on the interpolation of ultrasonic guided wave ... - Semantic Scholar
ON THE INTERPOLATION OF ULTRASONIC GUIDED WAVE SIGNALS Jennifer E. Michaels1, Ren-Jean Liou2, Jason P. Zutty1, and Thomas E. Michaels1 1
School of Electrical & Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
2
Department of Computers and Communications, National Pingtung Institute of Commerce, Pingtung, Taiwan
ABSTRACT. The application of ultrasonic guided wave methods to both nondestructive evaluation (NDE) and structural health monitoring (SHM) is becoming more prevalent as techniques to handle their multi-modal and dispersive nature are developed. There are several applications where it would be not only convenient but perhaps essential to interpolate arrays of measured guided wave signals. One application is that of linear spatial arrays acting as receivers, where it may be useful to interpolate signals in between array elements. Another application is interpolation of signals acquired as a function of a non-spatial variable such as temperature or applied load; this situation arises in SHM applications where it is desired to construct a baseline that is well-matched to the signal of interest. This problem is closely related to that of time domain up-sampling whereby signals are resampled at a higher rate, which can readily be performed using sinc interpolation when the Nyquist criterion is satisfied. In the spatial domain, there is an analogous spatial Nyquist criterion, and if it is satisfied, spatial signals can be similarly up-sampled. In this paper we derive a Nyquist criterion for sinc interpolation in the temperature domain. Examples are shown of sinc, linear and spline interpolation algorithms, and the efficacy of each is evaluated on both simulated and experimental data. Concluding remarks are made regarding both the usefulness and limitations of guided wave signal interpolation. Keywords: Lamb Waves, Arrays, Baseline Subtraction, Interpolation, Resampling PACS: 43.35.Zc, 43.60.-c
INTRODUCTION For structural health monitoring (SHM) systems employing guided ultrasonic waves, the measured signal is often evaluated by first subtracting a baseline signal that was recorded from a prior state. However, it is generally not possible to obtain baselines for all possible environmental conditions. One motivation for considering interpolation methods is to apply interpolation to a collected set of baselines to approximate the “real” baseline. Another motivation for investigating interpolation is that it is desirable to sample data no more finely than is actually needed for a specific application. In the time domain, the Nyquist criterion is well-known, which states that a signal must be sampled at more than twice the highest frequency present to prevent aliasing. However, for acquisition of most ultrasonic guided wave signals, this minimum allowable sampling frequency is typically exceeded to provide improved resolution in the time domain. Since higher frequency Review of Progress in Quantitative Nondestructive Evaluation AIP Conf. Proc. 1430, 679-686 (2012); doi: 10.1063/1.4716292 © 2012 American Institute of Physics 978-0-7354-1013-8/$30.00
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digitizers are readily available and memory is inexpensive, there are often no downsides to oversampling such signals. Oversampling in other domains, however, may not be so readily accomplished. For example, in the spatial domain, there are many more constraints regarding sampling. Sensor size can limit receiver spacing, and acquisition time requirements may dictate spatial increments for scanned systems. Interpolation for spatial up-sampling has been addressed for both geophysics [1] and biomedical ultrasonic applications [2], and has also been considered to address compensation for nonfunctioning receivers [3]. Spatial interpolation of ultrasonic guided wave signals is also of interest, particularly given the increasing use of scanning laser vibrometers for acquiring both 1D and 2D wavefield data. In addition, it is useful to consider interpolation in other domains, such as temperature or applied loads. These types of interpolation are applicable to structural health monitoring where it is desired to match current signals to baseline data that might have been acquired under different environmental and operational conditions. Existing methods to address temperature variations require stretching signals to compensate for temperature mismatch and thus achieve better baseline matching [4,5]. However, such methods are not generally applicable to other conditions (e.g., applied loads), and thus signal interpolation methods may be an effective alternative approach. In this paper the efficacy of sinc, linear and spline interpolated methods are investigated in the spatial and temperature domains. Both simulated and experimental data are considered. SAMPLING THEORY The traditional sampling theorem is based on the Nyquist criterion. A bandlimited temporal signal x(t) can be perfectly recovered from an infinite sequence of samples if the sampling rate exceeds 2 fmax, where fmax is the highest frequency of x(t). That is, if Fs is the sampling frequency, then
Fs 2 f max .
