Article

A History of Cepstrum Analysis and its Application to Mechanical Problems

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Abstract

It is not widely realised that the first paper on cepstrum analysis was published two years before the FFT algorithm, despite having Tukey as a common author, and its definition was such that it was not reversible even to the log spectrum. After publication of the FFT in 1965, the cepstrum was redefined so as to be reversible to the log spectrum, and shortly afterwards Oppenheim and Schafer defined the “complex cepstrum”, which was reversible to the time domain. They also derived the analytical form of the complex cepstrum of a transfer function in terms of its poles and zeros. The cepstrum had been used in speech analysis for determining voice pitch (by accurately measuring the harmonic spacing), but also for separating the formants (transfer function of the vocal tract) from voiced and unvoiced sources, and this led quite early to similar applications in mechanics. The first was to gear diagnostics (Randall), where the cepstrum greatly simplified the interpretation of the sideband families associated with local faults in gears, and the second was to extraction of diesel engine cylinder pressure signals from acoustic response measurements (Lyon and Ordubadi). Later Polydoros defined the differential cepstrum, which had an analytical form similar to the impulse response function, and Gao and Randall used this and the complex cepstrum in the application of cepstrum analysis to modal analysis of mechanical structures. Antoni proposed the mean differential cepstrum, which gave a smoothed result. The cepstrum can be applied to MIMO systems if at least one SIMO response can be separated, and a number of blind source separation techniques have been proposed for this. Most recently it has been shown that even though it is not possible to apply the complex cepstrum to stationary signals, it is possible to use the real cepstrum to edit their (log) amplitude spectrum, and combine this with the original phase to obtain edited time signals. This has already been used for a wide range of mechanical applications. A very powerful processing tool is an exponential “lifter” (window) applied to the cepstrum, which is shown to extract the modal part of the response (with a small extra damping of each mode corresponding to the window). This can then be used to repress or enhance the modal information in the response according to the application.

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... Smith et al. improved the identification efficiency of cepstrum-based OMA and reduced the complexity of the algorithm through optimization of the fitting method [24]. In recent years, with the increasingly widespread application of cepstrum analysis in modal analysis, cepstrum-based OMA has also been gradually refined [25,26]. Currently, cepstrum-based OMA is only applied in fault detection for simple rotating mechanical structures such as bearings and gears, with limited application in complex and large-scale mechanical structures like CNC machine tools [27,28]. ...
... However, the commonly encountered stationary signals in practical applications do not satisfy this requirement. Therefore, the complex cepstrum is usually obtained indirectly by first calculating the real cepstrum [26]. The relationship between the complex cepstrum and the real cepstrum can be expressed as: ...
... represents the impulse at f = i f t . Equation (26) shows that the sawtooth wave excitation, as a typical periodic signal, also has discrete characteristics in its amplitude spectrum. Moreover, there are significant differences in the amplitudes at the fundamental frequency and each harmonic frequency, and the spectral lines appear steep. ...
Article
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Conducting research on the dynamics of machine tools can prevent chatter during high-speed operation and reduce machine tool vibration, which is of significance in enhancing production efficiency. As one of the commonly used methods for studying dynamic characteristics, operational modal analysis is more closely aligned with the actual working state of mechanical structures compared to experimental modal analysis. Consequently, it has attracted widespread attention in the field of CNC machine tool dynamic characteristics research. However, in the current operational modal analysis of CNC machine tools, discrepancies between the excitation methods and the actual working state, along with unreasonable vibration response signal acquisition, affect the accuracy of modal parameter identification. With the development of specimen-based machine tool performance testing methods, the practice of identifying machine tool characteristics based on machining results has provided a new approach to enhance the accuracy of CNC machine tool operational modal analysis. Existing research has shown that vibration significantly influences surface topography in flank milling. Therefore, a novel operational modal analysis method is proposed for the CNC machine tool based on flank-milled surface topography. First, the actual vibration displacement of the tooltip during flank milling is obtained by extracting vibration signals from surface topography, which enhances the accuracy of machine tool operational modal analysis from both the aspects of the excitation method and signal acquisition. A modified window function based on compensation pulses is proposed based on the quefrency domain characteristics of the vibration signals, which enables accurate extraction of system transfer function components even when the high-frequency periodic excitation of the machine tool causes overlap between the system transfer function components and the excitation components. Experimental results demonstrate that the proposed method can obtain accurate operational modal parameters for CNC machine tools.
... For single input multiple output (SIMO) systems, any output signal ( ) is the convolution of the input signal ( ) (forcing function) with the impulse response function of the transmission path ( ) (transfer function), as given in Equation (5). The convolution ( * ) is converted to a multiplication in the frequency domain (Equation (6)) and to an addition process (Equation (7)) by the log function in the cepstrum operations as follows [35]: ...
... More comprehensive information and advice on the use of the cepstrum can be found in [36]. Cepstrum analysis has been widely used in a number of applications, e.g., radar signal and marine exploration [34], biomedical signal processing [37], speech analysis [38] and mechanics [35]. In speech analysis, the cepstrum is used to determine the voice pitch (performed by measuring the harmonic spacing) and to separate the formants, i.e., the transfer function of the vocal tract, from voiced and unvoiced sources [38]. ...
... In this sense, the cepstrum serves as an effective tool to characterize families of harmonics, sidebands and modulations in a clear and easy-to-interpret format, as these families become concentrated in rahmonics. This makes the cepstrum an important tool for gear health monitoring, as local faults in gears give an impulsive modulation of the gear mesh signals (both amplitude and frequency modulation) that results in sidebands spaced at the speed of the gear on which the local fault is located [35]. A number of researchers have made use of the cepstrum to detect and isolate the source of a local gear fault to a particular gear set, e.g., [40][41][42]. ...
Article
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Detecting gear rim fatigue cracks using vibration signal analysis is often a challenging task, which typically requires a series of signal processing steps to detect and enhance fault features. This task becomes even harder in helicopter planetary gearboxes due to the complex interactions between different gear sets and the presence of vibration from sources other than the planetary gear set. In this paper, we propose an effectual processing algorithm to isolate and enhance rim crack features and to trend crack growth in planet gears. The algorithm is based on using cepstrum editing (or liftering) of the hunting-tooth synchronous averaged signals (angular domain) to extract harmonics and sidebands of the planet gears and low-pass filtering and minimum entropy deconvolution (MED) to enhance extracted fault features. The algorithm has been successfully applied to a vibration dataset collected from a planet gear rim crack propagation test undertaken in the Helicopter Transmission Test Facility (HTTF) at DSTG Melbourne. In this test, a seeded notch generated by an electric discharge machine (EDM) was used to initiate a fatigue crack that propagated through the gear rim body over 94 load cycles. The proposed algorithm demonstrated a successful isolation of incipient fault features and provided a reliable trending capability to monitor crack progression. Results of a comparative analysis showed that the proposed algorithm outperformed the traditional signal processing approach.
... For single input multiple output (SIMO) systems, any output signal ( ( )) is the convolution of the input signal ( ) (forcing function) with the impulse response function of the transmission path ( ) (transfer function) as given in equation (5). The convolution (*) is converted to a multiplication in the frequency domain (equation 6) and to an addition process (equation (7) by the log function in the cepstrum operations as follows [35]: ...
... Cepstrum analysis has been widely used in a number of applications, e.g. radar signal and marine exploration [34], biomedical signal processing [37], speech analysis [38] and mechanics [35]. In speech analysis, the cepstrum was used to determine the voice pitch (done by measuring the harmonic spacing) and to separate the formants, i.e. the transfer function of the vocal tract, from voiced and unvoiced sources [38]. ...
... In this sense, the cepstrum serves as an effective tool to characterize families of harmonics, sidebands and modulations in a clear and easy to interpret format, as these families become concentrated in rahmonics. This makes the cepstrum an important tool for gear health monitoring, as local faults in gears give an impulsive modulation of the gear mesh signals (both amplitude and frequency modulation) that results in sidebands spaced at the speed of the gear on which the local fault is located [35]. A number of researchers have made use of the cepstrum to detect and isolate the source of a local gear fault to a particular gear set, e.g. ...
