aid of a wavelet transform, it is possible to better zoom in on very short-lived high frequency phenomena, like transients in signals or singularities in functions. Wavelets are mathematical microscopes that are able to magnify a given part of a function with a certain factor represented by a value of scale parameter.
This magnification is associated with the extraction of the information at a certain scale hidden in a local area of the analyzed function.
Figure 9.3: (a) Time domain plot of a synthetic test signal composed from summation of 1 Hz, 5 Hz, and 10 Hz frequency sinusoids. (b) Frequency domain plot (Fourier spectrum) of the test signal confirming the presence of three frequency components (1 Hz, 5 Hz, 10 Hz) in the signal.
0 1 2 3 4 5
10−1 100 101
Time (s)
scale (s)
(a)
−10
−5 0 5 10
0 1 2 3 4 5
10−1 100 101
Time (s)
scale (s)
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 105 −4
Figure 9.4: Plots of wavelet coefficients at different values of scale and time obtained from the Morlet wavelet transform showing : (a) amplitude, (b) wavelet energy spectrum of the test signal given in Figure 9.3(a).
possess lower energies. The maxima of the absolute values of|W(s, a)| are concentrated in the periods of the three components. Moreover, they are distributed in time according to the periodicity of the resulting wave field (Massel [172]). Three dominant frequencies were detected using the wavelet scale s=1.0, 0.2 and 0.1. The repetition of features of colour-coded contours ats=1.0, 0.2 and 0.1 for all time shifts indicate the periodicity of the signal with a time period of 1.0, 0.2 and 0.1 s. Hence all three time periods 1.0, 0.2 and 0.1 are observed, which may have signified a one-to-one relationship between scale and period, written as :
s=βT (9.19)
where β is a constant that depends on the analyzing wavelet. Seena & Sung [92] stated that for the complex Morlet wavelet,β= 1 whileWang [216] obtained the reciprocal relation between the temporal scaling factor and the equivalent Fourier frequency with constantβ = 0.97. The energy spectrum results in Figure 9.4 (b) show that as expected, the same amount of energy is contained in the three wavelet scales.
0 5 10 15 20 10−1
100 101
Time (s)
scale (s)
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10−4
Figure 9.5: (a) Synthetic test signal composed of 1 Hz, 5 Hz, and 10 Hz frequency sinusoids with different starting times (b) corresponding wavelet energy spectrum of the test signal.
Figure 9.5 (a) shows another synthetic test signal consisting of three sinusoids of amplitudes 2.0 but with different starting times. The 1 Hz signal appears for times between 0 and 8 s, the 5 Hz for times between 8 and 16s, while the 10 Hz component appears between 16 and 24 s. Both frequency and the beginnings of the signal components are clearly visible in the wavelet energy spectrum results shown in Figure 9.5 (b). Results show that the three sinusoids occur at the same scales of 1.0, 0.2 and 0.1 (as before) but this time starting at different times. As expected, equal amounts of energy are contained at these three spatial scales. Unlike results for the previously given test signal, the energy for this particular test signal do not appear for all the time shifts. From these results it can be concluded that the wavelet spatial scale holds a one-to-one relationship with the period of the spatial synthetic structures presented here.
Therefore the scales give information about the frequency. The finer the scale, the higher the frequency and vice-versa.
9.5.2 Energy contribution by the synthetic signal wavelet scales
The signal shown in Figure 9.6 (a) consisting of 1 Hz, 5 Hz and 10 Hz sinusoids were summed up and used to test the algorithm for examining the energy content in the various scales. The figure shows only up to a timeof 6 s. The three frequency components of the signal have relative amplitudes of 1, 2 and 4 respectively. Figure 9.6(b) and (c) show results of the wavelet analysis of the synthetic signal showing the amplitude and energy spectrum. These three figures show presence of signals centered
around three main scales of 0.1, 0.2 and 1.0 which correspond to the periods of 10 Hz, 5 Hz and 1.0 Hz components, respectively. Most of the wave energy in the signal is confined around the smallest scale of 0.1 that correspond to the largest frequency of 10 Hz in the signal, since the component had the largest amplitude.