(1)
This sampling theory is equally applicable to other domains, such as space and temperature, with Fs defined in its respective domain. However, Eq. (1) only applies to signals that are sampled for infinite time. A time-limited signal cannot be bandlimited, and thus the reconstruction (or interpolation) of a time-limited signal is by necessity an approximation, albeit often a very good one. Reconstruction of the original signal can be readily performed by sinc function multiplication and summing. It is equivalent to applying a perfect lowpass filter in the time domain. If x(nT) represents the samples of x(t), where T is the sampling period, the reconstructed signal x(t ) is obtained as x(t )
x(nT )
n
sin (t nT ) T
(t nT ) T
.
(2)
However, it can be problematic in practice to use the sinc function. The first difficulty is the computational requirements because summing a large number of samples is required. The primary difficulty, however, is related to edge effects because signals are windowed in all domains. The multiplicative windowing operation generates high frequency content, which means that the Nyquest criterion is not satisfied and interpolation performance may be poor. There are other interpolation methods available that have different characteristics, and two of the more commonly used schemes are considered here.
680 Downloaded 07 Jun 2012 to 130.207.50.37. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
For spatial sampling, consider a simple sinusoid that represents a traveling plane wave captured at a specific time s a function of the direction of propagation. The wellknown relationship between wavenumber k, wavelength λ, frequency f, and wave speed c is
k
2
2 f c and . c f
(3)
If x is the spatial sampling interval, then the Nyquist criterion for spatial sampling is
x
. (4) 2 This criterion only applies to signals that are sampled for an infinite spatial extent; a spacelimited signal cannot be band-limited and perfect reconstruction is therefore not achievable. Only an approximation can be obtained, which may (or may not) be sufficiently accurate. By examining the characteristics of ultrasonic signals recorded at different temperatures, it can be seen that multiple echoes are analogous to multiple waves propagating at different wave speeds. The so-called “temperature domain,” where signals are expressed as a function of temperature, is analogous to the spatial domain, where signals are a function of position. Since the speed of propagation changes linearly with the temperature change [4,5], individual echoes shift linearly as a function of temperature. We define a “wave speed” c T t in the temperature domain, where T is the temperature change (i.e., temperature sampling interval), f is frequency, and t is the time shift of an echo associated with the temperature change T; the corresponding wavelength is c f . The Nyquist criterion for temperature sampling thus becomes c T min , (5) 2 f max where cmin occurs at the maximum time tmax, and fmax is the maximum frequency. A simple illustration is depicted in Figure 1. INTERPOLATION METHODS Three different interpolation techniques are employed in this paper: sinc, linear and spline. Sinc interpolation, which is defined in Eq. (2), is “perfect” in the sense that in theory, a band-limited signal can be perfectly reconstructed if the Nyquist criterion is met.
FIGURE 1. Ultrasonic signals in the temperature domain.
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In practice, the Nyquist criterion is never perfectly met because signals in all domains are of finite length, causing the introduction of higher frequencies, which is called spectral leakage. Linear interpolation, which is the straight line between two samples, is the simplest form of interpolation and is not affected by windowing. However, it requires finer sampling to achieve acceptable results, and can also be very sensitive to additive noise. Cubic spline interpolation is another method whereby sampled points are exactly interpolated by piecewise continuous cubic polynomials [6]. Although an approximation, it has smoother results than linear interpolation but is affected by windowing. NUMERICAL RESULTS Spatial Domain Simulated data were generated in the spatial domain to study interpolation performance. Data correspond to signals received by a linear array of receivers after simultaneous excitation of three transmitters. The configuration is illustrated in Figure 2 where all transmitters and receivers lie in the xy plane and the origin is at the left-most receiver. The excitation was a 100 kHz Hanning windowed tone burst, the wave speed was 2.5 mm/s, signals were sampled at 2 MHz, and dispersion was not modeled. Almost all of the energy was below 200 kHz, so the minimum spatial wavelength was estimated to be 12.5 mm, which implies that the spatial sampling interval should be no larger than x = 6.25 mm (half of the smallest wavelength). Array signals were simulated for a receiver spacing of 0.5 mm (201 receivers). Figure 3(a) shows all of the received signals as an image, and Figure 3(b) shows a single signal for the receiver located at 20 mm.
FIGURE 2. Illustration of the geometry for spatial domain simulations (not to scale).
