Search results for: wavelet multi-resolution analysis

Authors: Subha D. Puthankattil, Paul K. Joseph

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Analyzing brain signals of the patients suffering from the state of depression may lead to interesting observations in the signal parameters that is quite different from a normal control. The present study adopts two different methods: Time frequency domain and nonlinear method for the analysis of EEG signals acquired from depression patients and age and sex matched normal controls. The time frequency domain analysis is realized using wavelet entropy and approximate entropy is employed for the nonlinear method of analysis. The ability of the signal processing technique and the nonlinear method in differentiating the physiological aspects of the brain state are revealed using Wavelet entropy and Approximate entropy.

Keywords: EEG, depression, wavelet entropy, approximate entropy, relative wavelet energy, multiresolution decomposition

Procedia PDF Downloads 332

Authors: N. Dahraoui, M. Boulakroune, D. Benatia

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In this paper, the improvement by deconvolution of the depth resolution in Secondary Ion Mass Spectrometry (SIMS) analysis is considered. Indeed, we have developed a new Tikhonov-Miller deconvolution algorithm where a priori model of the solution is included. This is a denoisy and pre-deconvoluted signal obtained from: firstly, by the application of wavelet shrinkage algorithm, secondly by the introduction of the obtained denoisy signal in an iterative deconvolution algorithm. In particular, we have focused the light on the effect of the iterations number on the evolution of the deconvoluted signals. The SIMS profiles are multilayers of Boron in Silicon matrix.

Keywords: DRF, in-depth resolution, multiresolution deconvolution, SIMS, wavelet shrinkage

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Authors: Bin Liu, Weijie Liu, Bin Sun, Yihui Luo

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In order to solve the problem of lower spatial resolution and block effect in the fusion method based on separable wavelet transform in the resulting fusion image, a new sampling mode based on multi-resolution analysis of two-channel non separable wavelet transform, whose dilation matrix is [1,1;1,-1], is presented and a multispectral image fusion method based on this kind of sampling mode is proposed. Filter banks related to this kind of wavelet are constructed, and multiresolution decomposition of the intensity of the MS and panchromatic image are performed in the sampled mode using the constructed filter bank. The low- and high-frequency coefficients are fused by different fusion rules. The experiment results show that this method has good visual effect. The fusion performance has been noted to outperform the IHS fusion method, as well as, the fusion methods based on DWT, IHS-DWT, IHS-Contourlet transform, and IHS-Curvelet transform in preserving both spectral quality and high spatial resolution information. Furthermore, when compared with the fusion method based on nonsubsampled two-channel non separable wavelet, the proposed method has been observed to have higher spatial resolution and good global spectral information.

Keywords: image fusion, two-channel sampled nonseparable wavelets, multispectral image, panchromatic image

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Authors: Mohamed Othmani, Yassine Khlifi

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This paper presents a technique for compact three dimensional (3D) object model reconstruction using wavelet networks. It consists to transform an input surface vertices into signals,and uses wavelet network parameters for signal approximations. To prove this, we use a wavelet network architecture founded on several mother wavelet families. POLYnomials WindOwed with Gaussians (POLYWOG) wavelet families are used to maximize the probability to select the best wavelets which ensure the good generalization of the network. To achieve a better reconstruction, the network is trained several iterations to optimize the wavelet network parameters until the error criterion is small enough. Experimental results will shown that our proposed technique can effectively reconstruct an irregular 3D object models when using the optimized wavelet network parameters. We will prove that an accurateness reconstruction depends on the best choice of the mother wavelets.

