**3. Description of processing techniques for combination. Expressions of SNR**

A number of distinct combination techniques to fuse several ultrasonic traces, coming from perpendicular transducers, have been proposed by the authors. There are two important parameters that define all these techniques: a) the initial type of the traces representation, and b) the particular operator utilized in their combination process.

To choose the best representation for the processing of signals is an open general problem with multiples solutions; the two most popular representations are in time or in frequency domains: a) the direct time domain is very useful for NDE problems because the spatial localization of possible defects or flaws (in the material under testing) is closely related with the apparition time of the echoes; b) the frequency domain is less used in this type of ultrasound based applications because does not permit a spatial localization; in addition, the spectrum of the ultrasonic information with interest for testing in some industrial applications, is almost coincident with the mean spectrum of the "grain" noise originated from the material texture, which some times appears corrupting the signals waveforms associated to the investigated reflectors.

An interesting possibility for introducing spectral information in these applications is the use of time-frequency representations (Cohen 1995) for the echo-graphic signals. This option shows in a 2D format the time information for the different frequency bands in which the received ultrasonic signals range. Therefore, each point of a 2D time-frequency representation corresponds with one spectral frequency and with one time instant. Two different time-frequency techniques, the wavelet transform (Daubechies 1992, Shensa 1992)

Comparative Analysis of Three Digital Signal Processing Techniques for 2D Combination of

transducers (V1, V2, V3 and V4) with vertical propagation beams.

the 2D ultrasonic representation after the combination process.

2

double of the initial SNR of the A-scans before combination (*SNRini*).

*SNR2Dtime*, can be expressed as a function of *SNRini* :

**3.2 Linear time-frequency combination technique** 

*SNR dB*

*L* is the length of the whole ultrasonic trace. The SNR of the final 2D representation is:

ultrasonic trace, respectively.

linear transformation.

Echographic Traces Obtained from Ultrasonic Transducers Located at Perpendicular Planes 83

ultrasonic transducers (H1, H2, H3 and H4) with horizontal propagation beams and four

Some theoretical characterizations of this method, including statistical distributions of the combined noise and some results about SRN enhancements were presented in (Rodríguez et al 2004). The more important result of that work is the expression of the resulting SNR for

2

(1)

(2)

2

2

2

<sup>2</sup>*Dtime*( )2 ( ) *SNR dB SNR dB ini* (3)

1

*i*

*M*

¦

<sup>1</sup> ( ( ))

*p i*

*n i*

1

*i*

2 2 1 1

*<sup>p</sup> i j <sup>M</sup>*

*n ij <sup>L</sup>*

*M M*

¦¦

*i j*

<sup>1</sup> ( ( , ))

*D*

*D*

2 2 1 1

¦¦

*i j*

( ) 10log <sup>1</sup> ( ( , ))

where, *p*2*<sup>D</sup>* and *n*2*<sup>D</sup>* denotes the 2D representation of the echo-pulse and of the grain noise; *M* and *L* are the dimensions of the 2D rectangular representations of the echo-pulse and of the

The SRN of the 2D representation obtained by using this time-domain combination method,

In consequence, the resulting SNR with this method, *SNR2Dtime* , expressed in dB, is the

The time-domain traces combination, previously described, works without any frequency consideration. In order to obtain a further improving of SNR, it would be necessary to use some type of processing in the frequency domain. Nevertheless, the ultrasonic echoes coming from flaws in some NDE applications, and the grain noise produced by the own material structure, have similar global mean spectra, which difficult the flaw discrimination in the frequency domain. But if these spectra are instantaneously analyzed, it can be observed that the instantaneous spectrum is more regular for echo-signal than for grain noise. The tools that permit the analysis of these differences between signal and noise are the time-frequency representations, which can be obtained by using a linear or also a non-

In this section, we will deal with the application of linear time-frequency representations to improve our signal-combination purpose. The two most popular linear time-frequency

¦

*L*

( ) 10log <sup>1</sup> ( ( ))

where, *p* denotes the echo-pulse and *n* represents the noise; *M* is the length of the pulse and

The SNR of the initial traces, *SNRini*, containing an echo-pulse and noise, is defined as:

*ini L*

*<sup>M</sup> SNR dB*

*D L L*

and the Wigner-Vile transform (Claasen and Mecklenbrauker 1980), will be applied in the following as complementary tools during the combination procedure.