Preprint
Full-text available
In this paper, an effectual processing algorithm to isolate and enhance rim crack features and to trend crack growth in planet gears is proposed. The algorithm is based on using cepstrum editing (or liftering) of the hunting tooth synchronous averaged signals (angular domain) to extract harmonics and sidebands of the planet gears, and low-pass filtering and minimum entropy deconvolution (MED) to enhance extracted fault features. The algorithm has been successfully applied to a vibration dataset collected from a planet gear rim crack propagation test undertaken in the Helicopter Transmission Test Facility (HTTF) at DSTG Melbourne. In this test, a seeded notch generated by an electric discharge machine (EDM) was used to initiate a fatigue crack that propagated through the gear rim body over 94 load cycles. The proposed algorithm demonstrated a successful isolation of incipient fault features and provided a reliable trending capability to monitor crack progression. Results of a comparative analysis showed that the proposed algorithm outperformed the traditional signal processing approach.
... where A(f) is the amplitude spectrum of the signal, iφ(f) is the phase spectrum of the signal, and F -1 is the inverse form of the Fourier transform (Randall, 2017). From the equations presented, it follows that the composite form of the cepstrum contains information about the phase of the signal. ...
... Successive values on the ordinate axis of the waveform can be interpreted as units of time, while values on the cut-off axis are related in some way to the autocorrelation and the fundamental tone can be determined from them. Cepstrum can be defined as information about the rate of change in the frequency bands of the frequency spectrum of the analyzed signal (Randall, 2017). Despite its age, Cepstrum is one of the less known and less used signal processing methods in terms of machine diagnostics. ...
... Despite its age, Cepstrum is one of the less known and less used signal processing methods in terms of machine diagnostics. Randall believes, however, that the capabilities of the method have not yet been fully exploited and it is potentially amenable to further development (Randall, 2017). ...
Article
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The paper presents the course of investigations and the analysis of the possibility of applying selected methods of time-frequency processing of non-stationary acoustic signals in the assessment of the technical condition of tram drive components, as well as a new combined method proposed by the authors. An experiment was performed in the form of a pass-by test of the acoustic pressure generated by a Solaris Tramino S105p tram. A comparative analysis has been carried out for an efficient case and a case with damage to the traction gear of the third bogie in the form of broken gear teeth. The recorded signal was analyzed using short-time Fourier transform (STFT) and continuous wavelet transform (CWT). It was found that the gear failure causes an increase in the sound level generated by a given bogie for frequencies within the range of characteristic frequencies of the tested device. Due to the limitations associated with the fixed window resolution in STFT and the inability to directly translate scales to frequencies in CWT, it was found that these methods can be helpful in determining suspected damage, but are too imprecise and prone to errors when the parameters of both transforms are poorly chosen. A new CWT-Cepstrum method was proposed as a solution, using the wavelet transform as a pre-filter before cepstrum signal processing. With a sampling rate of 8192 Hz, a db6 mother wavelet, and a scale range of 1:200, the new method was found to infer the occurrence of damage in an interpretation-free manner. The results were validated on an independent pair of trams of the same model with identical damage and as a reference on a pair of undamaged trams demonstrating that the method can be successfully replicated for different vehicles.
... Cepstrum is defined as the inverse Fourier transform of the logarithm of the estimated signal spectrum [19,20]. In terms of algorithm, it can be implemented as the logarithm of inverse fast Fourier transform (FFT) routine. ...
... A typical application case of cepstrum is in gearbox maintenance [19][20][21]; indeed, when gear faults are developing, sidebands increase their energy in structural response spectra at gear angular speeds. Such sidebands encountered in system frequency behavior exhibit peculiar spacing which may be identified more easily by cepstrum. ...
... In the same way that frequency in a complex spectrum gives no information regarding absolute times but just about recurrent temporal intervals (i.e., the periodic time), quefrency just gives indications regarding frequency spacing, but not about absolute frequencies. Indeed, many of the sidebands conveyed by a signal are better identifiable on a spectrum with a logarithmic amplitude scale; therefore, cepstral analysis is a tailored method to extract the (average) information hidden in a signal, as it converts numerous sidebands into a few significant "rahmonics" in signal cepstrum; indeed, only the first ones carry the main signal energy, thus allowing an easier identification of dominant components [20]. ...
Conference Paper
High-speed centrifugal compressors may be exploited to pressurize fuel cell systems. Nonetheless, due to fuel cells significant interposed volumes, compressor behavior can lead to severe vibrations related to fluid-dynamic instabilities during part load operating conditions. In particular, surge strongly limits centrifugal compressors’ stable operating region when moving towards low mass flow rates due to a change in system working point. Therefore, compressor dynamic response must be adequately characterized for early surge detection. To this aim, a dedicated experimental activity was conducted on a vaneless diffuser turbocharger coupled to a solid oxide fuel cell emulator plant; compressor evolution towards surge was investigated. Several signal processing techniques were applied to pressure signals as well as vibro-acoustic responses to better predict compressor behavior and classify its status as stable or unstable. Cepstrum, cross-correlation and wavelet transform have been identified as suitable techniques to define precursors able to detect incipient surge conditions early. By means of cross-correlation function, propagation phenomena in the ducts can be investigated to assess how they interact near compressor low-mass flow rate unstable conditions. Cepstrum provides a convenient way to determine pressure signal spectrum distortion in terms of further periodic components onset; they may be due to complex system responses generated by transient phenomena; indeed, it allows identification of hidden anomalous contributions in system response which may arise in incipient surge conditions. Wavelet transform was performed on both structural and pressure response signals to observe their dominant energy contents temporal evolution; indeed, such spectral pattern time-dependent variation can detect the rise of unstable conditions. By doing so, a complete system identification is performed which allows a deeper investigation of the physical phenomena involved; moreover, a more complete set of surge precursors extracted from different probes’ physical signals were defined. The results obtained provide original diagnostic insights for monitoring systems suited to perform early surge detection. Indeed, compressor instability prevention can extend its operating range, performance, and reliability to allow better integration with other plant components. Finally, cepstrum application for compressor instability identification can be regarded as a novel method in the fluid machinery field.
... So a wide range of methods for time series classification based on different invariances are available, offering a variety of performance characteristics suited to different applications [1], [2]. In the case of dynamic systems, it is also known that signal decomposition and interpretation methods such as spectral and cepstral analysis, and phase space reconstructions using Takens embedding theorem, provide useful features to interpret and compare the states of systems [3]- [5]. ...
... CEPS is not commonly used as a general time series similarity measure but it is effective when the series have an underlying regularity or cyclicity [5], [44] as with the problems here. The measure d CEPS captures information about the relative rates of change of the two signals across their frequency bands. ...
Preprint
Distinguishing between classes of time series sampled from dynamic systems is a common challenge in systems and control engineering, for example in the context of health monitoring, fault detection, and quality control. The challenge is increased when no underlying model of a system is known, measurement noise is present, and long signals need to be interpreted. In this paper we address these issues with a new non parametric classifier based on topological signatures. Our model learns classes as weighted kernel density estimates (KDEs) over persistent homology diagrams and predicts new trajectory labels using Sinkhorn divergences on the space of diagram KDEs to quantify proximity. We show that this approach accurately discriminates between states of chaotic systems that are close in parameter space, and its performance is robust to noise.
... Originally, it was applied for the echo detection in seismic signals. The cepstrum has been used in a wide variety of applications, such as pitch detection in acoustic signals [2], analysis of mechanical problems [3] and human activity recognition [4]. ...
... In the next Section, we will give some numerical illustrations and applications. 3 This is the troublesome step when m = l and there will not necessarily be a straightforward equivalence between det Φ H and det M M . A generalization of the multiplicative property, the Binet-Cauchy theorem [10], [11], may offer a solution, which we will not explore further here. ...
Preprint
This paper extends the concept of scalar cepstrum coefficients from single-input single-output linear time invariant dynamical systems to multiple-input multiple-output models, making use of the Smith-McMillan form of the transfer function. These coefficients are interpreted in terms of poles and transmission zeros of the underlying dynamical system. We present a method to compute the MIMO cepstrum based on input/output signal data for systems with square transfer function matrices (i.e. systems with as many inputs as outputs). This allows us to do a model-free analysis. Two examples to illustrate these results are included: a simple MIMO system with 3 inputs and 3 outputs, of which the poles and zeros are known exactly, that allows us to directly verify the equivalences derived in the paper, and a case study on realistic data. This case study analyses data coming from a (model of) a non-isothermal continuous stirred tank reactor, which experiences linear fouling. We analyse normal and faulty operating behaviour, both with and without a controller present. We show that the cepstrum detects faulty behaviour, even when hidden by controller compensation. The code for the numerical analysis is available online.
