Autoregressive modeling approach of vibration data for bearing fault diagnosis in electric motors

This study investigates the performance of autoregressive (AR) modeling method to detect bearing fault in induction motor. For this purpose, AR models of vibration signals which are acquired during the load performance tests of healthy and seven gradually aged cases of induction motor are constructed. The variation of AR coefficients with model order is compared for all cases of the motor from healthy to faulty. It is seen that sixtieth order model is adequate to reflect the progress of fault characteristics and the first AR coefficient gets bigger with aging. AR modeling error or residuals computed as the difference between original signals and their AR representation gets large in time domain which corresponds to decrease in modeling performance with aging. In addition the error computed as the absolute difference between spectra of the original signals and their AR models gets large in frequency domain and preserves bearing fault features as the energy increase in high frequencies above 1.5 kHz

Detection and identification of mechanical faults by Kalman filtering in electric machines

Kalman filtering technique is applied on vibration signals to detect eccentricity fault and bearing fault in induction motor. The vibration signals are acquired during load performance tests of healthy and faulty cases of induction motor. The state space equations needed for Kalman filter structure are constructed based on autoregressive model of the signals. The harmonics of fault related frequencies are identified from the spectrum of innovation signal, which is provided from the Kalman filter response, for vibration signal and also it is seen that Kalman filter reduce the noise level

Bearing fault diagnosis in electric motors by advanced signal processing techniques

Bu çalışmada asenkron motorlarda öngörülü bakım teknolojisine dayalı, rulman arıza gelişimi üzerine bir inceleme gerçekleştirilmiştir. Rulman arızası, hızlandırılmış eskitme süreçleri ile yapay olarak oluşturul-muştur. Motordan alınan titreşim ve akım işaretlerinin istatistiksel ve dalgacık analizi sonucu rulman arıza-sına ilişkin özellik çıkarımı yapılmıştır. Bu işaretlerin bir arada değerlendirilmesi ile koherens fonksiyonu tanımlanarak rotor eksenel bozukluğunu gösteren en baskın frekans değeri belirlenmiştir. Bu anlamda işa-retler arasındaki ilişki, bir yapay sinir ağına öğretilerek (YSA), arıza frekanslarının YSA ile belirlenebildiği gösterilmiştir. Genetik algoritma yaklaşımı da YSA çıkışlarındaki maksimum hatayı bulmak için kullanılmış-tır. Böylece YSA ve GA tabanlı hibrit bir yapay zeka izleme sistemi oluşturulmuştur. Anahtar Kelimeler: Asenkron motor, dalgacık, titreşim, yapay sinir ağları, arıza tanısı, öngörülü bakım.This paper presents to detect the motor bearing failures for predictive maintenance purpose using the accelerated aging studies based on bearing flutings in electric motors. In terms of the bearing damage, feature extraction from vibration signals in electric motors was realized using the wavelet analysis. And an early detection criterion was developed by means of the statistical analysis of the motor vibration signals. Also, considering both of the motor current and vibration signals, correlation between these signals is shown by the coherence function, which is defined in frequency domain. Hence, most dominant frequencies related to rotor eccentricity caused by the bearing failures can be easily determined. These computed coherence functions defined for each aging cycle form initial case to final case are used as training and recall data set for an Artificial Neural Network (ANN). With this way, it is shown that ANN provides a very good performance in terms of the motor failure detection. Also, standard Genetic Algorithm (GA) approach is used to find the maximum amplitude, which indicates the faulty case of the bearings, by error signals at output nodes of the ANN. Hence, a hybrid Artificial Intelligence technique, which is based on ANN and GA methodologies, is implemented for motor condition monitoring studies. Keywords: Induction motors, wavelet, vibration, artificial neural networks, fault detection, predictive maintenance

Prenatalna identifikacija izolirane aberantne potključne arterije: je li potrebna daljnja genetska obrada?

