·Table of Contents ·Methods and Instrumentation | Estimation of Differences in Event Arrival Times Through Wavelet Transform TechniquesJ. E. RuzzanteLaboratorio 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 |
Keywords: Wavelets, Undecimated Discrete Wavelet Transform, Pattern Recognition
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where s_{0} (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} s_{0}(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:
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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 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:
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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 t_{0} = 2 :7, t_{1} = 3:89 and t_{2} =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 t_{0} = 2:61, t_{1} =3:80 and t_{2} =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.
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