**2.3.1 Analysis or decomposition**

180 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

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>

> 2 , 00 <sup>0</sup> () ( ) *m*

2 <sup>0</sup> 0 0 ( , ) () ( )

0 0

*m n*

*m*

*R W mn a f t a t nb dt* 

The parameter *m* which is called level, determines the wavelet frequency, while the

( ) ( ,) ( )

where *k* is a constant that depends on the redundancy of the combination of the lattice with

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

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

),( )()( <sup>0</sup> <sup>0</sup>

*f*

2 <sup>0</sup> *nbkahkfanmW <sup>m</sup> m*

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

scales are a power of 2 and translations are proportional to powers of 2. Scaling by

*f t k W m n a a t nb*

*f*

2

*m*

0 0 0

*m*

(3)

(4)

 *t a a t nb* 

*m*

*m*

translation, respectively (Young, 1995). Then the wavelet can be represented by:

*m n* 

*f*

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

The inverse discrete wavelet transform is given by:

*<sup>m</sup> b nb a* , where <sup>0</sup>*a* and <sup>0</sup>*b* are the discrete steps of the scale and

(1)

(2)

translation becomes 0 0

The discrete wavelet transform is given by:

parameter *n* indicates its position.

the used mother wavelet (Young, 1995).

by (Young, 1995).

level *m* ,

**2.3 Multiresolution analysis** 

*<sup>m</sup> a a* and

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 (Mallat, 1989):

$$h(k) = (-1)^{1-k} \, \lg(1-k) \tag{5}$$

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

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 number of samples of the original signal.

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 decomposition.

Fig. 2. Structure of the multiresolution analysis

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.

Application of Wavelet Transform and Artificial Neural Network to

**3. Probabilistic neural network** 

as simple as desired (Specht, 1990).

**3.1 The Bayes strategy for pattern classification** 

categories in which the state of known nature

*x* . Then the Bayes decision rule is given by:

*Ah* is the a priori probability of category

 

and ( ) *<sup>B</sup> d x*

whether

 

where:

 *<sup>A</sup>* or 

priori probability that

decision ( ) *<sup>A</sup> d x*

Extract Power Quality Information from Voltage Oscillographic Signals in Electric Power Systems 183

The structure of a Probabilistic Neural Network (PNN) is similar to a feed forward network. The main difference is that the activation function is no longer the sigmoid; it is replaced by a class of functions which includes, in particular, the exponential function. The main advantage of PNN is that it requires only one step for training and that the decision surfaces are close to the contours of the Bayes optimal decision when the number of training samples increases. Furthermore, the shape of the decision surface can be as complex as necessary, or

The main drawback of PNN is that all samples used for the training process must be stored and used in the classification of new patterns. However, considering the use of high-density memories, problems with storage of training samples should not occur. In addition, the PNN processing speed in the classification of new patterns is quite satisfactory, and even several times faster than using back propagation algorithms as reported by (Maloney et al, 1989).

One of the traditionally accepted strategies or decision rules used to patterns classification is that they minimize the "expected risk." Such strategies are called Bayes strategies, and can

To illustrate the Bayes decision rule formalism, it is considered the situation of two

( ) () () ( ) () () *A AA A BB B B AA A BB B*

*dx if h l f x hl f x dx if h l f x hl f x*

, can be

 *<sup>A</sup>* or 

*<sup>B</sup>* . Then, the boundary between the regions in which the Bayes

(7)

*<sup>B</sup>* based on a measurements set represented by a *n* dimension vector

*<sup>B</sup>* . It is desired to decide

*<sup>A</sup>* and

when

*<sup>A</sup>* patterns occurrence, and 1 *B A h h* is the a

() () *A B f x Kf x* (8)

*h l* (9)

 *<sup>B</sup>* ,

*B*

when

be applied to problems containing any number of categories (Specht, 1988).

where ( ) *Af x* and ( ) *Bf x* are the probability density functions for categories

*<sup>A</sup>* ; *Bl* is the uncertainty function associated with the decision ( ) *<sup>A</sup> d x*

is given by:

respectively, *Al* is the uncertainty function associated with the decision ( ) *<sup>B</sup> d x*

*B B A A h l <sup>K</sup>*

It should be noted that, in general, the decision surfaces of two categories defined by Eq. (8) can be arbitrarily complex, since there are no restrictions on the densities except for those conditions to which all probability density functions must satisfy, namely that they must be always non-negative, and integrable and their integrals over all space be equal to unity.

Fig. 3. Schematic representation of a signal being decomposed at multiple levels.

Since the multiresolution analysis process is iterative, it can theoretically be continued indefinitely. In fact, the decomposition can proceed only up to 1 (one) detail, consisting of a single sample. The maximum number of decomposition levels for a signal having *N* samples is given by <sup>2</sup> log *N* .

#### **2.3.2 Synthesis or reconstruction**

The synthesis process or reconstruction is to obtain the original signal from the wavelet coefficients generated by the analysis or decomposition process. While the analysis process involves filtering and sub-sampling, the synthesis process performs a reverse sequence, over-sampling and filtering. The filters used in the synthesis process are called reconstruction filters, being *g k* ( ) the low pass filter, and *h k* ( ) the high pass filter. Figure 4 shows the reconstruction scheme from a single decomposition stage.

Fig. 4. Reconstruction scheme from a single decomposition stage.

It is observed from Figure 4 that to retrieve the original signal, it is necessary to reconstruct details and approximations. Details could be obtained with over-sampling of the *cD* coefficients, and a subsequent filtering with *h k* ( ). Approximations are obtained with oversampling of the coefficients *cA* , and a subsequent filtering with *g k* ( ) .The original signal is then obtained by:

$$S = A + D \tag{6}$$

The scheme presented in Figure 4 can be extended to a multi-level decomposition.