(a)
(b)
FIGURE 3. (a) Array signals at a spacing of 0.5 mm, and (b) signal for the receiver located at x = 20 mm.
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The three interpolation methods are evaluated by spatially down-sampling the signals shown in Figure 3(a), interpolating in between the down-sampled signals at an increment of 0.5 mm, and comparing the interpolated signals to the original ones prior to down-sampling. The interpolation error is calculated in dB based upon the maximum error for each signal relative to the peak amplitude for the entire data set. Prior work (e.g. [5]) indicates that an error of -40 dB is more than adequate to ensure that small, scattered signals from damage can be detected. Figure 4(a) shows results of sinc interpolation for down-sampled spatial increments of 1, 5 and 10 mm. Note that points that fall directly on receiver locations have zero error, and these points are omitted for clarity. As expected, the error increases as the spatial sampling increment increases, and the maximum error occurs near the ends of the receiver array. Even if the sampling interval is as small as 1 mm, the error is still well above the target of -40 dB near the edges. Figure 4(b) summarizes the maximum error for sampling increments of 1 to 10 mm. Linear and spline interpolation were applied to the same data, and results are shown in Figures 5 and 6. It is clear that sinc interpolation has the most severe edge effects whereas, as expected, linear interpolation has the least. Spline interpolation has the overall best performance, achieving errors less than -40 dB for spatial increments of 3 mm and smaller; sinc interpolation never achieves this goal, and linear does only for a 1 mm increment.
(a)
(b)
FIGURE 4. Sinc interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
(a)
(b)
FIGURE 5. Linear interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
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(a)
(b)
FIGURE 6. Spline interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
Temperature Domain The performance of interpolation in the temperature domain is investigated similarly to the spatial domain. Simulations were performed for the A0 guided wave mode propagating in a square aluminum plate of dimensions 610 mm × 610 mm × 2.0 mm. Theoretical dispersion curves were combined with ray tracing and the method of images to model signals traveling from a point transmitter to a point receiver separated by 336 mm. Varying temperatures were simulated by recalculating plate dimensions and dispersion curves using published values for the coefficient of thermal expansion and temperatureependent bulk wave speeds [4]. The excitation signal was a 100 kHz, Hanning windowed tone burst. Signals were simulated from 0 to 50°C at a 1°C increment and are shown in Figure 7. The expected interpolation performance is calculated by first estimating the minimum “wave speed” c T t in the temperature domain. The time shift for a particular echo and temperature change is estimated from the simulated data. Since time shifts are linear with time-of-flight, the time shift at the maximum time of 1000 μs is obtained by extrapolation. The result is cmin 4.17 C s and ΔT = 10.4°C for t = 1000 μs. If results from the spatial domain carry over to the temperature domain, then the expectation is that spline interpolation should be effective for temperature sampling intervals of about 5°C or less.
FIGURE 7. Guided wave signals at a temperature increment of 1°C.
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(a)
(b)
(c)
FIGURE 8. Maximum interpolation error for three interpolation methods. (a) Sinc, (b) linear, and (c) spline.
The three interpolation methods were applied to the temperature-dependent data for temperature increments of 2 to 10°C with a 1°C increment. Results are shown in Figure 8, and are similar to those obtained for interpolation in the spatial domain. In particular, the cubic spline method has the best performance and, as predicted, achieves a maximum error of -40 dB or less with temperature increments of 5°C and smaller. EXPERIMENTAL RESULTS Interpolation was performed on experimental guided wave signals that were recorded at unequally spaced temperatures for an aluminum plate specimen. The frequency of excitation was 250 kHz, and the S0 mode was dominant but with some A0 present. Figure 9(a) is a plot of the temperatures at which the 18 signals were collected, and Figure 9(b) shows an image of the signals for the time window of 0 to 500 μs. As was done for the simulated data, the theoretical temperature increment was calculated based upon measured time shifts. For the maximum recorded time of 1000 μs, this value was computed to be 3.33°C, so the expectation is that spline interpolation should be effective for half this amount, or 1.66°C. Linear and spline interpolation were performed on the 1000 μs duration signals by using the nine odd-numbered signals (1,3,…17) to interpolate the eight bracketed evennumbered signals (2,4,…16). From the data, the largest temperature increment between
(a)
(b)
FIGURE 9. (a) Temperatures at which signals were recorded, and (b) measured signals.
