Continuous Wavelet Transform Methods Research Articles

Application of wavelet and Fuorier transforms as powerful alternatives for derivative spectrophotometry in analysis of binary mixtures: A comparative study

Two signal processing methods, namely, Continuous Wavelet Transform (CWT) and the second was Discrete Fourier Transform (DFT) were introduced as alternatives to the classical Derivative Spectrophotometry (DS) in analysis of binary mixtures. To show the advantages of these methods, a comparative study was performed on a binary mixture of Naltrexone (NTX) and Bupropion (BUP). The methods were compared by analyzing laboratory prepared mixtures of the two drugs. By comparing performance of the three methods, it was proved that CWT and DFT methods are more efficient and advantageous in analysis of mixtures with overlapped spectra than DS. The three signal processing methods were adopted for the quantification of NTX and BUP in pure and tablet forms. The adopted methods were validated according to the ICH guideline where accuracy, precision and specificity were found to be within appropriate limits.

Co-movement of ASEAN stock markets: New evidence from wavelet and VMD-based copula tests

This paper aims to study the co-movement and the volatility fluctuation between stock markets in the Association of Southeast Asian Nations (ASEAN) countries from a new perspective. The analyses also delve more deeply into the effect of ASEAN trading link establishment on the short-term interdependency. By applying three-dimensional continuous wavelet transform (CWT) on daily returns of stock markets for the period 2009 to 2016, the interdependence level and lag-lead relationship among ASEAN trading link participants are estimated. The degree of interdependence in ASEAN stock markets is found to be stronger in the short term, especially following particular external shocks. A Variational Modes Decomposition (VMD)-based copula estimation shows that the effect of economic shock – in our case, ASEAN trading link establishment – on the stock markets’ level of comovement is only temporary and will progressively diminish within approximately two years. Only Indonesia and Malaysia display strong fundamental linkages between each other. Both the CWT and Copula methods consistently show that Vietnam (Indonesia) has the lowest (highest) interdependence with the rest of ASEAN trading link participants, as opposed to previous empirical evidence obtained from conventional methods. Investors who want to construct optimal portfolios and policymakers who aim to make effective macroeconomic policies should take these findings into account.

Time-Frequency Distribution of Seismocardiographic Signals: A Comparative Study.

Accurate estimation of seismocardiographic (SCG) signal features can help successful signal characterization and classification in health and disease. This may lead to new methods for diagnosing and monitoring heart function. Time-frequency distributions (TFD) were often used to estimate the spectrotemporal signal features. In this study, the performance of different TFDs (e.g., short-time Fourier transform (STFT), polynomial chirplet transform (PCT), and continuous wavelet transform (CWT) with different mother functions) was assessed using simulated signals, and then utilized to analyze actual SCGs. The instantaneous frequency (IF) was determined from TFD and the error in estimating IF was calculated for simulated signals. Results suggested that the lowest IF error depended on the TFD and the test signal. STFT had lower error than CWT methods for most test signals. For a simulated SCG, Morlet CWT more accurately estimated IF than other CWTs, but Morlet did not provide noticeable advantages over STFT or PCT. PCT had the most consistently accurate IF estimations and appeared more suited for estimating IF of actual SCG signals. PCT analysis showed that actual SCGs from eight healthy subjects had multiple spectral peaks at 9.20 ± 0.48, 25.84 ± 0.77, 50.71 ± 1.83 Hz (mean ± SEM). These may prove useful features for SCG characterization and classification.

Continuous wavelet transform methods for the simultaneous determinations and dissolution profiles of valsartan and hydrochlorothiazide in tablets

Continuous wavelet transform (CWT) was proposed for the simultaneous determination and dissolution profiles of valsartan (VAL) and hydrochlorothiazide (HCT) in tablets, without the use of a chemical separation procedure. The CWT approach was applied to the original UV spectra and their ratio spectra in the optimal wavelength ranges. After testing several wavelet families, Mexican hat function-CWT and Daubechies7-CWT (mexh-CWT and db7-CWT, respectively) were found to be suitable for the transformation of the original UV spectra. In the following procedure, mexh-CWT and Coiflets3-CWT (coif3-CWT) were found to be appropriate for the signal analysis of ratio spectra (RS) of VAL/HCT and HCT/VAL. Calibration graphs for VAL and HCT were obtained by measuring db7-CWT and mexh-CWT amplitudes in the transformation of the original absorption spectra and RS-coif-CWT and RS-mexh-CWT amplitudes in the transformation of the ratio spectra. The validity and applicability of the proposed CWT methods were evaluated through the analysis of an independent set of synthetic binary mixtures consisting of VAL and HCT. The proposed signal processing methods were then successfully applied to the simultaneous quantitative evaluation and simultaneous dissolution profiles of the related drugs in commercial tablets, with good agreement reported for the experimental results.

