Abstract of thesis

Introduction

Now the methods of the data processing are widely based on wavelet-transformations. Wavelet are mathematical functions allowing to analyse different frequency components of information. Wavelet possess substantial advantages as compared to the Fur'e transformation, because wavelet-transformation allows to see not only about the frequency spectrum of signal but also about that, one or another accordion appeared in what moment of time. With their help it is possible easily to analyse irregular signals, or signals with sharp splashes. In addition allow wavelet to analyse information in obedience to a scale, on one of the set levels (by the piece of chalk or large). Unique properties of wavelet allow to construct a base, in which presentation of information will be expressed only by a few unzeroing coefficients. This property does wavelet very attractive for the data packing, including video and audio information. The shallow coefficients of decomposition can be cast aside in accordance with the chosen algorithm without considerable influence on quality of the packed information. Wavelet found wide application in digital treatment of image, treatment of signals and data analysis.

Purpose of work

The search of application domain of wavelet in the telecomunication systems is the primary purpose of master's degree work. It is planned to explore advantages and lacks of this type of analysis of signals in wideband communication networks, at different values of noised communication channel.

Actuality

On today's moment in the telecommunication systems one of major problems, there is the fight against noises. Especially sharply this problem stands in telephone networks, the flow which lines of yet lay down in old times. Just in such lines and there is plenty of noises for diverse reasons, and just in order to not change these lines which lie by thousand kilometres through this country, and the effective enough method of filtration is needed. Represented method, is relatively new, and the scopes of application of this method until now are not certain. There is the great number of theoretical reflections in present time, however with every year wavelet-transformation finds all more and more of places for application.

Aims of analysis of signals

The "analysis" of signals is not only their cleanly mathematical transformations but also receipt on the basis of these transformations of conclusions about the specific features of the proper processes and objects.
Analysis of signals are usually used in:
- Determination or estimation of numerical parameters of signals (energy, middle power, mean quadratic value and other).
- Decomposition of signals on elementary constituents for comparison of properties of different signals.
- Comparison of degree of closeness, "alikeness", "relation" of different signals, including with certain quantitative estimations.

The mathematical vehicle of analysis of signals is very vast, and is widely used in practice in all regions without the exception sciences and techniques.

Methods of treatment of unstationary signals

Most signals have difficult frequency-temporal descriptions. As a rule, such signals consist of near at times, short time high-frequency component and of long durations, near on frequency of low frequency component.

For the analysis of such signals we need method able to provide good permission on frequency and time. The first is required for localization of low frequency constituents, second – for permission component of high-frequency.

Wavelet transformation swiftly conquers popularity in so different regions, as telecommunications, computer graphics, biology, astrophysics and medicine. Due to good adjusted to the analysis of unstationary signals it became a powerful alternative to the Fur'e transformation in a number of appendixes. Because many signals are unstationary, the methods of wavelet analysis are used for recognition and discovery of key diagnostic signs.

The Fur'e Transformation represents the signal set in a temporary realm, as decomposition on ortogonal base functions (to the sines and cosines), selecting frequency components thus. The lack of the Fur'e transformation consists that frequency components can not be noncommunicative in time, that imposes restraints on applicability of this method to the row of tasks (for example, in the case of study of dynamics of change of frequency parameters of signal on a temporal interval).

There are two approaches to the analysis of unstationary signals of such type.

First is the local transformation Fur'e (short-time Fourier transform). Following on this way, we work with an unstationary signal, as with stationary, preliminary breaking up it on segments (windows), statistics of which do not change in course of time.

The second approach is wavelet transformation. In this case an unstationary signal is analysed by decomposition on the base functions got from some prototype by compression, tensions and changes. A function prototype is named maternal, or analysing wavelet.

Brief review of the Fur'e transformation

The Fur'e transformation is the classic method of frequency analysis of signals, essence of which it is possible to express by a formula

Result of the Fur'e transformation – peak is frequency spectrum, on which it is possible to define the presence of some frequency in the explored signal.

In the case when a question about localization of temporal position of frequencies does not get up, the Fur'e method gives good results. But if it is necessary to define the temporal interval of presence of frequency it is necessary to apply other methods.

One of such methods is the generalized method Fur'e (local transformation Fur'e). This method consists of the following stages:
1. a “window” is temporal interval is created in the explored function, for which the function of f(x) is not equal 0, and f(x)=0 for the other values;
2. for this “window” the Fur'e transformation is calculated;
3. “a window” is moved, and for it the Fur'e transformation is also calculated.

