DOCSLIB.ORG

1 Basics of Wavelets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Basics of Wavelets ISyE8843A, Brani Vidakovic Handout 20 1 Basics of Wavelets The first theoretical results in wavelets are connected with continuous wavelet decompositions of L2 func- tions and go back to the early 1980s. Papers of Morlet et al. (1982) and Grossmann and Morlet (1985) were among the first on this subject. Let Ãa;b(x); a 2 Rnf0g; b 2 R be a family of functions defined as translations and re-scales of a single function Ã(x) 2 L2(R); µ ¶ 1 x ¡ b Ãa;b(x) = p à : (1) jaj a 1 Normalization by p ensures that jjÃa;b(x)jj is independent of a and b: The function à (called the jaj wavelet function or the mother wavelet) is assumed to satisfy the admissibility condition, Z jª(!)j2 Cà = d! < 1; (2) R j!j R ¡ix! where ª(!) = R Ã(x)e dx is the Fourier transformation of Ã(x): The admissibility condition (2) implies Z 0 = ª(0) = Ã(x)dx: R R ® Also, if Ã(x)dx = 0 and (1 + jxj )jÃ(x)jdx < 1 for some ® > 0; then Cà < 1: Wavelet functions are usually normalized to “have unit energy”, i.e., jjÃa;b(x)jj = 1: For any L2 function f(x), the continuous wavelet transformation is defined as a function of two variables Z CWT f (a; b) = hf; Ãa;bi = f(x)Ãa;b(x)dx: Here the dilation and translation parameters, a and b, respectively, vary continuously over Rnf0g £ R: Resolution of Identity. When the admissibility condition is satisfied, i.e., Cà < 1; it is possible to find the inverse continuous transformation via the relation known as resolution of identity or Calderon’s´ reproducing identity, Z 1 da db f(x) = CWT f (a; b)Ãa;b(x) 2 : Cà R2 a If a is restricted to R+; which is natural since a can be interpreted as a reciprocal of frequency, (2) becomes Z 1 jª(!)j2 Cà = d! < 1; (3) 0 ! 1 and the resolution of identity relation takes the form Z Z 1 1 1 1 f(x) = CWT f (a; b)Ãa;b(x) 2 da db: (4) Cà ¡1 0 a Next, we list a few important properties of continuous wavelet transformations. Shifting Property. If f(x) has a continuous wavelet transformation CWT f (a; b); then g(x) = f(x¡¯) has the continuous wavelet transformation CWT g(a; b) = CWT f (a; b¡ ¯): Scaling Property. If f(x) has a¡ continuous¢ wavelet transformation ¡ ¢ p1 x a b CWT f (a; b); then g(x) = s f s has the continuous wavelet transformation CWT g(a; b) = CWT f s ; s : Both the shifting property and the scaling property are simple consequences of changing variables under the integral sign. Energy Conservation. From (4), Z 1 Z 1 Z 1 2 1 2 1 jf(x)j dx = jCWT f (a; b)j 2 da db: ¡1 Cà ¡1 0 a p1 x0¡b Localization. Let f(x) = ±(x ¡ x0) be the Dirac pulse at the point x0: Then, CWT f (a; b) = a Ã( a ): Reproducing Kernel Property. Define K(u; v; a; b) = hÃu;v;Ãa;bi: Then, if F (u; v) is a continuous wavelet transformation of f(x), Z Z 1 1 1 1 F (u; v) = K(u; v; a; b)F (a; b) 2 da db; Cà ¡1 0 a i.e., K is a reproducing kernel. The associated reproducing kernel Hilbert space (RKHS)R isR defined as a 2 1 1 1 2 da db CWT image of L2(R) – the space of all complex-valued functions F on R for which jF (a; b)j 2 Cà ¡1 0 a is finite. R Characterization of Regularity. Let (1 + jxj) jÃ(x)j dx < 1 and let ª(0) = 0: If f 2 C®(Holder¨ space with exponent ®), then ®+1=2 jCWT f (a; b)j · Cjaj : (5) Conversely, if a continuous and bounded function f satisfies (5), then f 2 C®: Example 1.1 Mexican hat or Marr’s wavelet. The function 2 d 2 2 Ã(x) = [¡e¡x =2] = (1 ¡ x2)e¡x =2 dx2 is a wavelet [known as the “Mexican hat” or Marr’s