... [50,51,52] are not necessary, but could be useful for non-stationary applications. Lastly, no additional pre-processing methods are used before employing the ICS2 filter, however, future work could involve examining the influence of data cleaning techniques such as cepstrum editing or discrete-random separation [53,54]. ...
... This section investigates the performance of these standard processing tools and aims to provide an additional point of reference to compare against. Four popular signal processing methodologies are assessed: autopower spectrum [56], squared envelope spectrum [57], cyclic spectral coherence [58], and real cepstrum [53,59]. All four of these are spectral analysis techniques that allow for tracking the specific fault characteristic frequencies, i.e. harmonics or rahmonics related to the planet gear fault 505 meshing with the sun and ring gear. ...
... Hence, the cepstrum of the response signal ( ) is the sum of the forcing function's cepstrum and the transfer function's cepstrum, thus allowing the separation of the two. Note that this is applicable for SIMO systems, in the case of multiple input and multiple output systems, the response function is a sum of convolutions, requiring signal processing techniques, such as blind source separation [35], to process the signal to a single excitation prior to any further processing [36]. Since this topic falls outside the scope of this paper, the relevant research will be conducted in further work. ...
... The modal-property-dominant-generated layer[36]. ...
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Various deep learning methodologies have recently been developed for machine condition monitoring recently, and they have achieved impressive success in bearing fault diagnostics. Despite the capability of effectively diagnosing bearing faults, most deep learning methods are tremendously data-dependent, which is not always available in industrial applications. In practical engineering, bearings are usually installed in rotating machinery where speed and load variations frequently occur, resulting in difficulty in collecting large training datasets under all operating conditions. Additionally, physical information is usually ignored in most deep learning algorithms, which sometimes leads to the generated results of low compliance with the physical law. To tackle these challenges, a novel Physics-Informed Residual Network (PIResNet) is proposed for learning the underlying physics that is embedded in both training and testing data, thus providing a physical consistent solution for imperfect data. In the proposed method, a physical modal-property-dominant-generated layer is adopted at first to generate the modal-property-dominant feature. Then, a domain-conversion layer is constructed to enable the feasibility of extracting the discriminative bearing fault features under varying operating speed conditions. Lastly, a parallel bi-channel residual learning architecture that can automatically extract the bearing fault signatures is meticulously established to incorporate the bearing fault characteristics. Experimental datasets under variable operating speeds and loads, and time-varying operating speeds are utilized to demonstrate the superiority of the PIResNet under non-stationary operating conditions.
... Finding periodic patterns in state signals (especially vibrations) has been the subject of numerous previous theoretical and experimental approaches in signal processing techniques, particularly in the field of fault detection in rotating machinery. The inverse Fourier transform applied to the logarithm of the magnitudes of the power spectrum obtained by the direct Fourier transform of the state signal, known as the Cepstrum technique [16,17], or some other techniques derived from it have been applied in [12,[18][19][20]. Spectral correlation density [21], which examines the correlation between different components of a signal that are related to each other by frequency, or techniques derived from it, was applied in [22][23][24][25]. ...
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... 5 Advancements in condition monitoring and predictive maintenance techniques have been instrumental in the early detection and diagnosis of issues in rotating machinery, enabling timely interventions and minimizing unplanned downtime. [6][7][8][9] From a vast amount of monitored device data, advanced artificial intelligence algorithms are employed to extract diagnostic knowledge latent within the data, facilitating automated identification of the health status of the devices. This approach represents a prominent area of research, finding extensive application in the domain of fault diagnosis tasks for mechanical equipment. ...
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... We quickly summarize how the zero-latency filter is created, but point the reader to [54] for the full descrip-tion, motivation and derivation. The steps to compute the zero-latency filter are as follows • Compute the "cepstrum" time series of the PSD [55]; in our case, this is the inverse Fourier transform of the elementwise logarithm of the PSD. ...
Preprint
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... Beyond the more traditional signal processing techniques used by some authors, cepstral analysis has recently gathered interest among researchers. This type of technique involves calculating the inverse Fourier transform of the logarithm of a spectral estimate, which is known as cepstrum, and enables the investigation of periodic components, as well as abrupt changes, in the frequency spectrum [33][34][35]. Recent studies [36,37] indicate the significant potential of these features on structural damage detection, particularly the Mel-frequency cepstrum (MFC), which already has several applications in the field of audio processing and speech recognition [38][39][40][41]. ...
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... Decomposition and S-transform [15], and Hilbert-Huang Transform [47], among others. Then, features such as the root mean square of the time-domain signal [4], the kurtosis and peak value of the frequency-domain signal [22], and features from the cepstrum [30] are extracted from the processed signals. These feature information are then input into suitable machine learning models for classification. ...
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In conditions of multi-fault coupling, varying loads and speeds, as well as noise interference, bearing vibration signals present various complex issues, leading to difficulties in feature extraction and the need for a large number of training samples for diagnostic methods. This paper designs a multi-fault coupling experiment for rolling bearings under varying load and speed conditions and proposes a new fault diagnosis method that uses the power spectrum of the AR model and a convolutional neural network to diagnose complex multi-faults in rolling bearings. It takes the original vibration signal as input, uses the AR model to convert the time-domain signal into a power spectrum, and then classifies it using a convolutional neural network. To test the performance of the AR model power spectrum convolutional neural network, this method was compared with some fault diagnosis methods. The results show that this method can achieve higher diagnostic accuracy under varying loads and speeds, and requires fewer training samples. In addition, the noise resistance of this method is also superior to other fault diagnosis methods.
... From innovative data acquisition methods to sophisticated preprocessing techniques and the application of machine learning, the field demonstrates a blend of traditional signal processing principles and contemporary methodologies. The convergence of theoretical insights and practical implementations serves as a foundation for continued advancements in acoustic signal processing for bowel sound analysis (Poeppel, 2001;Poluboina et al., 2023;Randall, 2017;Ravanelli et al., 2018). ...
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... They prove effective in cases where the noise frequency range is known and the signal and noise frequency bands are distinct, but their denoising effectiveness diminishes when confronted with the prevalent white noise encountered in practical applications [21]. Inverse spectrum analysis emerges as a technique capable of accurately estimating the periodic components of a signal in the frequency domain, with noise typically manifesting as randomly distributed high-frequency components [22]. Consequently, effective suppression of high-frequency noise can be achieved by extracting the lowfrequency inverse spectrum components. ...
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The recognition of cavitation status is crucial in the state monitoring of centrifugal pumps. For improving the efficiency of identifying cavitation status in centrifugal pumps, a unique method is proposed based on signal demodulation and efficient neural network (DEN). Experimental investigations of cavitation phenomena were conducted on centrifugal pumps. Vibration signals at six distinct frequencies were collected from the pump casing under three different temperature conditions. Signal demodulation was used to extract the characteristic frequencies of the modulated components. The preprocessed data were then input into a deep learning model that integrates MBConv architecture. Subsequently, the researchers conducted parameter optimization and cross-validation to develop the final DEN cavitation status identification model. The research results indicate that this method achieved a successful cavitation state identification rate of 89.44%. Compared to using FFT-transformed frequency domain signals as model inputs, the recognition accuracy improved by 20.69%. Compared to an autoencoder model, the recognition precision enhanced by 25.28%. The results confirm the efficacy of integrating signal demodulation and deep learning for cavitation recognition, providing a new technological pathway for the monitoring of centrifugal pump conditions.
... The authors have cited additional references within the Supporting Information. [46][47][48] ...
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Polycyclic aromatic hydrocarbons and their nitrogen‐substituted analogues are of great interest for various applications in organic electronics. The performance of such devices is determined not only by the properties of the single molecules, but also by the structure of their aggregates, which often form via self‐aggregation. Gaining insight into such aggregation processes is a challenging task, but crucial for a fine‐tuning of the materials properties. In this work, an efficient approach for the generation and characterisation of aggregates is described, based on matrix‐isolation experiments and quantum‐chemical calculations. This approach is exemplified for aggregation of acridine. The acridine dimer and trimer are thoroughly analysed on the basis of experimental and calculated UV and IR absorption spectra, which agree well with each other. Thereby a novel structure of the acridine dimer is found, which disagrees with a previously reported one. The calculations also show the changes from excitonic coupling towards orbital interactions between two molecules with decreasing distance to each other. In addition, a structure of the trimer is determined. Finally, an outlook is given on how even higher aggregates can be made accessible through experiment.