Keywords: 3d object, optimization, parametrization, polywog wavelets, reconstruction, wavelet networks

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Authors: Seyed Sadegh Naseralavi, Sadegh Balaghi, Ehsan Khojastehfar

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Time history dynamic analysis of structures is considered as an exact method while being computationally intensive. Filtration of earthquake strong ground motions applying wavelet transform is an approach towards reduction of computational efforts, particularly in optimization of structures against seismic effects. Wavelet transforms are categorized into continuum and discrete transforms. Since earthquake strong ground motion is a discrete function, the discrete wavelet transform is applied in the present paper. Wavelet transform reduces analysis time by filtration of non-effective frequencies of strong ground motion. Filtration process may be repeated several times while the approximation induces more errors. In this paper, strong ground motion of earthquake has been filtered once applying each wavelet. Strong ground motion of Northridge earthquake is filtered applying various wavelets and dynamic analysis of sampled shear and moment frames is implemented. The error, regarding application of each wavelet, is computed based on comparison of dynamic response of sampled structures with exact responses. Exact responses are computed by dynamic analysis of structures applying non-filtered strong ground motion.

Keywords: wavelet transform, computational error, computational duration, strong ground motion data

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Authors: Geeta Kasana, Kulbir Singh, Satvinder Singh

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In this paper, a steganographic scheme for digital images using Integer Wavelet Transform (IWT) is proposed. The cover image is decomposed into wavelet sub bands using IWT. Each of the subband is divided into blocks of equal size and secret data is embedded into the largest and smallest pixel values of each block of the subband. Visual quality of stego images is acceptable as PSNR between cover image and stego is above 40 dB, imperceptibility is maintained. Experimental results show better tradeoff between capacity and visual perceptivity compared to the existing algorithms. Maximum possible error analysis is evaluated for each of the wavelet subbands of an image.

Keywords: DWT, IWT, MSE, PSNR

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Authors: Amanpreet Kaur

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Wavelet transform (WT) is a powerful statistical tool used in applied mathematics for signal and image processing. The different mother, wavelet basis function, has been compared to select the optimal wavelet function that represents the electromyogram signal characteristics of upper limb amputees. Four different EMG electrode has placed on different location of shoulder muscles. Twenty one wavelet functions from different wavelet families were investigated. These functions included Daubechies (db1-db10), Symlets (sym1-sym5), Coiflets (coif1-coif5) and Discrete Meyer. Using mean square error value, the significance of the mother wavelet functions has been determined for teres, pectorals, and infraspinatus around shoulder muscles. The results show that the best mother wavelet is the db3 from the Daubechies family for efficient classification of the signal.

Keywords: Daubechies, upper limb amputation, shoulder muscles, Symlets, Coiflets

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Authors: Goutham Kumar Dogiparti, D. R. Seshu

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In the present study, wavelet analysis was used for locating damage in simply supported and cantilever beams. Study was carried out varying different levels and locations of damage. In numerical method, ANSYS software was used for modal analysis of damaged and undamaged beams. The mode shapes obtained from numerical analysis is processed using MATLAB wavelet toolbox to locate damage. Effect of several parameters such as (damage level, location) on the natural frequencies and mode shapes were also studied. The results indicated the potential of wavelets in identifying the damage location.

Keywords: damage, detection, beams, wavelets

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Authors: A. Ghanbari Mardasi, N. Wu, C. Wu

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In this study, a spatial wavelet-based crack localization technique for a thick beam is presented. Wavelet scale in spatial wavelet transformation is optimized to enhance crack detection sensitivity. A windowing function is also employed to erase the edge effect of the wavelet transformation, which enables the method to detect and localize cracks near the beam/measurement boundaries. Theoretical model and vibration analysis considering the crack effect are first proposed and performed in MATLAB based on the Timoshenko beam model. Gabor wavelet family is applied to the beam vibration mode shapes derived from the theoretical beam model to magnify the crack effect so as to locate the crack. Relative wavelet coefficient is obtained for sensitivity analysis by comparing the coefficient values at different positions of the beam with the lowest value in the intact area of the beam. Afterward, the optimal wavelet scale corresponding to the highest relative wavelet coefficient at the crack position is obtained for each vibration mode, through numerical simulations. The same procedure is performed for cracks with different sizes and positions in order to find the optimal scale range for the Gabor wavelet family. Finally, Hanning window is applied to different vibration mode shapes in order to overcome the edge effect problem of wavelet transformation and its effect on the localization of crack close to the measurement boundaries. Comparison of the wavelet coefficients distribution of windowed and initial mode shapes demonstrates that window function eases the identification of the cracks close to the boundaries.