The objective of this study was to evaluate the association between aberrant right subclavian artery (ARSA) and chromosomal abnormalities. The study included 5211 women having attended our unit for fetal anatomic screening and fetal echocardiography from August 2016 until February 2019. After diagnosing ARSA, prenatal invasive testing was discussed with the patients. ARSA affected fetus was determined in 57 cases; of these, there were 38 cases of isolated ARSA and 19 cases of non-isolated ARSA but associated with soft markers and fetal anomalies. Nineteen patients underwent amniocentesis; Down syndrome was determined in two women, both of them from the non-isolated ARSA group, with fetal hydrops, atrioventricular septal defect and esophageal atresia. Fifteen of 38 patients who declined prenatal diagnostic testing, accepted karyotype analysis after delivery and none of these 15 cases had chromosomal abnormalities. Identification of ARSA should be followed by detailed ultrasound examination to ensure that there are no accompanying soft markers and/or structural defects. Isolated ARSA may not be an indication for karyotype analysis or 22q11.2 microdeletions. Non-ARSA implies a strong predictor of aneuploidy, and when additional findings are detected, invasive testing should be offered to the parents. The association between isolated ARSA and genetic disease should be evaluated in large powered prospective studies.Cilj ovoga istraživanja bio je procijeniti udruženost aberantne desne potključne arterije (ADPA) i poremećaja kromosoma. Istraživanje je obuhvatilo 5211 žena koje su posjetile našu jedinicu za anatomski probir i fetalnu ehokardiografiju od kolovoza 2016. do veljače 2019. godine. Nakon što je dijagnosticirana ADPA sa ženama se razgovaralo o prenatalnom invazivnom testiranju. Fetalna ADPA utvrđena je u 57 slučajeva, uključujući 38 slučajeva izolirane ADPA i 19 slučajeva ne-izolirane ADPA, ali udružene s „mekim“ biljezima i fetalnim anomalijama. Devetnaest žena podvrgnuto je amniocentezi. Downov sindrom utvrđen je kod dvije žene, obje iz skupine s ne-izoliranom ADPA, s fetalnim hidropsom, atrioventrikulskim septalnim defektom i atrezijom jednjaka. Petnaest od 38 žena koje su odbile prenatalno dijagnostičko testiranje prihvatilo je analizu kariotipa nakon porođaja i nijedna od njih nije imala kromosomne poremećaje. Nakon identificiranja ADPA treba uslijediti podroban ultrazvučni pregled kako bismo bili sigurni da ne postoje prateći „meki“ biljezi i/ili strukturni defekti. Izolirana ADPA ne mora biti indikacija za analizu kariotipa ili mikrodelecije 22q11.2. Ne-ADPA snažno upućuje na aneuploidiju, a kad se dobiju dodatni nalazi tada treba roditeljima ponuditi invazivno testiranje. Udruženost izolirane ADPA i genetske bolesti treba procijeniti u velikim i valjanim prospektivnim studijama

A case of Roberts syndrome: its ultrasonographic characteristics and genetic diagnosis

Objective: Roberts syndrome is a very rare genetic disease, and it has an autosomal recessive inheritance pattern. It develops as a result of the mutation in ESCO2 gene located in the 8th chromosome. In our study, we aimed to present a case which was found to have Roberts syndrome coexisting with multiple anomalies, particularly skeletal system anomaly, in the 17 weeks of gestation. Case(s): In the fetal ultrasonographic evaluation performed on the pregnant women who referred to our hospital for routine gestational examination in the 17 weeks of gestation, anomalies in the bilateral upper and lower extremities, contracted legs, bilateral cleft palate and lip, intrauterine growth restriction and cardiac anomaly were found in the fetus. Roberts syndrome was considered first with these ultrasonographic findings. The diagnosis of Roberts syndrome was confirmed by cytogenetic and molecular analyses. Early segregation of centromeres and early breaking up of heterochromatic regions near centromeres were found at metaphase stage. By cytogenetic and molecular analyses, homozygous mutation in ESCO2 gene of the fetus and heterozygous mutation in the parents were found. The termination of pregnancy was decided after the genetic consultation with the parents. Physical examination findings and prenatal ultrasound findings after termination were found similar. Conclusion: Many severe skeletal dysplasia cases can be diagnosed ultrasonographically before 20 weeks of gestation. Early diagnosis ensures to take necessary actions for medical support during postnatal period and in terms of labor if pregnancy continues as well as genetic consultation opportunity. If the genetic disease that causes skeletal dysplasia can be identified and parents are found to have this gene, healthy pregnancies can be achieved by obtaining normal embryos via pre-implantation genetic diagnosis in order to prevent the relapse of the disease

Fault detection based on coherence analysis for stator insulation in induction motors

Asenkron motorlar bütün endüstriyel süreçler içinde en sık kullanılan elemanlardır. Endüstriyel sistemlerin ekonomik ve daha güvenli çalışabilmeleri için arızaların erken tespit edilmesi ve izole edilmesi çok önemlidir. Bu nedenle, endüstrinin birçok alanında kritik sistemlerde öngörülü bakım amaçlı olarak durum izleme çalışmaları yapılmaktadır. Durum izleme sayesinde elde edilen verilerin kullanılması ile aniden ve plansız olarak meydana gelen kesintiler önlenebilmektedir. Bu çalışmada, asenkron motorlarda stator sargısı yalıtım arızası göz önünde tutulmuş, stator yalıtımında arıza özellikleri suni olarak yaşlandırma işlemi uygulanarak oluşturulmuş ve veriler spektral analiz yöntemi ile analiz edilmiştir. Stator yalıtım arızasının temel spektral özellikleri 5 BG gücünde bir  asenkron  motora hızlandırılmış yaşlandırma uygulanması ile elde edilmiştir. Hedefe ulaşmak üzere, tek başına akım ve titreşim işaretlerine ait verilerle yapılan spektral analizlerle sargı yalıtım arıza tanısına ait belirgin sonuçlar elde edilememiştir. Stator akımı ve titreşim işaretlerinin arasındaki koherens yaklaşımı ile ise çift harmonik değerlerinde kuvvetlenmeler olduğu belirlenmiş ve bunların stator yalıtımı arızasının göstergesi olarak kullanılabileceği belirlenmiştir. Bir motora ait akım ve titreşim verileri ile elde edilen çıkarımlar, aynı deneye tabi tutulan iki eşdeğer motora ait verilerle yapılan analizlerle de doğrulanmıştır. Moment bilgisinin stator akımı ve dolayısıyla magneto-motor kuvvet (mmk)  ile olan ilişkisine dayalı olarak titreşim işaretlerine yansıması dolayısıyla, akım-titreşim ilişkisinin daha belirleyici olacağı kararına varılmıştır. Bu çalışma, bu uygulamada 4. ve 16. harmonikler arasındaki çift harmoniklerin yalıtım arızasının en önemli imzası olduğunu göstermiştir. Anahtar Kelimeler: Asenkron motorlar, yaşlandırma işlemi, koherens, çift harmonikler, stator yalıtım arızası.Induction motors are the most frequently used components in almost every type of industrial processes. Early detection and isolation of faults that cause catastrophic system failure are crucial for more reliable and economic operation of industrial processes. For this reason, condition monitoring studies are performed for predictive maintenance purposes of the critical systems in most areas of the industry. Hence, sudden and unscheduled downtimes can be prevented by performing correct and timely maintenance activities using the information provided by condition monitoring studies. Through the statistical studies on induction motor failure distribution, it is reported that 40 % of all induction motor failures are caused by stator related faults such as insulation faults (Bonnett and Soukup, 1992). In the last two decades, the wide use of the motor drives based on the power electronics devices, caused current harmonics to occur and resulted in stator and bearing related problems. Hence, the use of motor drives creates some undesirable effects with the eventual failure of motors. In the related literature, there are several studies based on the stator degradation using the signal analysis methods for rotating machineries in recent years. Some of these investigators have reported the role of pulse-width modulated (PWM) devices on the stator winding insulation. They also indicate the effect of the even harmonics in the stator current of induction machines by using the spectral analysis approaches (Kueck, 2002; Filippetti, 2000). In this research, stator-winding insulation damage in induction motors is considered and fault features developed during the artificial aging process of the stator insulation are extracted from the collected data by the spectral analysis methods. Even harmonic effects are determined as a common feature between the phase currents and vibration signals using the coherence analysis and this feature indicates the stator insulation fault. This research is aimed to extract certain spectral features of the stator insulation damage based on an experimental study in the laboratory environment. This was determined by a signal processing application using the collected data. In order to get a database for induction motor failures from the experimental study, 5-HP, three phases, four poles induction motors are artificially aged by electrical methods. Several aging processes are performed. After each cycle of accelerated aging, the test motor was put on a motor performance test platform. From the experimental setup, high frequency data with a sampling frequency of 12 kHz were acquired and include 3-phase motor currents and voltages, rotor speed, torque, and six vibration measurements. The spectral analysis on the current signals is not so much informative. As an alternative approach, to determine the insulation failure, the cross approach which is defined between the current and vibration signals, is used. This new approach is the calculation of coherence function between the motor current and vibration signals regarding to the initial and aged cases. Hence the following functional relationship is calculated and it shows the effective harmonics with the huge amplitudes to extract the failure information related to motor insulation. Through the coherence analysis between the current signal and torque measurements for the initial and aged cases, the existence of all the harmonic components like the even and odd harmonics of the fundamental frequency at 60 Hz was clearly observed. Thereupon, the coherence application was used to reveal the relationship between motor current and vibration and then, as a result of this application, some special frequency values represented by the even harmonics of the line frequency at 60 Hz are determined at 240, 360, 480, 600, 720, 840 and 960 Hz. These even harmonics can be interpreted as the characteristic property of the insulation aging. These harmonics are present as a result of the stator current imbalances based upon the stator insulation damage occurring after the aging processes. These current imbalances act on the stator magneto motor force (mmf) and cause motor vibrations. Hence high correlated amplitudes in coherence variations as appeared in the figure 7(a) can be interpreted as indicators of the stator insulation damage. Results obtained by using time information of one of three motors have been analysed for remaining motors and similar results have also obtained for them. At the final part of the study, extracted failure features have been monitored by Artificial Neural Networks (ANN) method. Keywords: Induction motors, aging process, coherence, even harmonics, stator insulation faults

