Development of a wavelet-based algorithm to detect and determine ...
Development of a wavelet-based algorithm to detect and determine ...
Development of a wavelet-based algorithm to detect and determine ...
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6.2. BASIC IDEAS OF THE WAVELET TRANSFORM 35<br />
gorithms etc.<br />
Therefore, the situation is that the <strong>wavelet</strong> transform applications in power systems<br />
are very meaningful but just at the beginning, <strong>and</strong> there is a great deal <strong>of</strong> concerning<br />
research <strong>to</strong> do.<br />
The <strong>wavelet</strong> transform with its special characteristics, such as the time-frequency lo-<br />
cation, the au<strong>to</strong>-adaptivity <strong>to</strong> frequency, the flexible selection <strong>and</strong> the fast <strong>algorithm</strong>s<br />
etc., opens a door <strong>to</strong>wards a lot <strong>of</strong> possibilities <strong>and</strong> large developing field in power<br />
systems. However, it is not very easy <strong>to</strong> start underst<strong>and</strong>ing <strong>and</strong> using the <strong>wavelet</strong><br />
transform because the <strong>wavelet</strong> transform is built up on the theories <strong>of</strong> modern math-<br />
ematics <strong>and</strong> modern signal processing [7] <strong>and</strong> the most the <strong>of</strong> published books about<br />
<strong>wavelet</strong> transform are the contributions <strong>of</strong> mathematicians.<br />
In this chapter, the first section gives the basic ideas <strong>of</strong> the <strong>wavelet</strong> transform. In the<br />
second section, the continuous <strong>wavelet</strong> transform (CWT) is introduced. The discrete<br />
<strong>wavelet</strong> transform (DWT) as the key point will be introduced in the third section. The<br />
forth section displays how the <strong>wavelet</strong> transform works through an example. The fifth<br />
section then describes <strong>and</strong> discusses the general problems <strong>and</strong> corresponding solutions<br />
in practical applications.<br />
6.2 Basic ideas <strong>of</strong> the <strong>wavelet</strong> transform<br />
6.2.1 Representation <strong>of</strong> signals through transforms<br />
The idea in mathematics <strong>and</strong> in engineering is <strong>to</strong> represent <strong>and</strong> <strong>to</strong> analyze a signal<br />
or a system in different domains. The changes from on domain <strong>to</strong> another are called<br />
transforms.<br />
An useful transform is <strong>to</strong> decompose a signal in<strong>to</strong> elementary functions for a space.<br />
These elementary functions should be chosen with some care <strong>to</strong> be sure that the trans-<br />
form is invertible. Generally, they should form an orthogonal base or a base for the<br />
respective space.