Dalgacıklar ve elektrik mühendisliği'ndeki uygulamalar

Abstract

Geleneksel 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 < oo (3) -00 and ¥(0) = J V(t) dt = 0 (4) -CO Thus the transform is invertible and reconstruction of f(t) is obtained by f(t)= (V1 J J -^ Wf (a,b) M/ab (t) (5) n a -co 0 For large a, the basis function becomes a stretched version of a prototype wavelet, that is, a low-frequency function, while for small a, this basis function is a contracted version of the wavelet function, which is a short-time duration, high frequency function. Depending on the scaling parameter a, the wavelet function \j/(t) dilates or contracts in time, causing the corresponding contraction or dilation in the frequency domain. Thus the wavelet transform provides a flexible time-frequency resolution. Given the same fuction f(t) and two different mother wavelets, \|/ (t) and v|/2(t), the wavelet transform of f(t) with respect to \j/ (t) is not the same as the wavelet transform of f(t) with respect to \|/2(t). In much of wavelet transform literature, orthogonal, dyadic functions are chosen as the mother wavelet. Both the STFT and wavelet transform are highly redundant when the fequency - time parameters (Q,t) and scale and translation parameters (a,b) are continuous. Therefore the transforms are usually evaluated on a discrete grid on the time-frequency and time-scale plane, respectively, corresponding to a discrete set of continuous basis functions. There are various ways of discretizing time-scale parameters (a,b) each one yields a different type of wavelet transform. Time-scale parameters (a,b) are sampled on a so-called 'dyadic' grid in the time-scale plane (a,b). a= aom and b= n b0 aom m,n s Z ao > 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 <|)(t) such that for all m e Z 2~m/2 (j)(2'm t-n) is an orthonormal family, and if Vm is the vector spaces generated by this family of functions, then Vm, me Z is a regular multiresolution analysis of L2 (R). We show that the Fourier transform of n, and di,n can be achieved by up-sampling and convolution with the filters h0(n), and hi(n). This is expected, since the front end of the one-stage pyramid is simply the analysis section of a two-band, PR-QMF bank. The reconstruction therefore must correspond to the synthesis bank. Co,n =<f,= <fv\o,n> + <fw\= yv(n) + yw(n) (27) Synthesis equations are obtained by XIV yv (n) =4l Yj ci>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,