Keywords: edge effect, scale optimization, small crack locating, spatial wavelet

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Authors: Reza Rezaeipour Honarmandzad

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Wavelet analysis is an exciting method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, signal processing, image processing, pattern recognition, etc. Wavelets allow complex information such as signals, images and patterns to be decomposed into elementary forms at different positions and scales and subsequently reconstructed with high precision. In this paper a wavelet-based signal processing algorithm for airplane cable fault location is proposed. An orthogonal discrete wavelet decomposition and reconstruction algorithm is used to eliminate the noise in the aircraft cable fault signal. The experiment result has shown that the character of emission pulse and reflect pulse used to test the aircraft cable fault point are reserved and the high-frequency noise are eliminated by means of the proposed algorithm in this paper.

Keywords: wavelet analysis, signal processing, orthogonal discrete wavelet, noise, aircraft cable fault signal

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Authors: B. X. Tchomeni, A. A. Alugongo, T. B. Tengen

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In this paper, de Laval rotor system has been characterized by a hinge model and its transient response numerically treated for a dynamic solution. The effect of the ensuing non-linear disturbances namely rub and breathing crack is numerically simulated. Subsequently, three analysis methods: Orbit Analysis, Fast Fourier Transform (FFT) and Wavelet Transform (WT) are employed to extract features of the vibration signal of the faulty system. An analysis of the system response orbits clearly indicates the perturbations due to the rotor-to-stator contact. The sensitivities of WT to the variation in system speed have been investigated by Continuous Wavelet Transform (CWT). The analysis reveals that features of crack, rubs and unbalance in vibration response can be useful for condition monitoring. WT reveals its ability to detect non-linear signal, and obtained results provide a useful tool method for detecting machinery faults.

Keywords: Continuous wavelet, crack, discrete wavelet, high acceleration, low acceleration, nonlinear, rotor-stator, rub

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Authors: Saad Bakkali

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Morocco is a major producer of phosphate, with an annual output of 19 million tons and reserves in excess of 35 billion cubic meters. This represents more than 75% of world reserves. Resistivity surveys have been successfully used in the Oulad Abdoun phosphate basin. A Schlumberger resistivity survey over an area of 50 hectares was carried out. A new field procedure based on analytic signal response of resistivity data was tested to deal with the presence of phosphate deposit disturbances. A resistivity map was expected to allow the electrical resistivity signal to be imaged in 2D. 2D wavelet is standard tool in the interpretation of geophysical potential field data. Wavelet transform is particularly suitable in denoising, filtering and analyzing geophysical data singularities. Wavelet transform tools are applied to analysis of a moroccan phosphate deposit ‘disturbances’. Wavelet approach applied to modeling surface phosphate “disturbances” was found to be consistently useful.

Keywords: resistivity, Schlumberger, phosphate, wavelet, Morocco

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Authors: Nor Asrina Binti Ramlee

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Voltage sag, voltage swell, high-frequency noise and voltage transients are kinds of disturbances in power quality. They are also known as power quality events. Equipment used in the industry nowadays has become more sensitive to these events with the increasing complexity of equipment. This leads to the importance of distributing clean power quality to the consumer. To provide better service, the best analysis on power quality is very vital. Thus, this paper presents the events detection focusing on voltage sag and swell. The method is developed by applying time domain signal analysis using wavelet transform approach in MATLAB. Four types of mother wavelet namely Haar, Dmey, Daubechies, and Symlet are used to detect the events. This project analyzed real interrupted signal obtained from 22 kV transmission line in Skudai, Johor Bahru, Malaysia. The signals will be decomposed through the wavelet mothers. The best mother is the one that is capable to detect the time location of the event accurately.

