise cancellation filter (whose phase must be corrected by subtracting the time delay D) to be applied directly on the measured signal X ðnÞ: Although the impulse response will generally be larger than that of the SANC filter (MXN due to possible zero-padding), it is really the length N of the shorttime sequences that determines the amount of information in the filter (number of modes) and its frequency resolution. In this respect, N really compares with the filter length of the SANC and we shall actually refer to it as such in the following. Compared with the SANC algorithm, the frequency-domain (FD) algorithm has a much lower complexity. Taking the total number of multiplications involved in each implementation as a comparison basis, the FD algorithm requires about CFD ¼

2LM 2L ln M þ þ 3M Nð1  pÞ 1p

ð3Þ

operations, where L is the total signal length and p the percentage of overlap between two consecutive sequences of length N: The complexities are compared in Fig. 2 for M ¼ 2N and p ¼ 0: It is noticeable that the proposed FD algorithm is always faster than the SANC and the Fast-SANC—fastest possible implementation of SANC—by a factor of at least 2.5 with respect to the latter.

Fig. 2. Comparison of complexities.

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3. Theoretical performance analysis As previously mentioned, the apparent drawback of the proposed FD batch algorithm is that it introduces larger bias errors than the SANC due to the use of short-time sequences. However, we shall now demonstrate that these effects can be minimised by properly windowing the sequences, and even reduced in comparison with what is usually achievable with the classical H1 filter. Suppose as in Section 2.2 that the measured signal is made up of a sum of sinusoids with amplitudes Ai and frequencies fi ; plus a white-noise component of power s2 : When the minimum spacing jfi  fj j between two sinusoids is very small in regard to the effective bandwidth 1=N of the filter, it is easy to approximate it by a parallel network of single band-pass filters each centred on a single discrete component. The frequency gain of the single band-pass filter (centred on f ¼ 0) can be checked to have modulus jHðf Þj ¼ 1 2

1 2

rNjW ðf Þj2

rNjW ðf Þj2 þ 1

ð4Þ

where r ¼ 12 A2 =s2 is the signal-to-noise ratio (SNR) and W ðf Þ is the Fourier transform of window wðnÞ: This equation is to be compared with Eq. (10) of the SANC in part I. As an immediate consequence of formula (4), it is seen that the static gain of the FD filter equals that of the SANC (see Fig. 3 of part I), provided wðnÞ is designed such that jW ð0Þj ¼ 1: It is also easy to check that the position of the zeros and maxima in jHðf Þj are the same as in jW ðf Þj; thus giving an identical pedestal bandwidth. Finally, it can be proved that the slope of jHðf Þj decays twice as fast as that of the underlying function jW ðf Þj: Accordingly, the choice of wðnÞ directly determines the characteristics of the filter. Figs. 3 and 4 display the frequency gains jHðf Þj computed with conventional types of windows and compare them with the SANC frequency gain for a given filter length N ¼512. Two different SNRs are tested, namely r ¼102 and 1. From these figures, it is obvious that the rectangular window should be avoided since it gives rise to a family of extraneous side-lobes that increase dramatically with the SNR. The Hanning window displays a better behaviour while keeping a reasonable bandwidth. For large SNRs, it should however be replaced by the Parzen window in order to reduce side-lobe effects, but at the cost of an increased bandwidth. Note that the Parzen window produces a frequency gain approaching that of a nearly ideal band-pass filter. Other types of window would give comparable characteristics, such as the flattop window. Most of these results are consistent of course with classical spectral analysis [3,4] and are summarised in Table 1. 3.1. Bandwidth When compared with the SANC, the frequency gain characteristic of the FD algorithm has a pedestal larger by a factor 2 (Hanning window) to 4 (Parzen window). If the 3 dB bandwidth is considered, the enlargement is even more dramatic as the characteristic tends to a rectangular shape. This is shown in Fig. 5 for the Parzen window, where the 3 dB bandwidth is compared as a function of the filter length N and the SNRðrÞ: It appears that for large values of N; the bandwidth tends to be many times greater than the SANC bandwidth (e.g. factor 4.17 for r ¼1

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Fig. 3. Frequency gains of (a) the SANC, (b) the FD algorithm with the rectangular window, (c) the Hanning window and (d) the Parzen window. N ¼512 and r ¼102.