In relation to the other abovementioned parameter defining the combination techniques, several operators to make the trace combination have been used in previous author's works: maximum, minimum, mean, median, sum and product. Theoretical and experimental results obtained by applying these operators indicate that the best performances obtained, for all the combination methods, were produced when a product operator was employed.

For this reason, we have selected (among all the possible operators) the 2D product between echo-traces, in order to properly perform the comparison among all the methods considered in this paper. In the following, the three alternative processing techniques proposed for trace combination are described, showing their performance in relation to the resultant SNR.

#### **3.1 Time-domain combination technique**

This first technique performs the combination using the envelope of the ultrasonic traces. The first step in this method is the acquisition of the traces from the ultrasonic transducers involved, which are located over two perpendicular planes in the external part of the inspected piece. The following step is the matching in time of all the different pairs of traces, each one with echo-information corresponding to precisely the same volumetric elemental area, i.e. coming from the two specific transducers which projections define such area. To reduce problems due to no perfect synchronization of the two matched traces in those pairs, the signal envelopes are utilized instead of the original signals, because this option is less sensitive to little time-matching errors. These envelopes are obtained by means of applying them the Hilbert transform. The final step is the trace combination process, by using the mentioned 2D product operator.

Briefly, the method can be resumed in four successive steps: first, the collection of the traces from the different transducers; second, the traces envelope calculation; third, the matching between the information segments of each perpendicular transducers specifically related to the same inspection area; and fourth, the combination among all the segment couples. The corresponding functional scheme is presented in Figure 1 for the particular case of four

Fig. 1. Functional scheme of the time-domain echo-traces combination technique.

Some theoretical characterizations of this method, including statistical distributions of the combined noise and some results about SRN enhancements were presented in (Rodríguez et al 2004). The more important result of that work is the expression of the resulting SNR for the 2D ultrasonic representation after the combination process.

The SNR of the initial traces, *SNRini*, containing an echo-pulse and noise, is defined as:

$$SNR\_{ini}(d\mathcal{B}) = 10\log\frac{\frac{1}{M}\sum\_{i=1}^{M} (p(i))^2}{\frac{1}{L}\sum\_{i=1}^{L} (n(i))^2} \tag{1}$$

where, *p* denotes the echo-pulse and *n* represents the noise; *M* is the length of the pulse and *L* is the length of the whole ultrasonic trace.

The SNR of the final 2D representation is:

82 Applications of Digital Signal Processing

and the Wigner-Vile transform (Claasen and Mecklenbrauker 1980), will be applied in the

In relation to the other abovementioned parameter defining the combination techniques, several operators to make the trace combination have been used in previous author's works: maximum, minimum, mean, median, sum and product. Theoretical and experimental results obtained by applying these operators indicate that the best performances obtained, for all the combination methods, were produced when a product operator was employed. For this reason, we have selected (among all the possible operators) the 2D product between echo-traces, in order to properly perform the comparison among all the methods considered in this paper. In the following, the three alternative processing techniques proposed for trace combination are described, showing their performance in relation to the resultant SNR.

This first technique performs the combination using the envelope of the ultrasonic traces. The first step in this method is the acquisition of the traces from the ultrasonic transducers involved, which are located over two perpendicular planes in the external part of the inspected piece. The following step is the matching in time of all the different pairs of traces, each one with echo-information corresponding to precisely the same volumetric elemental area, i.e. coming from the two specific transducers which projections define such area. To reduce problems due to no perfect synchronization of the two matched traces in those pairs, the signal envelopes are utilized instead of the original signals, because this option is less sensitive to little time-matching errors. These envelopes are obtained by means of applying them the Hilbert transform. The final step is the trace combination process, by using the

Briefly, the method can be resumed in four successive steps: first, the collection of the traces from the different transducers; second, the traces envelope calculation; third, the matching between the information segments of each perpendicular transducers specifically related to the same inspection area; and fourth, the combination among all the segment couples. The corresponding functional scheme is presented in Figure 1 for the particular case of four

Fig. 1. Functional scheme of the time-domain echo-traces combination technique.

following as complementary tools during the combination procedure.

**3.1 Time-domain combination technique** 

mentioned 2D product operator.

$$SNR\_{2D}(d\mathcal{B}) = 10\log\frac{\frac{1}{M^2}\sum\_{i=1}^{M}\sum\_{j=1}^{M}(n\_{2D}(i,j))^2}{\frac{1}{L^2}\sum\_{i=1}^{L}\sum\_{j=1}^{L}(n\_{2D}(i,j))^2} \tag{2}$$

where, *p*2*<sup>D</sup>* and *n*2*<sup>D</sup>* denotes the 2D representation of the echo-pulse and of the grain noise; *M* and *L* are the dimensions of the 2D rectangular representations of the echo-pulse and of the ultrasonic trace, respectively.