wavelet. By direct calculation one may obtain Cà = 2¼: 2 a 6 r r r r r r r r r r r r r r r r rrr r r r r r r rrrrrrrrrrrrrrrrrrrrrrrrrr r r r r r r r r r r r r rrrrrrrrrrrrr r r r r r r rrrrrrr rrrr r r r rrr r - 0 b Figure 1: Critical Sampling in R £ R+ half-plane (a = 2¡j and b = k 2¡j). d 1 1 Example 1.2 Poisson wavelet. The function Ã(x) = ¡(1+ dx ) ¼ 1+x2 is a wavelet [known as the Poisson wavelet. The analysis of functions with respect to this wavelet is related to the boundary value problem of the Laplace operator. The continuous wavelet transformation of a function of one variable is a function of two variables. Clearly, the transformation is redundant. To “minimize” the transformation one can select discrete values of a and b and still have a transformation that is invertible. However, sampling that preserves all information about the decomposed function cannot be coarser than the critical sampling. The critical sampling (Fig. 1) defined by a = 2¡j; b = k2¡j; j; k 2 Z; (6) will produce the minimal basis. Any coarser sampling will not give a unique inverse transformation; that is, the original function will not be uniquely recoverable. Moreover under mild conditions on the wavelet j=2 j function Ã, such sampling produces an orthogonal basis fÃjk(x) = 2 Ã(2 x ¡ k); j; k 2 Zg: There are other discretization choices. For example, selecting a = 2¡j; b = k will lead to non- decimated (or stationary) wavelets. For more general sampling, given by ¡j ¡j a = a0 ; b = k b0 a0 ; j; k 2 Z; a0 > 1; b0 > 0; (7) numerically stable reconstructions are possible if the system fÃjk; j; k 2 Zg constitutes a frame. Here à ! ¡j j=2 x ¡ k b0 a0 j=2 j Ãjk(x) = a0 à ¡j = a0 Ã(a0x ¡ k b0); a0 is (1) evaluated at (7). Next, we consider wavelet transformations (wavelet series expansions) for values of a and b given by (6). An elegant theoretical framework for critically sampled wavelet transformation is Mallat’s Multiresolution Analysis (Mallat, 87; 89a, 89b, 98). 3 1.1 Multiresolution Analysis A multiresolution analysis (MRA) is a sequence of closed subspaces Vn; n 2 Z in L2(R) such that they lie in a containment hierarchy ¢ ¢ ¢ ½ V¡2 ½ V¡1 ½ V0 ½ V1 ½ V2 ½ ¢ ¢ ¢ : (8) The nested spaces have an intersection that contains the zero function only and a union that is dense in L(R); \nVj = f0g; [jVj = L2(R): [With A we denoted the closure of a set A]. The hierarchy (8) is constructed such that (i) V -spaces are self-similar, j f(2 x) 2 Vj iff f(x) 2 V0: (9) and (ii) there exists a scaling function Á 2 V0 whose integer-translates span the space V0; ( ) X V0 = f 2 L2(R)j f(x) = ckÁ(x ¡ k) ; k 1 and for which the set fÁ(² ¡ k); k 2 Zg is an orthonormal basis. R Mild technical conditions on Á are necessary for future developments. It can be assumed that Á(x)dx ¸ 0: Later, we will prove that this integral is in fact equal to 1. Since V0 ½ V1, the function Á(x) 2 V0 can be represented as a linear combination of functions from V1; i.e., X p Á(x) = hk 2Á(2x ¡ k); (10) k2Z for some coefficients hk; k 2 Z: This equation is called the scaling equation (or two-scale equation) and it is fundamental in constructing, exploring, and utilizing wavelets. ADD BASES FOR Vj. ANY L2 can be projected on Vj In the wavelet literature, the reader may encounter an indexing of the multiresolution subspaces, which is the reverse of that in (8), ¢ ¢ ¢ ½ V2 ½ V1 ½ V0 ½ V¡1 ½ V¡2 ½ ¢ ¢ ¢ : (11) FORM OF Ájk(x): 1It is possible to relax the orthogonality requirement. It is sufficient to assume