and N ¼102). Very similar results are obtained by using other definitions of the bandwidth (e.g. variance over peak value), which makes sense since the frequency gain is close to an ideal bandpass shape. Enlargement of the bandwidth was checked to be less dramatic with the Hanning window (e.g. factor 1.6 for r ¼1 and N ¼102). 3.2. Side-lobes A noteworthy feature of the frequency gain jHðf Þj is that it presents important side-lobes. Fig. 6 illustrates the rate of increase of the relative amplitude of the first side-lobes jHðfs Þj=jHðf Þj according to the SNR by filter-length product ðrNÞ: It is seen that the rectangular and Hanning windows produce side-lobes whose relative amplitude quickly exceeds that of the Dirichlet function. The overtaking occurs for rN approximately and, respectively, equal to 10 and 400. However, the overtaking of the Parzen window does not occur before rND13 200. On this basis, the Parzen window should be preferred to other alternatives, despite it producing a very large 3 dB bandwidth as just shown. Incidentally, the fact that the performance of the selecting filter depends on the product rN as demonstrated in Fig. 6 provides an interesting insight into the issue of rejecting broadband noise. It means that for high SNRs, the length of a well-behaved filter should be set as short as possible,

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Fig. 4. Frequency gains of (a) the SANC, (b) the FD algorithm with the rectangular window, (c) the Hanning window and (d) the Parzen window. N ¼512 and r ¼1. Table 1 Characteristics of the frequency gains used in the FD algorithm Type of underlying window

Position of the first zero ðf0 Þ

Rectangular

1 N

Hanning

2 N

Parzen

4 N

Position of the first side-lobe ðfs Þ

B

Value of K in 1 rNK 2 jHðfs Þj ¼1 2 2 jHð0Þj 2rNK þ 1

Roll-off slope of side-lobes (dB/octave)

3 2N

2 3p

12

5 2N

8 B 105p

36

6 N

16 81p

48

implying that we should accommodate ourselves with a poor bandwidth which anyway does not have much noise to reject. On the other hand, for low SNRs, the filter could be made much longer in order to reduce its bandwidth without exaggeratedly invalidating its performance.

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Fig. 5. Evolution of the 3 dB bandwidth (normalised frequency, Parzen window) in the FD algorithm and comparison with that in the SANC.

Fig. 6. Evolution of the peak to first side-lobe relative amplitude for different windows in the FD algorithm and comparison with that in the SANC.

3.3. Improving the characteristics We shall now propose a simple way for reducing the bandwidth of the FD filter and for diminishing the amplitude of the highest side-lobes for a given rN product. The idea comes by inspecting the structure of jHðf Þj as given by formula (4). If the window used for tapering the Xkd ðnÞ sequence is set shorted by a factor k (and divided by k to constrain the static gain to be close

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to unity) while the length of Xk ðnÞ is left unchanged, then jHðf Þj ¼

2 1 jW ðf Þj 2 rNjW ðf =kÞj : jW ðf =kÞj 12 rNjW ðf =kÞj2 þ 1

ð5Þ

In fact, this is the frequency gain produced by a window of length N=k and corrected by the weighting function jW ðf Þ=W ðf =kÞj: Because the underlying window is shorter, the highest sidelobes of the new jHðf Þj are smaller. However, the point is that the zeros are still the same as for an underlying window of length N due to the weighting by jW ðf Þ=W ðf =kÞj; thus preserving the bandwidth. Note that k must be chosen as an integer value in order to avoid singularities in the jW ðf Þ=W ðf =kÞj weighting function. Fig. 12 (c) and (d) illustrates the effect of the weighting function for k ¼2 and 4 in case of the Hanning and Parzen windows with N ¼512. The corresponding frequency gains are displayed in Fig. 7 (a) and (b) for r ¼1. Fig. 8 illustrates the reduction in the 3 dB bandwidth for k ¼1,2,3,4,5 in case of the Parzen window. It is seen that a substantial reduction in bandwidth appears when stepping from k ¼1 to 2. The absolute reduction then becomes marginal when continuing to increase k; which suggests that values of 2 or 3 should give sufficient improvement in practical applications. Note that for k >1, the product of N with the 3 dB bandwidth tends quickly to a horizontal asymptote. This

Fig. 7. Effect of using windows of different lengths (k ¼2, N ¼512, r ¼1). Weighting function with (a) the Hanning window and (b) the Parzen window. The modified frequency gains (c and d) are obtained by multiplying these weighting functions with, respectively, the gains in Fig. 4(c) and (d).