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odd-numbered signals was 3.4°C, which is larger than the expected required increment of 1.66°C. Not surprisingly, the resulting maximum errors were -25.8 dB and -30.5 dB for linear and spline interpolation, respectively, both of which are larger than -40 dB. Since performance should improve if signals are shorter in length, they were truncated to 500 μs. Interpolation performance improved slightly to -31.1 dB and -32.7 dB for linear and spline interpolation, respectively. The goal of -40 dB was not achieved, most likely because of either other temperature dependent effects (e.g., on the sensor itself) or noise, but performance of -30 dB may still be sufficient for many SHM applications. CONCLUSIONS This paper has considered sinc, linear and spline interpolation of ultrasonic guided wave signals in both spatial and temperature domains. A Nyquist sampling criterion was derived in the temperature domain, showing that the temperature sampling interval must decrease as the signal length increases because of both the change of velocity and structural dimensions with temperature. Simulated data in both the spatial and temperature domains were used to illustrate the negative impact of spectral leakage due to windowing. It was shown that spline interpolation achieved the best results for windowed signals, with a sampling interval of about half the theoretical value needed for satisfactory interpolation. Experimental results in the temperature domain were reasonable but not as good as expected from simulated data, suggesting that either noise or temperature-dependent effects not related to either velocity or dimensional changes are a contributing factor. Unlike the stretching/resampling methods currently in use for temperature compensation, interpolation methods offer a more general alternative for improving baseline matching, and thus merit further investigation. ACKNOWLEDGEMENTS The first author acknowledges partial support by the Air Force Office of Scientific Research under Grant Number FA9550-08-1-0241. REFERENCES 1. 2.
3.
4.
5.
6.
B. Liu and M. D. Sacchi, “Minimum weighted norm interpolation of seismic records,” Geophysics, 69(6), pp. 1560-1568, 2004. A. Trucco, S. Curletto, and M. Palmese, “Interpolation of medical ultrasound images from coherent and incoherent signals,” IEEE Transactions on Instrumentation and Measurement, 58(7), pp. 2048-2060, 2009. A. J. W. Duijndam, M. A. Schonewille, and C. O. H. Hindriks, “Reconstruction of band-limited signals, irregularly sampled along one dimension,” Geophysics, 64(2), pp. 524-538, 1999. Y. Lu and J. E. Michaels, “A methodology for structural health monitoring with diffuse ultrasonic waves in the presence of temperature variations,” Ultrasonics, 43, pp. 717-731, 2005. A. J. Croxford, J. Moll, P. D. Wilcox and J. E. Michaels, “Efficient temperature compensation strategies for guided wave structural health monitoring,” Ultrasonics, 50, pp. 517-528, 2010. C. Gerald and P. Wheatley, Applied Numerical Analysis, Addison-Wesley, 1994.
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School of Electrical & Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
2
Department of Computers and Communications, National Pingtung Institute of Commerce, Pingtung, Taiwan
ABSTRACT. The application of ultrasonic guided wave methods to both nondestructive evaluation (NDE) and structural health monitoring (SHM) is becoming more prevalent as techniques to handle their multi-modal and dispersive nature are developed. There are several applications where it would be not only convenient but perhaps essential to interpolate arrays of measured guided wave signals. One application is that of linear spatial arrays acting as receivers, where it may be useful to interpolate signals in between array elements. Another application is interpolation of signals acquired as a function of a non-spatial variable such as temperature or applied load; this situation arises in SHM applications where it is desired to construct a baseline that is well-matched to the signal of interest. This problem is closely related to that of time domain up-sampling whereby signals are resampled at a higher rate, which can readily be performed using sinc interpolation when the Nyquist criterion is satisfied. In the spatial domain, there is an analogous spatial Nyquist criterion, and if it is satisfied, spatial signals can be similarly up-sampled. In this paper we derive a Nyquist criterion for sinc interpolation in the temperature domain. Examples are shown of sinc, linear and spline interpolation algorithms, and the efficacy of each is evaluated on both simulated and experimental data. Concluding remarks are made regarding both the usefulness and limitations of guided wave signal interpolation. Keywords: Lamb Waves, Arrays, Baseline Subtraction, Interpolation, Resampling PACS: 43.35.Zc, 43.60.-c