Continuous Wavelet Transform Methods for the Simultaneous Quantification of Brimonidine Tartrate and Timolol Maleate in an Eye Drop Formulation

Continuous Wavelet Transform Methods for the Simultaneous Quantification of Brimonidine Tartrate and Timolol Maleate in an Eye Drop Formulation

Filtering methods in tidal-affected groundwater head measurements: Application of harmonic analysis and continuous wavelet transform

A new methodology has been developed to obtain tidal-filtered time series of groundwater levels in coastal aquifers. Two methods used for oceanography processing and forecasting of sea level data were adapted for this purpose and compared: HA (Harmonic Analysis) and CWT (Continuous Wavelet Transform). The filtering process is generally comprised of two main steps: the detection and fitting of the major tide constituents through the decomposition of the original signal and the subsequent extraction of the complete tidal oscillations. The abilities of the optional HA and CWT methods to decompose and extract the tidal oscillations were assessed by applying them to the data from two piezometers at different depths close to the shoreline of a Mediterranean coastal aquifer (Motril-Salobreña, SE Spain). These methods were applied to three time series of different lengths (one month, one year, and 3.7 years of hourly data) to determine the range of detected frequencies. The different lengths of time series were also used to determine the fit accuracies of the tidal constituents for both the sea level and groundwater heads measurements. The detected tidal constituents were better resolved with increasing depth in the aquifer. The application of these methods yielded a detailed resolution of the tidal components, which enabled the extraction of the major tidal constituents of the sea level measurements from the groundwater heads (e.g., semi-diurnal, diurnal, fortnightly, monthly, semi-annual and annual). In the two wells studied, the CWT method was shown to be a more effective method than HA for extracting the tidal constituents of highest and lowest frequencies from groundwater head measurements.

Time–Frequency Domain Analysis of Asphalt Longitudinal Strain

The application of time–frequency domain analysis methods, such as continuous wavelet transform (CWT) and short-time Fourier transform (STFT), is evaluated in the extraction of the dominant frequency from asphalt longitudinal strain signals. The pavement response data collected at the fully instrumented Integrated Road Research Facility in Edmonton, Alberta, Canada, was used in this study. Promising results were achieved when CWT and STFT were used to determine the dominant frequency of the measured longitudinal strain at the bottom of the asphalt layer at different vehicular speeds. The dominant frequencies obtained from these methods were compared with those from conventional time-to-frequency conversion methods. Results showed that a frequency calculation that used the inverse of tensile pulse duration led to noticeably larger frequencies compared with those associated with the CWT and STFT methods. The accuracy of the determined frequencies and the corresponding moduli of hot-mix asphalt were evaluated by predicting strains with the KENPAVE program. This analysis showed the advantage of using time–frequency domain methods because they led to more reasonable agreements between the measured and predicted responses. Finally, the impact of frequency calculation methods on the potential fatigue cracking life was assessed by taking the estimated moduli and strains of hot-mix asphalt into account. It was found that the fatigue cracking life can be overestimated by almost 45% when frequency is considered as a reciprocal of pulse duration compared with when prediction is based on time–frequency domain methods.