... CA is efficient for signals with many families of periodicities and can be used to detect and quantify families of periodically spaced spectral components such as harmonics, as well as equally spaced modulation sidebands. Cepstrum analysis presents a lack of sensitivity to the position of the accelerometer sensor on the gearbox case, as CA collects (and averages) information about the sidebands from the whole spectrum, reducing the same order of sidebands for various GMF harmonics into a single line in the cepstrum [15]. ...
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The current paper presents helical gearbox defect detection models built from raw vibration signals measured using a triaxial accelerometer. Gear faults, such as localized pitting, localized wear on helical pinion tooth flanks, and low lubricant level, are under observation for three rotating velocities of the actuator and three load levels at the speed reducer output. The emphasis is on the strong connection between the gear faults and the fundamental meshing frequency GMF, its harmonics, and the sidebands found in the vibration spectrum as an effect of the amplitude modulation (AM) and phase modulation (PM). Several sets of features representing powers on selected frequency bands or/and associated peak amplitudes from the vibration spectrum, and also, for comparison, time-domain and frequency-domain statistical feature sets, are proposed as predictors in the defect detection task. The best performing detection model, with a testing accuracy of 99.73%, is based on SVM (Support Vector Machine) with a cubic kernel, and the features used are the band powers associated with six GMF harmonics and two sideband pairs for all three accelerometer axes, regardless of the rotation velocities and the load levels.
... Wang et al., 2014 removed the smearing effect of varying speed to present fault characteristic frequency order and then used envelope order tracking to diagnose the rolling bearing fault. However, these methods required assistant devices such as a tachometer or encoder (Di Lorenzo et al., 2017;Randall, 2017). It will increase the measurement cost and make it difficult to install. ...
Article
Rolling bearings always operate under variable speed conditions, which poses a challenge for researchers in identifying and classifying bearing faults. In contrast to the stationary speed condition, the Fault Characteristic Frequency (FCF) under variable speed conditions exhibits a variable value that depends on the instantaneous shaft rotational speed (ISRS). The presentation of the FCFs in the frequency domain reveals overlapping patterns among them. To solve the mentioned problem, a method based on the Short-time Fourier Transform SynchroSqueezing Transform (FSST) and Principal Component Analysis (PCA) is proposed. By illustrating the envelope signal in time-frequency distribution using FSST, the FCF is highlighted in each ISRS value. Finally, this time-frequency distribution is used as input of PCA to classify rolling bearings. This method successfully diagnosed both inner race fault and outer race fault.
... Traditionally, the research on bearing fault detection is mainly focused on the field of signal analysis, which is mainly to obtain the time-domain, frequency-domain and time-frequency characteristics of the vibration signals, and the commonly used methods are power spectrum analysis [6], cepstrum analysis [7], envelope spectral analysis [8], wavelet analysis [9], continuous wavelet transform (CWT) [10], and empirical modal decomposition (EMD) [11]. Although they have achieved some success, these methods rely on manually designed features and have weak generalization ability, making them difficult to apply to new scenarios. ...
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Bearings are critical components of industrial equipment and have a significant impact on the safety of industrial physical systems. Their failure may lead to equipment shutdown and accidents, posing a significant risk to production safety. However, it is difficult to obtain a large amount of bearing fault data in practice, which makes the problem of small sample size a major challenge for bearing fault detection. In addition, some methods may overlook important features in bearing vibration signals, leading to insufficient detection capabilities. To address the challenges in bearing fault detection, this paper proposed a few sample learning methods based on the multidimensional convolution and attention mechanism. First, a multichannel preprocessing method was designed to more effectively utilize the information in the bearing vibration signal. Second, by extracting multidimensional features and enhancing the attention to important features through multidimensional convolution operations and attention mechanisms, the feature extraction ability of the network was improved. Furthermore, nonlinear mapping of feature vectors into the metric space to calculate distance can better measure the similarity between samples, thereby improving the accuracy of bearing fault detection and providing important guarantees for the safe operation of industrial systems. Extensive experiments have shown that the proposed method has good fault detection performance under small sample conditions, which is beneficial for reducing machine downtime and economic losses.
... While synchronous averaging retains the signal of the synchronous components, such as shafts and gears, Dephase filters them out. There are also alternative approaches, such as employing cepstrum analysis [42,43] for "liftering" out the synchronous components [44,45]. ...
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One of the common methods for implementing the condition-based maintenance of rotating machinery is vibration analysis. This tutorial describes some of the important signal processing methods existing in the field, which are based on a profound understanding of the component’s physical behavior. Furthermore, this tutorial provides Python and MATLAB code examples to demonstrate these methods alongside explanatory videos. The goal of this article is to serve as a practical tutorial, enabling interested individuals with a background in signal processing to quickly learn the important principles of condition-based maintenance of rotating machinery using vibration analysis.
... Including references e.g. [61], [62], [63], [64] ...
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Acoustic data serves as a fundamental cornerstone in advancing scientific and engineering understanding across diverse disciplines, spanning biology, communications, and ocean and Earth science. This inquiry meticulously explores recent advancements and transformative potential within the domain of acoustics, specifically focusing on machine learning (ML) and deep learning. ML, comprising an extensive array of statistical techniques, proves indispensable for autonomously discerning and leveraging patterns within data. In contrast to traditional acoustics and signal processing, ML adopts a data-driven approach, unveiling intricate relationships between features and desired labels or actions, as well as among features themselves, given ample training data. The application of ML to expansive sets of training data facilitates the discovery of models elucidating complex acoustic phenomena such as human speech and reverberation. The dynamic evolution of ML in acoustics yields compelling results and holds substantial promise for the future. The advent of electronic stethoscopes and analogous recording and data logging devices has expanded the application of acoustic signal processing concepts to the analysis of bowel sounds. This paper critically reviews existing literature on acoustic signal processing for bowel sound analysis, outlining fundamental approaches and applicable machine learning principles. It chronicles historical progress in signal processing techniques that have facilitated the extraction of valuable information from bowel sounds, emphasizing advancements in noise reduction, segmentation, signal enhancement, feature extraction, sound localization, and machine learning techniques. This underscores the evolution in bowel sound analysis. The integration of advanced acoustic signal processing, coupled with innovative machine learning methods and artificial intelligence, emerges as a promising avenue for enhancing the interpretation of acoustic information emanating from the bowel. This study initiates by introducing ML and subsequently delineates its developments within five key acoustics research domains: speech processing, ocean acoustics, bioacoustics, environmental acoustics, and Bowel Sound Analysis in everyday scenes.
... Details about the matrix-isolation setup and the experiments are included in the Supporting Information. The layer thicknesses of the matrices were obtained from near-infrared spectra using a cepstrum analysis [32] approach, which is also explained in detail in the Supporting Information. ...
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In this work, matrix‐isolation spectroscopy and quantum‐chemical calculations are used together to analyse the structure and properties of weakly bound dimers of the two isomers benzo[a]acridine and benzo[c]acridine. Our measured experimental electronic absorbance spectra agree with simulated spectra calculated for the equilibrium structures of the dimers in gas‐phase, but in contrast, disagree with the simulated spectra calculated for the structures obtained by optimising the experimental solid‐state structures. This highlights the sensitivity of the electronic excitations with respect to the dimer structures. The comparison between the solid‐state and gas‐phase dimers shows how far the intermolecular interactions could change the geometric and electronic structure in a disordered bulk material or at device interfaces, imposing consequences for exciton and charge mobility and other material properties.
... There are different methods available to analyze the dynamics of a rotating machine. The most commonly employed methods are the modal analysis [1], directional frequency analysis [2], order tracking [3], envelope analysis [4], and cepstrum transform [5]. These methods rely on frequency domain techniques to highlight the intrinsic dynamic characteristics of the rotating system, e.g. ...
... Randall [150] believed that cepstrum could be used to detect harmonic clusters and equidistant modulation sidebands in signals in the early stage. In the fault diagnosis of gearboxes, local faults modulate the frequency and amplitude of meshing signal, and a large number of sidebands representing faults are arranged at certain speed intervals. ...