INTRODUCTION For structural health monitoring (SHM) systems employing guided ultrasonic waves, the measured signal is often evaluated by first subtracting a baseline signal that was recorded from a prior state. However, it is generally not possible to obtain baselines for all possible environmental conditions. One motivation for considering interpolation methods is to apply interpolation to a collected set of baselines to approximate the “real” baseline. Another motivation for investigating interpolation is that it is desirable to sample data no more finely than is actually needed for a specific application. In the time domain, the Nyquist criterion is well-known, which states that a signal must be sampled at more than twice the highest frequency present to prevent aliasing. However, for acquisition of most ultrasonic guided wave signals, this minimum allowable sampling frequency is typically exceeded to provide improved resolution in the time domain. Since higher frequency Review of Progress in Quantitative Nondestructive Evaluation AIP Conf. Proc. 1430, 679-686 (2012); doi: 10.1063/1.4716292 © 2012 American Institute of Physics 978-0-7354-1013-8/$30.00
679 Downloaded 07 Jun 2012 to 130.207.50.37. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
digitizers are readily available and memory is inexpensive, there are often no downsides to oversampling such signals. Oversampling in other domains, however, may not be so readily accomplished. For example, in the spatial domain, there are many more constraints regarding sampling. Sensor size can limit receiver spacing, and acquisition time requirements may dictate spatial increments for scanned systems. Interpolation for spatial up-sampling has been addressed for both geophysics [1] and biomedical ultrasonic applications [2], and has also been considered to address compensation for nonfunctioning receivers [3]. Spatial interpolation of ultrasonic guided wave signals is also of interest, particularly given the increasing use of scanning laser vibrometers for acquiring both 1D and 2D wavefield data. In addition, it is useful to consider interpolation in other domains, such as temperature or applied loads. These types of interpolation are applicable to structural health monitoring where it is desired to match current signals to baseline data that might have been acquired under different environmental and operational conditions. Existing methods to address temperature variations require stretching signals to compensate for temperature mismatch and thus achieve better baseline matching [4,5]. However, such methods are not generally applicable to other conditions (e.g., applied loads), and thus signal interpolation methods may be an effective alternative approach. In this paper the efficacy of sinc, linear and spline interpolated methods are investigated in the spatial and temperature domains. Both simulated and experimental data are considered. SAMPLING THEORY The traditional sampling theorem is based on the Nyquist criterion. A bandlimited temporal signal x(t) can be perfectly recovered from an infinite sequence of samples if the sampling rate exceeds 2 fmax, where fmax is the highest frequency of x(t). That is, if Fs is the sampling frequency, then
Fs 2 f max .
(1)
This sampling theory is equally applicable to other domains, such as space and temperature, with Fs defined in its respective domain. However, Eq. (1) only applies to signals that are sampled for infinite time. A time-limited signal cannot be bandlimited, and thus the reconstruction (or interpolation) of a time-limited signal is by necessity an approximation, albeit often a very good one. Reconstruction of the original signal can be readily performed by sinc function multiplication and summing. It is equivalent to applying a perfect lowpass filter in the time domain. If x(nT) represents the samples of x(t), where T is the sampling period, the reconstructed signal x(t ) is obtained as x(t )
x(nT )
n
sin (t nT ) T
(t nT ) T
.
(2)
However, it can be problematic in practice to use the sinc function. The first difficulty is the computational requirements because summing a large number of samples is required. The primary difficulty, however, is related to edge effects because signals are windowed in all domains. The multiplicative windowing operation generates high frequency content, which means that the Nyquest criterion is not satisfied and interpolation performance may be poor. There are other interpolation methods available that have different characteristics, and two of the more commonly used schemes are considered here.