The application of continuous wavelet transform and least squares support vector machine for the simultaneous quantitative spectrophotometric determination of Myricetin, Kaempferol and Quercetin as flavonoids in pharmaceutical plants

Flavonoids are γ-benzopyrone derivatives, which are highly regarded in these researchers for their antioxidant property. In this study, two new signals processing methods been coupled with UV spectroscopy for spectral resolution and simultaneous quantitative determination of Myricetin, Kaempferol and Quercetin as flavonoids in Laurel, St. John’s Wort and Green Tea without the need for any previous separation procedure. The developed methods are continuous wavelet transform (CWT) and least squares support vector machine (LS-SVM) methods integrated with UV spectroscopy individually. Different wavelet families were tested by CWT method and finally the Daubechies wavelet family (Db4) for Myricetin and the Gaussian wavelet families for Kaempferol (Gaus3) and Quercetin (Gaus7) were selected and applied for simultaneous analysis under the optimal conditions. The LS-SVM was applied to build the flavonoids prediction model based on absorption spectra. The root mean square errors for prediction (RMSEP) of Myricetin, Kaempferol and Quercetin were 0.0552, 0.0275 and 0.0374, respectively. The developed methods were validated by the analysis of the various synthetic mixtures associated with a well- known flavonoid contents. Mean recovery values of Myricetin, Kaempferol and Quercetin, in CWT method were 100.123, 100.253, 100.439 and in LS-SVM method were 99.94, 99.81 and 99.682, respectively. The results achieved by analyzing the real samples from the CWT and LS-SVM methods were compared to the HPLC reference method and the results were very close to the reference method. Meanwhile, the obtained results of the one-way ANOVA (analysis of variance) test revealed that there was no significant difference between the suggested methods.

Using time-frequency analysis to determine time-resolved detonation velocity with microwave interferometry.

Two time-frequency analysis methods based on the short-time Fourier transform (STFT) and continuous wavelet transform (CWT) were used to determine time-resolved detonation velocities with microwave interferometry (MI). The results were directly compared to well-established analysis techniques consisting of a peak-picking routine as well as a phase unwrapping method (i.e., quadrature analysis). The comparison is conducted on experimental data consisting of transient detonation phenomena observed in triaminotrinitrobenzene and ammonium nitrate-urea explosives, representing high and low quality MI signals, respectively. Time-frequency analysis proved much more capable of extracting useful and highly resolved velocity information from low quality signals than the phase unwrapping and peak-picking methods. Additionally, control of the time-frequency methods is mainly constrained to a single parameter which allows for a highly unbiased analysis method to extract velocity information. In contrast, the phase unwrapping technique introduces user based variability while the peak-picking technique does not achieve a highly resolved velocity result. Both STFT and CWT methods are proposed as improved additions to the analysis methods applied to MI detonation experiments, and may be useful in similar applications.

Respiratory Rate Estimation from the Built-in Cameras of Smartphones and Tablets

This paper presents a method for respiratory rate estimation using the camera of a smartphone, an MP3 player or a tablet. The iPhone 4S, iPad 2, iPod 5, and Galaxy S3 were used to estimate respiratory rates from the pulse signal derived from a finger placed on the camera lens of these devices. Prior to estimation of respiratory rates, we systematically investigated the optimal signal quality of these 4 devices by dividing the video camera's resolution into 12 different pixel regions. We also investigated the optimal signal quality among the red, green and blue color bands for each of these 12 pixel regions for all four devices. It was found that the green color band provided the best signal quality for all 4 devices and that the left half VGA pixel region was found to be the best choice only for iPhone 4S. For the other three devices, smaller 50×50 pixel regions were found to provide better or equally good signal quality than the larger pixel regions. Using the green signal and the optimal pixel regions derived from the four devices, we then investigated the suitability of the smartphones, the iPod 5 and the tablet for respiratory rate estimation using three different computational methods: the autoregressive (AR) model, variable-frequency complex demodulation (VFCDM), and continuous wavelet transform (CWT) approaches. Specifically, these time-varying spectral techniques were used to identify the frequency and amplitude modulations as they contain respiratory rate information. To evaluate the performance of the three computational methods and the pixel regions for the optimal signal quality, data were collected from 10 healthy subjects. It was found that the VFCDM method provided good estimates of breathing rates that were in the normal range (12-24 breaths/min). Both CWT and VFCDM methods provided reasonably good estimates for breathing rates that were higher than 26 breaths/min but their accuracy degraded concomitantly with increased respiratory rates. Overall, the VFCDM method provided the best results for accuracy (smaller median error), consistency (smaller interquartile range of the median value), and computational efficiency (less than 0.5s on 1min of data using a MATLAB implementation) to extract breathing rates that varied from 12 to 36 breaths/min. The AR method provided the least accurate respiratory rate estimation among the three methods. This work illustrates that both heart rates and normal breathing rates can be accurately derived from a video signal obtained from smartphones, an MP3 player and tablets with or without a flashlight.