that the system of functions fÁ(² ¡ k); k 2 Zg constitutes a Riesz basis for V0: 4 Theorem 1.1 For the scaling function it holds Z Á(x)dx = 1; R or, equivalently, ©(0) = 1; R ¡i!x where ©(!) is Fourier transformation of Á, R Á(x)e dx: The coefficients hn in (10) are important in connecting the MRA to the theory of signal processing. The (possibly infinite) vector h = fhn; n 2 Zg will be called a wavelet filter. It is a low-pass (averaging) filter as will become clear later by considerations in the Fourier domain. To further explore properties of multiresolution analysis subspaces and their bases, we will often work in the Fourier domain. Define the function m0 as follows: X 1 ¡ik! 1 m0(!) = p hke = p H(!): (12) 2 2 k2Z The function in (12) is sometimes called the transfer function and it describes the behavior of the associated filter h in the Fourier domain. Notice that the function m0 is periodicp with the period 2¼ and that the filter taps fhn; n 2 Zg are the Fourier coefficients of the function H(!) = 2 m0(!): In the Fourier domain, the relation (10) becomes ³! ´ ³! ´ ©(!) = m © ; (13) 0 2 2 where ©(!) is the Fourier transformation of Á(x): Indeed, Z 1 ©(!) = Á(x)e¡i!xdx ¡1 X p Z 1 ¡i!x = 2 hk Á(2x ¡ k)e dx ¡1 k Z X h 1 = pk e¡ik!=2 Á(2x ¡ k)e¡i(2x¡k)!=2d(2x ¡ k) 2 k ¡1 X h ³! ´ = pk e¡ik!=2 © 2 2 k ³! ´ ³! ´ = m © : 0 2 2 By iterating (13), one gets Y1 ³ ! ´ ©(!) = m ; (14) 0 2n n=1 which is convergent under very mild conditions on rates of decay of the scaling function Á: There are several sufficient conditions for convergence of the product in (14).
Load more

Recommended publications
  • Haar Wavelet Based Numerical Method for the Solution of Multidimensional Stochastic Integral Equations
    International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 10 (2019) pp. 2507-2521 © Research India Publications. http://www.ripublication.com Haar Wavelet Based Numerical Method for the Solution of Multidimensional Stochastic Integral Equations S. C. Shiralashetti*, Lata Lamani1 P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, Karnataka, India. Abstract for numerical solution of multi-term fractional differential equations [5] and numerical solution of initial value problems In this paper, we develop an accurate and efficient Haar [6]. Solution of nonlinear oscillator equations, Stiff systems, wavelet based numerical method for the solution of Multi- regular Sturm-Liouville problems, etc. using Haar wavelets dimensional Stochastic Integral equations. Initially, we study were studied by Bujurke et al[14, 15]. In 2012, Maleknejad et the properties of stochastic integral equations and Haar al applied block-pulse functions for the solution of wavelets. Then, Haar wavelets operational matrix of multidimensional stochastic integral equations [13]. In this integration and Haar wavelets stochastic operational matrix of paper, we developed the numerical method for the solution of integration are developed. Convergence and error analysis of stochastic integral equations using Haar wavelets. Consider the Stochastic Haar wavelet method is presented for the the stochastic Ito-volterra integral equation, solution of Multi-dimensional Stochastic Integral equations. Efficiency of the proposed method is justified through the t U(t)=g(t) k s,t U(s)ds illustrative examples. 0 0 n t (1.1) Keywords: Haar wavelets, Multi-dimensional Stochastic k s, t U(s)dB (s), t [0,T) 0 ii Integral equations, stochastic operational matrix of i1 integration.