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Fig. 8. Reduction in the 3 dB bandwidth (D3 dB ) with respect to the filter length.

means that the FD algorithm then behaves like the SANC, where the 3 dB bandwidth is inversely proportional to the filter length. The asymptotic values for the Parzen window are 2.09 for k ¼2, 1.93 for k ¼3, 1.88 for k ¼4 and 1.86 for k ¼5. This is to be compared with value 0.886 of the Dirichlet function of the SANC filter. In other words, by using k >1 and a Parzen window, the bandwidth tends to a value twice that of the SANC. By using a Hanning window, it was checked that the ratio of bandwidths was only about 1.5 (the asymptotic values are 1.64 for k ¼2, 1.52 for k ¼3, 1.48 for k ¼4 and 1.47 for k ¼5). 3.4. Recommendations In conclusion, the performance of the FD algorithm was shown to be somewhat poorer than that of the SANC for an equivalent filter length by SNR product ðrNÞ: Depending on the choice of the underlying tapering window, the bandwidth of the frequency gain can be up to many times as large as that of the SANC. For instance, the 3 dB bandwidth with the Parzen window is easily increased by a factor 5 or 6. This means that the filter length should be increased accordingly to achieve the same frequency resolution as the SANC. The Hanning window would give a better frequency resolution, yet it should be avoided for large rN because it produces excessive sidelobes. Nonetheless, it was shown that the resolution could be improved considerably by using tapering windows of different lengths, the effect of which begin to reduce the bandwidth of the filter. For instance, by using a Parzen window and k >1, it was demonstrated that simply doubling the FDfilter length would give about the same frequency resolution as the SANC. By using a Hanning window, the FD and the SANC algorithms would give about the same resolution for the same filter length. In spite of doubling the filter length in case of a Parzen window, the FD algorithm still offers a significant gain in computational speed. This is illustrated in Fig. 9. Incidentally, this comparison does not account for the fact that the SANC algorithm often requires processing the same signal many times before reaching convergence. Hence, the computational superiority of the FD

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Fig. 9. Relative speed of the FD algorithm vs the fast-SANC algorithm for a given frequency resolution 1=N:

algorithm in addition to the fact that it gives the optimal filter in one run clearly justifies its use for practical applications. The next sections will now be dedicated to some simulations, and more specifically, to the application of the methods for separating periodic and non-deterministic signals in the specific case of rolling element bearing diagnostics. 3.5. Statistical performances This section is concerned with the statistical performance of the proposed filter when estimated on finite length signals. As a convenient way to investigate the statistical errors involved by the estimation of Hðf Þ; we propose to measure the (theoretical) mean square error induced by the denoising a pure sine of amplitude A and frequency f0 embedded in white additive noise of power s2 : In the Fourier domain, at positive frequency f ; this is ( 2 )   1 A # ÞXM ðf Þ ð6Þ E  dðf  f0 Þ  Hðf e2 ðf Þ ¼ lim  M-N 2M þ 1 2 # Þ is given by Eq. (2) (with the adequate phase correction) and XM ðf Þ is the M-long where Hðf Fourier transform of X ðnÞ; n ¼ M; y; M: Expansion of Eq. (6) is found to yield an expression # Þg and EfjHðf # Þj2 g: Evaluation of these two terms that depends on the expected values EfHðf requires distinguishing two cases, namely whether f ¼ f0 or f af0 : # 0 Þ is negligible compared to its variance so (a) When f ¼ f0 ; it can be shown that the bias on Hðf # that EfHðf0 ÞgCHðf0 Þ: With this assumption and using standard stochastic perturba# 0 Þj2 g ¼ jHðf0 Þj2 þ # 0 ÞgC½1  jHðf0 Þj2 =K so that EfjHðf tion calculus, one finds VarfHðf 2 ½1  jHðf0 Þj =K with K the number of averaged sequences. # Þg ¼ 0: Also Hðf # Þ has real (b) When f af0 ; then Hðf Þ is theoretically zero and EfHðf and imaginary random parts which are independent and asymptotically follow # Þg ¼ Gaussian distributions with mean zero and variance 1=ð2 KÞ: Therefore, VarfHðf 2 # EfjHðf Þj g ¼ 1=K .

