Short-time Fourier Transform Research Articles

Signal Enhancement Methods Based on Wavelet Transform, Fractional Fourier Transform and Short-time Fourier Transform

This review paper mainly focuses on different signal enhancement methods such as wavelet transform, fractional Fourier transform (FrFT) and short-time Fourier transform (STFT). First, this paper introduces the concept and importance of signal enhancement, as well as some current issues and challenges. Then, in recent years, the application of the three methods in wavelet transform, fractional Fourier transform and short-time Fourier transform is described. In terms of wavelet transform, this paper discusses the application of wavelet function to image enhancement and the specific steps, and analyzes its advantages and application in signal enhancement. In terms of fractional Fourier transform, this paper introduces its difference from traditional Fourier transform, and discusses its application in combination with adaptive filtering technology and the application of multi-level FrFT in the field of speech enhancement. Finally, in terms of short-time Fourier transform, this paper discusses its application in the fields of image enhancement and speech enhancement. The review paper finally discusses the characteristics, advantages and limitations of these three methods in signal enhancement, and looks forward to the future research directions. Through the research of this review paper, it can provide some reference and guidance for the research in the field of signal enhancement.

Radio Frequency Interference Mitigation in Synthetic Aperture Radar Data Based on Instantaneous Spectrum Forward Consecutive Mean Excision

Radio frequency interference (RFI) poses major threats to synthetic aperture radar (SAR) systems. Due to the suppression of useful target signals via high-power RFI, the SAR imaging quality is severely degraded. Nevertheless, existing studies on RFI mitigation mainly focus on narrowband filtering, while wideband RFI mitigation methods are relatively lacking and perform non-robustly. In this paper, an RFI mitigation scheme is proposed based on instantaneous spectrum forward consecutive mean excision (FCME), which is suitable for both narrowband and wideband RFI mitigation. The SAR echo signal is first transformed into a time–frequency (TF) domain through a short-time Fourier transform (STFT). On this basis, the instantaneous spectra polluted via RFI are detected via a kurtosis-based statistical test and then filtered via FCME to achieve RFI mitigation. Finally, connected component analysis is applied as a safety measure so as to avoid the unnecessary loss of useful target signal. The combination of FCME and connected component analysis enables the proposed method to thoroughly filter out RFI while retaining more useful target signals compared with other competing methods. The experimental results on real SAR raw data validate the effectiveness of the proposed method.

Hybrid Model for Motor Imagery Biometric Identification

Biometric systems are a continuously evolving and promising technological domain that can be used in automatic systems for the unique and efficient identification and authentication of individuals without necessitating users to carry or remember any physical tokens or passwords, in contrast to traditional methods such as password IDs. Biometrics are biological measurements or physical characteristics that can be used to ascertain and validate the identity of individuals. Recently, considerable interest has emerged in exploiting brain activity as a biometric identifier in automatic recognition systems, particularly focusing on data acquired through electroencephalography (EEG). Multiple research endeavors have indeed confirmed the presence of discriminative characteristics within brain signals recorded while performing specific cognitive tasks. However, EEG signals are inherently complex due to their nonstationary and high-dimensional properties, thus demanding careful consideration during both the feature extraction and classification processes. This study applied a hybridization technique integrating a pre-trained convolutional neural network (CNN) with a classical classifier and the short-time Fourier transform (STFT) spectrum. We used a hybrid model to decode two-class motor imagery (MI) signals for mobile biometric authentication tasks, which include subject identification and lock and unlock classification. To this purpose, nine potential classifiers (mostly classification algorithms) were utilized to build nine distinct hybrid models, with the ultimate goal of selecting the most effective one. Practically, six experiments were conducted in the experimental part of this study. The first experiment aims to develop a hybrid model for biometric authentication tasks. To do this, nine possible classifiers (mostly classification algorithms) were used to build nine hybrid models. It can be seen that the RF-VGG model achieved better performance compared with other models. Therefore, it was chosen to be utilized for mobile biometric authentication. The fourth experiment is to apply the RF-VGG model for doing the lock and unlock classification process, and their mean accuracy is 97.50%. Consequently, the fifth experiment was conducted to validate the RF-VGG model for the lock and unlock task, and their mean accuracy was 97.40%. Practically, the sixth experiment was to verify the RF-VGG model for the lock and unlock task over another dataset (unseen data), and their accuracy is 94.4%. It can be deduced that the hybrid model appraises the capability of decoding the MI signal for the left and right hand. Therefore, the RF-VGG model can contribute to the BCI-MI community by facilitating the deployment of the mobile biometric authentication task for (the subject identification and the lock and unlock classification).

