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Pulsed ultrasonic comb filtering effect and its applications in the measurement of sound velocity and thickness of thin plates
Ultrasonics 75 (2017) 199–208
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Pulsed ultrasonic comb filtering effect and its applications in the measurement of sound velocity and thickness of thin plates Jingfei Liu ⇑, Nico F. Declercq Georgia Institute of Technology, Laboratory for Ultrasonic Nondestructive Evaluation ‘‘LUNE”, Georgia Tech-CNRS UMI2958, Georgia Tech Lorraine, 2, rue Marconi, 57070 Metz, France
a r t i c l e
i n f o
Article history: Received 28 July 2016 Received in revised form 28 November 2016 Accepted 3 December 2016 Available online 8 December 2016 Keywords: Comb filter Pulsed ultrasonics Spectral peaks and notches Sound velocity Thin plates Comb filtering effect
a b s t r a c t An analytical and experimental study of the pulsed ultrasonic comb filtering effect is presented in this work intending to provide a fundamental tool for data analysis and phenomenon understanding in pulsed ultrasonics. The basic types of comb filter, feedforward and feedback filters, are numerically simulated and demonstrated. The characteristic features of comb filters, which include the formula for determining the locations of the spectral peaks or notches and the relationship between its temporal characteristics (relative time delay between constituent pulses) and its spectral characteristics (frequency interval between peaks or notches), are theoretically derived. To demonstrate the applicability of the comb filtering effect, it is applied to measuring the sound velocities and thickness of a thin plate sample. It is proven that the comb filtering effect based method not only is capable of accurate measurements, but also has advantages over the conventional time-of-flight based method in thin plate measurements. Furthermore, the principles developed in this study have potential applications in any pulsed ultrasonic cases where the output signal shows comb filter features. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction A comb filter refers to a filter whose frequency spectrum consists of a number of equally spaced elements resembling the tines of a comb [1]. As a physical tool, it has been used to improve the signal-to-noise ratio in pulsed radar systems [1], video signal transmission systems [2,3] as well as in biological signal detection [4] and speech transmission [5,6]. It has also been applied to designing gas sensors [7] and multi-wavelength fiber lasers [8–11]. In the auditory physiological and psycho-physical studies, it is even applied for generating noises [12]. As an algorithm of signal processing, its application for separating mixed signals [13], tracking slow changes in system frequencies [14,15], improving sensor performance [16] and GPS code tracking performance [17] have been investigated. The most recent research is focused on its application in optics [18], electronics [19] and photonics [20]. Within the scope of acoustics the comb filtering effect has been mostly investigated in room acoustics such as its role in the sound coloration of orchestra and opera halls [21], in musical acoustics such as its application in musical sound representation [22] and in speech acoustics such as speech enhancement [23] and pitch detection [24]. In audio engineering, the comb filtering effect, ⇑ Corresponding author. E-mail address: [email protected] (J. Liu). http://dx.doi.org/10.1016/j.ultras.2016.12.003 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.