Modal identification of structures by a novel approach based on FDD-wavelet method

An important application of system identification in structural dynamics is the determination of natural frequencies, mode shapes and damping ratios during operation which can then be used for calibrating numerical models. In this paper, the combination of two advanced methods of Operational Modal Analysis (OMA) called Frequency Domain Decomposition (FDD) and Continuous Wavelet Transform (CWT) based on novel cyclic averaging of correlation functions (CACF) technique are used for identification of dynamic properties. By using this technique, the autocorrelation of averaged correlation functions is used instead of original signals. Integration of FDD and CWT methods is used to overcome their deficiency and take advantage of the unique capabilities of these methods. The FDD method is able to accurately estimate the natural frequencies and mode shapes of structures in the frequency domain. On the other hand, the CWT method is in the time-frequency domain for decomposition of a signal at different frequencies and determines the damping coefficients. In this paper, a new formulation applied to the wavelet transform of the averaged correlation function of an ambient response is proposed. This application causes to accurate estimation of damping ratios from weak (noise) or strong (earthquake) vibrations and long or short duration record. For this purpose, the modified Morlet wavelet having two free parameters is used. The optimum values of these two parameters are obtained by employing a technique which minimizes the entropy of the wavelet coefficients matrix. The capabilities of the novel FDD-Wavelet method in the system identification of various dynamic systems with regular or irregular distribution of mass and stiffness are illustrated. This combined approach is superior to classic methods and yields results that agree well with the exact solutions of the numerical models.

Time–frequency analysis of the baroreflex control of renal sympathetic nerve activity in the rat

Cross-spectral analysis using the short-time Fourier transform (STFT) allows estimating the transfer function between spontaneous fluctuations of arterial pressure (AP) and renal sympathetic nerve activity (RSNA) at the heart rate (HR) frequency, which provides an index of sympathetic baroreflex sensitivity (sBRS) in rats. The method, however, cannot reliably compute more than one value per min. The goal of the present study was to achieve a better temporal resolution by using advanced methods. The first method is the continuous wavelet transform (CWT) using the Morlet wavelet. The second method is based on the empirical mode decomposition (EMD), a method that decomposes a signal into a sum of oscillating components. Using both methods, the transfer function was estimated over periods of 10 s. The two methods, together with STFT, were applied to AP and RSNA signals that were simultaneously recorded in conscious, freely behaving rats ( n = 10) during 1 h. When considering 1-h mean sBRS values obtained in each rat, both methods showed a strong correlation with STFT ( R = 0.96 and 0.91 for CWT and EMD, respectively, both P < 0.001). In each rat, sBRS values obtained by the CWT and EMD methods were tightly correlated ( R = 0.93 ± 0.01, n = 294 ± 13, P < 0.001). With both methods, high-frequency variations of sBRS (0.0083–0.5 Hz) accounted for about 40% of its overall variability. In urethane-anaesthetized rats ( n = 9), sBRS variability computed by either method was reduced by about two-thirds ( P < 0.001). Improving temporal resolution of sBRS computation reveals that in rats, sBRS exhibits fast, short-lasting fluctuations. These fluctuations largely depend on the state of vigilance.

Damping identification with the Morlet-wave

In the past decade damping-identification methods based on the continuous wavelet transform (CWT) have been shown to be some of the best methods for analyzing the damping of multi-degree-of-freedom systems. The CWT methods have proven themselves to be resistant to noise and able to identify damping at closely spaced natural frequencies. However, with the CWT-based techniques, the CWT needs to be obtained on a two-dimensional, time–frequency grid, and they are, therefore, computationally demanding. Furthermore, the CWT is susceptible to the edge effect, which causes a non-valid identification at the start and the end of the time-series. This study introduces a new method, called the Morlet-wave method, where a finite integral similar to the CWT is used for the identification of the viscous damping. Instead of obtaining the CWT on a two-dimensional grid, the finite integral needs to be calculated at one time–frequency point, only. Then using two different integration parameters, the damping ratio can be identified. A complete mathematical background of the new, Morlet-wave, damping-identification method is given and this results in a root-finding or a closed-form solution. The presented numerical experiments show that the new method has a similar performance to the CWT-based damping-identification methods, while the method is numerically, significantly less demanding, completely avoids the edge effect, and the procedure is straightforward to use.

