CHAPTER 3 BACKGROUND
3.2 Principle of Wavelet Analysis
The history of wavelets begins with the development of the traditional Fourier transform (FT) (Donald and Walden, 2000), which is widely applied in signal analysis and image processing. Fourier transform breaks down a signal into the sum of infinite series of sines and cosines of different frequencies, in another word; FT is a mathematical technique for transforming our view of the signal from time-based to frequency-based. Fourier transform is very effective in problems dealing with frequency location. However, time information is lost during the process of transforming to frequency domain. This means that although we might be able to determine all the frequencies present in a signal, we do not know when they are present. In the time series process data, the most important part of the signal is the transient characteristics: drift, trends, and abrupt changes, and FT is not suited to detect them.
In an effort to improve the performance of the FT, the short time Fourier transform (STFT) has been developed in signal analysis. STSF compromises between the time and frequency based views of a signal by examining a signal under a fixed time window. The drawback of STSF is that the time window is fixed and same for all the frequencies. Many signals require a more flexible approach; the window size is required to vary according to the frequency.
Wavelet analysis or wavelet transform is close in spirit to the Fourier transform, but has a significant advance. It applies a windowing technique with variable-sized regions, a shorter time interval is used to analyze the high frequency components of a signal and a longer one to analyze the low frequency components of the signal. Wavelet analysis is very effective for dealing with local aspects of a signal, like trends, breakdown points, and self similarity. Furthermore, wavelet analysis is capable of removing noise from signal and compress signal.
3.2.2 Wavelet Properties
domains, and are obtained from a single prototype wavelet, called mother wavelet or basic function Ψ (t), by scaling and translation (shifting). The wavelet family can be defined as
) ( 1 ) ( , a b t a t b a − Ψ = Ψ (3.2.1)
Where a and b represent the scale and translation parameters, respectively. In the discrete case, the scale and translation parameters are discretised as a 2= j and b=k2j. Ψ (t) can be rewritten as: ) 2 ( 2 ) ( ) ( /2 , , t t t k j j k j b a =Ψ = Ψ − Ψ − − (3.2.2)
where j and k denote the scale and translation parameters, respectively. The translation parameter determines the location of the wavelet in the time domain, while the scale parameter determines the location of the wavelet in the frequency domain.
Given a function x(t), the wavelet coefficients are obtained through the inner product
operation:
∫
∞ ∞ − Ψ = t x t dt b a W( , ) a,b( ) () (3.2.3)where j and k denote the scale and translation parameters, respectively.
In general, wavelet analysis uses the wavelet functions which can be stretched and translated with a flexible resolution in both frequency and time. The flexible windows are adaptive to the entire time-frequency domain, which narrows while focusing on high- frequency signals and widen while searching the low-frequency background. In this way, wavelet analysis allows the wavelets to be scaled to match most of the high and low frequency signal so as to achieve the optimal resolution with the least number of base functions.
continuous wavelets and discrete wavelets. The simplest discrete wavelet is the Haar wavelet, which is based on a box function. Daubechies wavelets with different orders are the most popular wavelets. In this study, Daubechies and Symlets wavelets are used for signal analysis. They are orthonormal with a compact support and capable of capturing smooth low-frequency features. Symlets are less asymmetric than Daubechies. There are other wavelet functions, including Coiflet, Mayer, Morlet, and Mexican Hat. The wavelets are chosen based on their shape and their ability to analyze the signal in a particular application. 3.2.3 Wavelet Transform Decomposition
The common application of wavelet transform is signal decomposition. Mallat (1989) developed a recursive algorithm to decompose and reconstruct a signal using wavelet transform method. This method connects the continuous time multi-resolution to the discrete time filters. As shown in Fig. 3.2, the wavelet transform can be used to decompose multivariate signals s into approximations a1 and details d1 coefficients at the first level.
Application of the same transform on the approximations a1 causes them to be decomposed
further into approximations a2 and details d2 coefficients at the second level. The
decomposition process can continue to a level L as long as the length of approximation coefficients in al is more than the length of coefficients in the wavelet filter. The wavelet
transform works like a filter. After passing a signal through a wavelet transform filter, wavelet coefficients are generated. The wavelet transform contains a low pass filter (only obtaining low frequencies), which is denoted by L0, and a high pass filter (only obtaining
high frequencies), which is denoted by H0. At each level, the original signal, s, passes
through two both low and high pass filters and emerges as two signals, which is detail coefficients dn, and approximation coefficients an. The terms “approximation” and “detail”
are named by the fact that an. is the approximation of an-1 corresponding to the “low
frequencies” of an-1, whereas the detail dn takes into account its “high frequencies”. Wavelet
coefficients at various frequencies reflect the signal variations at those frequencies and corresponding times. Clearly, at the kth step of this partitioning procedure, the original
signal, s, is expressed by Eq. 3.2.4. 1 1 d d d a s= k + k + k− +L+ (3.2.4)
With ak. representing the smooth signals referring to the time scale 2k and dn is the detail of
time series with the time scale located in the interval [2k-1, 2k].
Figure 3.2 Three level wavelet decomposition tree
Note: ↓2 means the number of coefficients is halved through the filters
One of the most important applications of wavelet analysis is the compression of signals. The “true” signal tends to dominate the low-frequency area. The common approach to filtering is to remove the high-frequency components above a certain level since they represent the detailed information in the signal. The approximation coefficients resemble a moving average. The lower frequency approximation coefficient gives the identity of the signal and reflects the signal trend. The number of wavelet coefficients decreases by a factor of 2 at coarser scales. In this way, the original signal vector is smoothed and halved through the low pass filters. This reaches the objective of signal compression.
The discrete wavelet transform can be used to analyze or decompose signals. The inverse process of the discrete wavelet transform is called signal reconstruction, how those components can be assembled back into the original signal without loss of information. Whilst wavelet analysis involves filtering and downsampling, the wavelet reconstruction
S H0 2 d1 L0 2 H0 2 d2 L0 2 H0 2 d3 L0 2 a3 a2 a1
process consists of filtering and upsampling. Upsampling is the process of lengthening a signal component by inserting zeros between samples. Wavelet analysis is still a new and emerging field; more possible unknown applications of wavelet analysis are waiting for exploring.
3.3 Pattern Matching in Historical Data