680 Downloaded 07 Jun 2012 to 130.207.50.37. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
For spatial sampling, consider a simple sinusoid that represents a traveling plane wave captured at a specific time s a function of the direction of propagation. The wellknown relationship between wavenumber k, wavelength λ, frequency f, and wave speed c is
k
2
2 f c and . c f
(3)
If x is the spatial sampling interval, then the Nyquist criterion for spatial sampling is
x
. (4) 2 This criterion only applies to signals that are sampled for an infinite spatial extent; a spacelimited signal cannot be band-limited and perfect reconstruction is therefore not achievable. Only an approximation can be obtained, which may (or may not) be sufficiently accurate. By examining the characteristics of ultrasonic signals recorded at different temperatures, it can be seen that multiple echoes are analogous to multiple waves propagating at different wave speeds. The so-called “temperature domain,” where signals are expressed as a function of temperature, is analogous to the spatial domain, where signals are a function of position. Since the speed of propagation changes linearly with the temperature change [4,5], individual echoes shift linearly as a function of temperature. We define a “wave speed” c T t in the temperature domain, where T is the temperature change (i.e., temperature sampling interval), f is frequency, and t is the time shift of an echo associated with the temperature change T; the corresponding wavelength is c f . The Nyquist criterion for temperature sampling thus becomes c T min , (5) 2 f max where cmin occurs at the maximum time tmax, and fmax is the maximum frequency. A simple illustration is depicted in Figure 1. INTERPOLATION METHODS Three different interpolation techniques are employed in this paper: sinc, linear and spline. Sinc interpolation, which is defined in Eq. (2), is “perfect” in the sense that in theory, a band-limited signal can be perfectly reconstructed if the Nyquist criterion is met.
FIGURE 1. Ultrasonic signals in the temperature domain.
681 Downloaded 07 Jun 2012 to 130.207.50.37. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
In practice, the Nyquist criterion is never perfectly met because signals in all domains are of finite length, causing the introduction of higher frequencies, which is called spectral leakage. Linear interpolation, which is the straight line between two samples, is the simplest form of interpolation and is not affected by windowing. However, it requires finer sampling to achieve acceptable results, and can also be very sensitive to additive noise. Cubic spline interpolation is another method whereby sampled points are exactly interpolated by piecewise continuous cubic polynomials [6]. Although an approximation, it has smoother results than linear interpolation but is affected by windowing. NUMERICAL RESULTS Spatial Domain Simulated data were generated in the spatial domain to study interpolation performance. Data correspond to signals received by a linear array of receivers after simultaneous excitation of three transmitters. The configuration is illustrated in Figure 2 where all transmitters and receivers lie in the xy plane and the origin is at the left-most receiver. The excitation was a 100 kHz Hanning windowed tone burst, the wave speed was 2.5 mm/s, signals were sampled at 2 MHz, and dispersion was not modeled. Almost all of the energy was below 200 kHz, so the minimum spatial wavelength was estimated to be 12.5 mm, which implies that the spatial sampling interval should be no larger than x = 6.25 mm (half of the smallest wavelength). Array signals were simulated for a receiver spacing of 0.5 mm (201 receivers). Figure 3(a) shows all of the received signals as an image, and Figure 3(b) shows a single signal for the receiver located at 20 mm.
FIGURE 2. Illustration of the geometry for spatial domain simulations (not to scale).
(a)
(b)
FIGURE 3. (a) Array signals at a spacing of 0.5 mm, and (b) signal for the receiver located at x = 20 mm.
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The three interpolation methods are evaluated by spatially down-sampling the signals shown in Figure 3(a), interpolating in between the down-sampled signals at an increment of 0.5 mm, and comparing the interpolated signals to the original ones prior to down-sampling. The interpolation error is calculated in dB based upon the maximum error for each signal relative to the peak amplitude for the entire data set. Prior work (e.g. [5]) indicates that an error of -40 dB is more than adequate to ensure that small, scattered signals from damage can be detected. Figure 4(a) shows results of sinc interpolation for down-sampled spatial increments of 1, 5 and 10 mm. Note that points that fall directly on receiver locations have zero error, and these points are omitted for clarity. As expected, the error increases as the spatial sampling increment increases, and the maximum error occurs near the ends of the receiver array. Even if the sampling interval is as small as 1 mm, the error is still well above the target of -40 dB near the edges. Figure 4(b) summarizes the maximum error for sampling increments of 1 to 10 mm. Linear and spline interpolation were applied to the same data, and results are shown in Figures 5 and 6. It is clear that sinc interpolation has the most severe edge effects whereas, as expected, linear interpolation has the least. Spline interpolation has the overall best performance, achieving errors less than -40 dB for spatial increments of 3 mm and smaller; sinc interpolation never achieves this goal, and linear does only for a 1 mm increment.
(a)
(b)
FIGURE 4. Sinc interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
(a)
(b)
FIGURE 5. Linear interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
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(a)
(b)
FIGURE 6. Spline interpolation results. (a) Error vs. position for three spatial sampling intervals, and (b) max error for sampling intervals ranging from 1 to 10 mm.
