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Estimation of Differences in Event Arrival Times Through Wavelet Transform Techniques J. E. Ruzzante Laboratorio de Emisiones Acusticas - Centro Atomico Constituyentes - CNEA - Argentine E.P.Serrano Escuela de Ciencia y Tecnologia - Universidad Nacional de San Martin -Argentine Email: eduser@dm.uba.ar Contact

Abstract:

The use of the Discrete Wavelet Transform is proposed for estimating differences in arrival times of consecutive events within the context of a physical signal. The method proposed is a combination of the filtering techniques characteristic of wavelet analysis with discrete correlation techniques. Some results are discussed in detail.

Keywords: Wavelets, Undecimated Discrete Wavelet Transform, Pattern Recognition

1. Introduction

When considering certain signals, the estimation of differences in arrival times of consecutive events is especially relevant. It is often necessary, for instance, to locate a source emitting elastic waves. Such is the case of seismic or acoustic emission signals. In other cases, the purpose is to determine the distance between certain objects, which requires the time elapsed between the emission of an acoustic wave and the reception of its echo to be calculated. This type of applications are frequent in robotics and materials characterization.
There are many techniques to approach the problem of estimating arrival times which cover a wide range of alternatives and derive, in general, from the specific features of the problem, the desired level of accuracy and, finally, the characteristics of the emission. Several relatively simple algorithms for determining differences in echo arrival times of ultrasonic signals have been proposed and discussed in a recent publication [1] :
• Overlapping
• The Hilbert Transform
• Phase Slope
• Cross Correlation
• Cepstrum
These methods offer an estimation of the required distances derived from the analysis of parameters which express the temporal localization of consecutive echoes. For further details, also refer to [2 - 4] .
Despite their simplicity, however, the effciency of the above mentioned techniques may be seriously affected by disturbances arising from the overlapping of contiguous events, from waveform distorsion in dispersive media, or from noise.
To cope with such situations, more refined and strong methods are required. Various alternatives are found in the existing literature. The technique proposed in [5] may be mentioned, por example, which implements Kalman filters to analyze and detect arrival times in well defined waveforms which are affected nevertheless by white noise.
Another approach proposes the use of filter banks and, more precisely, the application of the Wavelet Transform [3] , [6-8] as a tool for adapting estimation to noisy and dispersive media.
This paper presents a new estimation technique based on the application of the Discrete Wavelet Transform which is further explained in the following sections. The new method is computationally advantageous and guarantees reasonable efficiency through a wide range of applications.

2. DESCRIPTION OF THE PROBLEM

Let s (t) be a signal that may be considered as a sequence of regularly distanced transient events of decreasing amplitude. These transient events oscillate, decay rapidly, and show a common structure. The model for such a signal may be expressed as:

(1)

where s₀ (t) is the main event or pattern, with a waveform resulting from the excitation of the material tested. This form replicates along time: the consecutive t_k represent localization times and A_k are the corresponding amplitudes.
It should be noted that, for k ³ 1 eachcomponent s_k (t)=A_k s₀(t-t_k) represents the k - th event, esentially an attenuated replica of the pattern s ₀(t).
Ultrasound signals used in certain non-destructive techniques show, precisely, such characteristics. The first event represents the initial excitation of the test material, and replicas represent the consecutive echoes recorded by the transducer.
Certain discrete acoustic emission signals often show a burst sequence which also corresponds to this particular structure.
Estimating with accuracy enough the separation times t_k+1 - t_k between two consecutive events is especially important, since these parameters are related with the properties of the tested material. In the case of ultrasonic signals in particular, they are related to wave propagation through the material medium.
Under ideal conditions, we may assume that each echo or replica has exactly the same structure of the initial event, that separation times are constant, and that consecutive events are sufficiently distanced from each other. In other words, we may assume that they are perfectly distinguishable and there is no significant overlapping. In this case, the separation time may be readily estimated from the model here in proposed through usual correlation techniques. More precisely, in the local maxima of the following function:


(2)

covering the effective domain of the signal. Usually, an approximation to this correlation with a discrete correlation computed from a proper sequence of sample data {s_n = s(nDt)} should be provided, where Dt is the corresponding sampling time.
Such may be the ideal case, for example, of an ultrasonic signal composed of consecutive echoes of a relatively short initial impulse under ideal conditions of propagation through a non-dispersive medium. Precisely, the tecniques discussed in the introduction are supported by this model.
Nevertheless, these conditions are often disturbed by various factors which affect estimation to a greater or lesser extent,among which the following should be mentioned:

• The structure of consecutive events is affected in sample data by noise, either of physical, instrumental or computational origin.
  In this sense, it should be noted that such a situation occurs preferently in the case of acoustic emissions, since, as it is well-known, ultrasound experiments yield relatively low-noise signals.
• Separation times are not constant.
• Temporal distances between consecutive events are relatively short, and times are partially over-lapped, with a significant mutual interference.
• The medium is dispersive, and the wave propagation velocity is not uniform throughout the frequency spectrum. Thus, in the presence of a broad band transducer, the signal evidences the overlapping of several components of relatively high and low frequency differing in phase, which do not properly correspond to a defined waveform in propagation. Moreover, separation times between events usually vary throughout the spectrum.
• Such distortion factors, which usually work concurrently, demand an improved model and the implementation of more refined techniques.
The Discrete Wavelet Transform leads naturally to decompose the signal into ranges of frequencies or scales. Thus, the model discussed above may be further generalized as follows:

	(3)

where each component s_j (t) corresponds to a given range of frequencies or scale. The term º jk (t) represents the effect of further disturbances with respect to scale patterns s_jo(t) . The application of this technique thus involves decomposing the signal with a filter bank which properly prepares information for the estimation stage.
Precisely, the purpose of such a decomposition is to estimate the times t_jk by ranges of frequencies, applying at each level j a discrete correlation technique. Before describing the method proposed, a succintdescriptionofthe Discrete WaveletTransform tobe implementedisprovided.

3. UNDECIMATED DISCRETE WAVELET TRANSFORM

The Undecimated Discrete Wavelet Transform provides efficient numerical techniques to approach various problems arising from signal processing. Particularly, it enables the application of discrete correlation methods by operating with the values of the Transform themselves.
A brief description of this transform and its properties is provided here. Further details may be found in the existing literature and, particularly, in [6] and [8].
Assuming the signal to be analyzed is given by its sample values, with uniform sampling frequency Dt, we further assume it may be well represented by a cubic spline function having its nodes at the points ¿k = k :Dt .That is,the signals s(t) is a polynomial of degree º = 3 in each interval sufficiently smooth at the nodes. We further assume that the signal is of finite energy, i. e. : For simplicity, we may assume in the following that Dt =1.
The so-called analyzing wavelet consists in a symmetric spline function , centered at the point = :5 which oscillates for a brief period and is rapidly attenuated. Its Fourier transform is well localized within the frequency range . Through changes of scale, it generates a collection of replicas at differents scales:

(4)

Such replicas correpond to frequency ranges.For j £ - 1,the whole range j of frequencies for the signal is covered, according to Nyquist law. Therefore, each correlation:

(5)

for j £ - 1 filters the signal reflecting the information corresponding to the range of frequencies Wj.
On the other hand, since the functions are also splines, correlations are univocally characterized j by their discretization, i. e., by the values:

(6)

The whole sequence Ds={d_j(n)} is defined as the undecimated Discrete Wavelet Transform of the signal s (t). These values contain all the information of the signal, organized in time and frequency. More precisely, each value d_j(n) summarizes the information localized in the neighbourhood of node within the range of frequencies W_j.
Moreover, the transform provides an efficient numerically stable representation of the signal:

(7)

It is obvious that the Discrete Transform is invariant with respect to translations of the signal in the sampling set points, an indispensable condition for the implementation of correlation techniques. Particularly, it should be noted that if s₀ (t) is a segment of the signal corresponding to a well localized event within the interval [t₀ - ± ; t₀ + ±] , its information is properly characterized by the values d_j(n) corresponding with the indexes defining the pattern or local mask of the event.
Therefore, if the pattern is repeated along the signal, the mask is also repeated, correspondingly. This is precisely the basis of the detection method described in the following paragraphs.
In short, it may be concluded that the undecimated Discrete Wavelet Transform summarizes all the time-scale information in the signal. Particularly, it lays out the signal information in scales or ranges of frequencies, in an scheme appropriate to detect and characterize local phenomena.
Consequently, it is possible through the transform to detect similar patterns or structures combining correlation with an effective filtering.

4. ESTIMATION OF SEPARATION TIMES.

Considering that experimental data result in the spline signal s (t) which is assumed to satisfy the model discussed in section 2 .:
that is, it is a sequence of events having similar structure in each level of the range of frequencies.
The pattern or main event may thus be characterized as an overlapping of scale components:

(8)

It should be noted that the replicas or echoes have amplitudes and temporal localizations that vary according to the range of frequencies.
It may be assumed that A₀ = 1 and that the pattern is well localized in the neighbourhood of t₀ , that is, its energy is virtually contained within an interval [t₀ - ± ; t₀ + ±] . Echoes are respectively localized, therefore, depending on the scale, in the neighbourhood of intervals centered around the times t_jk , which are, precisely, the parameters to be estimated. The corresponding parameters A_jk express the relative amplitudes of the pattern.
These relationships are apparent in the Wavelet Transform of the signal, so that it is possible to formulate the discrete detection scheme as follows.
Let D_j(0) = (d_j(n )) be the pattern vector for the scale j < 0, for the indexes:

(9)

corresponding to the localization of the signal s₀ (t).
Likewise, let D_j(k )=(d_j (n + k )), for each j < 0 and each k > 0.
Let us first consider for each level j the normalized products:

(10)

provided that || D_j(k) || ³ ² , for a certain threshold ² > 0, it may be concluded that the vector D_j (k ) effectively contains the information. This operation may be interpreted as the correlation between the pattern and the signal for each range of frequencies. It is obvious that each |^âjk| is a value between 0 and 1 which strictly indicates the similarity between the pattern D_j (0) and the local vector D_j (k ).
For parameter values close to 1, the possible presence of a replica A_jkS_j(t - t_k) may be surmised. Therefore, according to this model, the following relation should hold:

(11)

and therefore:

(12)

Hence, the presence of an echo or replica in each range of frequencies j was decided upon, from the local maxima of the sequence:

(13)

Once the echoes or replicas are detected, the separation times between them may be immediately estimated. Note that the efficiency of the method is not affected by point distortions neither is it altered by the effects of dispersion in frequency. Besides, the effect of noise is significantly attenuated by the action of the Wavelet Transform itself and by correlations between information vectors.
However, certain uncertainty in decision may arise in the case of significant overlapping of events. The detection scheme may be refined adequately so as to cope with such cases.

5. EXAMPLES

The above mentioned methods have been implemented in the following examples. Frequency scales corresponding to actual sampling frequencies have been adapted accordingly.


Example 1. Figure 1. shows an ultrasound signal where three events may be clearly seen. The signal was sampled with a step Dt =0:01sec:. In this case, the above mentioned ideal conditions virtually hold and the separation time between events may be estimated directly from the signal itself. By inspection of the relative maxima of the signal, located at t₀ = 2 :7, t₁ = 3:89 and t₂ =5:08 respectively, the separation time may be immediately estimated in 1:19sec.
Applying the wavelet transform, we verify that 91:5 per cent of its energy is in the scale level j = - 1, i. e., within the range of frequencies between 25 and 5 H z. This componentisshown in figure 2. Note that it has virtually the same structure of the signal. Relatively low frequencies are filtered by the transducer and no dispersive phenomena may be observed.
Applying the estimation method upon this component, the maximum correlation values are obtained (figure 3.) at the points t₀ = 2:61, t₁ =3:80 and t₂ =4:99 respectively, which correspond to the same separation time of 1:19 sec. It should be noted that the location of the first point depends on the particular pattern selected.


Example 2.In order to illustrate the effect of noise disturbances, gaussian white noise was added to the sample data of the signal above, as shown in figure 4. This time, similarity structures are degraded, and itis notpossible to applymethods dependingonpointvalues ofsample data. The efficiencyof methods based on correlations is, however, preserved, since there is no overlapping of events and the medium is not dispersive.
The firstcomponentofthe wavelettransform is showninfigure 4. Theeffectoffiltering maybe noticed, as well as the removal of relatively high frequency components. Nevertheless, this component retains all the information needed to detect the events.
By applying the estimation method on this component, the profile of which is shown in figure 6. , the points t₀ =2:61, t₁ =3:80 and t₂ = 4 : 97 are respectively detected and separation times of 1:19 sec between the first and the second event, and 1:17 sec between the second and third event, may be estimated.

6. CONCLUSIONS

The use of the undecimated Discrete Wavelet Transform has been proposed in the context of cubic spline functions. For this purpose, a simple estimation scheme has been designed, and its efficiency in different cases has been illustrated.
Therefore, it is concluded that this kind of techniques may be highly useful for processing ultrasonic or acoustic emission signals in order to cope with different detection problems, structure characterization and comparison, and the estimation of parameters associated with these types of phenomena. Furthermore, the method here proposed is open to refining, further modifications, and extensions. The use of wavelet packets, natural extension of the wavelet transform, has proved particularly useful and should be highlighted as a means to attain higher resolution in frequency domains, as required.

References:

1. G. Leisk, A. Seigal, 'Digital Computer Algorithms to Calculate Ultrasonic Wave Speed', Materials Evaluation, Jul. 1996.
2. ] S. Ziola, M. Gorman, 'Source Location in Thin Plates Using Cross - Correlation', J. Acoust. Soc. Am. 90, 1991.
3. M. Peterson, 'A Method for Increased Accuracy of a Measurement of Relative Phase Velocity', Ultrasonics 35, 1997.
4. B. Audoin, J. Roux, 'An Innovative application of the Hilbert Transform to time delay estimation of overlapped ultrasonic echoes', Ultrasonics 34, 1996.
5. C. E. D' Attellis, L. V. Perez, D. Rubio, J. E. Ruzzante, 'Parameter Estimation in Acoustic Emission Signals', Journal of Acoustic Emission 10, 1992.
6. Y. Meyer, Wavelets, Algorithms & Applicattions , SIAM, Philadelphia, 1993.
7. S. Mallat, A Wavelet Tour of Signal Processing , Acadedmic Press, San Diego, 1998.
8. E. Serrano and M. Fabio, 'The Undecimated Wavelet Transform', in Wavelet Theory and Harmonic Analys is in Applied Sciences , Ed. Birkhauser, Boston, 1997.

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