Article
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Various mechanical equipment play a crucial role, and their health or status may affect efficiency and safety seriously. Spectrum analysis of the corresponding signal has been widely used to diagnose the fault in the past decades. The diagnosis method based on spectrum analysis technology covers almost all aspects of mechanical fault diagnosis. However, there is a lack of review of diagnostic methods of spectrum analysis technologies in the field of mechanical equipment fault diagnosis. In order to fill this gap, this paper reviews the spectrum analysis technology in mechanical equipment diagnosis in detail. First of all, in order to let the researchers who are in contact with spectrum analysis technology for the first time quickly understand this field, the principles of spectrum are systematically sorted out, including spectrum, cepstrum, energy spectrum, power spectrum, higher-order spectrum, Hilbert spectrum, marginal spectrum, envelope spectrum, singular spectrum and so on. Furthermore, the characteristics of corresponding spectrum analysis technologies are summarized, and their advantages and disadvantages are analyzed and compared. High-quality references in recent ten years are cited for illustration to enhance persuasiveness. Finally, the prospect of spectrum analysis technology is summarized, and the future development trend of spectrum analysis technology is pointed out. It is believed that the joint diagnosis of fault severity, variable speed fault diagnosis, combined with deep learning and multiple spectrum analysis technologies should be given more attention in the future. This paper is expected to provide a comprehensive overview of mechanical fault diagnosis based on spectrum analysis theory, and help to develop corresponding spectrum analysis technologies in practical engineering.
... Such a task can be solved in the time domain (using, e.g., the autocorrelation function), in the frequency domain, or in the cepstral domain. The proposed method is based on a cepstral analysis, as applied in the field of speech analysis for pitch estimation or in vibration analysis, e.g., for machine fault estimation [23]. Because fan tonal components overlap with BSN components (see Sec. II.B), they will not deteriorate this proposed second method. ...
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Aircraft noise emissions affect societies around the world by impacting the population’s health and land use planning. This calls for simulation tools able to predict these types of noise emissions with high accuracy. A crucial aircraft parameter to achieve satisfying precision is the rotating frequency of the low-pressure shaft of the turbofan engine, called [Formula: see text]. [Formula: see text] determines the engine’s power use and is here estimated acoustically from ground-based microphones. A new method for dynamic [Formula: see text] estimation is presented, which is more robust as compared to earlier approaches. It makes use of different aircraft sound characteristics and combines two methods. The first method tracks multiple fan tone harmonics over time within a de-Dopplerized sound pressure spectrogram. This frequency-tracking task is solved by dynamic programming to find the global optimum. The second method relates to buzz-saw noise, and is thus applied to departures only. The buzz-saw fundamental frequency is estimated in the cepstral domain. Both submethods are separately validated and assessed with concurrent sound pressure measurements and flight deck recording data of [Formula: see text]. The new robust [Formula: see text] estimation method will be applied in noise measurement campaigns with the goal of improving current aircraft noise emission models.
... Creating MFCCs uses fast Fourier transforms (FFTs) to transform signals from a time domain to a frequency domain. A cepstrum is defined as "the power spectrum of the logarithm of the power spectrum" (Randall (2017)). When creating MFCCs the audio must first be split into small overlapping sections and windowed (typically the Hanning function is applied). ...
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In this work we perform a scoping review of the current literature on the detection of throat cancer from speech recordings using machine learning and artificial intelligence. We find 22 papers within this area and discuss their methods and results. We split these papers into two groups - nine performing binary classification, and 13 performing multi-class classification. The papers present a range of methods with neural networks being most commonly implemented. Many features are also extracted from the audio before classification, with the most common bring mel-frequency cepstral coefficients. None of the papers found in this search have associated code repositories and as such are not reproducible. Therefore, we create a publicly available code repository of our own classifiers. We use transfer learning on a multi-class problem, classifying three pathologies and healthy controls. Using this technique we achieve an unweighted average recall of 53.54%, sensitivity of 83.14%, and specificity of 64.00%. We compare our classifiers with the results obtained on the same dataset and find similar results.
... where X f ð Þ= F½x t ð Þ, F denotes the Discrete Fourier Transform (DFT), and x(t) is the windowed signal. Cepstral coefficients are often referred to as the ''spectrum of a spectrum'' and contain information on the rate of change of a signal's spectral bands (Randall, 2017). LFCC's are a class of cepstral coefficients that are calculated using linearly spaced triangular band-pass filters. ...
Article
This paper proposes a new in-situ damage detection approach for wind turbine blades, which leverages blade-internal non-stationary acoustic pressure fluctuations caused by the mechanical loading as the main source of excitation. This acoustic excitation was leveraged for the detection of fatigue-related damage modes on a full-scale wind turbine blade undergoing edgewise fatigue testing. An unsupervised, data-driven structural health monitoring strategy was developed to learn the normal cavity-internal acoustic sequences generated by the blade’s load cycles and to detect damage-related anomalies in the context of those sequences. A linear cepstral-coefficient based feature set was used to characterize the cavity-internal acoustics and LSTM-autoencoders were trained to accurately reconstruct healthy-case sequences. The reconstruction error was then used to characterize anomalous acoustic patterns within the blade cavity. The technique was able to detect a damage event earlier than a strain-based system by 120,000 load cycles.
... There are four types of cepstrum, with power cepstrum being the most commonly used in machine diagnostics and monitoring. Cepstrum analysis has been used in gearbox diagnosis and monitoring, detection of friction in sliding bearings, and diagnosis of faults in a universal lathe machine [79][80][81][82][83]. The Cepstrum analysis can be sensitive to noise present in the vibration signals. ...
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Machine failure in modern industry leads to lost production and reduced competitiveness. Maintenance costs represent between 15% and 60% of the manufacturing cost of the final product, and in heavy industry, these costs can be as high as 50% of the total production cost. Predictive maintenance is an efficient technique to avoid unexpected maintenance stops during production in industry. Vibration measurement is the main non-invasive method for locating and predicting faults in rotating machine components. This paper reviews the techniques and tools used to collect and analyze vibration data, as well as the methods used to interpret and diagnose faults in rotating machinery. The main steps of this technique are discussed, including data acquisition, data transmission, signal processing, and fault detection. Predictive maintenance through vibration analysis is a key strategy for cost reduction and a mandatory application in modern industry.
Article
Optical noise detected by borehole distributed acoustic sensing (DAS) system exhibits various multiplicative characteristics, including non-uniform distribution and simultaneous occurrence with seismic signals. These characteristics are likely associated with instrument defects and warrant further investigation. However, research on this topic remains limited. This study aims to substantiate the multiplicative nature of optical noise using an enhanced cepstrum method. Cepstral line amplification is integrated into the conventional cepstrum method as a necessary step to suppress redundant information caused by additive seismic noise, while a pseudo-time constraint is imposed based on the propagation rules of signal waves. These operations ensure precision in the analysis results, thereby contributing to noise suppression and instrumentation improvement. A typical common-shot-point record, affected by significant optical noise, is employed for verification. The separated cepstral lines in the cepstrum results indicate that optical noise acts as a multiplicative interference in the acquired DAS records. Both continuous seismic traces and entire noisy area are utilized to establish the mathematical relationship between seismic signals and optical noise. Consistent conclusions emerge from the three-dimensional cepstrum, average cepstral amplitude, and specific trace distributions. A quantitative measurement has been developed to validate these findings. We desire that this study will serve as a reference for practical applications, such as improving acquisition instruments and designing denoising algorithms.
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The advent of Industry 5.0 envisages production systems that are more resilient, embrace human-machine collaboration and promote sustainability driven by technological research. The development of supervision solutions for industrial equipment fills in this picture as a basis for more proactive Condition-Based Maintenance strategies. The goal of this paper is to provide a self-contained set of guidelines to design such supervision solutions. With respect to existing literature on the topic, we provide a design process with a strong focus on experimental data collection and failure reproduction activities. Moreover, the connections between the steps of the proposed process are clearly highlighted to guide the user. First, the paper provides a set of tools to select the critical items and the methodological approaches for supervision. Then, these tools are used and referenced in the proposed design process. Finally, the proposed process is exemplified on two industrial case studies to show its effectiveness. Considerations, hints, and a user guidelines are given at the end of most sections.
Conference Paper
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The discrete short-time Fourier transformation (STFT) is widely used for the analysis of time-discrete acoustic signals. It describes the physical signal properties in the frequency domain but does not consider the physiology of the human ear, although frequency weighting of the amplitude spectrum is commonly performed, and the frequency axis is represented logarithmically. The cochlea nucleus is considered as an early processing stage in human hearing where e.g., onset and pitch detection, periodicity perception or binaural hearing take place. Based on a digital cochlear model (Lyon, 2017) the theory of auditory images (Licklider, 1951; Patterson et al., 1992) provides a representation of sound according to the first stage of the human hearing sense. We have successfully applied auditory images in several use cases in which conventional analysis methods have not delivered clear results. Auditory images have the advantage that they contain perception relevant information of a sound as well; phenomena such as roughness, fluctuation, periodicity, pitch (= tonality) can be found directly. We successfully used auditory images in a machine learning (ML) regression task to mimic human ratings. The resulting model is based on a recurrent neural network (RNN) and acts as a machine hearing model.