Automated ECG diagnostic P-wave analysis using wavelets

P-wave characteristics in the human ECG are an important source of information in the diagnosis of atrial conduction pathology. However, diagnosis by visual inspection is a difficult task since the P-wave is relatively small and noise masking is often present. This paper introduces novel wavelet characteristics derived from the continuous wavelet transform (CWT) which are shown to be potentially effective discriminators in an automated diagnostic process. Characteristics of the 12-lead ECG P-wave were derived using CWT and statistical methods. A normal control group and an abnormal (atrial conduction pathology) group were compared. The wavelet characteristics captured frequency, magnitude and variance components of the P-wave. The best individual characteristics (i.e. ones that significantly discriminated the groups) were entered into a linear discriminant analysis (LDA) for four different models: two-lead ECG, three-lead ECG, a derived three-lead ECG and a factor analysis solution consisting of wavelet characteristic loadings on the factors. A comparison was also made between wavelet characteristics derived form individual P-waves verses wavelet characteristics derived from a signal-averaged P-wave for each participant. These wavelet models were also compared to standard cardiological measures of duration, terminal force and duration divided by the PR segment. Results for the individual P-wave approach generally outperformed the standard cardiological measures and the signal-averaged P-wave approach. The best wavelet model on the basis of both classification performance and simplicity was the two-lead model that uses leads II and V1. It was concluded that the wavelet approach of automating classification is worth pursuing with larger samples to validate and extend the present study.

Detection of Demagnetization Faults in Permanent-Magnet Synchronous Motors Under Nonstationary Conditions

This paper presents the results of our study of the permanent-magnet synchronous motor (PMSM) running under demagnetization. We examined the effect of demagnetization on the current spectrum of PMSMs with the aim of developing an effective condition-monitoring scheme. Harmonics of the stator currents induced by the fault conditions are examined. Simulation by means of a two-dimensional finite-element analysis (FEA) software package and experimental results are presented to substantiate the successful application of the proposed method over a wide range of motor operation conditions. Methods based on continuous wavelet transform (CWT) and discrete wavelet transform (DWT) have been successfully applied to detect and to discriminate demagnetization faults in PMSM motors under nonstationary conditions. Additionally, a reduced set of easy-to-compute discriminating features for both CWT and DWT methods has been defined. We have shown the effectiveness of the proposed method by means of experimental results.

Estimation of Respiratory Rate From Photoplethysmogram Data Using Time–Frequency Spectral Estimation

We present a new method that uses the pulse oximeter signal to estimate the respiratory rate. The method uses a recently developed time-frequency spectral estimation method, variable-frequency complex demodulation (VFCDM), to identify frequency modulation (FM) of the photoplethysmogram waveform. This FM has a measurable periodicity, which provides an estimate of the respiration period. We compared the performance of VFCDM to the continuous wavelet transform (CWT) and autoregressive (AR) model approaches. The CWT method also utilizes the respiratory sinus arrhythmia effect as represented by either FM or AM to estimate respiratory rates. Both CWT and AR model methods have been previously shown to provide reasonably good estimates of breathing rates that are in the normal range (12-26 breaths/min). However, to our knowledge, breathing rates higher than 26 breaths/min and the real-time performance of these algorithms are yet to be tested. Our analysis based on 15 healthy subjects reveals that the VFCDM method provides the best results in terms of accuracy (smaller median error), consistency (smaller interquartile range of the median value), and computational efficiency (less than 0.3 s on 1 min of data using a MATLAB implementation) to extract breathing rates that varied from 12-36 breaths/min.