0 5 10 15 20
10−1 100 101
Time (s)
scale (s)
(b)
−6
−4
−2 0 2 4 6
0 5 10 15 20
10−1 100 101
Time (s)
scale (s)
(c)
2 4 6 8 10 12 14 16 x 10−4
Figure 9.6: (a) Synthetic signal consisting of 1 Hz, 5 Hz and 10 Hz components, used for checking energy contribution at different scales, (b) amplitude of the signal, (c) wavelet energy spectrum of the signal.
In the wavelet analysis of the synthetic signal, once again a total of 43 wavelet scales were used and have values ranging between 0.04 - 10.4. It is important, therefore, to note that the wavelet results of Figure 9.6 do not show all the scales for this signal. As in Longo [215], the scales in the signal were subdivided into 3 bands, namely : 0.04 - 0.155, 0.170 - 0.60 and 0.65 - 10.4, which are here namedmicro scales,mid scales andmacro scales respectively. The bands were intentionally set so that each frequency component in the test signal lies in at most one of the bands. The energies associated with the scales used here are labeledEmicro,Emid andEmacro. The total amount of energy in each of the three spatial scale bands was obtained by summing up the energy for all scales in the band. Table 9.1 summarizes
the characteristics of the three components that make up the signal under study, and also presents the expected energy versus the wavelet energy for each of the three components. The energy obtained by wavelet analysis is very close to the expected,Erms, but is a little less, because of the quantization errors involved in its calculation.
Table 9.1: Energy content of the signal shown in Figure 9.6(a).
f(Hz) scale band A Erms Wavelet energy,WE % of total energy
10.0 micro 4.0 8.0 7.96 76.7
5.0 mid 2.0 2.0 1.94 18.7
1.0 macro 1.0 0.5 0.48 4.6
Total 10.5 10.38 100
0 5 10 15 20
0 10 20 30 40 50 60 70 80 90 100
Time (s)
% of total energy
Emicro Emid Emacro
Figure 9.7: Variation of the energy contribution by different scale bands with time.
At each time, the energy available in each of the three scale bands was then normalized by the total available energy at that time, in order to get the percentage of the total energy existing at each scale.
Figure 9.7 shows the percentage of the total energy contributed by each scale band at different times. As expected, the macro scales are observed to make the least base contribution of about 4.7 % to the total energy available at each time, for all the times. The mid scales contribute 17.3 %, while the micro scales contribute 78.0 %. These results are consistent with the percentage of the total energy given in the last column of Table 9.1.
Another signal shown in Figure 9.8 was also used to test the algorithm for examining the energy content in the various scales. All three components have an amplitude of 4.0. The 10 Hz signal is available for times between 2- 6 s, the 5 Hz between 10 -14 s and the 1 Hz between 18 - 22 s. For all other times the signal components are zero. The computed Morlet energy spectrum for this test signal is shown in Figure 9.9 and clearly shows the duration of each of the three components. As can be seen, energy in the signal is available in equal amounts only at wavelet scales 0.10, 0.20 and 1.0 which correspond to the periods of the three components. This confirms the ability of the wavelet algorithm employed here, to determine the energy contribution by the various scales to the total energy in the signal.
Figure 9.8: Synthetic signal consisting of 10 Hz, 5 Hz and 1 Hz sinusoids used for checking energy contribution at different scales.
0 5 10 15 20
10−1 100 101
Time (s)
scale (s)
(a)
−20
−15
−10
−5 0 5 10 15 20
0 5 10 15 20
10−1 100 101
Time (s)
scale (s)
(b)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x 10−3
Figure 9.9: Wavelet analysis results of the test signal shown in Fig. 9.8 showing (a) amplitude and (b) energy spectrum.