The SRN of the 2D representation obtained by using this time-domain combination method, *SNR2Dtime*, can be expressed as a function of *SNRini* :

$$\text{SNR}\_{2Dtime} \text{(dB)} = \text{2} \cdot \text{SNR}\_{ini} \text{(dB)} \tag{3}$$

In consequence, the resulting SNR with this method, *SNR2Dtime* , expressed in dB, is the double of the initial SNR of the A-scans before combination (*SNRini*).

#### **3.2 Linear time-frequency combination technique**

The time-domain traces combination, previously described, works without any frequency consideration. In order to obtain a further improving of SNR, it would be necessary to use some type of processing in the frequency domain. Nevertheless, the ultrasonic echoes coming from flaws in some NDE applications, and the grain noise produced by the own material structure, have similar global mean spectra, which difficult the flaw discrimination in the frequency domain. But if these spectra are instantaneously analyzed, it can be observed that the instantaneous spectrum is more regular for echo-signal than for grain noise. The tools that permit the analysis of these differences between signal and noise are the time-frequency representations, which can be obtained by using a linear or also a nonlinear transformation.

In this section, we will deal with the application of linear time-frequency representations to improve our signal-combination purpose. The two most popular linear time-frequency

Comparative Analysis of Three Digital Signal Processing Techniques for 2D Combination of

each band are independent and perfectly synchronized (Rodríguez et al 2004b):

being, *L,* the number of the selected frequency bands.

**3.3 Wigner-Ville Transform (WVT) combination technique** 

equipment can be approximately modelled by the following expression:

The WVT of the ultrasonic pulse modelled by (6) is (Rodríguez 2003):

pulse (*a*>0), and *ǚ*0 is the central frequency of its spectrum.

linear time-frequency transform.

pulses coming from real flaws.

Echographic Traces Obtained from Ultrasonic Transducers Located at Perpendicular Planes 85

The final global SNR, after the combination of all the 2D displays belonging to the different frequency bands, *SNR2DTFlinear*, can be obtained supposing that the 2D representations for

Consequently, in this case, the resulting *SNR2DTFlinear* presents an important factor of improvement over the *SNRini* . This factor is the double of the number of frequency bands used in the combination. It must be noted that comparing expressions (5) and (3), the SNR improvements is multiplied by *L,* but the computational complexity of the algorithm is also multiplied by the same factor *L*. In the experimental results section of this chapter, the accuracy of this expression will be confirmed comparing (5) with simulations using as linear time-frequency tool the undecimated wavelet packet transform (Shensa 1992, Coifman and Wickerhauser 1992). In any case, it must be noted that this expression is also valid for any

The non-linear time-frequency distributions present some advantages over linear transforms, but some non-linear terms ("cross-terms") appear degrading the quality of the distributions and usually the computational cost is incremented. One of the most popular non-linear time-frequency representations is the Wigner-Ville transform (WVT) (Claasen and Mecklenbrauker 1980), which has been previously utilized in ultrasonic applications with good results (Chen and Guey 1992, Malik and Saniie 1996, Rodríguez et al 2004a). The WVT presents an useful property for dealing with ultrasonic traces: its positivity for some kind of signals (Cohen 1995). In order to illustrate the suitability of this transform for the processing of the ultrasonic pulses typical in NDE applications, we will show that they fulfil that property. For it, an ultrasonic pulse-echo like to those acquired in such NDE

> <sup>2</sup> ( /2) <sup>0</sup> ( ) cos( ) *at <sup>p</sup> t Ae*

where *A* is the pulse amplitude, *a* is a constant related to the duration and bandwidth of the

The expression (7) shows that the WVT of an ultrasonic pulse with Gaussian envelope has only positive values. The chirp with Gaussian envelope is the most general signal for which the WVT is positive through-out the time-frequency plane (Cohen 1995). The ultrasonic grain noise does not carry out this property, so resulting that the sign of the WVT values represents a useful option to discriminate this type of difficult-to-eliminate noise of the echo

*a <sup>A</sup> tWVT* /)()2/(- 2 1

= ),(

Z

)(

S

*at <sup>a</sup> <sup>p</sup> e*

<sup>2</sup> <sup>2</sup> <sup>0</sup> <sup>2</sup>

ZZ

The combination method begins in this case by calculating the WVT in all the ultrasonic traces. After the band selection is performed, the negative values (that correspond mainly

Z

*t* (6)

(7)

<sup>2</sup>*DTFlinear*( ) 2 ( ) *SNR dB L SNR dB ini* (5)

representations are the Short-Time Fourier Transform and the Wavelet transform (Hlawatsch and Boudreaux-Barlets 1992). Both types of transforms can be implemented in an easy way by means of linear filter banks.