Combined application of continuous wavelet transform-zero crossing technique in the simultaneous spectrophotometric determination of perindopril and indapamid in tablets

Signal processing methods based on the combined use of the continuous wavelet transform (CWT) and zero-crossing technique were applied to the simultaneous spectrophotometric determination of perindopril (PER) and indapamide (IND) in tablets. These signal processing methods do not require any priory separation step. Initially, various wavelet families were tested to identify the optimum signal processing giving the best recovery results. From this procedure, the Haar and Biorthogonal1.5 continuous wavelet transform (HAAR-CWT and BIOR1.5-CWT, respectively) were found suitable for the analysis of the related compounds. After transformation of the absorbance vectors by using HAAR-CWT and BIOR1.5-CWT, the CWT-coefficients were drawn as a graph versus wavelength and then the HAAR-CWT and BIOR1.5-CWT spectra were obtained. Calibration graphs for PER and IND were obtained by measuring the CWT amplitudes at 231.1 and 291.0 nm in the HAAR-CWT spectra and at 228.5 and 246.8 nm in BIOR1.5-CWT spectra, respectively. In order to compare the performance of HAAR-CWT and BIOR1.5-CWT approaches, derivative spectrophotometric (DS) method and HPLC as comparison methods, were applied to the PER-IND samples. In this DS method, first derivative absorbance values at 221.6 for PER and 282.7 nm for IND were used to obtain the calibration graphs. The validation of the CWT and DS signal processing methods was carried out by using the recovery study and standard addition technique. In the following step, these methods were successfully applied to the commercial tablets containing PER and IND compounds and good accuracy and precision were reported for the experimental results obtained by all proposed signal processing methods.

Recognition of Milankovitch cycles in the stratigraphic record: application of the CWT and the FFT to well-log data

The authors applied a the combination of Continuous Wavelet Transform (CWT) and Fast Fourier Transform (FFT) methods to gamma ray well-log data from the Q3, G1 and D2 wells. This high-resolution stratigraphic study was based on Milankovitch's orbital cycle theory. It was found that the CWT scale factors, ‘a,’ of 12, 24 and 60 match the ratios of the periodicities of precession, obliquity and eccentricity very well. Nine intervals of the Permo-carboniferous strata were recognized to have Milankovitch cycles in them. For example, section A of well Q3 has 29 precession cycles, 15 obliquity cycles and 7 short eccentricity cycles. The wavelengths are 2.7, 4.4 and 7.8 m for precession, obliquity and eccentricity, respectively. Important geological parameters such as the stratigraphic completeness and the accumulation rate were also estimated. These results provide basic information for further cyclostratigraphic correlation studies in the area. They are of great significance for the study of ancient and future climate change.

Refractive index and extinction coefficient determination of an absorbing thin film by using the continuous wavelet transform method

We present the continuous wavelet transform (CWT) method for determining the dispersion curves of the refractive index and extinction coefficient of absorbing thin films by using the transmittance spectrum in the visible and near infrared regions at room temperature. The CWT method is performed on the transmittance spectrum of an a - Si(1-x)C(x):H film, and the refractive index and extinction coefficient of the film are continuously determined and compared with the results of the envelope and fringe counting methods. Also the noise filter property of the method is depicted on a theoretically generated noisy signal. Finally, the error analyses of the CWT, envelope, and fringe counting methods are performed.

Wavelet analysis of bileaflet mechanical valve sounds

It has been reported that asynchronous leaflet closure in a bileaflet mechanical valve causes a split in the valve closing sound. We have previously reported that the continuous wavelet transform (CWT) with the Morlet wavelet as modified by Ishikawa (the Morlet wavelet) is the most suitable method among the CWTs for detecting a split in the bileaflet mechanical valve sound because this method can detect the highest frequency signal among the CWT methods with higher time resolution. This is the first article which discusses the acoustic properties of five types of bileaflet valves using the Morlet CWT. Similar behavior of the valve sound split intervals with wide fluctuations over consecutive heartbeats was found to be the common finding for all the bileaflet valves. This result suggests that fluctuation of the split interval proves the normal movement of both leaflets without movement disturbance. The mean differences in the split interval between these bileaflet valves were statistically significant, and the wavelet coefficients of the CWT showed characteristic scalographic patterns, such as a teardrop shape or a triangle beneath the split. However, these two findings gave no valuable information for the diagnosis of bileaflet valve malfunction. A split in the valve closing sound with a fluctuating interval was the common finding in these five normally functioning bileaflet valves, and careful observation of the split's behavior may be a key to diagnosis of bileaflet valve malfunction.