**2. Background theory**

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

only "average" results being obtained for each analysed time segment, requiring short

the shorter the analysed time segment is, the coarser the resulting frequency resolution

A more rigorous explanation of the latter is the Uncertainty Principle or Bandwidth-Time product that can be easily proved in [1] using the Parseval theorem and Schwartz inequality.

<sup>4</sup> *f t*

where *f* is the frequency resolution expressed in Hertz and *t* is the time resolution expressed in seconds. It can be easily understood that Eq. 1 points to a limitation in STFT analysis methods: fine resolution in both time and frequency domains cannot be obtained at

Several techniques have been developed [2][3] to overcome this problem and to analyse

As is reported in [2], one can distinguish between three important classes of non-stationary

Evolutionary Harmonic Signals related to a periodic phenomenon (i.e. rotation) of

Evolutionary Broadband Signals with a broadband spectrum with spectral content

 Transient Signals which show a very short time segment of a wholly evolving nature (i.e. door-slam acoustic response and diesel engine irregularity within one combustion

Another important class of non-stationary signals is represented by Cyclostationary Signals which are not described here. Since this study deals with Transient signals, Wavelet

In general, each type of fault produces a different vibration signature which might be detected by means of suitable signal processing techniques. Concerning i.c. engines, fault detection and diagnosis can be carried out using different strategies. One strategy can consist in modelling the whole mechanical system using lumped or finite element methods in order to simulate several faults and compare the results with the experimental data [4][5]. Another strategy is to adopt signal processing techniques in order to obtain features or maps that can be used to detect the presence of the defect [6][7]. Regarding the latter, a decision algorithm is require for a visual or automatic detection procedure. Moreover, maps can also be analysed for diagnostic purposes [8]. This method is used most commonly and is well

The latter strategy involves the application of time-frequency distribution techniques which are well suited for the analysis of non-stationary signals and have been widely applied to

Transforms (WT) have been proposed as an appropriate analysis tool.

1

(1)

analysis segments for good time resolution;

will be.

the same time.

cycle).

signals:

This Principle states that:

different types of non-stationary signals.

evolving over time (i.e. road noise);

suited to judgements involving expert technicians.

varying frequency;

engine monitoring [9]-[11].

This paragraph introduces the theory of fundamental background in order to understand achievements concerning the application of CWT and DWTs on real signals.