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Bearings are crucial in electromechanical equipment, and the state significantly affects operational efficiency and industrial income. Therefore, efficient and accurate monitoring of bearing health and fault diagnosis are imperative for intelligent equipment operation and maintenance. The Kurtogram is widely used for bearing fault diagnosis as it adeptly extracts faulty information from the frequency domain. However, this approach imposes limitations on the center frequency and bandwidth, preventing the association of specific frequency groups with the spectrum. Furthermore, the Kurtogram exhibits poor robustness and cannot support its use with new equipment or under extreme working conditions. To address these shortcomings, this paper introduces a novel technique known as Cepsogram. First, a new cepstrum reconstruction spectral trend estimation method is designed to distinguish different modal information in the frequency domain. In this method, the number of reconstruction iterations is increased to improves the complexity of spectrum trends and expand the diversity of modular segmentation. Additionally, to capture the cyclic transient characteristics of the signal, spectral negentropy is employed to reduce interference and obtain more obvious fault characteristics compared to steepness. The use of engineering data is evaluated to substantiate the effectiveness and stability of this proposed method.
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In measurements and numerical modelling of wave propagation, undesired interference between direct and multipath arrivals can be reduced using Fourier-based signal processing methods. Existing methods, such as cepstral analysis and time-signal gating, are not applicable to all cases. Here, an alternative Fourier-based signal processing method is presented, called spectrum-of-spectrum (SoS) filtering. Its main advantage over existing methods is its ability to extract single direct or multipath arrivals for relatively short propagation distances even when subsequent arrivals do not become successively weaker. The method is based on the following steps: •Apply a lowpass filter to the real and imaginary parts of an input frequency spectrum individually, using a digital finite impulse response (FIR) filter in the frequency domain. •Recombine the filtered real and imaginary parts of the frequency spectrum to get the frequency spectrum of the direct arrival. •For extraction of the first multipath arrival, subtract the filtered frequency spectrum from the input frequency spectrum and repeat the previous steps. Repeat multiple times to extract subsequent multipath arrivals.
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High-speed centrifugal compressors may be exploited to pressurize fuel cell systems. Nonetheless, due to fuel cells significant interposed volumes, compressor behavior can lead to severe vibrations related to fluid-dynamic instabilities during part load operating conditions. In particular, surge strongly limits centrifugal compressors stable operating region when moving towards low mass flow rates due to a change in system working point. Therefore, compressor dynamic response must be adequately characterized for early surge detection. To this aim, a dedicated experimental activity was conducted on a vaneless diffuser turbocharger coupled to a solid oxide fuel cell emulator plant; compressor evolution towards surge was investigated. Several signal processing techniques were applied to pressure signals as well as vibro-acoustic responses to better predict compressor behavior and classify its status as stable or unstable. Cepstrum, cross-correlation and wavelet transform have been identified as suitable techniques to define precursors able to early detect surge. By means of cross-correlation function, propagation phenomena in the ducts can be investigated to assess how they interact near compressor low-mass flow rate unstable conditions. Cepstrum provides a convenient way to determine pressure signal spectrum distortion in terms of further periodic components onset. These harmonic components are due to complex system responses generated by transient phenomena; indeed, cepstrum allows to identify hidden anomalous contributions in system response spectra which may arise in incipient surge conditions. Wavelet transform was performed on both structural and pressure response signals to observe their dominant energy contents temporal evolution.
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All vibration-based machine condition monitoring uses response signals, which are a mixture of forcing function and transfer function effects, and often an assumption is made as to whether a change is due to one or the other. Operational modal analysis (OMA) is one way of determining the dynamic properties of a structure or machine from the response signals only, while in operation, and thus provides the potential to separate them from the forcing functions. It then becomes a powerful machine diagnostic tool. The cepstrum has the property that forcing functions and transfer functions are additive, at least for single input, multiple output (SIMO) situations, and moreover they are often located in disparate regions in the cepstrum, allowing them to be separated. In recent years, new cepstral analysis methods have been developed to assist OMA in two ways: 1) As a pre-processing tool to enhance the modal properties and remove other effects, such as forcing functions (often at discrete frequencies) to simplify the application of standard OMA techniques. 2) To perform the OMA by fitting a pole/zero model of the structural dynamic properties to the enhanced response data. The current paper shows the success of both these methods in the case of simulated signals from a variable speed gearbox, by vastly reducing the effects of the very complicated forcing function. An extension from earlier proposed approaches is the determination of zeros (normally masked by noise in response signals) using transmissibilities, measured between pairs of responses, the noise being reduced by averaging. These new results confirm that limited, and only partially successful, earlier results were contaminated by nonlinearities in the support of the test object.
Conference Paper
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All vibration-based machine condition monitoring uses response signals, which are a mixture of forcing function and transfer function effects, and often an assumption is made as to whether a change is due to one or the other. Operational modal analysis (OMA) is one way of determining the dynamic properties of a structure or machine from the response signals only, while in operation, and thus provides the potential to separate them from the forcing functions. It then becomes a powerful machine diagnostic tool. The cepstrum has the property that forcing functions and transfer functions are additive, at least for single input, multiple output (SIMO) situations, and moreover they are often located in disparate regions in the cepstrum, allowing them to be separated. In recent years, new cepstral analysis methods have been developed to assist OMA in two ways: 1) As a pre-processing tool to enhance the modal properties and remove other effects, such as forcing functions (often at discrete frequencies) to simplify the application of standard OMA techniques. 2) To perform the OMA by fitting a pole/zero model of the structural dynamic properties to the enhanced response data. The current paper shows the success of both these methods in the case of simulated signals from a variable speed gearbox, by vastly reducing the effects of the very complicated forcing function. An extension from earlier proposed approaches is the determination of zeros (normally masked by noise in response signals) using transmissibilities, measured between pairs of responses, the noise being reduced by averaging. These new results confirm that limited, and only partially successful, earlier results were contaminated by nonlinearities in the support of the test object.
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Discrete frequency components such as machine shaft orders can disrupt the operation of normal Operational Modal Analysis (OMA) algorithms. With constant speed machines, they have been removed using time synchronous averaging (TSA). This paper compares two approaches for varying speed machines. In one method, signals are transformed into the order domain, and after the removal of shaft speed related components by a cepstral notching method, are transformed back to the time domain to allow normal OMA. In the other simpler approach an exponential shortpass lifter is applied directly in the time domain cepstrum to enhance the modal information at the expense of other disturbances. For simulated gear signals with speed variations of both ±5% and ±15%, the simpler approach was found to give better results The TSA method is shown not to work in either case. The paper compares the results with those obtained using a stationary random excitation.
Conference Paper
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Cepstrum-based operational modal analysis (OMA) relies on identifying the transfer function poles and zeros in the response measurement, and then summing (in log magnitude) the individual pole/zero contributions to regenerate the corresponding frequency response function (FRF). Yet this regenerated FRF will be subject to magnitude distortion from the effects of truncation, i.e., from the residual effects of out-of-band poles and zeros. As long as a reference FRF is available – for example from conventional experimental modal analysis or from a finite element model – this distortion can be corrected for using a magnitude equalisation curve. This paper discusses the nature of this equalisation curve, and gives recommendations on how best to obtain it. Also discussed are a number of observations relating to the FRF regeneration process, as well as some broader points explaining FRFs from a pole-zero perspective. It is hoped that the discussion will assist in the application of cepstrum-based OMA methods and will lead to improved understanding of the FRF regeneration process and of frequency response functions more broadly.
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Operational modal analysis (OMA) seeks to determine a structure׳s dynamic characteristics from response-only measurements, which comprise both excitation and transmission path effects. The cepstrum has been used successfully in a number of applications to separate these source and path effects, after which the poles and zeros of the transfer function can be obtained via a curve-fitting process. The contributions from the individual poles and zeros can then be added (in log magnitude) to regenerate the frequency response function (FRF). Cepstrum-based OMA was originally developed in the 1980s and 90s, but there have been a number of recent developments that warrant discussion and explanation, and this is the basis of the present paper, which focusses on the FRF regeneration process and on a number of broader points explaining FRFs from a pole–zero perspective.