    [Show full text]
  • Discrete Wavelet Transforms of Haar's Wavelet
    INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 9, SEPTEMBER 2014 ISSN 2277-8616 Discrete Wavelet Transforms Of Haar’s Wavelet Bahram Dastourian, Elias Dastourian, Shahram Dastourian, Omid Mahnaie Abstract: Wavelet play an important role not only in the theoretic but also in many kinds of applications, and have been widely applied in signal processing, sampling, coding and communications, filter bank theory, system modeling, and so on. This paper focus on the Haar’s wavelet. We discuss on some command of Haar’s wavelet with its signal by MATLAB programming. The base of this study followed from multiresolution analysis. Keyword: approximation; detail; filter; Haar’s wavelet; MATLAB programming, multiresolution analysis. ———————————————————— 1 INTRODUCTION Definition 4. Let 퐻 be a Hilbert space i) A family 풙 ⊂ 푯 is an orthonormal system if Wavelets are a relatively recent development in applied 풋 풋∈푱 mathematics. Their name itself was coined approximately a decade ago (Morlet, Arens, Fourgeau, and Giard [11], Morlet 1 푖 = 푗, ∀푖, 푗 ∈ 퐽 ∶ 푥 , 푥 = [10], Grossmann, Morlet [3] and Mallat[9]); in recently years 푖 푗 0 푖 ≠ 푗. interest in them has grown at an explosive rate. There are several reasons for their present success. On the one hand, ii) A family 풙 ⊂ 푯 is total if for all 풙 ∈ 푯 the following the concept of wavelets can be viewed as a synthesis of ideas 풋 풋∈푱 which originated during the last twenty or thirty years in implication holds engineering (subband coding), physics (coherent states, renormalization group), and pure mathematics (study of ∀푗 ∈ 퐽 ∶ 푥, 푥푗 = 0 ⇒ 푥 = 0.
    [Show full text]
  • SOHO: Orthogonal and Symmetric Haar Wavelets on the Sphere
    SOHO: Orthogonal and Symmetric Haar Wavelets on the Sphere CHRISTIAN LESSIG and EUGENE FIUME University of Toronto We propose the SOHO wavelet basis – the first spherical Haar wavelet basis that is both orthogonal and symmetric, making it particularly well suited for the approximation and processing of all- frequency signals on the sphere. We obtain the basis with a novel spherical subdivision scheme that defines a partition acting as the domain of the basis functions. Our construction refutes earlier claims doubting the existence of a basis that is both orthogonal and symmetric. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice. Categories and Subject Descriptors: I.3.0 [Computing Methodologies]: Computer Graph- ics—General; G.1.0 [Numerical Analysis]: General—Numerical Algorithms; G.1.2 [Numerical Analysis]: Approximation—Nonlinear Approximation General Terms: Theory, Measurement Additional Key Words and Phrases: wavelet transform, spherical signals 1. INTRODUCTION Many signals are naturally parametrized over the sphere S2. Examples from com- puter graphics include bidirectional reflectance distribution functions (BRDFs), ra- diance, and visibility. Spherically parametrized signals can also be found in many other fields, including astronomy, physics, climate modeling, and medical imaging. An efficient and distortion free representation of spherical signals is therefore of importance. Of particular interest are the ability to approximate a wide range of signals accurately with a small number of basis function coefficients, and the pos- sibility of obtaining computationally efficient algorithms to process a signal in its basis representation. A variety of representations for spherical signals has been proposed in the literature.
    [Show full text]
  • Haar Wavelet Operational Matrix Method for Solving Fractional Partial Differential Equations
    Copyright © 2012 Tech Science Press CMES, vol.88, no.3, pp.229-243, 2012 Haar Wavelet Operational Matrix Method for Solving Fractional Partial Differential Equations Mingxu Yi1 and Yiming Chen1 Abstract: In this paper, Haar wavelet operational matrix method is proposed to solve a class of fractional partial differential equations. We derive the Haar wavelet operational matrix of fractional order integration. Meanwhile, the Haar wavelet operational matrix of fractional order differentiation is obtained. The operational matrix of fractional order differentiation is utilized to reduce the initial equation to a Sylvester equation. Some numerical examples are included to demonstrate the validity and applicability of the approach. Keywords: Haar wavelet, operational matrix, fractional partial differential equa- tion, Sylvester equation, numerical solution. 1 Introduction Wavelet analysis is a relatively new area in different fields of science and engi- neering. It is a developing of Fourier analysis. Wavelet analysis has been ap- plied widely in time-frequency analysis, signal analysis and numerical analysis. It permits the accurate representation of a variety of functions and operators, and establishes a connection with fast numerical algorithms [Beylkin, Coifman, and Rokhlin (1991)]. Functions are decomposed into summation of “basic functions”, and every “basic function” is achieved by compression and translation of a mother wavelet function with good properties of smoothness and locality, which makes people analyse the properties of locality and integer in the process of expressing functions [Li and Luo (2005); Ge and Sha (2007)]. Consequently, wavelet analysis can describe the properties of functions more accurate than Fourier analysis. Fractional differential equations are generalized from classical integer order ones, which are obtained by replacing integer order derivatives by fractional ones.