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Putting all results together, it follows that the mean square error on denoising a pure sine is   A2 rN þ 1 s2 2 ð7Þ 1 þ e ðf ÞCdðf  f0 Þ þ K K 4ð1=2rN þ 1Þ2 where we have used the fact that Hðf0 Þ ¼ 1=2rN=ð1=2rN þ 1Þ as given by Eq. (4). Eq. (7) has an important meaning. It says that two sources of error are expected when denoising # 0 Þ which does a pure sine in broadband noise. The first one stems from the frequency gain Hðf not exactly equal unity at f ¼ f0 : This error decomposes as the sum of a ‘‘systematic error’’ b ¼ A2 =½4ð1=2rN þ 1Þ2 and a ‘‘stochastic error’’ equal to bðrN þ 1Þ=K: The former is reduced by increasing the filter length N whereas the latter is reduced by increasing the number of averages K: This means that on a finite length signal where N and K are related, a compromise must be chosen between reducing the systematic or the stochastic error. The second source of error stems from the additive noise and is equal to s2 =K: Here again, this error is reduced by increasing K which places a similar trade-off as before.

4. Performance assessment through simulations Before applying the proposed algorithm on real vibration signals, it was instructive to test it on simulated signals. The first experiment consisted in extracting a pure sine (period of 10 samples for a total length of L ¼2 104 samples) embodied in white Gaussian noise with an SNR r ¼1. The filter length was set to N ¼512. The estimated frequency gains are displayed in Fig. 10 for k ¼1 and 2 and are in good agreement with the theoretical results of Figs. 4 and 7. Note the presence of a background power density due to estimation errors, the level of which theoretically equals 0.05 according to the equations derived in Section 3.4. For comparison with the SANC procedure, the next experiment consisted in extracting a slowly modulated sine as done in part I of this paper [1]. Fig. 11 displays the estimated frequency gain of

Fig. 10. Frequency gain of the FD algorithm for a pure sine (N ¼512, r ¼1).

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Fig. 11. Frequency gain of the FD algorithm for a modulated sine wave (N ¼2048, r ¼1).

the FD algorithm, for N ¼2048 and k ¼2 with a Parzen window. The FD gain turned out to be almost identical to the SANC gain (Fig. 5 of part I), as expected with a filter length twice as large. Consequently, the FD algorithm could be checked to have similar behaviour to that of SANC on this recognised difficult case.

5. Application to the separation of gear and bearing signals The noise cancellation principle has been found a useful technique for separating gear and bearing signals, to vastly improve the diagnostics of the latter, often masked by the strong gear signals (cf. Ref. [2] of part I). This is particularly the case in helicopter gearboxes, where shaft speeds range from several hundred Hertz at the input to a few Hertz at the rotor output. The associated gearmesh-related frequencies encompass the whole of the audible range and beyond, thus making it difficult to find a range dominated by the bearing signal. Gear signals are composed primarily of sinusoidal components phase-locked to shaft speeds, whereas bearing signals are rather pseudo-periodic, with a small random variation in the apparent period because of inevitable combination of rolling and slipping of the elements. Hence, for constant speed, the gear signals give rise to a discrete spectrum, whereas the small randomness in the bearing signals is enough to produce a continuous spectrum. This is very obvious for the higher harmonics in the bearing signals, where they smear over each other. On the basis of different autocorrelation times, the two signals can therefore be separated as explained in Section 2.1. The fact that the signal of interest here is the broadband noise requires some recommendations. Firstly, because it is important to completely remove the contribution of the periodic signals, it is conservative to use a noise cancellation filter (or more exactly the inverse of it) with a large bandwidth. Secondly, the effect of the side-lobes has to be minimised as they distort the recovered signal. The Parzen window of the FD algorithm turns out to be a satisfactory candidate with respect to these