Conference Paper
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This paper discusses the regeneration of frequency response functions (FRFs) based on a previously-proposed cepstrum-based operational modal analysis (OMA) technique. OMA differs from experimental modal analysis (EMA) in that it seeks to determine a structure’s dynamic characteristics from response-only measurements, without precise knowledge of excitation forces. Response measurements, however, comprise both excitation and transmission path effects, which must be separated before the structural properties can be determined. The method employed in this paper achieves source-path separation with the cepstrum, which is able to deal with ‘frequentially smooth’ (not just frequentially white) inputs. After separation, the poles and zeros of the transfer function can be obtained by curve-fitting the transfer path cepstrum. But the FRF regenerated from these poles and zeros corresponds to a truncated model of the system, covering only a limited frequency range. The out-of-band poles and zeros affect the magnitude and phase of the in-band FRFs, and this distortion must be corrected in the FRF regeneration process. This paper focuses on that correction using data from a steel beam experiment. To do this, an ‘equalisation curve’ is used, based on a comparison of the regenerated FRF with a ‘reference FRF’, found with EMA or FEM techniques. The paper proposes a polynomial-fit approach to obtain the equalisation curves, resulting in excellent agreement between measured and OMA-regenerated FRFs. The techniques outlined in the paper have a number of potential applications, particularly in the fault diagnostics and structural health monitoring fields, where damage is often detectable by changes in the structure’s FRFs.
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This paper presents a response-only structural health monitoring (SHM) technique that utilises cepstrum analysis and artificial neural networks (ANNs) for the identification of damage in civil engineering structures. The method begins by applying cepstrum-based operational modal analysis (OMA), which separates source and transmission path effects to determine the structure’s frequency response functions (FRFs) from response measurements only. Principal component analysis (PCA) is applied to the obtained FRFs to reduce the data size, and structural damage is then detected using a two-stage ensemble of ANNs. The proposed method is verified both experimentally and numerically using a laboratory two-storey framed structure and a finite element (FE) representation, both subjected to a single excitation. The laboratory structure is tested on a large-scale shake table generating ambient loading of Gaussian distribution. In the numerical investigation, the same input is applied to the FE model, but the obtained responses are polluted with different levels of white Gaussian noise to better replicate real-life conditions. The damage is simulated in the experimental and numerical investigations by changing the condition of individual joint elements from fixed to pinned. In total, four single joint changes are investigated. The results of the investigation show that the proposed method is effective in identifying joint damage in a multi-storey structure based on response-only measurements in the presence of a single input. Because the technique does not require a precise knowledge of the excitation, it has the potential for use in online structural health monitoring. Recommendations are given as to how the method could be applied to the more general multiple-input case.
Conference Paper
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This paper presents a damage identification technique based on response-only data utilising cepstrum analysis and artificial neural networks (ANNs) for the identification of added mass in a two-storey framed structure. The proposed technique applies cepstrum-based operational modal analysis (OMA) for the regeneration of frequency response functions (FRFs), and added mass is detected through the combined use of principal component analysis (PCA) for data compression and ANNs for feature extraction and pattern recognition. In particular, different treatments of the zeros in the curve-fitting of the transfer function cepstrum are investigated to improve the automation potential of the method for application in continuous online structural health monitoring (SHM). The proposed technique is validated on a laboratory structure tested on a large-scale shake table with ambient base loading. The results of the investigation show that the method is effective in identifying added mass based on response-only measurements.
Article
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In the current investigation a procedure is proposed to extract poles and zeros of transfer functions from response vibrations. In this procedure use is made of the deconvolution properties of cepstral analysis, that is, in the cepstrum domain, source and path effects are not only additive but also separated into different quefrency regions. The source effect is excluded and the complex or differential cepstra of the path are curve-fitted to extract poles and zeros. The Levenberg-Marquardt and Ibrahim time domain methods are adapted for the curve-fitting purpose. In the Levenberg-Marquardt method path dominated complex or differential cepstra (after the source effect is removed) are curve-fitted to their corresponding analytical expressions, while the Ibrahim time domain method differential cepstra are treated as free response data. The advantages and disadvantages of the two methods are compared. The validation of this procedure is demonstrated by using the response of a free–free beam to impact and double impact excitations.
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A method for updating modal models from response measurements is extended from the case of a single impulsive excitation to a more general broadband excitation in the presence of secondary excitations. The original technique was based on analysis of the cepstrum of the response, as forcing function and transfer function effects are additive in the response cepstrum, and also separated if the force log spectrum is reasonably smooth and flat. Use is made of principal components analysis by singular value decomposition to separate the autospectrum of the response at each point to the dominant excitation, which is then curve fitted in the cepstral domain for its poles and zeros to give updated estimates of the FRFs. The resulting FRFs are scaled (because of including the information on zeros) and in the study gave reasonable estimates of the mode shapes when the dominant force was four times larger than the next largest.
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This paper presents a technique to differentially diagnose two localized gear tooth faults: a spall and a crack in the gear tooth fillet region. These faults could have very different prognoses, but existing diagnostic techniques only indicate the presence of local tooth faults without being able to differentiate between a spall and a crack. The effects of spalls and cracks on the behavior/response of gear assemblies were studied using static and dynamic simulation models. Changes in the kinematics of a pair of meshing gears due to a gear tooth root crack and a tooth flank spall were compared using a static analysis model. The difference in the variation of the transmission error caused by the two faults reveals their characteristics. The effect of a tooth crack depends on the change in stiffness of the tooth, while the effect of a spall is predominantly determined by the geometry of the fault. The effect of the faults on the gear dynamics was studied by simulating the transmission error in a lumped parameter dynamic model. A technique had previously been proposed to detect spalls, using the cepstrum to detect a negative echo in the signal (from entry into and exit from the spall). In the authors’ simulations, echoes were detected with both types of fault, but their different characteristics should allow differential diagnosis. These concepts are presented prior to experimental validation in hopes that the diagnostic techniques will be useful in the failure analysis community prior to the validation by ongoing experimental testing of the concepts and the evaluation of how metallurgical defects may influence fault development and detection.
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The idea of the log spectrum or cepstral averaging has been useful in many applications such as audio processing, speech processing, speech recognition, and echo detection for the estimation and compensation of convolutional distortions. To suggest what prompted the invention of the term cepstrum, this article narrates the historical and mathematical background that led to its discovery. The computations of earlier simple echo representations have shown that the spectrum representation domain results does not belong in the frequency or time domain. Bogert et al. (1963) chose to refer to it as quefrency domain and later termed the spectrum of the log of a time waveform as the cepstrum. The article also recounts the analysis of Al Oppenheim in relation to the cepstrum. It was in his theory for nonlinear signal processing, referred to as homomorphic systems, that the realization of the characteristic system of homomorphic convolution was reminiscent of the cepstrum. To retain both the relationship to the work of Bogart et al. and the distinction, the term power cepstrum was eventually applied to the nonlinear mapping in homomorphic deconvolution . While most of the terms in the glossary have faded into the background, the term cepstrum has survived and has become part of the digital signal processing lexicon.
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A new procedure is proposed that uses the real cepstrum to localize and edit the log amplitude of the original signal, removing unwanted discrete frequency components, and then combines the edited amplitude with the original phase spectrum to return to the time domain. This cepstral editing procedure (CEP) is used to remove discrete frequency components from signals measured on two machines with a faulty bearing, and then perform envelope analysis on the residual signal to diagnose the bearing fault.
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For complete models, there is an exact equivalence between pole/zero and pole/residue representations of transfer functions, but in practice truncation is almost inevitably necessary, in which case the equivalence breaks down. This paper discusses how extra zeros are typically added into a truncated model to compensate for the effects of out-of-band modes, and illustrates their effects. Because compensation is primarily required on the magnitude of the FRF, these extra zeros, named 'phantom zeros', are typically arranged in pairs around the frequency axis, so that half of them have maximum phase properties even when the physical model is minimum phase. The number of phantom zeros required depends on the separation of the excitation and response points. For a driving point measurement, where there are virtually as many zeros as poles, the effects of truncation are very small, whereas at the other extreme, with no actual zeros, a correspondingly greater number of phantom zeros is required to correct the slope of the magnitude of the FRF.