In the present linear technique, the combination process begins with the time-frequency representation of the all the acquired ultrasonic traces. A linear time-frequency transform is applied and the frequency bands with maximum ultrasonic energy are selected in each trace. The number of selected bands will be denoted as *L*. At this point, we have to resolve *L* problems similar to that presented in the previous time-domain combination method. In this way, *L* separate 2D displays are produced, one for each frequency band. The final step is the combination of these 2D displays by using a point-to-point product of them. A simple case, where combination is performed by selecting the same frequency bands for all the transducers, was presented in (Rodríguez et al 2004b), but also it could be possible to make the combination by using different bands for each transducer. The method scheme is presented in the Figure 2 for 4 horizontal and 4 vertical transducers.

Here, the combination for each frequency band is similar to the case of the time-domain technique. Then, it will be necessary to make the following steps: a) to match in time the common information of the different transducer pairs (for each frequency band), b) to calculate the time-envelope for all the bands selected in each trace, c) to perform the combinations obtaining several one-band 2D representations, and d) to fuse all these 2D displays, so resulting the final 2D representation.

Fig. 2. Functional scheme of the linear time-frequency traces combination technique

The matching process can be common for all the frequency bands if the point number of the initial traces is maintained and if the delays of the filtering process are compensated in each band. The SNR of the 2D representation of each individual band, ( ) 2 *band i DTFlinear SNR* is obtained from expression (3).

$$\text{SNR}^{(band-i)}\_{\text{2DTFlinear}} \text{(dB)} = \text{2} \cdot \text{SNR}\_{ini} \text{(dB)} \tag{4}$$

The final global SNR, after the combination of all the 2D displays belonging to the different frequency bands, *SNR2DTFlinear*, can be obtained supposing that the 2D representations for each band are independent and perfectly synchronized (Rodríguez et al 2004b):

$$\text{SNR}\_{2DTFlinear} \text{(dB)} = \text{2} \cdot L \cdot \text{SNR}\_{ini} \text{ (dB)} \tag{5}$$

being, *L,* the number of the selected frequency bands.

84 Applications of Digital Signal Processing

representations are the Short-Time Fourier Transform and the Wavelet transform (Hlawatsch and Boudreaux-Barlets 1992). Both types of transforms can be implemented in

In the present linear technique, the combination process begins with the time-frequency representation of the all the acquired ultrasonic traces. A linear time-frequency transform is applied and the frequency bands with maximum ultrasonic energy are selected in each trace. The number of selected bands will be denoted as *L*. At this point, we have to resolve *L* problems similar to that presented in the previous time-domain combination method. In this way, *L* separate 2D displays are produced, one for each frequency band. The final step is the combination of these 2D displays by using a point-to-point product of them. A simple case, where combination is performed by selecting the same frequency bands for all the transducers, was presented in (Rodríguez et al 2004b), but also it could be possible to make the combination by using different bands for each transducer. The method scheme is

Here, the combination for each frequency band is similar to the case of the time-domain technique. Then, it will be necessary to make the following steps: a) to match in time the common information of the different transducer pairs (for each frequency band), b) to calculate the time-envelope for all the bands selected in each trace, c) to perform the combinations obtaining several one-band 2D representations, and d) to fuse all these 2D

Fig. 2. Functional scheme of the linear time-frequency traces combination technique

band. The SNR of the 2D representation of each individual band, ( )

( )

The matching process can be common for all the frequency bands if the point number of the initial traces is maintained and if the delays of the filtering process are compensated in each

<sup>2</sup> ( )2 ( ) *band i SNR dB SNR dB DTFlinear ini*

2 *band i*

(4)

*DTFlinear SNR* is obtained

presented in the Figure 2 for 4 horizontal and 4 vertical transducers.

an easy way by means of linear filter banks.

displays, so resulting the final 2D representation.

from expression (3).

