Cepstral Removal of Periodic Spectral Components from Time Signals

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1 Cepstral Removal of Periodic Spectral Components from Time Signals Robert B. Randall 1, Nader Sawalhi 2 1 School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney 252, Australia b.randall@unsw.edu.au 2 School of Mechanical Engineering, Prince Mohammad Bin Fahd University, Al Khober 31952, Kingdom of Saudi Arabia nsawalhi@pmu.edu.sa Abstract The use of the cepstrum for removing components from a signal which manifest themselves as periodic spectral components has previously been described. These include discrete frequency components with uniform spacing such as families of harmonics and modulation sidebands, but also narrow band noise peaks coming from slight random modulation of almost periodic signals, such as higher harmonics of bladepass frequencies. The removal is effected by applying a notch lifter to the real cepstrum of the signal, thus removing the targeted components from the log amplitude spectrum, and then combining the modified amplitude spectrum with the original phase spectrum. Not much attention was previously paid to the type of notch lifter, but two different situations occurring in conjunction with analysis of signals from wind turbines showed that different lifters have advantages in different situations. This paper describes two different approaches, illustrating them with the two examples of application. Keywords: cepstrum, harmonic removal, sideband removal, notch lifter, wind turbines 1 Introduction The cepstrum is the inverse Fourier transform of a log spectrum, and thus concentrates periodic spectrum components, such as families of equally spaced harmonics and sidebands into a small number of components called rahmonics in

2 2 the cepstrum. There are a number of types of cepstrum, but the most common are the complex cepstrum, formed from the complex spectrum, with both log amplitude and phase, and the real cepstrum (or power cepstrum) formed from the log amplitude only [Childers 1977]. Because the complex cepstrum includes the phase of the spectrum, it is possible to reverse the whole procedure back to the time domain after editing (or liftering ) in the cepstrum. However, to calculate the complex cepstrum requires that the phase be unwrapped to a continuous function of frequency, and this is only possible for well-behaved functions such as frequency response functions, for which the phase is continuous and then only if the log amplitude stays within the dynamic range of the measurement. Phase unwrapping is not possible for stationary signals, either forcing functions or responses, since by definition these consist of (mixtures of) discrete frequency components, for which the phase is undefined between the components, and stationary random components for which the phase varies randomly from one frequency to the next. However, it was recently shown [Randall, Sawalhi 211] that the log amplitude spectrum of stationary signals can be edited using the real cepstrum, and then the edited spectrum amplitudes combined with the original phase spectrum to return to the time domain. The procedure is illustrated in Figure 1.This can be done for example to remove a family of discrete frequency components, since these occupy only a small proportion of the total spectrum. The phase will be incorrect at the frequencies where the amplitudes are edited, but the latter will typically be greatly reduced to the level of adjacent noise components, and are much fewer in number, so the effects of such disturbance are often negligible. Fig. 1 Schematic diagram of the cepstral method for removing selected families of harmonics and/or sidebands from time signals

3 3 Figure 2 shows a typical result [Randall, Sawalhi 211] using the described method to remove all shaft harmonics (including gearmesh components) from the signals from a gearbox with a faulty bearing, where the result is almost as good as that obtained by removing the synchronously averaged signal, synchronized with the shaft rotational speed (only one average was required because the gear ratio was 1:1). Even though the residual signal containing the effects of the bearing fault was slightly noisier for the cepstral method, the envelope analysis for diagnosis of the bearing fault was equally good. 2 (a) Acceleration (m/s 2 ) (b) 2 (c) Shaft rotation Fig. 2: Time domain signals for gearbox test rig (a) Raw signal (b) Residual signal (after removing the synchronous average) (c) Residual signal after editing the Cepstrum to remove the shaft rahmonics In order to remove the harmonic/sideband family from the log amplitude spectrum, a notch lifter must be applied to the cepstrum, and this paper discusses the different ways in which this can be done. 2 Applying a Notch Lifter in the Cepstrum Notch Lifter Type 1 One of the first applications of a notch lifter to the cepstrum was to remove rahmonics corresponding to an echo. Figure 3 shows a typical lifter used to re-

