**2.2 Discrete wavelet transform**

In the discrete wavelet transform the term "discrete" applies only to the parameters in the transformed domain, that is, scales and translations, and not to the independent variable time, of the function being transformed. The discrete wavelet transform provides a set of coefficients corresponding to points on a grid or two-dimensional lattice of discrete points in the time-scale domain. This grid is indexed by two integers, the first, denoted by *m* , corresponds to the discrete steps of the scale, while the second, denoted by *n* , corresponds to the discrete steps of translation (time displacement). The scale *a* becomes <sup>0</sup> *<sup>m</sup> a a* and translation becomes 0 0 *<sup>m</sup> b nb a* , where <sup>0</sup>*a* and <sup>0</sup>*b* are the discrete steps of the scale and translation, respectively (Young, 1995). Then the wavelet can be represented by:

$$
\psi\_{m,n}(t) = a\_0^{\frac{-m}{2}} \psi \left( a\_0^{-m} t - n b\_0 \right) \tag{1}
$$

Application of Wavelet Transform and Artificial Neural Network to

**2.3.1 Analysis or decomposition** 

number of samples of the original signal.

Fig. 2. Structure of the multiresolution analysis

(Mallat, 1989):

decomposition.

Extract Power Quality Information from Voltage Oscillographic Signals in Electric Power Systems 181

than the original one. Over-sampling by a factor of 2, consists of inserting zeros between

The structure of the multiresolution analysis is shown in Figure 2. The original signal passes through two filters, a low pass filter *g k*( ) , the function scale, and a high pass filter *h k*( ), the mother wavelet. The impulse response of *h k*( ) is related to the impulse response of *g k*( ) by

In the structure presented in Figure 2, the input signal is convolved with the impulse response of *h k*( ), and *g k*( ) , obtaining two output signals. The low pass filter output represents the low frequency content of the input signal or an approximation of it. The high pass filter output represents the high frequency content of the input signal or a detail of it. It should be noted in Figure 2 that the output provided by the filters has together twice the

This drawback is overcome by the process of decimation performed on each signal, thereby obtaining the signal *cD* , the wavelet coefficients that are the new signal representation in the wavelet domain, and the signal *cA* , the approximation coefficients which are used to feed the next stage of the decomposition process in an iterative manner resulting in a multi-level

The decomposition process in Figure 2 can be iterated with successive approximations being decomposed, then the signal being divided into several resolution levels. This scheme is called "wavelet decomposition tree" or "pyramidal structure" (Young, 1995 and Misit et al, 2000). Figure 3 shows the schematic representation of a signal being decomposed at multiple levels.

<sup>1</sup> ( ) ( 1) (1 ) *<sup>k</sup> hk g k* (5)

each two samples resulting in a signal with twice the elements of the original one.

Filter *h k*( ) is the mirror of filter *g k*( ) and they are called quadrature mirror filters.

The discrete wavelet transform is given by:

$$\mathcal{W}\_f(m, n) = a\_0^{\frac{-m}{2}} \int\_{\mathbb{R}} f(t) \varphi(a\_0^{-m}t - nb\_0) dt \tag{2}$$

where, *mn Z* , , and Z is the set of integer numbers.

The parameter *m* which is called level, determines the wavelet frequency, while the parameter *n* indicates its position.

The inverse discrete wavelet transform is given by:

$$f(t) = k \sum\_{m=0}^{\infty} \sum\_{n=0}^{\infty} \mathcal{W}\_f(m, n) a\_0^{\frac{-m}{2}} \,\psi(a\_0^{-m} t - n b\_0) \tag{3}$$

where *k* is a constant that depends on the redundancy of the combination of the lattice with the used mother wavelet (Young, 1995).

Along with the time-scale plane discretization, the independent variable (time) can also be discretized. The sequence of discrete points of the discretized signal can be represented by a discrete time wavelet series DTWS. The discrete time wavelet series is defined in relation to a discrete mother wavelet, *h k*( ). The discrete wavelet time series maps a discrete finite energy sequence to a discrete grid of coefficients. The discrete time wavelet series is given by (Young, 1995).

$$W\_f(m, n) = a\_0^{\frac{-m}{2}} \sum f(k)h(a\_0^{-m}k - nb\_0) \tag{4}$$

#### **2.3 Multiresolution analysis**

Multiresolution Analysis - MRA, aims to develop a signal *f t*( ) representation in terms of an orthogonal basis which is composed by the scale and wavelets functions. An efficient algorithm for this representation was developed in 1988 by Mallat (Mallat, 1989) considering a scale factor 0*a* 2 and a translation factor <sup>0</sup> *b* 1 . This means that at each decomposition level *m* , scales are a power of 2 and translations are proportional to powers of 2. Scaling by powers of 2 can be easily implemented by decimation (sub-sampling) and over-sampling of a discrete signal by a factor of 2. Sub-sampling by a factor of 2, involves taking a signal sample from every two available ones, resulting in a signal with half the number of samples than the original one. Over-sampling by a factor of 2, consists of inserting zeros between each two samples resulting in a signal with twice the elements of the original one.
