**6. References**

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

**Case Excitation F1 F2 F3 F4 F5 F6 F7 F8**

Found specialised excitation improved efficiency of analysed single parametric (soft) faults. Utilisation of wavelet transform brought further improvements: i.a. there has been reached 100% proper location of faults F7 and F8. Improvement of fault diagnosis was also obtained

Utilisation of a wavelet transform as a feature extraction from CUT responses and in building fault dictionary resulted in general improvement of diagnosis efficiency. There have been investigated single catastrophic (hard) and parametric (soft) faults of passive and active analogue electronic circuits. It must be emphasized that the last faults are much more difficult to diagnose, because their influence on circuit behaviour (e.g. transfer function) is much weaker than catastrophic ones. It must be also noted that fault location is more difficult diagnostic goal than fault detection ("just" a differentiation between healthy and faulty circuits). Wavelet transform has been found useful tool in diagnosis of analogue electronic circuits, both in reference cases of simple excitations (step function, real Dirac pulse, linear function) and in cases when excitation has been designed by genetic algorithm. In every case, combination of specialised excitation and wavelet transform resulted in

x1(n) 0.68 0.63 0.58 0.63 0.74 0.68 1.00 1.00 Step 0.47 0.42 0.79 0.63 0.37 0.47 0.74 0.89

x2(n) 0.58 0.63 0.68 0.47 0.89 0.74 1.00 1.00 Step 0.58 0.63 0.53 0.63 0.21 0.53 1.00 1.00

1

2

Table 10. Fault location efficiency

Fig. 22. Found specialised excitation for case 2

**5. Conclusions** 

highest efficiency of fault diagnosis.

in testing using simple step excitation (tab. 10, case 2).


**11** 

*Iran* 

**Application of Wavelet** 

**Analysis in Power Systems** 

Reza Shariatinasab1 and Mohsen Akbari2 and Bijan Rahmani2 *1Electrical and Computer Engineering Department, University of Birjand* 

*2Electrical and Computer Engineering Department, K.N. Toosi University of Technology* 

When you capture and plot a signal, you get only a graph of amplitude versus time. Sometimes, you need frequency and phase information, too. However, you need to know whenever in a waveform the certain characteristics occur. Signal processing could help, but you need to know which type of processing to apply to solve your data-analysis problem. Many books and papers have been written that explain WT of signals and can be read for further understanding of the basics of wavelet theory. The first recorded mention of what we now call a "wavelet" seems to be in 1909, in a thesis by A. Haar. The concept of wavelets in its present theoretical form was first proposed by J. Morlet, a Geophisicist, and the team at the Marseille Theoretical Physics Center working under A. Grossmann, a theoretical phisicist, in France. They provided a way of thinking for wavelets based on physical intuition. They also proved that with nearly any wave shape they could recover the signal exactly from its transform (Graps, 1995). In other words, the transform of a signal does not

The wavelet functions are created from a single charasteristic shape, known as the mother wavelet function, by dialating and shifting the window. Wavelets are oscillating transforms of short duration amplitude decaying to zero at both ends. Like the sine wave in Fourier transform (FT), the mother wavelet *ψ*(*t*) is the basic block to represent a signal in WT. However, unlike the FT whose applications are fixed as either sine or cosine functions, the mother wavelet, *ψ*(*t*), has many possible functions. Fig. 1 shows some of the popular wavelets including Daubechies, Harr, Coiflet, and Symlet. Dilation involves the stretching and compressing the mother wavelet in time. The wavelet can be expanded to a coarse scale to analyze low frequency, long duration features in the signal. On the other hand, it can be shrunk to a fine scale to analyze high frequency, short duration features of a signal. It is this ability of wavelets to change the scale of observation to study different scale features is its

The WT of a signal is generated by finding linear combinations of wavelet functions to represent a signal. The weights of these linear combinations are termed as wavelet coefficients. Reconstruction of a signal from these wavelet coefficients arises from a much

older theory known as Calderon's reproducing activity (Grossmann & Morlet, 1984).

change the information content presented in the signal.

**1. Introduction** 

hallmark.