which normally results from the interference of sound with its delayed duplicates, is either deliberately avoided to eliminate undesired colored sound, or used to create flanging effects. All these acoustic investigations are performed in the low frequency regime, and to the authors’ knowledge no investigation has been done in the high frequency regime such as in ultrasonics. In ultrasonic applications, the comb filtering effect occurs in various cases such as the reflection and transmission of the pulses normally incident unto thin plates (which is studied in this work). If this effect can be well understood, it will definitely be helpful for extracting desired information from ultrasonic measurements or properly interpreting experimental observations. The current work serves as a fundamental study of the comb filtering effect in pulsed ultrasonics. The theoretical investigation of the pulsed ultrasonic comb filtering effect starts with a numerical model of a pulse with a Gaussian spectrum. Two basic forms of comb filter, namely feedforward and feedback, are first theoretically described using a block diagram and the difference equation; and are then numerically simulated based on the defined pulse. The important features of the comb filtering effect in pulsed ultrasonics, namely the locations of spectral amplitude enhancing and reducing effects and the relationship between the temporal and spectral characteristic features, are analytically derived. Then, a special case of the feedback comb filter, in which all the pulses overlap in the time domain, is introduced to demonstrate the comb
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filtering effect in extreme situations. The pulsed ultrasonic comb filtering effect is further demonstrated experimentally based on a series of pulse-echo and through-transmission measurements on thin plate samples. In addition to the theoretical analysis and experimental demonstration, and in order to show its practical usefulness in ultrasonics, the characteristic relationship of the comb filtering effect is applied to measuring the sound velocity and thickness of thin plate samples as the solution of a common problem in pulsed ultrasonics. For sound velocity measurement, the method based on the comb filtering effect not only is capable of simultaneously obtaining both longitudinal and shear velocities, but is even capable of doing so in cases where the conventional time-of-flight (TOF) based ultrasonic method fails in the case of very thin plates where an overlap of ultrasonic echoes occurs in addition to the inherent reduction in temporal resolution [25]. Furthermore, the same principle derived from the comb filtering effect appears even applicable to acquire thickness measurement of thin plates with high accuracy. The rest of this paper is outlined as follows. The comb filtering effect in pulsed ultrasound is theoretically described in Section 2 and experimentally demonstrated in Section 3. Section 4 shows the application of this effect in measuring the sound velocity and thickness of thin plate samples as a case study of potential application of the study presented in this work. Section 5 provides the concluding remarks. 2. Pulsed ultrasonic comb filtering effect
amplitude A0 n and the ‘phase’ of the phase spectrum refers to the angle of the complex amplitude obtained in the fast Fourier transform (FFT) calculation. An example of a Gaussian amplitude spectrum and the resultant pulse waveform as well as its phase spectrum is shown in Fig. 1. If more than one pulse is defined in one wave, their phase difference is realized by choosing a proper pulse phase u0 for each pulse in addition to a relative time delay t0. 2.2. Feedforward comb filter The block diagram and difference equation of the feedforward comb filter are shown in Fig. 2 and Eq. (3), respectively. It is easy to see that the output of this filter is a linear combination of the direct and delayed input signals. From the perspective of pulsed ultrasonics, this filter is the physical model of the reality in which a pulse is superposed onto a delayed and attenuated version of itself. The reason of delay can be either extra distance or different materials that the input pulse propagates through, or both. And the source of the attenuation can possibly be absorption, diffraction or spherical spreading loss.
yðnÞ ¼ b0 xðnÞ þ bM xðn MÞ
ð3Þ
To demonstrate the effect of this feedforward comb filter in pulsed ultrasonics, a wave containing a pulse defined in Section 2.1 and its delayed and attenuated version is assumed as shown in Fig. 3(a). Compared with the first pulse, the second pulse is delayed by 1 ls and attenuated to 80% of the amplitude. Because the second pulse is the delayed and attenuated version of the first pulse,
There are two basic types of comb filters: feedforward filter and feedback filter [26]. In this section, we will first define an ultrasonic pulse commonly appearing in pulsed ultrasonic technology, and then numerically demonstrate these two types of comb filters in both the time domain and the frequency domain. Together with the relationship for determining the frequencies of the peaks/ notches, the relationship between the time domain characteristics (relative time delay between pulses) of the output of these filters and their frequency domain characteristics (distance between the spectral peaks/notches) will be analytically derived. A special case, in which the output pulses of these filters overlap in the time domain, will also be demonstrated. 2.1. Defining a pulse In most cases of ultrasonic testing the spectra of ultrasonic pulses are not exactly Gaussian per se, but they are to a good approximation Gaussian-like. For the convenience of theoretical analysis a Gaussian spectrum is assumed for all the ultrasonic pulses in this study. The amplitude of the Gaussian spectrum is defined as
Aðf Þ ¼ A0 e
ðf f 0 Þ2 2D2
ð1Þ
in which A0 is the maximum amplitude, f0 the center frequency, and D the width deviation of the spectrum. An ultrasonic pulse corresponding to this spectrum is defined by the following Fourier series, ! ! K K X X 1 1 0 i2pf n t ið2pf n ðtt 0 Þþu0 Þ xðtÞ ¼ Re An e An e ¼ Re ð2Þ K K n¼1 n¼1 in which x(t) is the pulse waveform, K the total number of component harmonics, An the complex amplitude, fn the frequency, u0 the pulse phase, i.e. a phase offset of all the harmonic components representing the given pulse, and t0 is the time delay of the pulse. Unless otherwise clearly stated, in what follows, the ‘amplitude’ of an amplitude spectrum means the absolute value of a complex
Fig. 1. (a) A Gaussian amplitude spectrum with unity maximum amplitude, center frequency of 10 MHz and width deviation of 2 MHz, (b) the resultant ultrasonic pulse defined on this spectrum and (c) the phase spectrum of the defined pulse. Zero phase offset and zero time delay are chosen for all the component harmonics in this pulse.
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Fig. 2. Block diagram of the feedforward comb filter.
Fig. 4. Illustration of the feedforward comb filter using a time domain signal containing a pulse and its delayed (by 1 ls) and attenuated (80% in amplitude) variation with a phase offset of p: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
Fig. 3. Illustration of the feedforward comb filter using a time domain signal containing a pulse and its delayed (by 1 ls) and attenuated (80% in amplitude) version: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
there is no phase shift between them. The amplitude spectrum of this wave, which is compared with that of its first pulse, is shown in Fig. 3(b). This figure clearly shows the comb shape of this filter: compared with its first pulse, the spectral amplitude of this twopulse wave is enhanced at some frequencies (such as 6, 7, 8, 9, 10, 11, 12, 13 and 14 MHz) and reduced at other frequencies (such as 6.5, 7.5, 8.5, 9.5, 10.5, 11.5, 12.5, and 13.5 MHz). The amplitude enhancement and reduction result from the constructive and destructive interference of the two pulses of the wave, respectively. It is interesting to observe from Fig. 3(c) that phase shifts occur at the frequencies of the notches in Fig. 3(b), while no phase shifts can be observed at the frequencies of the peaks. This observation is a confirmation of the occurrence of constructive and destructive interferences: amplitude is enhanced or reduced at the frequencies with a phase difference smaller or larger than p/2, respectively. Besides delay and attenuation, the second pulse of this twopulse model of the feedforward comb filter could also have phase difference from the first pulse. As a special case, a phase difference of p is assumed for the second pulse in addition to the 1 ls delay and 80% amplitude attenuation shown in Fig. 4. The purpose of showing this case is that such a model is often encountered in
reality such as the second pulse being the reflection of the first pulse from a rigid interference (causing a phase difference of p). Similar to Fig. 3, a comb shape of amplitude spectrum can be observed in Fig. 4(b) and the phase shifts occurring at the notch frequencies can also be observed in Fig. 4(c). Different from Fig. 3, the spectral locations for amplitude enhancement (peaks) and reduction (notches) are shifted by 0.5 MHz as shown in Fig. 4(b), which is the result of the phase offset (p) between the two pulses in the waveform (Fig. 4(a)). 2.3. Feedback comb filter The block diagram and difference equation of the feedback comb filter are shown in Fig. 5 and Eq. (4), respectively. The output of this filter is a linear combination of the input signal and a series of exponentially decaying versions of the input signal, which are uniformly spaced in time domain. This filter can work as the
Fig. 5. Block diagram of the feedback comb filter.
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physical model of a wave bouncing back and forth within two parallel surfaces while being received at one side of the parallel surfaces.
yðnÞ ¼ b0 xðnÞ aM xðn MÞ
ð4Þ
In pulsed ultrasonics, the effect of this feedback comb filter can be demonstrated with a time domain signal shown in Fig. 6(a), which contains six pulses that are in phase, equally spaced in time but exponentially decayed in amplitude by a factor of 0.6. Similar to the amplitude spectrum of the feedforward comb filter (Fig. 4(b)), the spectral amplitude of the feedback filter shown in Fig. 6(b) are also enhanced and reduced, but the effect of enhancement is much more visible. This is because all the six pulses are in phase, and each one starting from the second one enforces the effect of the amplitude enhancement at a specific set of frequencies (such as 6, 7, 8, 9, 10, 11, 12, 13 and 14 MHz), while in between these frequencies the amplitude is generally reduced. Fig. 6(c) also shows a series of phase shifts that are corresponding to local notches (less visible) in Fig. 6(b). To clearly demonstrate the amplitude reducing effect of the feedback filter, the signal chosen is shown in Fig. 7(a), which is similar to wave in Fig. 6(a) except that the phase offset between the first pulse and the other five pulses is p (while these five pulses are in phase). Due to the phase offset of p, the spectral amplitudes at the same set of frequencies, at which the amplitude enhancement is observed in Fig. 6(b), are reduced, while at other frequencies the spectral amplitudes are somehow enhanced as shown in Fig. 7(b). Again, phase shifts occur in Fig. 7(c) at the frequencies where the amplitude reducing effect occurs in Fig. 7(b). Fig. 7. Illustration of the feedback comb filter using a time domain signal containing a pulse and five of its variations. Each of the variations has a 1 ls delay and 60% amplitude compared with its previous one. All these five variations are in phase and they are all p out of phase with the first pulse. (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
2.4. Analytical derivation of important features of pulsed comb filtering effect After demonstrating the two types of comb filters in Sections 2.2 and 2.3, an important question arises: how to determine the frequencies at which spectral amplitude enhancement or reduction occurs. In this section, we will analytically investigate this question. Taking a signal containing two pulses (feedforward filter) as an example, the spectral minima or maxima are related to the relative time delay (Dt0) and the initial phase shift (Du0) of the two pulses. Assuming that the harmonic components of a pulse take the following general form in terms of its particle displacement (u):
u ¼ A expði2pf ðt t 0 Þ þ iu0 Þ
ð5Þ
with A as the amplitude, f the frequency, t0 the time delay and u0 the initial phase. The effective phase shift (Du) caused by Dt0 and Du0 that possibly leads to spectral amplitude reduction (notch) is
Du ¼ 2pf m Dt0 Du0 ¼ ð2m þ 1Þp
ð6Þ
where m is an integer, and fm is the notch frequency corresponding to the value m. This expression can be rewritten as
fm ¼ Fig. 6. Illustration of the feedback comb filter using a time domain signal containing a pulse and five of its duplicates, each has a 1 ls delay and 60% amplitude compared with its previous one: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
2m þ 1 Du0 þ 2 Dt 0 2p
ð7Þ
For the case shown in Fig. 3, the time interval between two pulses (Dt0) is 1 ls and the phase offset (Du0) is 0, thus Eq. (7) can be simplified as
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f m ¼ m þ 0:5
ð8Þ
which explains why the spectral notches appear at the frequencies of 6.5, 7.5, 8.5, 9.5, 10.5, 11.5, 12.5, and 13.5 MHz. In the same manner, Eq. (7) can also explain the case shown in Fig. 4. For spectral amplitude enhancement (peaks), the effective phase shift (Du0) should take the form:
Du ¼ 2pf n Dt 0 Du0 ¼ 2np
ð9Þ
with n being an integer. And the frequency for amplitude enhancement fn can be expressed as
fn ¼
n Du 0 þ Dt 0 2p
ð10Þ
This equation explains the amplitude enhancement shown in Figs. 3 and 4 if considering their respective values for Dt0 and Du0. Although Eqs. (7) and (10) are derived from the feedforward comb filter (a two-pulse model), it also applies to the feedback comb filter (multiple-pulse model) if we treat all the pulses except for the first one as one when considering the initial phase shift (Du0). One thing that should be noted is that only one type of spectral amplitude changing effect, either amplitude enhancement or reduction, can be distinctly identified in the feedback comb filter. This is the consequence of the accumulation of one type of spectral amplitude changing effect caused by the multiple in-phase pulses (except for the first one) in the output signal of the filter. In the preceding investigations (Sections 2.2 and 2.3), an interesting relationship is observed between the time domain and frequency domain characteristic features of the signals: the time delay Dt0 between adjacent pulses and the frequency interval (Df) between adjacent spectral peaks or notches. For all the cases shown in Figs. 3, 4, 6 and 7, Df is 1 MHz, and it is exactly the reciprocal of Dt0 (1 ls). In other words, the time domain characteristics (Dt0) of the filter output and their frequency domain characteristics (Df) satisfy the following relationship:
Df ¼
1 Dt 0
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constituent pulses are close enough or even overlap with each other to appear as a single wave. In this section, we take the feedback comb filter as an example to show the cases with pulses overlapping in the output signal in order to demonstrate a more complete picture of the comb filtering effect. In the examples of the feedback comb filter shown in Figs. 6 and 7, the time delay between pulses is 1 ls. To show the case in which the pulses are overlapping, this time delay is reduced to 0.25 ls, and the six pulses in Figs. 6(a) and 7(a) become a single wave as shown in Fig. 8(a) and (c), respectively. According to Eq. (11), as the time delay Dt0 decreases, the frequency interval Df becomes larger. As a consequence, in Fig. 8(b) and (d) only two, instead of 9 as in Figs. 6(b) and 7(b), spectral peaks and notches respectively appear in the spectra of the corresponding waves. Although both waveform and amplitude spectrum in this case have distinct change compared with those with output pulses clearly separated, the characteristic features of the comb filter, i.e. the relationships shown in Eq. (11), Eqs. (7) and (10) or Eqs. (13) and (14), hold.
ð11Þ
This relationship actually can be derived from both Eqs. (7) and (10) as shown in Eqs. (12a) and (12b), respectively. This means that this relationship is the intrinsic properties of the pulsed comb filters.
Df ¼ f mþ1 f m ¼ Df ¼ f nþ1 f n ¼
1 Dt 0
1 Dt 0
ð12aÞ ð12bÞ
Considering the relationship in Eq. (11), the frequency for the occurrence of spectral amplitude enhancing and reducing effects can also be determined using Eqs. (13) and (14), which are the variation of Eqs. (7) and (10), respectively.
1 Du 0 fm ¼ m þ Df þ 2 2p f n ¼ nDf þ
Du0 2p
ð13Þ
ð14Þ
2.5. A special case of temporally overlapping pulses In order to clearly demonstrate the constituent pulses of the output of the comb filters, the time delay (Dt0) between adjacent pulses in the waves shown in Sections 2.2 and 2.3 is chosen to be large enough for the pulses being distinctly separated. However, this is not always the case in reality because in many cases the
Fig. 8. Illustration of the cases of the feedforward comb filter in which the pulses are overlapping with each other. (a) and (c) are the variations (with a 0.25 ls time delay between pulses) of Figs. 6(a) and 7(a), respectively. (b) and (d) are the corresponding variations of Figs. 6(b) and 7(b), respectively.
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3. Experimental demonstration of the pulsed ultrasonic comb filtering effect To experimentally demonstrate the comb filtering effect in pulsed ultrasonics, a series of measurements are performed on thin plate samples using immersion transducers. In addition, these experimental results will also be used in the next section for demonstrating the application of the comb filtering effect in measuring the sound velocity and thickness of thin plates.