Musculoskeletal Complaints: When Should We Consult a Pediatric Rheumatologist?

This review aimed to summarize key points that might suggest rheumatologic diseases to physicians dealing with musculoskeletal  (MSK) complaints. Evaluation of a child presenting with MSK findings requires a comprehensive, multidisciplinary, and systematic approach. In children with MSK complaints, detailed anamnesis, appropriate physical examination and joint examination, and the use of correct laboratory tests will be helpful for accurate diagnosis. The algorithm we have suggested for MSK complaints of children will be a guide for the physicians

FARKLI NANE (MENTHA X PİPERİTA L. VE MENTHA PULEGİUM L.) TÜRLERİNİN DOĞAL VE DOKU KÜLTÜRÜ YARDIMIYLA GELİŞTİRİLİP SEKONDER METABOLİT VE UÇUCU BİLEŞENLERİNİN TESPİTİ

Bu çalışmada, nanenin iki türünün (Mentha x piperita L.,Mentha pulegium L.) mikroçoğaltım yöntemiyle ve doğal ortamda geliştirilip uçucu bileşenlerinin ve sekonder metabolitlerinin tespit edilerek karşılaştırılması amaçlanmıştır. Bu bağlamda, her iki bitki mikroçoğaltım yöntemi ve doğal ortamda yetiştirilerek uçucu bileşenleri araştırılmıştır. Aynı zamanda mikroçoğaltım yöntemiyle geliştirilen bitkiler morfolojik ve fizyolojik olarak incelenmiştir. Ek olarak her iki türde de farklı hormon konsantrasyonları kullanılarak kallus gelişimleri değerlendirilmiştir. Bitkilerin yaprak numunelerindeki uçucu bileşenleri Tepe Boşluğu-Katı Faz Mikro Ekstraksiyonu (HS-SPME) tekniği ile ekstraksiyon işlemi yapıldıktan sonra GC-MS kullanılarak analiz yapılmıştır. Mikroçoğaltım için her iki türde de 100 μl/L NAA (Naphthylacetic Asit) ve 600μL/L IBA ( Indol Butyric Asit) hormon seviyelerinde kök ve gövde oluşumu tespit edilmiştir.Mentha pulegium türünün kallus gelişimi için 100 μL/L NAA ve 250μL/L BAP (Benzylaminopurine) hormon seviyelerinin uygun olduğu görülmüştür.Biyokimyasal analizlerden, mikroçoğaltım ile geliştirilen Mentha x piperita türürün GC-MS analizleri sonucunda, limonen (%7.22), karvon (%76.16), caryophyllene (%0.99) ve beta-myrcene (%4.08) oranlarında olduğu anlaşılmaktadır, Aynı türün doğal ortamda geliştirilmesinde ise limonen (%2.61), menthol (%22.28), p-menthon (% 17.34), caryophyllene (% 6.23), karvon (%15.28) ve beta-myrcene (%1.20) oranlarında olduğu tespit edilmiştir. Mentha pulegium L. türünün mikroçoğaltım yöntemi ile geliştirilen yaprak numunelerinde pulegon (% 68.16), karvon (% 7.24), piperitenone (% 3.47), 3-octanone (%1.99) ve menthalactone (2.00) oranlarında gözlemlenmiştir. Aynı türün doğal ortamda geliştirilen yaprak numunelerinde pulegon (% 90.75), piperitenone (%1.00), menthalactone (% 0.02) ve 3-octanone (% 0.03) oranlarında bulunmuştur. Mikroçoğaltım yöntemi ile geliştirilen Mentha x piperita da, doğal ortamda yetiştirilen bitkiye nazaran miktarları karvon 5 kat, limonen 3 kat ve beta myrcene 4 kat daha fazla tespit edilmiştir. Mentha pulegium’un doğal ortama nazaran mikroçoğaltım yöntemiyle geliştirilmesinde miktarları karvon 7 kat, piperiton 3 kat, 3-octanone 2 kat ve menthalactone 2 kat olarak tespit edilmiştir. Bu sonuçlar bitkilerin mikroçoğaltım ile geliştirilmesinde alternatif tıpta kullanılan bazı sekonder metabolitlerin daha fazla olduğunu göstermiştir. Bu bileşenlerin mikroçoğaltım yöntemiyle fazla miktarlarda elde edilmesi ekonomik alanda yarar sağlayacaktır