Consequently, in this case, the resulting *SNR2DTFlinear* presents an important factor of improvement over the *SNRini* . This factor is the double of the number of frequency bands used in the combination. It must be noted that comparing expressions (5) and (3), the SNR improvements is multiplied by *L,* but the computational complexity of the algorithm is also multiplied by the same factor *L*. In the experimental results section of this chapter, the accuracy of this expression will be confirmed comparing (5) with simulations using as linear time-frequency tool the undecimated wavelet packet transform (Shensa 1992, Coifman and Wickerhauser 1992). In any case, it must be noted that this expression is also valid for any linear time-frequency transform.

#### **3.3 Wigner-Ville Transform (WVT) combination technique**

The non-linear time-frequency distributions present some advantages over linear transforms, but some non-linear terms ("cross-terms") appear degrading the quality of the distributions and usually the computational cost is incremented. One of the most popular non-linear time-frequency representations is the Wigner-Ville transform (WVT) (Claasen and Mecklenbrauker 1980), which has been previously utilized in ultrasonic applications with good results (Chen and Guey 1992, Malik and Saniie 1996, Rodríguez et al 2004a).

The WVT presents an useful property for dealing with ultrasonic traces: its positivity for some kind of signals (Cohen 1995). In order to illustrate the suitability of this transform for the processing of the ultrasonic pulses typical in NDE applications, we will show that they fulfil that property. For it, an ultrasonic pulse-echo like to those acquired in such NDE equipment can be approximately modelled by the following expression:

$$p(t) = A \cdot e^{-(at^2/2)} \cos(a\_0 t) \tag{6}$$

where *A* is the pulse amplitude, *a* is a constant related to the duration and bandwidth of the pulse (*a*>0), and *ǚ*0 is the central frequency of its spectrum.

The WVT of the ultrasonic pulse modelled by (6) is (Rodríguez 2003):

$$WVT\_{\mathbb{P}}(t,\alpha) = \frac{A^2}{\frac{1}{(a\pi)^2}} \cdot e^{-(a^2/2) - (\alpha - \alpha\_0)^2/a} \tag{7}$$

The expression (7) shows that the WVT of an ultrasonic pulse with Gaussian envelope has only positive values. The chirp with Gaussian envelope is the most general signal for which the WVT is positive through-out the time-frequency plane (Cohen 1995). The ultrasonic grain noise does not carry out this property, so resulting that the sign of the WVT values represents a useful option to discriminate this type of difficult-to-eliminate noise of the echo pulses coming from real flaws.

The combination method begins in this case by calculating the WVT in all the ultrasonic traces. After the band selection is performed, the negative values (that correspond mainly

Comparative Analysis of Three Digital Signal Processing Techniques for 2D Combination of

**4. Protocols used in the different testing experiments** 

**4.1 Experiments type-I based on simulated noisy traces** 

these registers to evaluate those expressions.

Rodríguez et al 2004b).

Echographic Traces Obtained from Ultrasonic Transducers Located at Perpendicular Planes 87

Two types of experiments (I and II) have been designed with the purpose of evaluating and comparing the three trace combination methods presented in the previous section. The comparison will be performed over the same set of ultrasonic traces for the three cases. The type-I experiments are based on simulated noisy ultrasonic traces and those of type-II use experimentally acquired echo-traces. The protocols used in these experiments are an extension of those we have planned in references (Rodríguez et al 2004a, Rodríguez 2003,

Type-I experiments were carried out with simulated signal registers. They provide adequate calculation results to confirm the accuracy of the expressions estimated from the theoretical models of the processing techniques proposed in the equations (3), (5) and (8) to predict the distinct SNRs (*SNR2Dtime*, *SNR2DTFlinear* and *SNR2DWVT*). So, those expressions could be validated for an ample range of values in *SNRini* with perfectly controlled characteristics in echo-signals and their associated grain noises. Some results, in a similar context, using these same rather simple simulated registers, have been compared in a previous work (Rodríguez et al 2004a) with the obtained results when a more accurate ultrasonic trace generator was used. A very close agreement between them was observed, which confirms the suitability of

The testing case proposed to attain this objective is the location of a punctual reflector into a rectangular parallelepiped from 2 external surfaces, perpendicular between them, and using 4 transducers by surface. The general scheme of these experiments, with 4 horizontal (H1, H2, H3, H4) and 4 vertical (V1, V2, V3, V4) transducers is depicted in the Figure 4. Transducers H3 and V2 receive echoes from the reflector whereas the other transducers (H1, H2, H4, V1, V3 and V4) only receive grain noise. To assure compatibility of experiments type-I with experiments type-II, ultrasonic propagation in a piece of 24x24 mm has been simulated assuming for calculations a propagation velocity 2670 m/s very close to that

corresponding to methacrylate material. The sampling frequency was 128 MHz.