4 4 move the effects of a double hit with an impact hammer [Gao 1996]. In principle, the effects of an echo in the log spectrum are to give an additive periodic component to the log spectrum, which translates into a series of discrete rahmonics in the cepstrum. However, these would only occupy single lines if the frequency range of the whole (2-sided) spectrum corresponded to an integer number of periods of this frequency spacing (the reciprocal of the echo delay time T), and in general it doesn t. Consequently, the cepstrum is convolved with the Fourier transform of a rectangular window, (a sinc function) and to remove all discrete rahmonics a notch window with a width corresponding to the effective width of this sinc function must be used. For this reason, the width of all notches in the lifter type 1 shown in Fig. 3 is made the same. Note that the overall lifter shown there (for that application) includes a highpass lifter, which at the same time removes the very low quefrency components corresponding to the forcing function, a short impact force with a smooth flat log spectrum. T T T Fig. 3: Lifter to remove rahmonics corresponding to an echo ( double hit hammer blow) The same reasoning applies to the use of a notch lifter to remove families of discrete harmonics from the log spectrum, where the windowing effect in the spectrum comes not only from a non-integer number of harmonics, but also from the fact that not all members of the harmonic family extend from the noise level by the same amount. Thus the family can be interpreted as multiplied by an amplitude modulating function in the frequency domain, whose Fourier transform is likewise convolved with the discrete components in the cepstrum. The (constant) width of the notch lifter is typically selected visually based on inspection of the first rahmonic(s). Notch Lifter Type 2 However, not all periodic spectrum components have this property of constant width in the spectrum, for example those resulting from a pseudo-cyclostationary

5 5 signal such as that from local faults in a rolling element bearing. Because of random slip, the series of impulse responses resulting from impacts of the rolling elements on the fault are not uniformly spaced, and the intervals Ti (= Ti + 1 Ti ) [Antoni 23]. This are random and result from a stationary Markov process { T i } has been studied in detail in [Antoni 23], where it was shown that the uncertainty of the arrival time of future impulses increases with time (measured from the one defined as the first), meaning that the probability density (of arrival time) increases in width, but decreases correspondingly in height, with increasing time from the reference point. The Ti only vary by a small amount of the order of 1 or 2%, and in this case the probability density can be modelled as Gaussian around the mean arrival time of each impulse, with mean spacing T. With this assumption the spectrum of a series of unit impulses was shown [4] to be given by: [ ] ( j T ) F ( α ) = 1 Φ *( α ) where Φ ( α ) = exp σ α α (1) α is the radian frequency, and σ is a measure of the random variation ( σ T is 1% for a standard deviation of 1%). Figure 4(a) shows the log amplitude spectrum for a 1% variation, and it is seen that the width of the harmonics also increases with increasing frequency at the same time as the height decreases. Figure 4(b) shows the real cepstrum corresponding to Fig. 4(a), and it is seen that the rahmonics also increase with increasing quefrency, though not proportionally to the latter Frequency Quefrency Fig. 4: Log spectrum and cepstrum for a pseudo-cyclostationary series of impulses with 1% random variation in pulse spacing (a) log spectrum (b) cepstrum

6 6 Another case where this type of spectrum structure occurs, with uniformly spaced spectral components but of gradually increasing width, is for the bladepassing harmonics of bladed machines such as turbines and fans, where there is a small amount of random frequency modulation of the perceived blade passage intervals, because the only connection between the blades and fixed points on the casing is via a turbulent fluid. Although coming from a much smaller percentage variation, the higher harmonics still gradually become broader. This led to the definition of a notch lifter Type 2, where the notch width was made proportional to quefrency. Even though not strictly necessary for the type of signal just discussed, it had the practical advantage that the notches of increasing width eventually merge into continuous zeros, giving a superimposed lowpass lifter, which tends to reduce noise in the high quefrency part and smooth the corresponding log spectrum to some extent. Until recently, this notch lifter Type 2 was used for all our applications with the new cepstral method, and gave excellent results in a number of different applications. These include the removal of fan blade passage harmonics as well as other discrete shaft harmonics to reveal a bearing fault [Randall, Sawalhi 211], where it was found that time synchronous averaging (TSA) removed the low harmonics of shaft speed but not all of the bladepass harmonics of a fan, because only the centre frequency of each harmonic peak was removed by TSA. It was also very successful in removing harmonics of the rotor bladepass frequency from response signals measured on a helicopter in flight, to facilitate performing operational modal analysis on the residual broadband signals [Randall, Peeters 212]. In that case, a slightly improved result was achieved by combining the notch lifter with an additional exponential lowpass lifter in any case, to give even more smoothing of the spectrum. 3 Application to Wind Turbines Two separate applications to wind turbines gave an appreciation of the different properties of lifters Types 1 and 2, so they are presented here. Removal of one set of harmonics The authors took part in a Round Robin organized by National Renewable Energy Laboratory (NREL), Golden CO, USA, where a wind turbine gearbox was analysed for faults that had occurred due to a short term loss of lubricant. Signals measured on the gearbox by NREL were initially analysed blind by the participants, and then again after they had received an inspection report on the dismantled gearbox. It was initially difficult to comment on the condition of gears, because it was only at the time of receiving the inspection report that spectra from the gearbox in healthy condition were provided. Many of the faults on the failed gears could then be detected and diagnosed, and most could in fact have been diagnosed blind if the reference data had been provided earlier. The results of analysis by all Round Robin teams have been published in an NREL Report [Sheng