Wavelets and application to electrical engineering

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1997Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1997Geleneksel olarak işaret işlemede kullanılan Fourier dönüşümünün bazı yetersizlikleri işaret işlemecileri yeni dönüşüm teknikleri geliştirmeye itmiştir. İşaret analizinde kullanılan ayrıştırma tekniklerinin bir kısmı yalnızca zaman domeninde ya da yalnızca frekans domeninde düşünülür. Zaman ve frekans domenleri arasında bağlantı olmasına rağmen bilinen ayrıştırma teknikleri bu domenlerin birisiyle sınırlanır. İşaretin birarada zaman ve frekansta içerdiği bilgiyi ortaya çıkarmak ve durağan olmayan işaretleri inceleyebilmek için geliştirilen dalgacık dönüşüm tekniği işaretin dalgacık fonksiyonları üzerine izdüşümü alınarak elde edilir. Dalgacık fonksiyonları tek bir ana dalgacığın ötelenmesi ve yayılması (ya da sıkıştırılması) ile elde edilirler. Pratikte sık karşılaşılan gerçek zamanda işaretlerin işlenmesinde kullanılan dalgacık dönüşüm tekniği Elektrik Mühendisliği' nin bazı alanlarında da uygulanmıştır. Ortonormal dalgacık bazlarına dayanan çoklu çözünürlük analizinde işaretin, dalgacıkla farklı çözünürlüklerde kendi içinde birbirine dik bileşenlerine ayınlması yoluyla içerdiği bilgi incelenir. Dolayısıyla işaretin zaman-frekans analizinde herhangi bir andaki ani frekans bileşeni karakterize edilebilir. Ayrık dalgacık dönüşümü sayısal uygulamalar açısından filtre takımı ile temsil edildiğinden bilgisayar uygulamaları için verimlidir. Tezde birinci bölümde zaman-frekans analizi ve bu analiz için kullanılan yöntemler kısaca tanıtılmıştır. İkinci bölümde dalgacık dönüşümünün tanımı ve matematiksel özellikleri verilerek filtre takımı yapısı anlatılmıştır. Üçüncü bölüm dalgacık dönüşümünün bilgisayar uygulaması sonuçlarını içerir.A basic objective in signal analysis is to extract relevant information from a signal by transforming it or is to devise a transformation that represents the signal features simultaneously in time and frequency. Standard Fourier analysis decomposes a signal into frequency components and determines the relative strength of each components. It does not tell when the signal exhibited the particular frequency characteristic. If the frequency content of a signal were to vary drastically from interval to interval, the Fourier transform, would sweep over the entire time axis and wash out any local anomalies (e.g. bursts of high frequency) in the signal. Therefore it is not suitable for such non-stationary signals. Wavelet transforms have become well known as useful tools for various signal processing applications. Signals that are not stationary decompose into linear combinations of wavelets. The purpose of the Short-Time Fourier transform (STFT) is to capture the time variation of the frequency contents of the signal. In particular, the wavelet transform is of interest for the analysis of non-stationary signals, because it provides an alternative to the classical STFT or Gabor Transform. The basic difference is as follows: In contrast to the STFT, which uses a single analysis window, the wavelet transform uses short windows at high frequencies and long windows at low frequencies. The wavelet transform is an operation that transforms a function by integrating it with modified versions of some kernel function. The kernel fuction is called the mother wavelet, and the modifications are translations and compressions (or dilations) of the mother wavelet. The mother wavelet can be thought of as a bandpass filter. The bandpass filters have constant relative bandwidth or constant-Q. Given a time-varying signal f(t), wavelet transforms consist of computing coefficient that are inner product of the signal and a family of wavelets. The wavelet family is defined by scale and shift parameters a, b, as V«b(t) =-t= V 1 ft-bN Va I a ) (1) where \|/(t) is the wavelet "prototype". The prototype wavelet i|/(t) can be thought of as a bandpass filter, and the constant-Q properties of the other bandpass filters (wavelets) follows because they are scaled versions of the prototype. The continuous wavelet transform is defined by Wf(a,b)= J