Keywords: power quality, voltage sag, voltage swell, wavelet transform

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Authors: T. Bensana, S. Mekhilef

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The methodology of vibration based condition monitoring technology has been developing at a rapid stage in the recent years suiting to the maintenance of sophisticated and complicated machines. The ability of wavelet analysis to efficiently detect non-stationary, non-periodic, transient features of the vibration signal makes it a demanding tool for condition monitoring. This paper presents a methodology for fault diagnosis of rolling element bearings based on wavelet envelope power spectrum technique is analysed in both the time and frequency domains. In the time domain the auto-correlation of the wavelet de-noised signal is applied to evaluate the period of the fault pulses. However, in the frequency domain the wavelet envelope power spectrum has been used to identify the fault frequencies with the single sided complex Laplace wavelet as the mother wavelet function. Results show the superiority of the proposed method and its effectiveness in extracting fault features from the raw vibration signal.

Keywords: preventive maintenance, fault diagnostics, rolling element bearings, wavelet de-noising

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Authors: Tri Wijayanti Septiarini, Agus Maman Abadi, Muhammad Rifki Taufik

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The exchange rate of IDR/USD can be the indicator to analysis Indonesian economy. The exchange rate as a important factor because it has big effect in Indonesian economy overall. So, it needs the analysis data of exchange rate. There is decomposition data of exchange rate of IDR/USD to be frequency and time. It can help the government to monitor the Indonesian economy. This method is very effective to identify the case, have high accurate result and have simple structure. In this paper, data of exchange rate that used is weekly data from December 17, 2010 until November 11, 2014.

Keywords: the exchange rate, fuzzy mamdani, discrete wavelet transforms, fuzzy wavelet

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Authors: Mahacine Amrani

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This paper aims at reviewing some Moroccan industrial applications of wavelet especially in the dynamic identification of a process model using Haar wavelet mother response. Two recent Moroccan study cases are described using dynamic data originated by a distillation column and an industrial polyethylene process plant. The purpose of the wavelet scheme is to build on-line dynamic models. In both case studies, a comparison is carried out between the Haar wavelet mother response model and a linear difference equation model. Finally it concludes, on the base of the comparison of the process performances and the best responses, which may be useful to create an estimated on-line internal model control and its application towards model-predictive controllers (MPC). All calculations were implemented using AutoSignal Software.

Keywords: process performance, model, wavelets, Haar, Moroccan

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Authors: Syed Jawad Hussain Shahzad, Muhammad Zakaria, Mobeen Ur Rehman, Saniya Khaild

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This study examines the co-movement between the Pakistan, Indian, S&P 500 and Nikkei 225 stock markets using weekly data from 1998 to 2013. The time-frequency relationship between the selected stock markets is conducted by using measures of continuous wavelet power spectrum, cross-wavelet transform and cross (squared) wavelet coherency. The empirical evidence suggests strong dependence between Pakistan and Indian stock markets. The co-movement of Pakistani index with U.S and Japanese, the developed markets, varies over time and frequency where the long-run relationship is dominant. The results of cross wavelet and wavelet coherence analysis indicate moderate covariance and correlation between stock indexes and the markets are in phase (i.e. cyclical in nature) over varying durations. Pakistan stock market was lagging during the entire period in relation to Indian stock market, corresponding to the 8~32 and then 64~256 weeks scale. Similar findings are evident for S&P 500 and Nikkei 225 indexes, however, the relationship occurs during the later period of study. All three wavelet indicators suggest strong evidence of higher co-movement during 2008-09 global financial crises. The empirical analysis reveals a strong evidence that the portfolio diversification benefits vary across frequencies and time. This analysis is unique and have several practical implications for regional and international investors while assigning the optimal weightage of different assets in portfolio formulation.