    [Show full text]
  • An Introduction to Wavelets for Economists
    Bank of Canada Banque du Canada Working Paper 2002-3 / Document de travail 2002-3 An Introduction to Wavelets for Economists by Christoph Schleicher ISSN 1192-5434 Printed in Canada on recycled paper Bank of Canada Working Paper 2002-3 January 2002 An Introduction to Wavelets for Economists by Christoph Schleicher Monetary and Financial Analysis Department Bank of Canada Ottawa, Ontario, Canada K1A 0G9 The views expressed in this paper are those of the author. No responsibility for them should be attributed to the Bank of Canada. iii Contents Acknoweldgements. iv Abstract/Résumé. v 1. Introduction . 1 2. Wavelet Evolution . 3 3. A Bit of Wavelet Theory . .5 3.1 Mallat’s multiscale analysis . 10 4. Some Examples. 16 4.1 Filtering. 18 4.2 Separation of frequency levels . 20 4.3 Disbalancing of energy . 21 4.4 Whitening of correlated signals . 23 5. Applications for Economists. 24 5.1 Frequency domain analysis. 24 5.2 Non-stationarity and complex functions. 25 5.3 Long-memory processes . 26 5.4 Time-scale decompositions: the relationship between money and income . 27 5.5 Forecasting . .28 6. Conclusions . 28 7. How to Get Started. 29 Bibliography . 30 iv Acknowledgements I would like to thank Paul Gilbert, Pierre St-Amant, and Greg Tkacz from the Bank of Canada for helpful comments. v Abstract Wavelets are mathematical expansions that transform data from the time domain into different layers of frequency levels. Compared to standard Fourier analysis, they have the advantage of being localized both in time and in the frequency domain, and enable the researcher to observe and analyze data at different scales.
    [Show full text]
  • Efficient Implementations of Discrete Wavelet Transforms Using Fpgas Deepika Sripathi
    Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2003 Efficient Implementations of Discrete Wavelet Transforms Using Fpgas Deepika Sripathi Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING EFFICIENT IMPLEMENTATIONS OF DISCRETE WAVELET TRANSFORMS USING FPGAs By DEEPIKA SRIPATHI A Thesis submitted to the Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master of Science Degree Awarded: Fall Semester, 2003 The members of the committee approve the thesis of Deepika Sripathi defended on November 18th, 2003. Simon Y. Foo Professor Directing Thesis Uwe Meyer-Baese Committee Member Anke Meyer-Baese Committee Member Approved: Reginald J. Perry, Chair, Department of Electrical and Computer Engineering The office of Graduate Studies has verified and approved the above named committee members ii ACKNOWLEDGEMENTS I would like to express my gratitude to my major professor, Dr. Simon Foo for his guidance, advice and constant support throughout my thesis work. I would like to thank him for being my advisor here at Florida State University. I would like to thank Dr. Uwe Meyer-Baese for his guidance and valuable suggestions. I also wish to thank Dr. Anke Meyer-Baese for her advice and support. I would like to thank my parents for their constant encouragement. I would like to thank my husband for his cooperation and support. I wish to thank the administrative staff of the Electrical and Computer Engineering Department for their kind support. Finally, I would like to thank Dr.
    [Show full text]
  • Haar Wavelets, Fluctuations and Structure Functions
    1 Haar wavelets, fluctuations and structure functions: convenient choices for geophysics S. Lovejoy, Physics, McGill University, 3600 University st., Montreal, Que. H3A 2T8, Canada [email protected] D. Schertzer, LEESU, École des Ponts ParisTech, Université Paris-Est, 6-8 Av. B. Pascal, 77455 Marne-la-Vallée Cedex 2, France [email protected] 2 Abstract: Geophysical processes are typically variable over huge ranges of space-time scales. This has lead to the development of many techniques for decomposing series and fields into fluctuations Δv at well defined scales. Classically, one defines fluctuations as differences: (Δv)diff = v(x+Δx)-v(x) and this is adequate for many applications (Δx is the “lag”). However, if over a range one has scaling Δv ∝ Δx H , these difference fluctuations are only adequate when 0<H<1, hence the need for other types of fluctuations. In particular, atmospheric processes in the range ≈ 10 days to 10 years generally have -1 < H < 0 so that a definition valid over the range -1< H <1 would be very useful for atmospheric applications. The general framework for defining fluctuations is wavelets. However the generality of wavelets often leads to fairly arbitrary choices of “mother wavelet” and the resulting wavelet coefficients may be difficult to interpret. In this paper we argue that a good choice is provided by the (historically) first wavelet, the Haar wavelet (Haar, 1910) which is easy to interpret and - if needed - to generalize, yet has rarely been used in geophysics. It is also easy to implement numerically: the Haar fluctuation (Δv)Haar at lag Δx is simply proportional to the difference of the mean from x to x+Δx/2 and from x+Δx/2 to x+Δx.
