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 45<br />
Scale <strong>and</strong> Frequency Scaling a function simply means stretching or compressing it.<br />
Figure 6.3 shows the effect <strong>of</strong> the scale a. The larger scales correspond <strong>to</strong> the more<br />
"stretched" waves. It is clear that, for a sinusoidal function, the scale a is inverse pro-<br />
portion <strong>to</strong> the frequency ω.<br />
There is a similar relation between the scale <strong>and</strong> the frequency in the <strong>wavelet</strong> trans-<br />
form. The more stretched the <strong>wavelet</strong>, the longer the portion <strong>of</strong> the signal will be<br />
analyzed, <strong>and</strong> thus the coarser or lower frequency features <strong>of</strong> the signal will be mea-<br />
sured by the <strong>wavelet</strong> coefficients. Therefore, there is a corresponding relation between<br />
the <strong>wavelet</strong> scale <strong>and</strong> frequency as following:<br />
• Small scale ⇒ compressed <strong>wavelet</strong> ⇒ rapidly changing detail ⇒ high frequency.<br />
• Large scale ⇒ stretched <strong>wavelet</strong> ⇒ slowly changing, coarse feature ⇒ low fre-<br />
quency.<br />
In other words, frequency is inverse proportional <strong>to</strong> scale. However, the proportion<br />
ratio depends on the mother <strong>wavelet</strong>, sampling frequency <strong>and</strong> the <strong>wavelet</strong> transform<br />
type (CWT or DWT). Figure 6.8(a) shows the relation between frequency <strong>and</strong> scale<br />
with a given sampling frequency. The sampling frequency leads <strong>to</strong> the corresponding<br />
frequency b<strong>and</strong> in different scale levels. Figure 6.8(b) is an example for a sampling<br />
frequency <strong>of</strong> 4,8 kHz. Of course the sampling frequency should fulfil the Nyquist<br />
criteria which depends on the highest frequency <strong>of</strong> the signal.<br />
For obtaining a fixed relation, there is a simple method. [9] The relation can be derived<br />
for a particular <strong>wavelet</strong> function by making the <strong>wavelet</strong> transform <strong>of</strong> a cosine wave<br />
function with a known frequency <strong>and</strong> computing the scale a at which the <strong>wavelet</strong> co-<br />
efficient reaches its maximum. Following this method, the relation between frequency<br />
<strong>and</strong> scale is obtained for the Db4 <strong>wavelet</strong> in form ψ(t) = e −t2 /2 cos(5t):<br />
1<br />
f = kfs<br />
a<br />
1<br />
= 0, 8fs<br />
a<br />
(6.8)<br />
Where f <strong>and</strong> a are corresponding <strong>to</strong> frequency <strong>and</strong> scale, fs denotes sampling fre-<br />
quency in Hz <strong>and</strong> k is a constant which changes solely with function form <strong>of</strong> each