Keywords: co-movement, Pakistan stock exchange, S&P 500, Nikkei 225, wavelet analysis

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Authors: Somkait Udomhunsakul

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In this paper, a multi-focus image fusion method using Spatial Frequency Measurements (SFM) and Wavelet Packet was proposed. The proposed fusion approach, firstly, the two fused images were transformed and decomposed into sixteen subbands using Wavelet packet. Next, each subband was partitioned into sub-blocks and each block was identified the clearer regions by using the Spatial Frequency Measurement (SFM). Finally, the recovered fused image was reconstructed by performing the Inverse Wavelet Transform. From the experimental results, it was found that the proposed method outperformed the traditional SFM based methods in terms of objective and subjective assessments.

Keywords: multi-focus image fusion, wavelet packet, spatial frequency measurement

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Authors: Prabhakar Sathujoda

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Vibration analysis of a misaligned rotor coupling bearing system has been carried out while decelerating through its critical speed. The finite element method (FEM) is used to model the rotor system and simulate flexural vibrations. A flexible coupling with a frictionless joint is considered in the present work. The continuous wavelet transform is used to extract the misalignment features from the simulated time response. Subcritical speeds at one-half, one-third, and one-fourth the critical speed have appeared in the wavelet transformed vibration response of a misaligned rotor coupling bearing system. These features are also verified through a parametric study.

Keywords: Continuous Wavelet Transform, Flexible Coupling, Rotor System, Sub Critical Speed

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Authors: C. S. Seema, P. R. Prince

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The characterization of spatio-temporal inhomogeneities occurring in the ionospheric F₂ layer is remarkable since these variations are direct consequences of electrodynamical coupling between magnetosphere and solar events. The temporal and spatial variations of the F₂ layer, which occur with a period of several days or even years, mainly owe to geomagnetic and meteorological activities. The hourly F₂ layer critical frequency (foF2) over 23rd solar cycle (1996-2008) of three ionosonde stations (Wakkanai, Kokunbunji, and Okinawa) in northern hemisphere, which falls within same longitudinal span, is analyzed using continuous wavelet techniques. Morlet wavelet is used to transform continuous time series data of foF2 to a two dimensional time-frequency space, quantifying the time evolution of the oscillatory modes. The presence of significant time patterns (periodicities) at a particular time period and the time location of each periodicity are detected from the two-dimensional representation of the wavelet power, in the plane of scale and period of the time series. The mean strength of each periodicity over the entire period of analysis is studied using global wavelet spectrum. The quasi biennial, annual, semiannual, 27 day, diurnal and 12 hour variations of foF2 are clearly evident in the wavelet power spectra in all the three stations. Critical frequency oscillations with multi-day periods (2-3 days and 9 days in the low latitude station, 6-7 days in all stations and 15 days in mid-high latitude station) are also superimposed over large time scaled variations.

Keywords: continuous wavelet analysis, critical frequency, ionosphere, solar cycle

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Authors: Ajish Sreedharan

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With the fast evolution of digital data exchange, security information becomes much important in data storage and transmission. Due to the increasing use of images in industrial process, it is essential to protect the confidential image data from unauthorized access. As encryption process is applied to the whole image in AES ,it is difficult to improve the efficiency. In this paper, wavelet decomposition is used to concentrate the main information of image to the low frequency part. Then, AES encryption is applied to the low frequency part. The high frequency parts are XORed with the encrypted low frequency part and a wavelet reconstruction is applied. Theoretical analysis and experimental results show that the proposed algorithm has high efficiency, and satisfied security suits for image data transmission.

Keywords: discrete wavelet transforms, AES, dynamic SBox

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Authors: Reza Bazargan lari, Mohammad H. Fattahi

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Natural resources management including water resources requires reliable estimations of time variant environmental parameters. Small improvements in the estimation of environmental parameters would result in grate effects on managing decisions. Noise reduction using wavelet techniques is an effective approach for pre-processing of practical data sets. Predictability enhancement of the river flow time series are assessed using fractal approaches before and after applying wavelet based pre-processing. Time series correlation and persistency, the minimum sufficient length for training the predicting model and the maximum valid length of predictions were also investigated through a fractal assessment.