    [Show full text]
  • Wavelet Variance Analysis for Random Fields
    1 Wavelet Variance Analysis for Random Fields Debashis Mondal and Donald B. Percival Abstract There has been considerable recent interest in using wavelets to analyze time series and images that can be regarded as realizations of certain one- and two-dimensional stochastic processes. Wavelets give rise to the concept of the wavelet variance (or wavelet power spectrum), which decomposes the variance of a stochastic process on a scale-by-scale basis. The wavelet variance has been applied to a variety of time series, and a statistical theory for estimators of this variance has been developed. While there have been applications of the wavelet variance in the two-dimensional context (in particular in works by Unser, 1995, on wavelet-based texture analysis for images and by Lark and Webster, 2004, on analysis of soil properties), a formal statistical theory for such analysis has been lacking. In this paper, we develop the statistical theory by generalizing and extending some of the approaches developed for time series, thus leading to a large sample theory for estimators of two-dimensional wavelet variances. We apply our theory to simulated data from Gaussian random fields with exponential covariances and from fractional Brownian surfaces. We also use our methodology to analyze images of four types of clouds observed over the south-east Pacific ocean. Index Terms analysis of variance, cloud data, Daubechies filters, fractional Brownian surface, semi-variogram. I. INTRODUCTION Two-dimensional versions of the wavelet transform have been used extensively in the past to address problems such as image compression and segmentation, edge detection, deconvolution and denoising; see, for example, [12].
    [Show full text]
  • An Implementation of Haar Wavelet Based Method for Numerical
    IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 177 An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger¨ and Coupled Schrodinger¨ Systems Najeeb Alam Khan and Tooba Hameed Abstract—The objective of this paper is to solve the time- equations, such as the variational iteration method [3], differ- fractional Schrodinger¨ and coupled Schrodinger¨ differential ential transform method [4], homotopy analysis method [5], equations (TFSE) with appropriate initial conditions by using the Jacobi spectral tau and collocation method [6]¡[8], Laplace Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method transform method [9], homotopy perturbation method [10], to solve a coupled system of time-fractional partial differential Adomian decomposition method [11], [12], high-order finite equations. As a general rule, piecewise constant approximation of element methods [13] and many others [14]¡[19]. The scope a function at different resolutions is presentational characteristic and distinct aspects of fractional calculus have been written of Haar wavelet method through which it converts the differential by many authors in [20]¡[22]. equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research In the past few years, there has been an extensive attraction and it also helps in the development of automation science, in employing the spectral method [23]¡[25] for numerically physics, and engineering as well. Illustratively, several test prob- solving the copious type of differential and integral equations. lems are discussed to draw an effective conclusion, supported The spectral methods have an exponential quota of conver- by the graphical and tabulated results of included examples, to gence and high level of efficiency.
    [Show full text]
  • Lecture 6 – Orthonormal Wavelet Bases
    Lecture 6 – Orthonormal Wavelet Bases David Walnut Department of Mathematical Sciences George Mason University Fairfax, VA USA Chapman Lectures, Chapman University, Orange, CA 6-10 November 2017 Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Outline Wavelet systems Example 1: The Haar Wavelet basis Characterizations of wavelet bases Example 2: The Shannon Wavelet basis Example 3: The Meyer Wavelet basis Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Wavelet systems Definition A wavelet system in L2(R) is a collection of functions of the form j/2 j D2j Tk j,k Z = 2 (2 x k) j,k Z = j,k j,k Z { } 2 { − } 2 { } 2 where L2(R) is a fixed function sometimes called the 2 mother wavelet. A wavelet system that forms an orthonormal basis for L2(R) is called a wavelet orthonormal basis for L2(R). Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases If the mother wavelet (x) is concentrated around 0 then j j,k (x) is concentrated around 2− k. If (x) is essentially supported on an interval of length L, then j,k (x) is essentially j supported on an interval of length 2− L. Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Since (D j T ) (γ)=D j M (γ) it follows that if is 2 k ^ 2− k concentrated on the interval I then j,k is concentrated on the interval 2j I. b b d Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Dyadic time-frequency tiling Walnut (GMU) Lecture 6 – Orthonormal Wavelet Bases Example 1: The Haar System This is historically the first orthonormal wavelet basis, described by A.