7 7 212]. The analysis was made more difficult than usual by the fact that the design was far from hunting tooth, considered good design practice, where the numbers of teeth in mating pairs have no common factors, so that each tooth on one gear contacts every tooth on the other gear. In this case the final drive had a ratio of exactly 4:1 (88:22), meaning that the 22 teeth on the pinion always had contact with the same four groups of 22 teeth on the wheel. The drive from the sun gear to the intermediate shaft did however have a hunting tooth design (82:23). In our contribution to the NREL report, we performed a time synchronous average (TSA) over the rotation period of the intermediate shaft (IS), which turned out to be dominated by the high speed (HS) gearmesh (harmonics 88, 176..) and strong sidebands around them at the HS shaft speed of approx. 3 Hz (4 IS speed). When compared with the spectra in original condition, the HS toothmesh harmonics had increased indicating general wear, and the strong sidebands which had appeared were compatible with the severe scuffing that was reported around both pinion and wheel, but dominated by the location on the pinion. The four sections of 22 teeth were then averaged together to get the best estimate of the contribution of the pinion, this being subtracted from the total signal to get a residual dominated by effects other than from the HS pinion. This result is shown in Figure 5, and it is seen that the spectrum of the residual is dominated by the harmonics of the IS pinion gearmesh (23 teeth), but without modulation sidebands, although the noise level is quite high because of the small number of averages. These mesh harmonics were also much higher compared with the original condition, and the lack of (a) (b) Harmonics of ISS Harmonics of ISS Figure 5. Spectra of IS TSA average (a) Original including four rotations of the HSS. Harmonics at 88 (HS mesh), sidebands at 4 (HS shaft) (b) Residual after removal of the HSS average. Harmonics at 23 (IS pinion mesh). From [Sheng 212]

8 8 sidebands is compatible with the fact that the wear was distributed uniformly by the hunting tooth design, a point remarked on in the inspection report. After the NREL report was completed it was decided to try the cepstral method to separate the effects of the two sets of gears, using an appropriate notch lifter. Normally, even without a hunting tooth design, the tooth ratio is not an exact integer, so the two shaft speeds will have separate sets of rahmonics in the cepstrum. In this case, because of the exact 4:1 ratio, every fourth rahmonic of the HS shaft corresponded with a rahmonic of the IS shaft, so the code had to be modified to leave every fourth rahmonic. Complete separation cannot be achieved as it can be appreciated that remnants of the HS rahmonics are left in the IS rahmonics, but the same applied to the TSA method of Fig. 5.The cepstra before and after the application of the notch lifter are shown in Fig. 6, where it is seen that the Lifter Type 2 has been used, with increasing notch width. Note that the cepstra were obtained from the raw signals, without the requirement for order tracking or TSA, because the speed was reasonably stable..2 (a).1 HSS ISS (b) Quefrency (samples) Quefrency (samples) Figure 6 Cepstra of the IS shaft signal before (a) and after (b) the application of a notch lifter Type 2 to remove HSS rahmonics. Figure 7 shows the log spectra corresponding to the cepstra of Fig. 6, and can be compared with Fig. 5. It is scaled in Hz rather than harmonic order, but the HS shaft speed is about 3 Hz (and IS shaft speed 7.5 Hz). It is seen that the results are very similar, but the cepstral method gives better resolution and better definition of the base noise level.

9 9 Power Spectrum Magnitude (db) Power Spectrum Magnitude (db) (a) (b) Frequency (Hz) Frequency (Hz) Figure 7 Log spectra corresponding to the cepstra of Fig. 6 (a) Original signal. Harmonics at HS gearmesh. Sidebands at HS shaft speed. (b) Liftered signal. Harmonics at IS pinion mesh frequency. Removal of sidebands One of the advantages of the cepstral method is its ability to remove families of uniformly spaced sidebands, something that cannot be done by TSA (unless the sidebands are also harmonics). A study has been made by the company Mechanical Solutions Inc. NJ, USA, of the possibilities for making diagnosis of wind turbine machinery from measurements at the base of the tower. The initial measurements on a turbine with a squirrel cage induction generator were reported in [Randall et al. 211] and were very promising. Later measurements were made on a variable speed machine with doubly fed induction generator, where the rotor is supplied with a varying frequency field to change the speed. It appears that the variable frequency supply is transmitted via the tower, so it was found that the acceleration measurements made at the base of the tower were corrupted by electrical components with a spacing of 12 Hz (twice mains frequency). These did not appear in measurements made directly on the machines in the nacelle. The carrier frequencies for the modulation are unknown, but it is clear that they are not multiples of 12 Hz, so the sidebands are not harmonics.