**H4** 

**H3** 

**H2** 

**H1** 

Fig. 4. Geometry of the inspection case planned to evaluate the different combination methods: detection of a single-flaw in a 2D arrangement with 16 elemental-cells.

**V1 V2 V3 V4** 

flaw

with noise) are set to cero. For each frequency band, a combination is made by using the 2D product operator, like as it was used in the time-domain combination technique. The final 2D representation is obtained with a point to point product of all the 2D displays related to the different frequency bands. A functional scheme of this WVT based combination method is presented in the Figure 3, for the case of eight transducers considered in this section.

Fig. 3. Functional scheme of the WVT traces combination method.

A good estimation of the resulting SNR for the 2D representation in this WVT case, *SNR2DWVT*, can be obtained from the results presented in (Rodríguez 2003):

$$\text{SNR}\_{2D\text{WWT}}\text{(dB)} \equiv \text{3} \cdot L \cdot \text{SNR}\_{\text{ini}} \quad \text{(dB)} \tag{8}$$

Therefore, the improvement factor of the SNR, expressed in dB, which can be obtained by this WVT method, is the triple of the number of frequency bands that had been selected.

In consequence, the theoretic improvement levels in the SNR provided by the three alternative techniques for combining ultrasonic traces coming from two perpendicular transducers, (i.e., the basic option using traces envelope product, and the others two options based on linear time-frequency and WVT trace decompositions), are quite different.

So, the quality of the resulting 2D combinations, in a SNR sense, is predicted to be quite better when time-frequency decompositions are chosen, and the best results must be expected for the Wigner-Ville option, which in general seems to be potentially the more effective processing. Nevertheless, in spite of these good estimated results for the WVT case, it must be noted that in general this option supposes higher computational cost. Therefore, the more effective practical option should be decided in each NDE situation depending on the particular requirements and limitations in performance and cost being needed. In the following sections, the confirmation of these predictions will be carried out, by means of several experiments from simulated and measured ultrasonic traces.

### **4. Protocols used in the different testing experiments**

86 Applications of Digital Signal Processing

with noise) are set to cero. For each frequency band, a combination is made by using the 2D product operator, like as it was used in the time-domain combination technique. The final 2D representation is obtained with a point to point product of all the 2D displays related to the different frequency bands. A functional scheme of this WVT based combination method is presented in the Figure 3, for the case of eight transducers

Fig. 3. Functional scheme of the WVT traces combination method.

several experiments from simulated and measured ultrasonic traces.

*SNR2DWVT*, can be obtained from the results presented in (Rodríguez 2003):

A good estimation of the resulting SNR for the 2D representation in this WVT case,

Therefore, the improvement factor of the SNR, expressed in dB, which can be obtained by this WVT method, is the triple of the number of frequency bands that had been selected. In consequence, the theoretic improvement levels in the SNR provided by the three alternative techniques for combining ultrasonic traces coming from two perpendicular transducers, (i.e., the basic option using traces envelope product, and the others two options

So, the quality of the resulting 2D combinations, in a SNR sense, is predicted to be quite better when time-frequency decompositions are chosen, and the best results must be expected for the Wigner-Ville option, which in general seems to be potentially the more effective processing. Nevertheless, in spite of these good estimated results for the WVT case, it must be noted that in general this option supposes higher computational cost. Therefore, the more effective practical option should be decided in each NDE situation depending on the particular requirements and limitations in performance and cost being needed. In the following sections, the confirmation of these predictions will be carried out, by means of

based on linear time-frequency and WVT trace decompositions), are quite different.

<sup>2</sup>*DWVT* ( ) 3 ( ) *SNR dB L SNR dB* # *ini* (8)

considered in this section.

Two types of experiments (I and II) have been designed with the purpose of evaluating and comparing the three trace combination methods presented in the previous section. The comparison will be performed over the same set of ultrasonic traces for the three cases. The type-I experiments are based on simulated noisy ultrasonic traces and those of type-II use experimentally acquired echo-traces. The protocols used in these experiments are an extension of those we have planned in references (Rodríguez et al 2004a, Rodríguez 2003, Rodríguez et al 2004b).
