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Research of a sound processing ways. Wavelet transform


The author's abstract to the master's work

Viacheslav Maidan


Introduction

Wavelets is very wide theme for researches in general. Among people who work first of all above processing graphic and video of the documentation, waveleting is very widespread practice. Now its{her} circle exists even the certain agiotage. And it only in wide sphere of using.

Wavelet (wavelet-transformation, waves, wave transformations). All wavelet-transformations consider function (taken as time- function) in terms of the fluctuations localized on time (space) and frequency. Localness in space means that energy of waves (wavelets) concentrated on a separate interval, so-called function on compact carriers. Frequency localization means that Fourier transform of a wave is localized. Frequency localization of function is reduced to concepts of smoothness and quantity of the disappearing moments. Wavelet-transformation usually divides on discrete (DWT) and continuous (CWT).

Discrete wavelet-transformation (DWT) is usually used for the signal coding, whereas CWT - for the signal analysis. For this reason DWT is widely applied in engineering and computer sciences, and CWT - in scientific researches of physical processes. Wavelet-transformation now is taken on arms for huge quantity of versatile applications, quite often replacing usual Fourier transform into many applied problems. This change of a paradigm is observed in many areas of physics, including molecular dynamics, astrophysics, the quantum mechanics, geophysics, optics, mechanics of a liquid, and in many other things areas, including processing of images, the analysis of a blood pressure, pulse and an electrocardiogram, the analysis of DNA, research of fibers, studying of a climate, the general signal processing, recognition of language, computer to the schedule and the multifractal analysis. Such wide use wavelet-transformation provide an opportunity to construct on their basis methods that will demand a minimum of operations, in a counterbalance of methods of Fourier-transform.


The basic purpose and separate tasks of work

Such though and compressed, but detailed enough explanation wavelet-transformations speaks that in this work I consider as the main task the most detailed and as it is possible the most simple analysis and the description of the process of wavelet-transformation. On the basis of the collected information in the long term - creation of the training program on wavelet - to transformations for the most simple and the most economical on time of mastering by knowledge from the given theme.


Potential scientific novelty

Essentially not the new idea of creation of the curriculum for wavelet-transformations as those is perspective in case of the most detailed studying and the analysis of a theme. Besides parallel researches of separate directions of the given work connected with practical application of wavelets, such as, for example, researches of new graphic standard JPEG2000, that behind the potential should take a place of existing standard JPEG in the future, potentially have greater scientific value.


Expected practical value

Essentially not new idea of creation of the curriculum for wavelet-transformation as those is perspective in case of the most detailed studying and the analysis of a theme. But in case of incompleteness and imperfection of system there is an opportunity of completion of system that is usual practice, especially in such wide themes for research as wavelet. Therefore the given theme in any case remains enough perspective.


The existing researches review

Last researches concerning continuous wavelet-transformation (CWT) are presented below.

Any law describing certain process, is limited both in time, and on size. From a rate of the mathematical analysis it is known, that such law can be presented in the form of the sum of harmonious fluctuations of different frequency and intensity (amplitude).


Harmonious making increases in value of process

Thus fluctuations which have low frequency, are responsible for slow, smooth, large-scale changes of the described size, and high-frequency - for short, small-scale changes. The more strongly the size described by given law on the given scale changes, the components on corresponding frequency have greater amplitude. Thus, any process can be considered as in time area (that is development of process in time), and in frequency area (that is by way of scale of changes of the investigated size). Speak also about behaviour of process in frequency-hour area - that is about law that describes process in dependence both on time, and on frequency (scale of changes).

Wavelet is the certain function well localized (that is concentrated in small area of the certain point, and such, that sharply decreases to zero moderately distances from it) both in time, and in frequency area refers to. We shall note, that wavelet is not any concrete mathematical dependence or the formula, and any object or process that owns the specified features. There are wavelets having the diversified properties and such, that approach for the decision of a wide spectrum of problems.


Haar and Gauss Wavelets

It is possible to apply two operations to the wavelet:
• shift, that is moving of area of its localization to time;
• scaling (extension or compression), that is moving of area of its localization on frequency.


Shift and scaling of wavelet

Use of these operations, in view of property of wavelet localness in frequency-hour area, allows to analyze data on different scales and precisely to define position of their prominent features in time.

The idea of continuous wavelet-transformation consists in calculation of scalar multiplication (size that shows the size of "similarity" of two laws) investigated data with different shifts of certain wavelet on different scales. As a result there is a set of factors that show, how much behaviour of process in the given point is similar on "behaviour" of wavelet on the given scale. The closer kind of analyzed dependence in the field of the given point to a kind of wavelet, it has the greater absolute size corresponding factor. Negative factors show, that dependence similar to “mirror display” of wavelet.


Formation of factors of wavelet-transformation

The received factors can be presented in a graphic kind, if on one îñå to arrange shift of wavelet (an axis of time), and on another - scales (an axis of scales), and "to paint" points of the received scheme depending on size of corresponding factors: the more values of factor, the more brightly color. The received image name a card of factors of transformation, or simply card of transformation. The card of transformation is something like « x-ray to a picture » investigated process. On it all is visible its prominent features: scale and intensity of periodic changes, a direction and size of process importance, presence, an arrangement and duration of jumps, etc.


Conclusion

In the long term the future wavelet and wavelet-transformation researches it is necessary to concentrate attention to, so to say, practical application of that knowledge, that already received and turned out on the given theme. Exception of an opportunity of creation of the on duty knowledge base Means, that them now is a plenty., thus, it is enough of them for creation program object which main features would become novelty, utility and a practicality.


References

1. A.Jakovlev " Basis of wavelet-transformation of signals », a Series « Abstracts of lectures on radio engineering disciplines », release ¹10. Moscow: “Ôèçìàòëèò”, 2003, 80 pages. (http://www.math.spbu.ru/user/dmp/BookYakov.html)
2. V.Vorobev, V.Gribunin . «The theory and practice of wavelet-transformation» Moscow, 1999. ( http://lib.mexmat.ru/books/10151)
3. BaseGroup Company, Technologies of the data analysis (http://www.basegroup.ru/)
4. R.Polikar «The Wavelet Tutorial» (http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html)

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