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concerns. Also, better results are obtained if the measured vibration signal is first-order-tracked in order to make its deterministic part perfectly periodic. Experiments on a parallel-shaft spur-gear rig (1:1 ratio) were conducted at the Acoustics and Vibration Laboratories of the University of New South Wales. Different types of faults were inserted into the bearing supporting the output shaft (Bearing Koyo 1250, doube row, selfaligning). Results are presented here for an outer race fault, an inner race fault (notches generated by spark erosion) and a ball fault (filed line). The signals were obtained from a transducer placed close to the output bearing. They were sampled at 12 kHz during 4.17 s, and resampled according to a tacho signal giving one pulse per revolution. In order to resolve the output shaft modulations running at 6 Hz, the window length was set to N=8192 samples and k=1 was chosen for better extraction of the discrete components. Fig. 12(a) shows the vibration signal for the case of an outer race fault. It is very typical of gearbox vibration where the periodic gearmesh signal dominates over other phenomena. Fig. 12(c) shows the extracted broadband components after applying the FD algorithm. Here the presence of impacts on the faulty outer race becomes evident. These were buried in the strong periodic components of the gear signal and could not be detected before processing the signal. Similar results were obtained for other types of bearing fault. For example, Fig. 13 displays the results in the case of a ball fault. It was observed in these examples that the overall shape of the frequency gain naturally turned itself into a low-pass filter, due to the fact that the bearing faults prevailed over the bearing signal in the high-frequency range, whereas the reverse is true in the middle and low-frequency ranges. However, noise cancellation was able to extract the bearing signal even in the middle frequency

Fig. 12. Example of an outer-race fault in a gearbox: (a) measured vibration signal, (b) extracted periodic part, (c) extracted non-deterministic part.

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Fig. 13. Example of a ball fault in a gearbox: (a) measured vibration signal; (b) extracted periodic part; (c) extracted non-deterministic part.

Fig. 14. Example of an inner race fault: magnitude spectrum of the squared envelope in the demodulation band 4–7 kHz: (a) before noise cancellation; (b) after noise cancellation.

range where it was very weak. This is shown in Fig. 14, which compares the spectrum of the (non-standardised) squared envelope in the demodulation band 4–7 kHz, in the case of an inner race fault, before and after applying the FD algorithm. The first spectrum really only exhibits

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peaks related to the output shaft rotation (order 1), i.e. to periodic components, including the gearmesh frequency at order 32. On the other hand, the second spectrum clearly reveals the ballpass frequency of the inner race fault (about 7.1 orders), all the interfering periodic components being now removed. Note also the modulation sidebands and low harmonics from the shaft rotation, as expected for an inner race fault. Incidentally, it was observed in these examples that it was difficult for the Fast-SANC to converge to an accurate solution, even after processing the same signal several times. The reason is believed to come from an excessively small value of the forgetting factor m; necessary to ensure mean-square convergence but not strong enough to give fast convergence. 6. Conclusion This first part of this paper stated the problem of separating periodic signals from broadband signals by means of only one sensor or, in other words, to decompose a frequency spectrum into a discrete and a continuous part. A review was given of the existing techniques with a special emphasis on self-adapative noise cancellation, which has long proved to be efficient. However, a number of drawbacks exist with the SANC procedure when applied to actual vibration signals, mainly related with making the adaptation converge. In order to avoid the need for setting parameters and having an adaptation phase, it was proposed to directly estimate the noise cancellation frequency gain in the frequency domain, by following the classical approach of the H1 frequency response used in experimental modal analysis. This led to a new algorithm based on FFTs, with computational advantages. However, the price to be paid is a significant loss in frequency resolution. A simple refinement was proposed to largely solve this problem with no countereffects. The analytical performance of the proposed frequency-domain algorithm was analysed in detail and supported by simulation results. Evaluation of the statistical performance was also investigated. The last section of the paper illustrates the application of the noise cancellation principle to the separation of gear and bearing signals in gearboxes. The proposed frequency-domain algorithm proved to be simple to use and gave satisfying results for this purpose, thus making the diagnostics of faulty systems particularly easy. Indeed, the scope of applications in vibration-based diagnostics is likely to be very large. References [1] J. Antoni, R.B. Randall, this issue, Unsupervised noise cancellation for vibration signals. Part I—evaluation of adaptive algorithms, Mechanical Systems and Signal Processing (2003) 89–101. [2] B. Widrow, et al., Adaptive noise cancelling: principles and applications, Proc. IEEE 63 (1975) 1692–1716. [3] L.D. Mitchell, Improved methods for the FFT calculation of the frequency response function, Journal of Mechanical Design 104 (1982) 277–279. [4] R.B. Randall, Frequency Analysis, 3rd Edition, Bruel and Kjaer, Naerum, Denmark, 1987.