Research on Role Orientation and Situation Construction of Contextualized Music Performance in the Background of Artificial Intelligence

Abstract In this paper, the Skip-gram model is used to process the main melody information of contextualized music, combining feature extraction and Schenkel analysis to extract the main melody note sets and vectors. By applying the short-time Fourier transform (STFT) to the audio signal, the spectral center of mass and irregularities can be calculated to represent the emotional features of the music. To obtain music features based on wavelet coefficients, the music signal is decomposed, and each scale’s signal features are counted individually. Finally, by taking the musical elements as independent variables and the perceived results of pleasure and activation in the contextual model as dependent variables, we compared the variability of the dynamic associations between emotional perceptions and the musical elements among different contextualized music pieces. The results showed that the overall differences in emotional cognition across musical performances ranged between ±0.5, with the mean short-term energy amplitude of relatively cheerful music ranging between ±0.2, which was smaller than that of angry (±0.3) and sad (±0.5) music. In this study, musical works were accurately characterized, and their expressive and infectious power was effectively enhanced through contextual construction.

A cylinder block dynamic characteristics-based data augmentation method for wear state identification under data imbalance condition

The deterioration of the wear state in the key friction pairs will degenerate the performance of the axial piston pump. The wear rates differ under different wear states in practical applications. Therefore, the amount of monitoring data is significantly different under different wear states, which causes data imbalance problem. In addition, the distribution characteristics of the external signals are often drowned by environmental noise. To address these issues, a novel cylinder block dynamic characteristics-based data augmentation method is presented for wear state identification. Firstly, a cylinder block’s dynamic characteristics state perception method is proposed to obtain the time domain data. Then, to reduce the computing costs, the raw data is fused into multi-channel time–frequency domain data by the short-time Fourier transform (STFT) based method. Finally, the minority time–frequency data is augmented by the designed deep convolutional generative adversarial network (DCGAN). To verify the wear state identification performance of the proposed method, a convolutional neural network (CNN) is designed. Wear states injection experiments are adopted to verify the feasibility of the proposed method. The advantage of the developed method is that it obtains prominent data distribution characteristics from internal signals to enhance the performance of the data augmentation process.

Time-frequency analysis on flat tori and Gabor frames in finite dimensions

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori TN2=R2/(Z×NZ)=[0,1]×[0,N] act as phase spaces. We work on an N-dimensional subspace SN of distributions periodic in time and frequency in the dual S0′(R) of the Feichtinger algebra S0(R) and equip it with an inner product. To construct the Hilbert space SN we apply a suitable double periodization operator to S0(R). On SN, the STFT is applied as the usual STFT defined on S0′(R). This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

Application of Short Time Fourier Transform (STFT) For Diagnosing Rolling Bearing Faults

A fan is crucial for maintaining airflow in industries. Bearings in fans prevent friction and must be robust to function effectively. Damage to the bearings can diminish machine performance. Predictive maintenance is essential for early detection of faults. One way to analyze bearing faults is by using the Short Time Fourier Transform (STFT), as it excels in analyzing non-stationary signals. Experiments were conducted under normal conditions and with inner race faults in bearings at a shaft speed of 1162.5 Hz. Vibration detection was done using an accelerometer sensor, and Matlab analysis was employed. The data was processed using the Fourier Transform (FT) method through both time and frequency domains, as well as the STFT through spectrograms. In the spectrum plot, there is still a significant amount of noise present. This high amplitude of noise from other frequencies obscures the bearing fault amplitudes. Furthermore, the Fourier Transform (FT) is only suitable for analyzing stationary signals. To address this, an envelope analysis was used to filter out the noise. The STFT analysis method provides simultaneous frequency and time information. This reveals that the spectrogram results for inner race faults depict three high amplitude peaks at harmonic frequencies. This indicates that the signal is non-stationary due to fluctuating amplitudes, making bearing fault analysis more accessible.