Keywords: wavelet, de-noising, predictability, time series fractal analysis, valid length, ANN

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Authors: Alsaidi M. Altaher, Mohd Tahir Ismail

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Wavelet thresholding has been a power tool in curve estimation and data analysis. In the presence of outliers this non parametric estimator can not suppress the outliers involved. This study proposes a new two-stage combined method based on the use of the median filter as primary step before applying wavelet thresholding. After suppressing the outliers in a signal through the median filter, the classical wavelet thresholding is then applied for removing the remaining noise. We use automatic boundary corrections; using a low order polynomial model or local polynomial model as a more realistic rule to correct the bias at the boundary region; instead of using the classical assumptions such periodic or symmetric. A simulation experiment has been conducted to evaluate the numerical performance of the proposed method. Results show strong evidences that the proposed method is extremely effective in terms of correcting the boundary bias and eliminating outlier’s sensitivity.

Keywords: boundary correction, median filter, simulation, wavelet thresholding

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Authors: Danladi Ali, Abdullahi Mukaila

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In this work, GSM signal strength (power) was monitored in an indoor environment. Samples of the GSM signal strength was measured on mobile equipment (ME). One-dimensional multilevel wavelet is used to predict the fading phenomenon of the GSM signal measured and neural network clustering to determine the average power received in the study area. The wavelet prediction revealed that the GSM signal is attenuated due to the fast fading phenomenon which fades about 7 times faster than the radio wavelength while the neural network clustering determined that -75dBm appeared more frequently followed by -85dBm. The work revealed that significant part of the signal measured is dominated by weak signal and the signal followed more of Rayleigh than Gaussian distribution. This confirmed the wavelet prediction.

Keywords: decomposition, clustering, propagation, model, wavelet, signal strength and spectral efficiency

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Authors: Alla Levina, Sergey Taranov

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The classical approach to the providing noise immunity and integrity of information that process in computing devices and communication channels is to use linear codes. Linear codes have fast and efficient algorithms of encoding and decoding information, but this codes concentrate their detect and correct abilities in certain error configurations. To protect against any configuration of errors at predetermined probability can robust codes. This is accomplished by the use of perfect nonlinear and almost perfect nonlinear functions to calculate the code redundancy. The paper presents the error-correcting coding scheme using biorthogonal wavelet transform. Wavelet transform applied in various fields of science. Some of the wavelet applications are cleaning of signal from noise, data compression, spectral analysis of the signal components. The article suggests methods for constructing linear codes based on wavelet decomposition. For developed constructions we build generator and check matrix that contain the scaling function coefficients of wavelet. Based on linear wavelet codes we develop robust codes that provide uniform protection against all errors. In article we propose two constructions of robust code. The first class of robust code is based on multiplicative inverse in finite field. In the second robust code construction the redundancy part is a cube of information part. Also, this paper investigates the characteristics of proposed robust and linear codes.

Keywords: robust code, linear code, wavelet decomposition, scaling function, error masking probability

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Authors: A. Benyahia, M. Zergoug, M. Amir, M. Fodil

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The main objective of this work is to enhance the Pulsed Eddy Current (PEC) response from the aluminum structure using signal processing. Cracks and metal loss in different structures cause changes in PEC response measurements. In this paper, time-frequency analysis is used to represent PEC response, which generates a large quantity of data and reduce the noise due to measurement. Power Spectral Density (PSD) after Wavelet Decomposition (PSD-WD) is proposed for defect detection. The experimental results demonstrate that the cracks in the surface can be extracted satisfactorily by the proposed methods. The validity of the proposed method is discussed.

Keywords: DT, pulsed eddy current, continuous wavelet transform, Mexican hat wavelet mother, defect detection, power spectral density.

