Chapter 2. Signal Analysis Methodology
2.1 The Continuous Wavelet Transform
2.1.1 The Continuous Wavelet Transform
The continuous wavelet transform (CWT) is a linear transformation, represents the signal x(t) as a sum of dilated and time-shifted wavelets in the form:
in which (t)L2() is called the mother wavelet and * represents a complex conjugate.
The mother wavelet (t) is dilated by various a, which are the scale parameters defining
the analysis window stretching, and shifted parameters b, which localize the wavelet
W . By this approach, one can examine the signal at different time window and frequency band by controlling the wavelet’s translation and dilatation. The higher the value of wavelet coefficient, the more similar the wavelet basis is to the original signal.
One can observe the variation of wavelet coefficients from the scalogram, which take the modulus of wavelet coefficients and show in time-scale plane with different color patterns
wavelet will provide different result of CWT. Therefore, the selection of an appropriate mother wavelet is an important issue which will be discussed in section 2.1.2.
The mother wavelet ψ(t) should satisfied the following admissibility condition to ensure existence of the inverse wavelet transform such as
2
There are countless mother wavelets used in practice for CWT. The wavelet analysis with real wavelet function like Mexican Hat unveils discontinuities or isolated peaks in the signal. For those who want to observe the phase variation, an analytical wavelet like the complex Morlet and Shannon can satisfy such requirement [6]. The scaling function
the infinite regularity permits an exact reconstruction.
Figure 2-1 shows the comparison between these three wavelets. All the wavelets have the same center frequency of 5 Hz. The result reveals that the Mexican Hat is simple zero phase wavelet, the complex Morlet wavelet has side lobes on both sides, and the Shannon wavelet is a leggy wavelet with a number of side lobes that die out on both sides.
One might suspect that the Morlet and Shannon wavelets will have somewhat lower temporal resolution due to their side lobes, in contrast to the Mexican Hat wavelet which should exhibit higher temporal resolution. But in Fourier Transform, Mexican Hat shows worse frequency resolution than other two’s and the complex Morlet wavelet shows best shape which matches the sharp peaks of signal in frequency domain. Therefore, for this data processing, the modified complex Morlet wavelet (MCMW) is used as the mother wavelet. The MCMW is defined as:
2 2
0 ( /2 )
( )t e eit t
(2.1.6)which is essentially a complex exponential modulated by a Gaussian envelope and
is a measure of the spread in time.The MCMW has Fourier transform
02 2
1
ˆ ( ) 2 e 2
(2.1.7)
The scaled wavelet and its Fourier Transform are
2 2
0( ) ( ) /2
1 i t b t b
*
(b,a)( ) e i b ˆ(a )
(2.1.9)For the Morlet wavelet, there is a unique relationship between the scale parameter a and Fourier frequency
, at which the wavelet is focus. The scale parameter a is inversely proportional to frequency
a 0
(2.1.10)
Figure 2-2 shows the relationship between scale parameters a and analysis frequencies
for using central frequency 01Hz and also take into consider the
term for choosing 1 and 2. From the Figure, we can see the uncertainty principle of signal processing. As the scale parameter a1 (0), it shows that an increase in
time resolution results in a decrease in frequency resolution, and vice versa. For the modified Morlet wavelet ( 2), it enhances the frequency resolution at the expense of time resolution.
2.1.3 Variable Central Frequency
Facing the uncertainty principle of signal processing, we change the general way of choosing different scales ( 0 const. ) but to choose different central frequency
(aconst.) through the analysis frequencies.
0 1
a
(2.1.11)
Using this method, we can keep the resolution the same and change the coefficients
from time-scale analysis to time-frequency analysis:
Compare with Figure 2-2, the resolution remain the same through the analysis frequency for using proposed variable central frequency. And Figure 2-4 shows the relationship between
, time duration and frequency bandwidth under a1. We can see that if we choose larger
, the time duration gets longer and the frequency bandwidth becomes narrower which will provide smoother wavelet spectrum in time-axis and sharper spectrum in frequency-axis, and vice versa. It is optional for the user to choose proper
, for example, if we care more about the frequency domain than time domain like white noise data, we can choose larger sigma.2.1.4 Edge effect melioration: Signal padding
The loss of considerable regions of a signal is the unfortunate consequence of edge effects. One possible solution to this problem is padding the beginning and end of the signal with real signal and leaving these values at the both sides to be corrupted by edge effect. Using this method, the characteristics of the signal can be preserved [11]. Figure 2-5 shows an example of signal padding operation for using 2. From Figure 2-4,
reflect a portion of the signal (4.9 seconds) about its beginning and end to compute the wavelet coefficients.
2.1.5 The Parseval Equality and Computation of Wavelet Transform via FFT The Parseval equality for the inner product of two functions f and g is [22]
1 ˆ ˆ
2.1.6 Example 1 (Simulation)
variable central frequencies (MCMW+VCF), a nonlinear time function is generated and
Application of the general CWT and the proposed MCMW+VCF to the nonlinear time function, Figure 2-7 shows the modulus of wavelet coefficients from these two methods in scalogram and the instantaneous frequency (IF) using Hilbert transform of the signals is also shown. We can see that using general CWT, the time resolutions are poor in lower frequencies, but using variable central frequencies, we can fix the resolution and provide better time-frequency representation.
2.1.7 Example 2 (Experiment)
The proposed method is also applied to the shaking table test of a SDOF RC frame [4] . The test specimen was a one-story two-bay frame (Figure 2-8) with overall height of 2 m and an approximate weight of 6454 kg, the dimension of the frame is shown in Figure 2-9. Take the average of the measurements from accelerometers A1, A4, and A7.
The results of the acceleration using general CWT and the proposed method using
different sigma ( 1,2) are shown in Figure 2-10. From these three scalograms, it’s
hard to compare the pros and cons of the two methods, it can only see that the bright regions under 2 are sharper and longer than other two. Using the scalogram to extract the ridge as instantaneous frequency (Figure 2-11), which will be describe in section 4.1.1, we can see that the general CWT provide poorer instantaneous frequency
in the initial small amplitude data because the energy preservation factor a1/2 provides larger weight for the analysis frequencies which are smaller than central frequency, it may also enlarge the frequency given by earthquake. And the difference between using sigma equals to 1 and 2 is that 2 provides more stable instantaneous frequency though the time of frequency shift is earlier than 1 due to its longer time duration.