A noval pulmonary function evaluation method based on ResNet50 + SVR model and cough

Traditionally, the clinical evaluation of respiratory diseases was pulmonary function testing, which can be used for the detection of severity and prognosis through pulmonary function parameters. However, this method is limited by the complex process, which is impossible for patients to monitor daily. In order to evaluate pulmonary function parameters conveniently with less time and location restrictions, cough sound is the substitute parameter. In this paper, 371 cough sounds segments from 150 individuals were separated into 309 and 62 as the training and test samples. Short-time Fourier transform (STFT) was applied to transform cough sound into spectrogram, and ResNet50 model was used to extract 2048-dimensional features. Through support vector regression (SVR) model with biological attributes, the data were regressed with pulmonary function parameters, FEV1, FEV1%, FEV1/FVC, FVC, FVC%, and the performance of this models was evaluated with fivefold cross-validation. Combines with deep learning and machine learning technologies, the better results in the case of small samples were achieved. Using the coefficient of determination (R2), the ResNet50 + SVR model shows best performance in five basic pulmonary function parameters evaluation as FEV1(0.94), FEV1%(0.84), FEV1/FVC(0.68), FVC(0.92), and FVC%(0.72). This ResNet50 + SVR hybrid model shows excellent evaluation of pulmonary function parameters during coughing, making it possible to realize a simple and rapid evaluation for pneumonia patients. The technology implemented in this paper is beneficial in judge the patient's condition, realize early screening of respiratory diseases, evaluate postoperative disease changes and detect respiratory infectious diseases without time and location restrictions.

A new method using deep transfer learning on ECG to predict the response to cardiac resynchronization therapy

Background: Cardiac resynchronization therapy (CRT) has emerged as an effective treatment for heart failure patients with electrical dyssynchrony. However, accurately predicting which patients will respond to CRT remains a challenge. This study explores the application of deep transfer learning techniques to train a predictive model for CRT response. Methods: In this study, the short-time Fourier transform (STFT) technique was employed to transform ECG signals into two-dimensional images. A transfer learning approach was then applied to the MIT-BIT ECG database to pre-train a convolutional neural network (CNN) model. The model was fine-tuned to extract relevant features from the ECG images and then tested on our dataset of CRT patients to predict their response. Results: Seventy-one CRT patients were enrolled in this study. The transfer learning model achieved an accuracy of 72% in distinguishing responders from non-responders in the local dataset. Furthermore, the model showed good sensitivity (0.78) and specificity (0.79) in identifying CRT responders. The performance of our model outperformed clinic guidelines and traditional machine learning approaches. Conclusion: The utilization of ECG images as input and leveraging the power of transfer learning allows for improved accuracy in identifying CRT responders. This approach offers potential for enhancing patient selection and improving the outcomes of CRT.

A Robust Approach Assisted by Signal Quality Assessment for Fetal Heart Rate Estimation from Doppler Ultrasound Signal.

Fetal heart rate (FHR) monitoring, typically using Doppler ultrasound (DUS) signals, is an important technique for assessing fetal health. In this work, we develop a robust DUS-based FHR estimation approach complemented by DUS signal quality assessment (SQA) based on unsupervised representation learning in response to the drawbacks of previous DUS-based FHR estimation and DUS SQA methods. We improve the existing FHR estimation algorithm based on the autocorrelation function (ACF), which is the most widely used method for estimating FHR from DUS signals. Short-time Fourier transform (STFT) serves as a signal pre-processing technique that allows the extraction of both temporal and spectral information. In addition, we utilize double ACF calculations, employing the first one to determine an appropriate window size and the second one to estimate the FHR within changing windows. This approach enhances the robustness and adaptability of the algorithm. Furthermore, we tackle the challenge of low-quality signals impacting FHR estimation by introducing a DUS SQA method based on unsupervised representation learning. We employ a variational autoencoder (VAE) to train representations of pre-processed fetal DUS data and aggregate them into a signal quality index (SQI) using a self-organizing map (SOM). By incorporating the SQI and Kalman filter (KF), we refine the estimated FHRs, minimizing errors in the estimation process. Experimental results demonstrate that our proposed approach outperforms conventional methods in terms of accuracy and robustness.

