Development of a wavelet-based algorithm to detect and determine ...

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