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Example 1-Top: The low-pass filter in (30) and the high-pass filters in Table I. Middle: the iterated low-pass filter h (n). Bottom: two of the numerous discrete-time analysis/synthesis functions at level j = 6. Compare to
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![Fig. 6. Example from [5] of minimal-length filters for an orthonormal...](profile/Ilker-Bayram/publication/224345311/figure/fig3/AS:349259007315970@1460281244549/Example-from-5-of-minimal-length-filters-for-an-orthonormal-rational-discrete-wavelet_Q320.jpg)
This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the time-frequency plane. A straightforward algorithm is...
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Citations
... Every PLL function is carried out by the program using the SPLL algorithm. Due to its faster operation, stability to environmental changes, accuracy, and reconfigurability, SPLL offers more benefits than hardware PLL [30]. The output error signal from the SPLL is evaluated using several signal processing methods. ...
... For ω l ∈ [0, π ], w 1 {t} G 1 *X 1 , X 1 fft (x). Equation (22) may be used to compute the total energy e 1 (t) of the RDWT signal with respect to the t th sub-band [28][29][30] ...
Early fault detection in brushless direct current (BLDC) motors used in electric vehicles is crucial for ensuring the reliability, safety, and performance of the vehicle. Cost-effective fault diagnostics and non-invasive techniques help to enhance industrial practices and promote sustainable and efficient technology. The vibration of a BLDC motor is an important feature for diagnosing various faults. The most common causes of BLDC motor failure are mechanical and electrical faults, which can result in expensive maintenance and downtime. The existing methods for diagnosing BLDC motor faults using invasive or non-invasive methods have many limitations. This work proposes a technique for diagnose the BLDC motor faults non-invasively by projecting high-frequency signal onto the motor and recorded using SIGVIEW software. This technology uses portable ultra-wideband (UWB HB-100) radar, high-frequency noise filtering by software phase-locked loops (SPLL), and rational dilation wavelet transforms (RDWT) for signal analysis to identify BLDC motor faults. The RDWT output is divided into 12 sub-bands, and the energy distribution of each sub-band is calculated using MATLAB-R2023b. As the number of bearing faults increases from 2 to 3, the RDWT output sub-band energy at levels 4 and 5 increases by 17.06%–18.59% and 131.73%–179.10%, respectively, indicating a significant increase in energy due to the faults. By detecting motor faults early on, this technique reduces the likelihood of unexpected defects and provides proactive maintenance. Overall, the proposed approach offers an effective method for identifying defects in BLDC motors.
... In [9], for the orthonormal wavelet with rational dilation factor, a perfect reconstruction condition is given. An overcomplete discrete wavelet transform with rational dilation factor was developed in [4]. In [8], a discrete wavelet transform fast algorithm with dilation factor 3/2 is proposed, which overcomes the drawback of frequency distortion in the high frequency subband in the process of decomposition of Mallat discrete wavelet transform (DWT). ...
Dr. Auscher first proposed the concept of rational multiresolution analysis in his doctoral dissertation in 1989. However, the orthonormal basis of rational Littlewood-Paley wavelet first proposed by Dr. Auscher is incorrect, and the proof that rational Littlewood-Paley wavelet family is orthonormal basis in was not given in Dr.Auscher's doctoral dissertation. In this paper, firstly, the corrected orthonormal basis of rational Littlewood-Paley wavelet is presented. Then, it is completely proved that the corrected rational Littlewood-Paley wavelet family is an orthonormal wavelet basis of .
... During the reconstruction phase of the algorithm, the subband wavelet can be extracted at the interested scale by equating the coefficients of other filters to 0. It is based on the rational-dilation factor [52] in which a flexible time-frequency window-frame is used. Thus the reconstruction at different scales increases the signal-to-noise ratio. ...