Chirp Rate Estimation of LFM Signals Based on Second-Order Synchrosqueezing Transform

For the problem of low time-frequency aggregation of the short-time Fourier transform (STFT), which causes the parameter estimation performance degradation of linear frequency modulation (LFM) signals at low signal-to-noise ratio (SNR), second-order synchrosqueezing transform (SSST) is proposed based on the square of STFT amplitude. The time-frequency resolution and energy aggregation are improved by means of squeezing and reassigning the time-frequency spectrum. Meanwhile, in order to decrease the calculation of classical parameter estimation methods, the Hough transform is used for rough estimation, and then the fractional Fourier transform (FRFT) is used for accuracy estimation based on the Renyi entropy. The simulation result shows that higher estimation accuracy is obtained at low SNR, and it has better robustness.

Time-frequency enhanced characterization method based on asymmetric image reconstruction autoencoder

The vibration signals of mechanical equipment are subject to the influence of complex and variable working conditions, often exhibiting non-smooth and non-linear characteristics. The conventional time-frequency (TF) analysis (TFA) method, which relies on energy concentration, is susceptible to noise and impact, making it challenging to accurately extract fault characteristics. To overcome these limitations, this paper proposes an innovative approach. In this paper, we introduce an asymmetric image reconstruction autoencoder model, which is based on two well-known TFA methods, namely, short-time Fourier transform (STFT) and synchroextracting transform (SET), effectively reducing noise and improving the TF energy concentration process through learning the mapping relationship between STFT and SET. To address the clarity issue in the reconstructed TF images, the paper incorporates a channel attention mechanism known as SE Block into the encoding-decoding structure. Additionally, a skip connection structure is introduced to aid in restoring the structural details of the reconstructed TF images. Moreover, an improved weighted joint loss function is proposed to adaptively enhance various types of TF distribution features. This enhancement ensures that different characteristics of TF distribution are adequately addressed during the reconstruction process. The proposed method is put to the test using both simulated signals and experimental signals from gearbox rolling bearing faults. The results demonstrate that compared to traditional TFA and post-processing methods, the proposed model exhibits superior capabilities in enhancing the TF characterization of multi-source time-varying signals. Furthermore, it demonstrates remarkable robustness to noise and can accurately extract instantaneous frequency. These findings point to the promising potential of this method for mechanical fault identification and diagnosis applications.

DeepCilia: Automated, deep-learning based engine for precise ciliary beat frequency estimation

Technology advancements over recent decades have greatly impacted medical treatments and diagnostic methods. These advancements could be useful in nasal cytology, which is becoming increasingly critical in diagnosing nasal conditions. In this study we propose DeepCilia, a Ciliary beating frequency (CBF) decay estimation engine that autonomously detects ciliated strias using YOLOv8 to perform the detection task and is able to produce accurate estimations even for low FPS videos, by using the Short-Time Fourier Transform (STFT). DeepCilia achieves very good results with an average computation time below 2 s for each video in the dataset, and an RMSE of less than 1.5 Hz. These achievements make DeepCilia an ideal tool in the field of ciliary motility analysis to estimate the time of death in the context of forensics medicine.

Enabling Safe ITS: EEG-Based Microsleep Detection in VANETs

Researchers nowadays are particularly focusing on the interpretation of EEG signals to understand and exploit the information they provide for brain activities. Deep learning architectures performing sleep staging have recently grown to their full potential with their ability to learn and interpret highly complex mathematical contexts. This has been catered to owing to the increasing availability of large EEG data sets. In this paper, we describe how sleep staging differs from microsleep prediction. We also provide a fresh methodology for the microsleep classification job that works with even less training data. Our proposed model exploits the attention-based mechanism that clubs the advantages available in Wavelet transform with Short Time Fourier Transform(STFT) Spectrogram. We also put forward a robust deep learning model that contains separate “time-dependent” and “time-independent” parts, which can record contexts from the sequence of features and simultaneously learn intra-epoch relations. A single-electrode EEG signal was employed for our analysis to accommodate such procedures’ social acceptance. For the task of microsleep detection on the MWT dataset, our model achieves fairly high accuracy rates (92% training and 89.9% testing accuracy), and an overall improvement in the kappa value by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\approx$</tex-math> </inline-formula> 42%, as compared to prior novel approaches.

