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 39<br />
translation b is just a representation <strong>of</strong> time <strong>and</strong> scale a influences the duration so that<br />
has a direct relation with frequency.<br />
Figure 6.3: Mother <strong>wavelet</strong> <strong>and</strong> <strong>wavelet</strong> family<br />
A <strong>wavelet</strong> family forms a base or an orthogonal base <strong>of</strong> the general space L 2 (R). There-<br />
fore, nearly every interesting function could be represented as a linear combination <strong>of</strong><br />
a <strong>wavelet</strong> family <strong>and</strong> also could be reconstructed. A space base formed by a <strong>wavelet</strong><br />
family is also called a <strong>wavelet</strong> base. Such representation with a <strong>wavelet</strong> base is called<br />
the <strong>wavelet</strong> transform.<br />
Special characteristics <strong>of</strong> a <strong>wavelet</strong> family: As other transforms, the characteristic<br />
<strong>of</strong> the <strong>wavelet</strong> transform is decided by the properties <strong>of</strong> the space base formed by the<br />
<strong>wavelet</strong> family, i.e. the <strong>wavelet</strong> base.<br />
Figure 6.4(a) shows an orthogonal <strong>wavelet</strong> base in the time-frequency plane. The<br />
<strong>wavelet</strong> base is like a music note, as shown in Figure 6.4(b), which tells when <strong>and</strong><br />
which <strong>to</strong>ne (frequency) is played. the <strong>wavelet</strong> base forms such windows, which are
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