Features describing the state of industrial gearboxes and their extraction from the mixed noisy signal are always an issue of concern. Unfortunately, traditional feature extraction methods are not equally useful for both adaptive and non-adaptive decomposition techniques. In the present study, a novel feature extraction method named weighted multi-scale fluctuation-based dispersion entropy (wtMFDE) is proposed for the condition monitoring of planetary gearbox (PGB). It is equally useful for adaptive (ensemble empirical mode decomposition) and non-adaptive (flexible analytical wavelet transform) signal denoising techniques. Four state-of-the-art classifiers viz. support vector machine (SVM), random forest (RF), K-nearest neighbor (KNN), and multilayer perceptron (MLP), which have extensive applications in machine condition monitoring, are used for the classification of the chipped sun gear, pitted planet gear as well as the combined effect for these faults. It is observed that the extracted features are highly sensitive and effective in a way that they can recognize faults with comparable accuracy with all the selected classifiers.
... Table 1 shows that each iteration of the Kurt-WATV includes the inner product, vector multiplication, and vector addition, as well as an ORDWT and its inverse transform. According to the literature [26,31,32], the complexity of calculating a DFT of a discrete sequence of length N is O (NlogN), whereas the ORDWT is a set consisting of discrete Fourier transforms. Its computational complexity is O(rNlogN), where r ...
As one of the essential parts of the mechanical transmission system, rolling bearing is vital to ensure the safe operation of mechanical equipment. The rolling bearing goes through four stages from its installation to the end of its life: normal operation, early weak failure, serious failure, and failure. If faults can be found in the early failure stage of the whole life cycle and maintenance strategies can be adopted in time, the safe and trouble-free operation of the equipment can be guaranteed. However, the fault features are not apparent in the early failure stage of the bearing’s full life cycle. Moreover, being completely submerged in strong background noise can easily occur, making early fault diagnosis challenging. This study presents a new sparse enhancement model based on kurtosis-wavelet total variation denoising (Kurt-WATV) for early fault feature extraction. Firstly, a sparse optimization model is constructed to extract the early fault feature, and the original signal is decomposed by the over-complete rational discrete wavelet transform (ORDWT). Then a fast iterative algorithm is deduced to solve the established sparse optimization model, and the optimal wavelet subband is selected by Kurt-WATV, which is reconstructed to the fault signal. Finally, the bearing test data from bearing’s full-life cycle are adopted to illustrate the effectiveness and robustness of the proposed method. Results confirm that the established method can achieve excellent performance in early fault feature extraction.
... Equations (12) and (13) give the central frequency f c and bandwidth B W of the filter bank in the j-th layer, respectively [41]: ...
Fault diagnosis of rolling bearings is not a trivial task because fault-induced periodic transient impulses are always submerged in environmental noise as well as large accidental impulses and attenuated by transmission path. In most hybrid diagnostic methods available for rolling bearings, the problems lie in twofolds. First, most optimization indices used in the individual signal processing stage do not take the periodical characteristic of fault transient impulses into consideration. Second, the individual stages make use of different optimization indices resulting in inconsistent optimization directions and possibly an unsatisfied diagnosis. To solve these problems, a multistage fault feature extraction method of consistent optimization for rolling bearings based on correlated kurtosis (CK) is proposed where maximum correlated kurtosis deconvolution (MCKD) is employed to attenuate the influence of transmission path followed by tunable Q factor wavelet transform (TQWT) to further enhance fault features by decomposing the preprocessed signals into multiple subbands under different Q values. The major contribution of the proposed approach is to consistently use CK as an optimization index of both MCKD and TQWT. The subband signal with the maximum CK which is an index being able to measure the periodical transient impulses in vibration signal yields an envelope spectrum, from which fault diagnosis is implemented. Simulated and experimental signals verified the effectiveness and advantages of the proposed method.
... Thus, this paper uses wavelet transforms which have an inherent property of dynamic window size. Wavelet packets and dyadic wavelet (DyWT) [61], Cosine modulated filter banks are a few examples of the available wavelet transforms for vibrating signal analysis. DyWT is an easily invertible "constant-Q" transform that is very effective for the sparse representation of piecewise smooth signals, but has shortcomings like poor frequency resolution and low-Q factor. ...
... In the wavelet output plots, the total signal power is considered 100% and a sub-band signal power is calculated as a percentage of the total power [42]. The RDWT consists of a filter bank with a low-pass component H XL (ω XL ) and high-pass component G XH (ω XL ) given as equation (27) and (28), respectively [61,62]. ...
Vibration is the vital feature of the motor to diagnose various faults. The existing methods have many limitations. Thus, this work proposes a non-invasive Software Phase Locked Loop (SPLL) technique to diagnose the multiple rotor faults. A high-frequency (10.525GHz) microwave signal is projected on the motor and the reflected signal is captured by the handheld ultra-wide band (UWB) radar. Hardware filter is replaced by a software phase locked loop filter to remove high-frequency noise content in the signal. The output of the SPLL is analysed with Rational Dilation Wavelet Transforms (RDWT) and compared the change in the signal energy of the wavelet sub-band 6 for the multiple rotor bar faults and sub-bands 6 and 7 for the combination of rotor bar and bearing faults, respectively. The signal energy at sub-band 6 increased from 2.19% without fault to 3.99%, 6.03% and 13.66% with an increase in the number of rotor bars with faults. The variation of the experimental values of the signal energy for rotor bar faults compared with the theoretical values found that the error is reduced and accuracy is increased with the number of faults. For simultaneous rotor bar and bearing faults, the signal energy at sub-bands 6 and 7also increases. The proposed SPLL-based method that is able to diagnose the individual faults and combination of the faults and has advantages such as simple, cost-effective and non-invasive technique compared with the existing methods.
... WT can be implemented as critically sampled and overcomplete. The over complete WT has advantages like minimal-length perfect reconstruction filters, shift-invariant and smooth analysis/ synthesis function [53][54][55] over the critically sampled WT. The various over complete WT are double density Wavelet transform [56], higher density Wavelet transform [57][58][59], dense framelets [60], Mband flexible Wavelet transform [61] and the RDWT. ...
The vibration is an important feature of the motor to diagnose the different faults. The existing invasive and non-invasive methods to capture and analysis vibration signals have many limitations. Thus, this work proposes a technique to capture and process the motor vibrations non-invasively to diagnose the multiple bearing faults using a microwave signal and software low pass filter respectively. The proposed method uses a high-frequency signal from microwave sensor (handheld Ultra-Wide Band (UWB) radar) projected on the Squirrel Cage Induction Motor (SCIM) and the reflected signal captured. The signal obtained is filtered with Software Phase Locked Loop (Low pass filter (SPLL)) and analyzed with a signal processing algorithm like Wavelet Transform to identify the faults in the motor. In this paper multiple bearing faults under no-load and full-load and a combination of bearing and rotor bar faults are diagnosed with the proposed method using Rational Dilation Wavelet Transforms (RDWT). The various bearing fault signal’s energy at the sub-band-7 compared under normal and fault conditions. The signal energy at the fault frequency sub-band under no-load increases by 2.11%, 23.5% and 42.5% compared with the no-fault condition with the increase in the number of bearing faults from 1 to 3. The signal energy variation indicates the severity of the defects and the accuracy of the proposed method is verified with the contact method using a vibration sensor (accelerometer). The other faults analyzed are the combination of the bearing and rotor bar faults with the variation of the signal energy at sub-band 7 & 6. The variation of the signal energy for bearing and rotor bar faults are verified with the theoretical calculation and the proposed method detects the faults with the accuracy of approximately 93%. On the other hand, the proposed method is simple and costeffective compared with the existing methods.
... Tunable Q-factor wavelet transform RSSD utilizes tunable Q-factor wavelet transform (TQWT) [19][20][21] which achieves high Q-factor constant-Q (wavelet) transforms for the sparse representation of high-resonance component and low Q-factor constant-Q (wavelet) transforms for the sparse representation of low-resonance component with the aid of high Q-factor analysis and low Q-factor analysis, respectively. On the other hand, tunable Q-factor wavelet transformation which is developed in terms of iterated two-channel filter banks and can be implemented efficiently with fast Fourier transform (FFT) has a series of advantages, such as it is fully discrete has the perfect reconstruction property and can be modestly overcomplete. ...
The main purpose of the paper is to propose a new method to achieve separating periodic impulse signal among multi-component mixture signal and its application to the fault detection of rolling bearing. In general, as local defects occur in a rotating machinery, the vibration signal always consists of periodic impulse components along with other components such as harmonic component and noise; impulse component reflects the condition of rolling bearing. However, different components of multi-component mixture signal may approximately have same center frequency and bandwidth coincides with each other that is difficult to disentangle by linear frequency-based filtering. In order to solve this problem, the author introduces a proposed method based on resonance-based sparse signal decomposition integrated with empirical mode decomposition and demodulation that can separate the impulse component from the signal, according to the different Q-factors of impulse component and harmonic component. Simulation and application examples have proved the effectiveness of the method to achieve fault detection of rolling bearing and signal preprocessing.
... RDWT can be implemented as critically sampled and overcomplete RDWT. The overcomplete RDWT provides several advantages compared to the critically sampled RDWT, such as minimal-length perfect reconstruction filters, shift-invariant and smooth analysis/synthesis functions [33]. ...
... Percentage of each sub band signal power is measured as the ratio of the particular level signal power to the total power. The RDWT satisfy the Parseval property, i.e., the distribution of the power across the subbands reflects the frequency content of the signal [31], [33]. So, the percentage power of each subband gives the percentage of frequency content in the total signal. ...
This study analyses the vibration of Squirrel Cage Induction Motor (SCIM) with bearing faults using high-frequency signal (Handheld UWB radar) and Software Phase Locked Loop algorithm (SPLL). The contact or non-contact methods perform condition monitoring of the SCIM. The proposed method is a non-contact technique to perform condition monitoring of the SCIM. The contact methods execute via vibration, instantaneous frequency, rotor speed and flux signals analysis; whereas non-contact methods accomplish via acoustic, current and stray flux measurement. The existing techniques suffer from the influence of adjoining electrical machines; require human expertise to mount sensors and analysing the signals. In this paper, a new, non-contact method proposed for bearing fault identification in the SCIM. The proposed method uses a high-frequency signal projected on the motor and the reflected signal captured. The signal obtained is analysed with an advanced signal processing algorithm like Rational Dilation Wavelet Transform (RDWT) to identify the faults in the SCIM. The signal energy at the fault frequency level increases from 4.72 % to 5.82 % with the increase in the number of the faults. The signal energy variation indicates the severity of the faults. From the experimental results, the bearing fault of the motor identified in the beginning stage of the fault.
DOI: 10.5755/j01.eie.25.1.22735
... RDWT can be implemented as critically sampled and overcomplete RDWT. The overcomplete RDWT provides several advantages compared to the critically sampled RDWT, such as minimal-length perfect reconstruction filters, shift-invariant and smooth analysis/synthesis functions [33]. ...
... Percentage of each sub band signal power is measured as the ratio of the particular level signal power to the total power. The RDWT satisfy the Parseval property, i.e., the distribution of the power across the subbands reflects the frequency content of the signal [31], [33]. So, the percentage power of each subband gives the percentage of frequency content in the total signal. ...
This study analyzes the vibration of Squirrel Cage Induction Motor (SCIM) with bearing faults using high-frequency signal (Handheld UWB radar) and Software Phase Locked Loop algorithm (SPLL). The contact or non-contact methods perform condition monitoring of the SCIM. The proposed method is a non-contact technique to perform condition monitoring of the SCIM. The contact methods execute via vibration, instantaneous frequency, rotor speed and flux signals analysis; whereas non-contact methods accomplish via acoustic, current and stray flux measurement. The existing techniques suffer from the influence of adjoining electrical machines; require human expertise to mount sensors and analyzing the signals. In this paper, a new, non-contact method proposed for bearing fault identification in the SCIM. The proposed method uses a high-frequency signal projected on the motor and the reflected signal captured. The signal obtained is analyzed with an advanced signal processing algorithm like Rational Dilation Wavelet Transform (RDWT) to identify the faults in the SCIM. The signal energy at the fault frequency level increases from 4.72% to 5.82% with the increase in the number of the faults. The signal energy variation indicates the severity of the faults. From the experimental results, the bearing fault of the motor identified in the beginning stage of the fault.