An Experimental Study of the Effects of Cylinder Lubricating Oils on the Vibration Characteristics of a Two-Stroke Low-Speed Marine Diesel Engine

Abstract Two-stroke, low-speed diesel engines are widely used in large ships due to their good performance and fuel economy. However, there have been few studies of the effects of lubricating oils on the vibration of two-stroke, low-speed diesel engines. In this work, the effects of three different lubricating oils on the vibration characteristics of a low-speed engine are investigated, using the frequency domain, time-frequency domain, fast Fourier transform (FFT) and short-time Fourier transform (STFT) methods. The results show that non-invasive condition monitoring of the wear to a cylinder liner in a low-speed marine engine can be successfully achieved based on vibration signals. Both the FFT and STFT methods are capable of capturing information about combustion in the cylinder online in real time, and the STFT method also provides the ability to visualise the results with more comprehensive information. From the online condition monitoring of vibration signals, cylinder lubricants with medium viscosity and medium alkali content are found to have the best wear protection properties. This result is consistent with those of an elemental analysis of cylinder lubrication properties and an analysis of the data measured from a piston lifted from the cylinder after 300 h of engine operation.

Oil-Water Flowing Experiments and Water-Cut Range Classification Approach Using Distributed Acoustic Sensing

Summary The accurate measurement of dynamic water cut is of great interest for analyzing reservoir performance and optimizing oilwell production. Downhole water-cut measurement is a very challenging work. Moreover, the surface-measured water cut is a comprehensive indicator of commingled producing well and it is difficult to use this parameter to deduce the downhole water cut of each contributing layer. In this paper, we propose to use distributed fiber-optic acoustic sensing (DAS) technology for the classification of water-cut range. DAS can dynamically monitor the entire wellbore by “listening” to the acoustic signals during flow. A large number of laboratory experimental data from DAS have been collected and analyzed using wavelet time scattering transform and short-time Fourier transform (STFT). The extracted low-variance scattering feature, short time-frequency feature, and fusion feature (combination of two extracted features) were learned with backpropagation (BP) neural network, decision tree (DT), and random forest (RF) algorithm. Then, a classification method of water-cut range in oil-water flow was established with machine learning. Field DAS data were collected from two oil wells to verify the effectiveness of the proposed method. The classification accuracies for the vertical well (Well A) are 92.4% and 87.4% by DT and RF model, respectively. For the horizontal well (Well B), the average classification accuracy exceeds 90% for all three methods. Water shutoff measure was conducted in Well B, and an obvious water decrease was realized. The result shows that the fusion feature overweighs single feature in machine learning with DAS data. This study provides a novel way to identify downhole water-cut range and detect water entry location in horizontal, vertical, and deviated oil-producing wells.

Understanding Dog Behavior through Visual and Auditory Sensing Using Machine Learning

This work aims to understand a dog’s behavior towards environmental stimuli. Different from previous works, we collect multi-modality data including both video and audio data observed from the dog’s egocentric perspective. We propose to model the association between a dog’s reaction and the visual and auditory stimuli perceived by the dog using machine learning, in particular through an extended Convolutional Neural Network (eCNN). The eCNN model takes colored images, Short Time Fourier Transform (STFT) of audio, and motion fields extracted from image sequences as input, and outputs a prediction of the dog’s reaction, classified as Sit, Stand, Walk, or Smell. Our proposed model achieves promising prediction results, with an average accuracy of 79.02% over all four classes. We also evaluate model performance by separately using one of image, audio, and motion information. Our results show that the dog responds strongly to low-frequency sounds and various color differences in its field of view. These research findings provide valuable insights to understanding animal behavior and intelligence as well as insights for building robotic companion dogs.