10 1 A typical section of a spectrum is shown in Figure 8, where groups of sidebands spaced at 12 Hz can be seen in the spectrum. As can be seen in Fig. 9(a), the rahmonics resulting from these sidebands are so spread by weighting in the Hz Frequency (Hz) Figure 8 Zoomed spectrum showing groups of sidebands spaced at 12 Hz frequency domain that it was not possible to select the first manually, and so a different approach was used. The spacing of the rahmonics was known exactly, so the optimum width of notch lifter was found by trial and error. The result, using a notch Type 1 is shown in Figure 9, with a notch width of ±15% of the spacing. (a) Quefrency (sample no.).2 (b) Quefrency (sample no.) Figure 9 Cepstrum of whole spectrum (a) Original (b) After liftering by a Type 1 lifter

11 11 The original and liftered spectra are compared in Figure 1, and it can be seen that the remaining spectrum is little affected by the operation (a) Power Spectrum Magnitude (db) (b) Frequency (Hz) Figure 1 Log spectra corresponding to the cepstra of Fig. 9 (a) Original signal (b) After liftering Initially a Type 2 notch lifter was used, as this had become standard practice, but this had such a strong lowpass effect that the resulting liftered spectrum was more distorted. It has since been realized that a good first estimate for the notch width can be obtained by considering the windowing of the relevant components in the frequency domain. From Figures 8 and 1(a), it can be seen that the groups of sidebands cover 6-8 sideband spacings, and can be considered as equivalent to a weighting by a rectangular window of this width. Thus the equivalent window in the quefrency domain would be 1/8 to 1/6 of the spacing, ie 12.5% to 16.7%, where trial and error gave 15%. The reason for the sideband removal in this application was to remove extraneous effects from the spectra measured at the base of the tower, so that spectral changes would be more likely to indicate a change in condition of the machinery.

12 12 5 Conclusion A new method has become available to edit time signals to remove components that have a periodic structure in the frequency domain, such as families of harmonics and sidebands. This requires a notch lifter to be applied in the real cepstrum to edit the spectrum amplitudes, which are then combined with the original phase to obtain edited time signals. Two different types of notch lifters have been found to have different advantages in different situations and these are illustrated by two applications to wind turbine machinery. 6 Acknowledgements This work is partially supported by Australia s Defence Science and Technology Organisation (DSTO). Signals used for demonstration purposes were provided by National Renewable Energy Laboratory, CO, USA, and Mechanical Solutions Inc., NJ, USA, the latter obtained under a research project funded by NYSERDA. NYSERDA has not reviewed the information contained herein and the opinions expressed in this paper do not necessarily reflect those of NYSERDA or the State of New York. 7 References Antoni J, Randall, RB (23) A Stochastic Model for Simulation and Diagnostics of Rolling Element Bearings with Localised Faults. ASME J. Vib. Acoust., 125: Childers, DG, Skinner, D.P. Kemerait, R.C. (1977) The Cepstrum: a Guide to Processing, Proc. IEEE, 65(1): Gao Y, Randall, RB (1996) Determination of Frequency Response Functions from Response Measurements. Part I: Extraction of Poles and Zeros from Response Cepstra, Mech. Systems and Signal Processing, 1(3): Randall, RB, Peeters, B, Antoni, J, Manzato, S (212) New cepstral methods of signal preprocessing for operational modal analysis. ISMA212, Leuven, Belgium, Randall RB, Sawalhi N (211) Editing Time Signals using the Real Cepstrum, MFPT Conference, Virginia Beach, May. Randall RB, Sawalhi N, Marscher W, Walter T, Weiss J (211) Decomposition of structural vibration for wind turbine diagnosis, MFPT Conference, Virginia Beach, May. Sheng, S (Ed.) (212) Wind Turbine Gearbox Condition Monitoring Round Robin Study Vibration Analysis. National Renewable Energy Laboratory, Technical Report NREL/TP , July 212. doi:

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