yab(t)f (t)dt= (2) -00 ' IX where f* (t) is the conjugate of f(t). \|/(t) must be short and oscillatory, i.e., it must have zero average and decay quickly at both ends. \|/(t) satisfies the admissibility condition defined by ^v j Cv = J | T(Q)r \Q.r dO 1, bo > 0 are fixed The wavelets are in this case Vmn(t) = ao-m/2 ie(a0-mt-nb0) (6) 2 Then f(t)eL (R) is expressed as the superposition f(t)= X X dT V»»(t) (7) m n where the wavelet coefficient dmn is the inner product. m/2 J dmn = = - ^72 J f(t) V (ao"m t-n) (8) a0 Both time and time-scale perameters are discrete in discrete wavelet transform (DWT). As far as the structure of computations is concerned, the DWT is in fact the same as an octave-band filter bank, ao and bo are selected for octave or dyadic grid as ao=2, bo -1. Thus wavelets obtained by this way satisfy orthonormality condition. y m n (t) = ao-"1"2 \|/ (ao"m t-nb0 ) m,n eZ (9) Multiresolution signal analysis provides the vehicle for the link which is between these wavelet families and the pyramid-dyadic tree expansions of a signal. In 2 this representation, we express a function f gL(R) as a limit of successive approximations, each of which is a smoothed version of f(t). These successive approximations correspond to different resolutions-much like a pyramid. This smoothing accomplished by convolution with a low-pass kernel called the scaling function (j)(t). A multiresolution analysis consists of a sequence of closed subspaces {Vm| meZ} of L (R) which have the some properties. The multiresolution approach 2 to wavelets enable us to characterize the class of functions \j/(t)eL~( R) that generate an orthonormal basis. V +, = V © W which means that the space Wm contains the 'detail' information needed to go from an approximation at resolution m to an approximation at resolution m+1. Consequently, © Wj = L2 (R) where the symbol © stands for j direct sum. "W. denote a space complementing V in V +, Since § e Vo c Vi, a square summable sequence (h0(n)} exists such that the scaling function satisfies. (2t-n) (10) XI This functional equation is also called the refinement equation, dilation equation or two-scale difference equation. The scaling function (t) dt =1 (11) The coefficient set {hQ (n)} are the interscale basis coefficients. This is the low-pass unit sample response of the two-band paraunitary filter bank. Since \|/(t)eWo C Vi wavelet bandpass function can be expressed as a linear combination of translates of (j>(2t) W (t)= 2^ h, (n) (2t-n) (12) This is the fundamental wavelet equation. The coefficient {h.x (n)} is high-pass branch in the two-band PR (perfect reconstruction) filter bank structure. There exists a unique function c|)(t)eL (R) such that for any m,n e Z 2~m/2(j)(2"mt-n) is a wavelet orthonormal basis of Vm. We study the functions + k h0 (n-2k) (28) yw (n) = V2 2 di,k hi (n-2k) (29) c(0,n »fhn- ?U2>^ hfl- »(g)^^- > hn c(2-n) KT2 ?hf da.nl ?©- ?E f^*(§Kh Kt2->h S4°'n) Figure 1 Multiresolution (pyramid) decomposition and reconstitution structure for a two-level dyadic subband tree. We can extrapolate these results for the multiscale decomposition and reconstitution for the dyadic subband tree as shown in figure 1. The gain of\/2 associated with each filter is not shown explicitly. We have therefore shown that orthonormal wavelets of compact support imply FIR PR-QMF filter banks. f(t)= £ c(L,n)2"L/2 + Z Z d(m,n)2 -m/2 ¥ m=l n=-oo V2n (30) Eq.(28) shows that f(t) can be represented as a low-pass approximation at scale L plus the sum of L detail (wavelet) components at different resolutions. The first chapter of this thesis introduces the time-frequency analysis and what important of this is. And the second chapter expresses multiresolution approximations of wavelet orthonormal basis and gives mathematical formulation of wavelet transform. Chapter three includes implementation of wavelet transform to pump vibration data and a signal with noise decomposes this data to components. The last chapter includes the conclusion and further work of these topics, especially concerning the power engineering part of Electrical Engineering discipline.</fw\</fv\</f,Yüksek LisansM.Sc