Temperature Domain The performance of interpolation in the temperature domain is investigated similarly to the spatial domain. Simulations were performed for the A0 guided wave mode propagating in a square aluminum plate of dimensions 610 mm × 610 mm × 2.0 mm. Theoretical dispersion curves were combined with ray tracing and the method of images to model signals traveling from a point transmitter to a point receiver separated by 336 mm. Varying temperatures were simulated by recalculating plate dimensions and dispersion curves using published values for the coefficient of thermal expansion and temperatureependent bulk wave speeds [4]. The excitation signal was a 100 kHz, Hanning windowed tone burst. Signals were simulated from 0 to 50°C at a 1°C increment and are shown in Figure 7. The expected interpolation performance is calculated by first estimating the minimum “wave speed” c T t in the temperature domain. The time shift for a particular echo and temperature change is estimated from the simulated data. Since time shifts are linear with time-of-flight, the time shift at the maximum time of 1000 μs is obtained by extrapolation. The result is cmin 4.17 C s and ΔT = 10.4°C for t = 1000 μs. If results from the spatial domain carry over to the temperature domain, then the expectation is that spline interpolation should be effective for temperature sampling intervals of about 5°C or less.
FIGURE 7. Guided wave signals at a temperature increment of 1°C.
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(a)
(b)
(c)
FIGURE 8. Maximum interpolation error for three interpolation methods. (a) Sinc, (b) linear, and (c) spline.
The three interpolation methods were applied to the temperature-dependent data for temperature increments of 2 to 10°C with a 1°C increment. Results are shown in Figure 8, and are similar to those obtained for interpolation in the spatial domain. In particular, the cubic spline method has the best performance and, as predicted, achieves a maximum error of -40 dB or less with temperature increments of 5°C and smaller. EXPERIMENTAL RESULTS Interpolation was performed on experimental guided wave signals that were recorded at unequally spaced temperatures for an aluminum plate specimen. The frequency of excitation was 250 kHz, and the S0 mode was dominant but with some A0 present. Figure 9(a) is a plot of the temperatures at which the 18 signals were collected, and Figure 9(b) shows an image of the signals for the time window of 0 to 500 μs. As was done for the simulated data, the theoretical temperature increment was calculated based upon measured time shifts. For the maximum recorded time of 1000 μs, this value was computed to be 3.33°C, so the expectation is that spline interpolation should be effective for half this amount, or 1.66°C. Linear and spline interpolation were performed on the 1000 μs duration signals by using the nine odd-numbered signals (1,3,…17) to interpolate the eight bracketed evennumbered signals (2,4,…16). From the data, the largest temperature increment between
(a)
(b)
FIGURE 9. (a) Temperatures at which signals were recorded, and (b) measured signals.
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odd-numbered signals was 3.4°C, which is larger than the expected required increment of 1.66°C. Not surprisingly, the resulting maximum errors were -25.8 dB and -30.5 dB for linear and spline interpolation, respectively, both of which are larger than -40 dB. Since performance should improve if signals are shorter in length, they were truncated to 500 μs. Interpolation performance improved slightly to -31.1 dB and -32.7 dB for linear and spline interpolation, respectively. The goal of -40 dB was not achieved, most likely because of either other temperature dependent effects (e.g., on the sensor itself) or noise, but performance of -30 dB may still be sufficient for many SHM applications. CONCLUSIONS This paper has considered sinc, linear and spline interpolation of ultrasonic guided wave signals in both spatial and temperature domains. A Nyquist sampling criterion was derived in the temperature domain, showing that the temperature sampling interval must decrease as the signal length increases because of both the change of velocity and structural dimensions with temperature. Simulated data in both the spatial and temperature domains were used to illustrate the negative impact of spectral leakage due to windowing. It was shown that spline interpolation achieved the best results for windowed signals, with a sampling interval of about half the theoretical value needed for satisfactory interpolation. Experimental results in the temperature domain were reasonable but not as good as expected from simulated data, suggesting that either noise or temperature-dependent effects not related to either velocity or dimensional changes are a contributing factor. Unlike the stretching/resampling methods currently in use for temperature compensation, interpolation methods offer a more general alternative for improving baseline matching, and thus merit further investigation. ACKNOWLEDGEMENTS The first author acknowledges partial support by the Air Force Office of Scientific Research under Grant Number FA9550-08-1-0241. REFERENCES 1. 2.
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