<< Chapter < Page Chapter >> Page >

This book develops the ideas behind and properties of wavelets and shows how they can be used as analytical tools for signal processing,numerical analysis, and mathematical modeling. We try to present this in a way that is accessible to the engineer, scientist, and appliedmathematician both as a theoretical approach and as a potentially practical method to solve problems. Although the roots of this subject goback some time, the modern interest and development have a history of only a few decades.

The early work was in the 1980's by Morlet, Grossmann, Meyer, Mallat, and others, but it was the paper by Ingrid Daubechies [link] in 1988 that caught the attention of the larger applied mathematics communities insignal processing, statistics, and numerical analysis. Much of the early work took place in France [link] , [link] and the USA [link] , [link] , [link] , [link] . As in many new disciplines, the first work was closely tied to a particular application or traditional theoreticalframework. Now we are seeing the theory abstracted from application and developed on its own and seeing it related to other parallel ideas. Ourown background and interests in signal processing certainly influence the presentation of this book.

The goal of most modern wavelet research is to create a set of basis functions (or general expansion functions) and transforms that will givean informative, efficient, and useful description of a function or signal and allow more effective and efficient processing. If the signal is represented as a function of time, wavelets provideefficient localization in both time and frequency or scale. Another central idea is that of multiresolution where the decomposition of a signal is in terms of the resolution of detail.

For the Fourier series, sinusoids are chosen as basis functions, then the properties of the resulting expansion are examined. For wavelet analysis,one poses the desired properties and then derives the resulting basis functions. An important property of the wavelet basis is providing amultiresolution analysis. For several reasons, it is often desired to have the basis functions orthonormal. Given these goals, you will seeaspects of correlation techniques, Fourier transforms, short-time Fourier transforms, discrete Fourier transforms, Wigner distributions, filterbanks, subband coding, and other signal expansion and processing methods in the results.

Wavelet-based analysis is an exciting new problem-solving tool for the mathematician, scientist, and engineer. It fits naturally withthe digital computer with its basis functions defined by summations not integrals or derivatives. Unlike most traditional expansion systems, thebasis functions of the wavelet analysis are not solutions of differential equations. In some areas, it is the first truly new tool we have had inmany years. Indeed, use of wavelets and wavelet transforms requires a new point of view and a new method of interpreting representations that weare still learning how to exploit.

Work by Donoho, Johnstone, Coifman, and others have added theoretical reasons for why wavelet analysis is so versatile and powerful,and have given generalizations that are still being worked on. They have shown that wavelet systems have some inherent generic advantages and arenear optimal for a wide class of problems [link] . They also show that adaptive means can create special wavelet systems forparticular signals and classes of signals.

Questions & Answers

a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
f(n)= 2n + 1
Samantha Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?