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Authors: Dorota Witkowska, Paweł Gosek, Lukasz Swiecicki, Wojciech Jernajczyk, Bruce J. West, Miroslaw Latka

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In clinical practice, the selection of an antidepressant often degrades to lengthy trial-and-error. In this work we employ a normalized wavelet power of alpha waves as a biomarker of antidepressant treatment response. This novel EEG metric takes into account both non-stationarity and intersubject variability of alpha waves. We recorded resting, 19-channel EEG (closed eyes) in 22 inpatients suffering from unipolar (UD, n=10) or bipolar (BD, n=12) depression. The EEG measurement was done at the end of the short washout period which followed previously unsuccessful pharmacotherapy. The normalized alpha wavelet power of 11 responders was markedly different than that of 11 nonresponders at several, mostly temporoparietal sites. Using the prediction of treatment response based on the normalized alpha wavelet power, we achieved 81.8% sensitivity and 81.8% specificity for channel T4.

Keywords: alpha waves, antidepressant, treatment outcome, wavelet

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Authors: Chun-Wei Huang, Jyh-Cherng Gu, Ming-Ta Yang

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Non-synchronous breakage or line failure in power systems with light or no loads can lead to core saturation in transformers or potential transformers. This can cause component and capacitance matching resulting in the formation of resonant circuits, which trigger ferroresonance. This study employed a wavelet transform for the detection of ferroresonance. Simulation results demonstrate the efficacy of the proposed method.

Keywords: ferroresonance, wavelet transform, intelligent electronic device, transformer

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Authors: Ritu Rani

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In this paper, we apply minimalistically upheld linear semi-orthogonal B-spline wavelets, exceptionally developed for the limited interim to rough the obscure function present in the integral equations. Semi-orthogonal wavelets utilizing B-spline uniquely developed for the limited interim and these wavelets can be spoken to in a shut frame. This gives a minimized help. Semi-orthogonal wavelets frame the premise in the space L²(R). Utilizing this premise, an arbitrary function in L²(R) can be communicated as the wavelet arrangement. For the limited interim, the wavelet arrangement cannot be totally introduced by utilizing this premise. This is on the grounds that backings of some premise are truncated at the left or right end purposes of the interim. Subsequently, an uncommon premise must be brought into the wavelet development on the limited interim. These functions are alluded to as the limit scaling functions and limit wavelet functions. B-spline wavelet method has been connected to fathom linear and nonlinear integral equations and their systems. The above method diminishes the integral equations to systems of algebraic equations and afterward these systems can be illuminated by any standard numerical methods. Here, we have connected Newton's method with suitable starting speculation for solving these systems.

Keywords: semi-orthogonal, wavelet arrangement, integral equations, wavelet development

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Authors: Bülent Kantar, Numan Ünaldı

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This paper presents a new method which includes robust and invisible digital watermarking on images that is colored. Colored images are used as watermark. Frequency region is used for digital watermarking. Discrete wavelet transform and discrete stationary wavelet transform are used for frequency region transformation. Low, medium and high frequency coefficients are obtained by applying the two-level discrete wavelet transform to the original image. Low frequency coefficients are obtained by applying one level discrete stationary wavelet transform separately to all frequency coefficient of the two-level discrete wavelet transformation of the original image. For every low frequency coefficient obtained from one level discrete stationary wavelet transformation, watermarks are added. Watermarks are added to all frequency coefficients of two-level discrete wavelet transform. Totally, four watermarks are added to original image. In order to get back the watermark, the original and watermarked images are applied with two-level discrete wavelet transform and one level discrete stationary wavelet transform. The watermark is obtained from difference of the discrete stationary wavelet transform of the low frequency coefficients. A total of four watermarks are obtained from all frequency of two-level discrete wavelet transform. Obtained watermark results are compared with real watermark results, and a similarity result is obtained. A watermark is obtained from the highest similarity values. Proposed methods of watermarking are tested against attacks of the geometric and image processing. The results show that proposed watermarking method is robust and invisible. All features of frequencies of two level discrete wavelet transform watermarking are combined to get back the watermark from the watermarked image. Watermarks have been added to the image by converting the binary image. These operations provide us with better results in getting back the watermark from watermarked image by attacking of the geometric and image processing.

Keywords: watermarking, DWT, DSWT, copy right protection, RGB

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