    [Show full text]
  • 1 Introduction 2 the 1-D Discrete Haar Wavelet Transform
    c 2011 by Duraisamy Sundararajan Fundamentals of the Discrete Haar Wavelet Transform Duraisamy Sundararajan Abstract–The discrete wavelet transform, a generalization of the Fourier analysis, is widely used in several signal and image processing applications. In this paper, the fundamentals of the discrete Haar wavelet transform are presented from signal processing and Fourier analysis point of view. Fast algorithms for the implementation of Haar discrete wavelet transform, for both 1-D and 2-D signals, are presented. 1 Introduction Fourier analysis is the approximation of an arbitrary signal by a sum of sinusoidal waveforms. Therefore, the study of the Fourier analysis begins with the study of the sinusoidal wave- forms. The sinusoids used in the Fourier analysis have constant amplitude and are referred as constant-amplitude sinusoids. Expressing an arbitrary signal in terms of constant-amplitude sinusoids provides efficient methods to solve many practical problems. However, exponentially unbounded signals cannot be expressed in terms of constant-amplitude sinusoids. To analyze this class of signals, the Fourier analysis is generalized by including varying-amplitude sinu- soids in the set of basis functions. This generalization is the Laplace transform for continuous signals and the z-transform for discrete signals. It is well-known that Fourier analysis is used for steady-state and spectral analysis while the Laplace and z-transforms are used for the de- sign, transient, and stability analysis of systems. The point is that transformation of a signal (changing the form) is the fundamental method for efficient signal and system analysis. Sev- eral transforms are used in signal and system analysis and each transform is more suitable for some applications.
    [Show full text]
  • EE269 Signal Processing for Machine Learning Lecture 17
    EE269 Signal Processing for Machine Learning Lecture 17 Instructor : Mert Pilanci Stanford University March 13, 2019 I Continuous Wavelet Transform 1 W (s, ⌧)= f(t) ⇤ dt = f(t), s,⌧ h s,⌧ i Z1 I Transforms a continuous function of one variable into a continuous function of two variables : translation and scale I For a compact representation, we can choose a mother wavelet (t) that matches the signal shape I Inverse Wavelet Transform 1 1 f(t)= W (s, ⌧) s,⌧ d⌧ds Z1 Z1 Continuous Wavelet Transform I Define a function (t) I Create scaled and shifted versions of (t) 1 t ⌧ = ( − ) s,⌧ ps s I Inverse Wavelet Transform 1 1 f(t)= W (s, ⌧) s,⌧ d⌧ds Z1 Z1 Continuous Wavelet Transform I Define a function (t) I Create scaled and shifted versions of (t) 1 t ⌧ = ( − ) s,⌧ ps s I Continuous Wavelet Transform 1 W (s, ⌧)= f(t) ⇤ dt = f(t), s,⌧ h s,⌧ i Z1 I Transforms a continuous function of one variable into a continuous function of two variables : translation and scale I For a compact representation, we can choose a mother wavelet (t) that matches the signal shape Continuous Wavelet Transform I Define a function (t) I Create scaled and shifted versions of (t) 1 t ⌧ = ( − ) s,⌧ ps s I Continuous Wavelet Transform 1 W (s, ⌧)= f(t) ⇤ dt = f(t), s,⌧ h s,⌧ i Z1 I Transforms a continuous function of one variable into a continuous function of two variables : translation and scale I For a compact representation, we can choose a mother wavelet (t) that matches the signal shape I Inverse Wavelet Transform 1 1 f(t)= W (s, ⌧) s,⌧ d⌧ds Z1 Z1 Wavelets Continuous Wavelet
    [Show full text]