Signal processing offers a large variety of tools, the most popular technique, Fourier analysis, breaks down a signal to sinusoidal constituents of several frequencies, transforming a signal in the time domain into its counterpart in the frequency domain. For signals with well-defined frequencies and where this information meets the requirements, Fourier analysis is more than satisfactory. However, there are some disadvantages, especially when it is used with signals that undergo sudden changes, fluctuations, discontinuities or localized phenomena, among others. The wavelet theory has been used in recent years to cover some of the limitation in Fourier analysis. it is based on windowing the signal into variable sized regions, using long time intervals for information on low frequencies and shorter intervals for high frequencies, thus allowing analysis of localised areas without losing the time information, furthermore, in opposition to the smooth and continuous shape of harmonic function, the wavelets my present peaks and discontinuities and improve the sensibility for abrupt phenomena, the mother wavelet db3, used in this work, is orthogonal and asymmetrical and present irregular shape as shown in Figure 1 .

The wavelet transform breaks down a signal into components (wavelets) which very “in scale” (stretched or compressed) and in location (shifted). Maintaining the Integrity of the Specifications [3] .

Wavelet analysis is becoming a common tool for analysing localised variations of power within a time series. By decomposing a time series into a frequency space, one is able to determine both the dominant modes of variability and how those modes vary in time. A wavelet is wave like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times wavelets are termed a “brief oscillation”, a taxonomy of wavelets has been established, based on the number and direction of its pulses [4] [5] .

To meet the needs for adaptive time-frequency analysis in applied mathematics, physiques and engineering, the wavelet theory was developed in the late 1980 by Morlet and Daubechies (see Mallat, 1990). In wavelet transform (WT), a signal of interest is decomposed by examining the coefficients of the wavelet transform is a convenient tool for processing non-stationary signals. It has been widely used to analyse machining signals for tool condition monitoring. Let ${\psi }_{a,b}\left(t\right),a,b\int R,a\ne 0$, be a family of functions defined as translations and rescales of single function ${\psi }_{a,b}\left(t\right)ϵL^2\left(R\right)$ (Mallat, 1989):

${\psi }_{a,b}\left(t\right)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$ (1)

With the following basic properties

${\displaystyle \underset{R}{\int }{\psi }_{a,b}\left(t\right)\text{d}t}=0$ and ${\displaystyle \underset{R}{\int }{\psi }_{a,b}^2\left(t\right)\text{d}t=1}$ (2)

These properties indicate that the wavelet is small wave oscillating around zero and diminishing quickly (finite energy). The fundamental idea of wavelet transform is that the transformation should allow only changes in time extension, but not shape, imposing a restriction on choosing suitable basis functions. Changes in the time extension are expected to conform to the corresponding analysis frequency of the bases function. Based on the uncertainty principle of signal processing Δt*Δw ≥ (1/2), where t represents time and w angular frequency. The higher the required resolution in time, the lower the resolution in frequency has to be. The larger the extension of the analysis windows is chosen, the larger is the value of Δt [5] [6] . When Δt is large:

1) Bad time resolution

2) good frequency resolution

3) Low frequency, large scaling factor

when Δt is small

1) Good time resolution

2) Bad frequency resolution

3) High frequency, small scaling factor.

In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is a wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution, it capture both frequency and location information (location in time). The DWT of a signal x is calculated by passing it through a series of filters. First the samples are passed through a low-pass filter with impulse response g resulting in a convolution of two [5] [6] :

$y=\left[n\right]=\left(x\ast g\right)\left[n\right]\underset{k=-\infty }{\overset{+\infty }{{\displaystyle \sum }}}x\left[k\right]g\left[n-k\right]$ (3)

The signal is also decomposed simultaneously using a high-pass filter h. The outputs give the detailed coefficients (from the high-pass filter) and approximation coefficients (from the low-pass). It is important that the two filters are related to each other and they are known as a quadrature mirror. However, since half the frequencies of the signal have now been removed, half the sample can be discated according to Nyquist’s rule [4] [5] ( Figure 2 ).

The ultrasonic signals obtained during a measurement are always enriched with information on the characterized environment, sometimes this information is overlapped between them, or invisible, drowned in the high frequencies for this reason it is necessary to decompose the signals obtained in order to obtain the maximum information on the tested environment. Figures 5-12 below show the decomposition of the measured ultrasonic signal obtained by the prism technique.

In order to test the degree of continues wavelet we applied different degrees db1, db2, …, db6. According to the results shown in the figures, we see that db3 is the most reliable in our case, also we see also that the continuous wavelet transform allows the signal to be decomposed into eight separate signals, each signal can be information on the analyzed medium. Figure 11 and Figure 12 is another example of separation of ultrasonic measurement signal, we see the different staps of separations using continuous wavelet.

The majority of measurement signals are affected by noise, Therefore, applying a

filter to minimize or remove noise is crucial task throughout the measurement chain. In this section, we will apply the continuous wavelet transform to remove noise. The bending of signal is primordial problem during experimental measurements, Figure 13 is an example of the treatment of ultrasonic signals by the wavelet transformations.

Figure 14 and Figure 15 show the image of ultrasonic measurement in two dimension presentation and three presentation respectively using wavelet transform.