“Passing” by such “window” along all signal, some three-dimensional function depending on position “windows” and frequencies turns out.

This approach allows to define the fact of presence in the signal of any frequency, and interval of its presence. It considerably extends possibilities of method as compared to the classic transformation Fur'e, but there are the certain failings. In obedience to the consequences of principle of the Geyzenberga vagueness in this case it is impossible to assert the fact of presence of frequency of w0 in a signal in the moment of time of t0 - it is possible only to define that the spectrum of frequencies (w1,w2) is in an interval (t1,t2). Thus permission on frequency (at times) remains permanent without depending on the region of frequencies (times), research is produced in which. Therefore, if, for example, a high-frequency constituent is substantial in a signal only, multiplying permission is possible only changing the parameters of method. As a method, not possessing similar family by failings, the vehicle of wavelet analysis was offered.

Substantive provisions of wavelet - analysis

The discrete and continuous wavelet - analysis is distinguished, the vehicle of which can be applied both for continuous and for discrete signals.

A signal is analysed by decomposition on the base functions got from some prototype by compression, tensions and changes. A function-prototype is named analysing (maternal) wavelet.

Wavelet function must fulfil conditions:
1. Mean value (integral on all line) equal 0.
2. A function quickly decreases at time aspiring to endlessness.

In general case wavelet transformation of function of f(t) looks:

where t is axis of time, x is moment of time, s is parameter reverse to frequency, and (*) – means complex-attended.

In the wavelet analysis a function is a staple - wavelet. Use most popularity following wavelets:

So, we have some function f(t), depending on time. By the result of it wavelet - analysis there will be some function W(x,s), which depends already on two variables: from time and from frequency (in inverse ratio). For every pair the x and s recipe of calculation of wavelet transformation is following:
1. The function of wavelet stretches in s times on a horizontal line and in 1/s times on a vertical line.
2. Further it is moved pithily x. Got veyvlet is designated psy(x,s).
3. We using middle result in neighbouring of point of s through psy(x,s).

A fully evident picture illustrating frequency-temporal descriptions of signal “appears” as a result. On abscising axis time is put aside, on a y-axis is frequency (sometimes the dimension of y-axis gets out so: log(1/s), where s-frequency is), and the absolute value of wavelet transformation for the concrete pair of x and s determines a color which this result will be represented (what in a greater degree one or another frequency is in a signal in the concrete moment of time, the darker there will be a tint).

In the case of wavelet analysis (decomposition) of process (signal) due to the change of scale of wavelets able to expose distinction in descriptions of process on different scales, and by means of change it is possible to analyse properties of process in different points on all explored interval.

Studying these properties, some components can be deleted, that is widely used for the delete of noises.

As a rule, for the delete of noise the reception is delete of high-frequency constituents from the spectrum of signal known from the technique of filtration is used well. However as it applies to wavelet there is another way is limitation of level of going coefficients into detail.

Brief features of signal – making detail coefficients with high maintenance of noises. Setting some threshold for their level, and cutting away some coefficients on a level, it is possible to decrease the level of noises. But most important, that the level of limitation can be set for every coefficient separately, that allows to build the systems of cleaning of signals adaptive to the changes from noise on the basis of wavelets.

Basic difference of wavelet - to filtration from the traditional methods of selection of useful signals from hindrances and noises consists that the choice of parameters of wavelet filter poorly enough depends on descriptions of spectrum of the analysed signal. It allows to avoid those difficulties which usually accompany the choice of parameters of frequency-transmission function of traditional filter, when a too narrow frequency window results in distortion of form of useful signal and worsening of settling ability of the system, and too wide window — to uneffective of process of filtration from the large level of noises in an output signal.

Got results

On today's moment a programmatic method realizes the process of wavelet transformation. The receipt of spectrums of different signals became possible, and also cleaning of signals from some types of noises. In future is planned on the example of model of the ADSL communication channel, in different places to apply this method of filtration and compare it to already existing.

Conclusion

On today's moment wavelet transformation conquers everything large popularity, that gives a wide demesne in practical realization of wavelet transformation, in master's degree work it is planned to apply wavelet transformation for filtration of signals in wideband communication networks. ADSL will undertake as such technology. In which it is realized simultaneous transmission of voice and information on ordinary telephone lines, which the great number of different noises is in. As a result of research the model of communication channel will be created and it will be certain how expediently to use this method in this technology.

Literature: