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Wavelet Transform Done By: S.Satya Phanindra Kumar Roll No:10881A04A1 Project Supervisor: K.Durga Prasad Associate Professor

Wavelets Prsnt

Jun 03, 2018



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Wavelet Transform

Done By:S.Satya Phanindra KumarRoll No:10881A04A1

Project Supervisor:K.Durga PrasadAssociate Professor

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IntroductionFourier transformLimitationsShort time fourier transformSTFT DisadvantagesPrinciples of Wavelet transformExamples of WaveletsHAAR Wavelets

Applications Of Wavelets


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Introduction to the world oftransform

• What are transforms :-

• A mathematical operation that takes a function orsequence and maps into another form.

• Examples : Laplace, Fourier, DTFT, DFT, FFT, z-transform

• Transformation takes place in laplace transform.In this thetransformation takes place in between time signal and frequency


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Fourier Transform

df e f X t x

dt et x f X

ft j

ft j





Mathematical Form :-

Notes :-

• A Fourier transform is based on the principle of fourierseries.•Fourier series decomposes periodic signals into oscillatingfunctions namely sine and cosine.•They are used for calculating the frequency spectrum ofsignals that change over a time.

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It only determines the frequencycomponents of that signal.

It shows what frequency exists in that signal.

The time and frequency in formation cannotbe seen at the same time.

The time – frequency representation of thesignal is needed .

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Short-time Fourier Transform

• Also known as a STFT

• It is used to analyse only a small section ofa signal at a time.

• This technique is called Windowing TheSignal.

• .It has some disadvantges also.

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STFT :: Disadvantages

• Wider windowGood frequency resolution , Poor time

resolution.• Narrower window• Good time resolution , Poor frequency


• Heisenberg principle exists in this case which statesthat frequency is unknown at their respective timeintervals.

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Wavelet Transform

• Overcomes the shortcoming of STFT by using variablelength windows.

• Narrower window for high frequency thereby givingbetter time resolution and Wider window at lowfrequency thereby giving better frequency resolution.

• Heisenberg’s Principle still holds.

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Principles Of Wavelet Transforms

Splits up the signals into bunch of signals.

Provides what frequency bands exist at what timeintervals.

It represents the same signal in correspondance todifferent frequency bands.

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Examples of wavelets

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• It is the simplest possible wavelet.

• It is a sequence of rescaled square shapedfunctions which together form a wavelet.It was

proposed in 1909 by Alfred Harr.

• Haar used these functions to give an example of anorthonormal system for the space of square-

integrable functions on the interval [0,1].

• The technical disadvantage of this is it is notcontinuous,and therefore not differentiable.

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The haar function is given by:-

h(x)= 1 0<x<1/2-1 1/2<x<10 otherwise


h jk (x)=h(2 j x-k)

and j is a non-negative integer and 0<k<2 j-1

It is done for some values of j and k and theirrespective graphs are shown.

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Applications of wavelets

• There are a lots of uses ofwavelets .... The mostprominent application ofwavelets are

• Computer and Human Vision

• FBI Finger Print compression• Image compression• Denoising Noisy data• Detecting self-similar

behavior in noisy data

• Musical Notes synthesis• Animations

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o They are also used inComputer graphics andmultifractal analysis.

o Their usage in geographic study is also verysignoificant.

o Orthonormal wavelets are used in the study ofatmospheric layer turbelance.


It is also used in another geo physical applicationssuch as analysis of marine seismic data.

o It is mainly used for finger print verification .

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There are many applications of wavelets in variousfields and their functionality is discussed.I also tried tocompare wavelet transforms and fourier transforms bygiving the drawbacks of the fourier transforms.Besidesthis,I also listed the advantages of wavelettransforms.From the discussion it is clear that wavelettransforms approach is better than existing methodsand it takes less responsive time with highaccuracy.Finally,we can say that wavelet transform is

more reliable technique than that of any othertechniques.

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[1]Paul S. Addison, The Illustrated Wavelet TransformHandbook , Institute of Physics, 2002.[2]Ali Akansu and Richard Haddad, Multiresolution Signal

Decomposition: Transforms, Subbands, Wavelets , AcademicPress, 1992.

[3]B. Boashash, editor, "Time-Frequency Signal Analysis andProcessing – A Comprehensive Reference", Elsevier Science,Oxford, 2003.

[4]Tony F. Chan and Jackie (Jianhong) Shen, Image Processingand Analysis – Variational, PDE, Wavelet, and Stochastic

Methods , Society of Applied Mathematics, 2005.[5]Ingrid Daubechies, Ten Lectures on Wavelets , Society forIndustrial and Applied Mathematics, 1992.

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