Wind turbine blade damage detection and classification based on sound feature signal using machine learning

Wind turbine blades experience variable environmental conditions and are prone to failure due to fatigue and weather-induced damage. In this study, the aerodynamic sound signal of wind turbine blades is collected while in operation. Short-Time Fourier Transform method identified and used to analyze the periodic signal of the rotation of the wind turbine blades. The spectral centroid of the pulse signal and the variation of the corresponding three blades spectral centroid are extracted as the input features for supervised machine learning. Gaussian mixture model algorithm classified features and used with blades damage trend for structural health monitoring. Through the Gaussian mixture model established by using the mean value and standard deviation of the peak value of the spectral centroid, the results after the model test show the accuracy of the damaged blade judgment can reach 98 %. In addition, from the visual classification chart, it can be known that the blade will enter the damage state, and maintenance should be carried out in advance to avoid damage to the blade and maintain the stable operation of the wind turbine.

The Correlation between Ionospheric Electron Density Variations Derived from Swarm Satellite Observations and Seismic Activity at the Australian–Pacific Tectonic Plate Boundary

Swarm electron density (Ne) observations from the Langmuir probe (LP) can detect ionospheric disturbances at the altitude of a satellite. Along-track satellite observations provide a large number of very short observations of different places in the ionosphere, where Ne is disturbed. Moreover, different perturbations occupy various Ne signal frequencies. Therefore, such short signals are more recognizable in two dimensions, where aside from their change in time, we can observe their diversity in the frequency domain. Spectral analysis is an essential tool applied here, as it enables signal decomposition and the recognition of composite patterns of Ne disturbances that occupy different frequencies. This study shows a high-resolution application of short-term Fourier transform (STFT) to Swarm Ne observations in the Papua New Guinea region in the vicinity of earthquakes, tsunamis, and related general seismic activity. The system of tectonic plate junctions, including the Pacific–Australian boundary, is located orthogonally to Swarm track footprints. The selected wavelengths of seismically induced ionospheric disturbances detected via Swarm are compared with the three sets of three-month records of seismic activity: in the winter solstice of 2016/2017, when seismic activity was highest, and in the summer solstice and vernal equinox of 2016, which were calmer. Moreover, more Swarm data records are analyzed at the same latitudes for validation purposes, in a place where there are no tectonic plate boundaries that are orthogonal to the Swarm orbital footprint. Additional validation is supplied through Swarm Ne observations from completely different latitudes, where the Swarm orbital footprint orthogonally crosses a different subducting plate boundary. Aside from the seismic energy, the solar radio flux (F10.7), equatorial plasma bubbles (EPBs), and geomagnetic ap and Dst indices are also reviewed here. Their influence on the ionospheric Ne is also found in Swarm observations. Finally, the Pearson correlation coefficient (PCC), applied to the pairs of 3-month time series created from Swarm Ne variations, seismic energy, ap, Dst, and F10.7, summarizes the graphical inspection of mutual correlations. It points to the predominant correlation of Swarm Ne disturbances with seismicity, especially during nighttime. We show that most of the Ne disturbances at a selected wavelength of 300 km correlate more with seismicity than with geomagnetic and solar indices. Therefore, Swarm LP can be assessed as being capable of observing the lithosphere–atmosphere–ionosphere coupling (LAIC) from the orbit.

Baby cry recognition based on SLGAN model data generation and deep feature fusion

Deep learning models have been applied in baby cry recognition to enhance the recognition accuracy. However, the current research still suffers from data imbalance problem, which leads to bias in model learning. Sparse Autoencoder Long Short-Term Memory based Generative Adversarial Network (SLGAN) is proposed to solve the data imbalance problem. The proposed SLGAN model generates new baby cry data to ensure the number of samples for every cry class is equal. Speech features are extracted using Mel-spectrograms and Short-Time Fourier Transform (STFT). Two deep learning models, i.e. VGG16 and VGG19 are used to extract the deep features. The deep features are then dimensionally reduced by using Principal Component Analysis (PCA). A sparse autoencoder model is used to fuse the deep features. Finally, the fused features are trained and classified using the Deep Neural Network. The experimental results show that the proposed method outperforms the existing methods.