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This paper presents a new technique for scaling mode shapes, obtained from cepstrum-based operational modal analysis (OMA) techniques, such as that described in the companion paper, using finite element model updating. This OMA technique estimated frequency response functions (FRFs) between a cyclostationary input and response measurements. If the input is frequentially white, the resulting FRFs can be obtained up to an overall scaling constant using the in-band poles and zeros identified in the OMA process and employing the response autospectrum as a reference to correct for the effect of out of band modes. In this way, the mode shapes would be scaled correctly relative to each other but would still have arbitrary overall magnitude. If the input is not white, then no reference is available to correct FRF regenerated from in-band poles and zeros, and so these FRFs will exhibit both an overall slope resulting from the effect of out-of-band poles and zeros, and an arbitrary magnitude. This overall slope will differ between measurement locations so even the relative scaling between the mode shapes will be lost. This paper describes a simple technique for recovering both the relative and overall scaling of the FRFs, and hence the mode shapes, based on finite element model updating.
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This paper presents a new technique for operational modal analysis (OMA) of multiple input multiple output (MIMO) systems excited by at least one cyclostationary input with a unique cyclic frequency. The technique is based on two signal separation steps; the cyclostationary properties of the input are exploited to estimate the cyclic spectral density, effectively reducing the system from a MIMO to a single input multiple output (SIMO) situation, and curve-fitted in the cepstrum domain, which allows for the separation of the input and transfer function. This technique is demonstrated using measurements taken on a steel beam test rig and a passenger rail vehicle. The performance of this technique is discussed and compared to traditional input/output modal analysis and an existing cepstrum-based OMA technique. It is shown that the technique is able to correctly identify modal parameters, but like other spectrum-based OMA techniques, long time records are required in order to obtain both smooth cyclic spectrum estimates and sufficient resolution for accurate damping estimates. The nature of the input may also inhibit its performance in the very low-frequency region.
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Cepstral Analysis is an example of nonlinear filtering that has been applied to extracting the properties of transmission path and source characteristics in acoustics. To see why this is so, we review some of the properties of linear windowing in the time and frequency domains with a view to revealing the limitations that these methods have. We then describe the cepstrum and the conditions under which it can be helpful in separating source and path characteristics. The method is illustrated by describing some applications. Finally, research directions that may help to extend the applicability of cepstral analysis to structural vibration transmission are discussed.
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A new approach to separating convolved signals, referred to as homomorphic deconvolution, is presented. The class of systems considered in this report is a member of a larger class called homomorphic systems, which are characterized by a generalized principle of superposition that is analogous to the principle of superposition for linear systems. A detailed analysis based on the z-transform is given for discrete-time systems of this class. The realization of such systems using a digital computer is also discussed in detail. Such conputational realizations are made possible through the application of high-speed Fourier analysis techniques. As a particular example, the method is applied to the separation of the components of a convolution in which one of the components is an impulse train. This class of signals is representative of many interesting signal-analysis and signal-processing problems such as speech analysis and echo removal and detection. It is shown that homomorphic deconvolution is a useful approach to either removal or detection of echoes.
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An efficient method for the calculation of the interactions of a 2' factorial ex- periment was introduced by Yates and is widely known by his name. The generaliza- tion to 3' was given by Box et al. (1). Good (2) generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an N-vector by an N X N matrix which can be factored into m sparse matrices, where m is proportional to log N. This results inma procedure requiring a number of operations proportional to N log N rather than N2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of N. It is also shown how special advantage can be obtained in the use of a binary computer with N = 2' and how the entire calculation can be performed within the array of N data storage locations used for the given Fourier coefficients. Consider the problem of calculating the complex Fourier series N-1 (1) X(j) = EA(k)-Wjk, j = 0 1, * ,N- 1, k=0
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A new algorithm of matrix spectral factorization is proposed which can be applied to compute an approximate spectral factor of any positive definite matrix function which satisfies the Paley-Wiener condition.
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Spectral kurtosis (SK) represents a valuable tool for extracting transients buried in noise, which makes it very powerful for the diagnostics of rolling element bearings. However, a high value of SK requires that the individual transients are separated, which in turn means that if their repetition rate is high their damping must be sufficiently high that each dies away before the appearance of the next. This paper presents an algorithm for enhancing the surveillance capability of SK by using the minimum entropy deconvolution (MED) technique. The MED technique effectively deconvolves the effect of the transmission path and clarifies the impulses, even where they are not separated in the original signal. The paper illustrates these issues by analysing signals taken from a high-speed test rig, which contained a bearing with a spalled inner race. The results show that the use of the MED technique dramatically sharpens the pulses originating from the impacts of the balls with the spall and increases the kurtosis values to a level that reflects the severity of the fault. Moreover, when the algorithm was tested on signals taken from a gearbox for a bearing with a spalled outer race, it shows that each of the impulses originating from the impacts is made up of two parts (corresponding to entry into and exit from the spall). This agrees well with the literature but is often difficult to observe without the use of the MED technique. The use of the MED along with SK analysis also greatly improves the results of envelope analysis for making a complete diagnosis of the fault and trending its progression.
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The kurtogram is a fourth-order spectral analysis tool recently introduced for detecting and characterising non-stationarities in a signal. The paradigm relies on the assertion that each type of transient is associated with an optimal (frequency/frequency resolution) dyad {f,Δf} which maximises its kurtosis, and hence its detection. However, the complete exploration of the whole plane (f,Δf) is a formidable task hardly amenable to on-line industrial applications. In this communication we describe a fast algorithm for computing the kurtogram over a grid that finely samples the (f,Δf) plane. Its complexity is on the order of , similarly to the FFT. The efficiency of the algorithm is then illustrated on several industrial cases concerned with the detection of incipient transient faults.
Article
The cepstrum, defined as the power spectrum of the logarithm of the power spectrum, has a strong peak corresponding to the pitch period of the voiced‐speech segment being analyzed. Cepstra were calculated on a digital computer and were automatically plotted on microfilm. Algorithms were developed heuristically for picking those peaks corresponding to voiced‐speech segments and the vocal pitch periods. This information was then used to derive the excitation for a computer‐simulated channel vocoder. The pitch quality of the vocoded speech was judged by experienced listeners in informal comparison tests to be indistinguishable from the original speech.
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  • J W Tukey
B.P. Bogert, M.J.R. Healy, J.W. Tukey, The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum and Saphe Cracking in: M. Rosenblatt (Ed.) Time Series Analysis, 1963, pp. 209-243.
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Cepstrum Analysis and Gearbox Fault Diagnosis, Brüel and Kjaer Application Note No. 13-150
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R.B. Randall, Cepstrum Analysis and Gearbox Fault Diagnosis, Brüel and Kjaer Application Note No. 13-150, Copenhagen, 1973.
Gearbox Fault Diagnosis using Cepstrum Analysis
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R.B. Randall, Gearbox Fault Diagnosis using Cepstrum Analysis, Proc. IVth World Congress on the Theory of Machines and Mechanisms, Newcastle, UK, Vol. 1 (1975) pp. 169-174.
Separating excitation and structural response effects in gearboxes
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R.B. Randall, Separating excitation and structural response effects in gearboxes, Third International Conference on Vibrations in Rotating Machinery, Yorkshire, UK, (1984) pp. 101-107.
Global curve-fitting of frequency response measurements using the rational fraction polynomial method
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M.H. Richardson, D.F. Formenti, Global curve-fitting of frequency response measurements using the rational fraction polynomial method, The 3rd International Modal Analysis Conference (IMAC), Orlando, Florida, (1985) pp. 390-397.
Updating modal properties from response-only measurements on a rail vehicle
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  • R Randall
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R. Ford, R. Randall, T. Wardrop, Updating modal properties from response-only measurements on a rail vehicle, ISMA2002 Conference, Leuven, Belgium, (2002).
Multiple-Input Multiple-Output Blind System Identification for Operational Modal Analysis using the Mean Differential Cepstrum
  • W L Chia
W.L. Chia, Multiple-Input Multiple-Output Blind System Identification for Operational Modal Analysis using the Mean Differential Cepstrum, University of New South Wales, 2007. Electronic copy available through UNSW Library.
New cepstral methods of signal preprocessing for operational modal analysis
  • R B Randall
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  • S Manzato
R.B. Randall, B. Peeters, J. Antoni, S. Manzato, New cepstral methods of signal preprocessing for operational modal analysis, ISMA2012 International Conference on Noise and Vibration Engineering, Leuven, Belgium, (2012) pp. 755-764.
Cepstral Removal of Periodic Spectral Components from Time Signals
  • R B Randall
  • N Sawalhi
R.B. Randall, N. Sawalhi, Cepstral Removal of Periodic Spectral Components from Time Signals, 3rd International Conference on Condition Monitoring of Machinery in Non-Stationary Operations (CMMNO2013), Ferrara, Italy, (2013).