3.1. Experiments Five thin plates with different thickness are chosen for ultrasonic measurements. Among the five samples, three are glass plates (Glass I, II and III) and the other two are silicon wafers (Silicon I and II) with [100] as the thickness direction. The thicknesses of the samples are measured using a Mitutoyo digital caliper (Model No. CD-15DC) and their average values (A) and the ratios of standard deviation to average (SD/A) obtained from 10 measurements for each sample are listed in Table 1. Four pairs of immersion transducers are used in the ultrasonic measurements and their basic information is listed in Table 2. These transducers are fabricated by either Valpey Fisher (VP) or NDT Systems (NDTS). For each transducer, the center frequency (f0) is provided by the manufacturer, while the temporal pulse length (Ds) and the spectral pulse bandwidth (B) are measured using an echo reflected from a flat steel surface in a pulse-echo test. The thresholds for measuring Ds and B are 5% of the maximum amplitude of the echo waveform and 10% of the maximum value of echo spectrum, respectively. The schematic of the ultrasound signal propagation is shown in Fig. 9(a), in which an ultrasound pulse is perpendicularly incident onto Medium II (thin plates) and the reflected waves (solid lines with arrows) and the transmitted waves (dashed lines with arrows) are received in the pulse-echo mode (Fig. 9(b)) and the through-transmission mode (Fig. 9(c)), respectively. All the measurements are performed in water, and so Medium I in Fig. 9 is water. In addition, a photo (Fig. 9(d)) is also provided for the through-transmission configuration in order to clearly display the experimental setup. Experimental measurements are performed in water using a customer-designed scanner (Inspection Technology Europe BV, Netherlands). This scanner has an angular resolution of 0.01°, which facilitates the accurate positioning of the transducer and accordingly the incident beams. The experimental data are acquired by the data acquisition software ‘Winspect’ using a
Table 1 Thickness of the samples: average value (A) and ratio of standard deviation to average (SD/A). Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
A (mm) SD/A (%)
2.84 0.11
1.42 0.57
0.97 0.59
0.48 0.87
0.38 1.12
Table 2 Basic information of the immersion transducers used in the ultrasonic measurements: manufacturer, center frequency (f0) and temporal pulse length (Ds) and spectral pulse bandwidth (B). Manufacturer
VF
NDTS
NDTS
VF
f0 (MHz) Ds (ls) B (MHz)
5 0.71 8.01
10 0.36 12.78
20 0.26 44.77
50 0.08 58.16
sampling frequency of 100 MHz. The spectra of the received signals are obtained using the Fast Fourier Transform (FFT) functions in ‘Matlab’.
3.2. Demonstration of the comb filtering effect It is not hard to see that the physical model of the ultrasound measurements in Fig. 9 is the feedback comb filter. To demonstrate the comb filtering effect in these measurements, the transmission and reflection measurements of Glass I obtained using 10 MHz transducers are shown in Fig. 10. From the comparison of the amplitude spectra of the reflection and transmission in Fig. 10(b) we can see that transmitted and reflected signals give the same results concerning the locations for spectral enhancements (peaks) and reductions (notches). An interesting observation can be obtained from Fig. 10(b) that there are two small spectral anomalies appearing at 10.36 MHz and 13.40 MHz. This is actually the comb filtering effect of the shear waves generated in the plate samples due to the coexistence of longitudinal and shear waves in the experimental measurements. The fact that the anomalies at 10.36 MHz and 13.40 MHz are caused by shear wave can be confirmed by comparing the measured frequency interval (3.04 MHz) with the frequency interval Df, which can be obtained by applying the relationship in Eqs. (11) and (15) using the measured thickness (Table 1) and shear velocity (Table 3) of Glass I. Theoretically, for the normal incidence of plane waves onto a plate with two parallel surfaces such as the samples in this work, the coexistence of longitudinal and shear waves in the reflection and transmission is impossible. But in this work, longitudinal and shear waves are observed as coexistent, especially for thinner plate samples. Fig. 11 is an example of the coexistence of the longitudinal and shear waves. In the reflection waveform of Silicon I shown in Fig. 11(a), the longitudinal and shear waves are mainly contained in the main bang (before 150 ls) and tail (after 150 ls), respectively. The reason that the wave in the tail of the reflection is identified as shear wave is that the speed of the wave in the tail obtained using the method proposed in this study (Section 4) matches the shear wave speed of the tested material with very high accuracy (Table 4). As to the mechanism of the shear wave generation in the current experimental configuration, a mode conversion effect must take place because all the measurements are performed in water (in which only longitudinal waves is supported), although the exact mechanism is not clear to us. Since the incident wave is a bounded beam, the incident wavefront is not a perfect plane, which leads to the fact that a portion of energy in the normally incident beam will hit the interfaces at slightly small angles and this portion of obliquely incident energy will probably lead to the generation of shear waves (first mode conversion). When the shear wave in the solid plates encounters the interfaces again, it will be converted to longitudinal wave again (the second mode conversion) and received as either the reflected wave (in pulse-echo mode) or the transmitted wave (in throughtransmission mode). Moreover, it is also observed from the spectrogram such as Fig. 11(b) that the longitudinal wave has large frequency coverage (as the emitting transducer), decays faster and dominates in the earlier part of the received signal; while the shear wave normally contains several discrete frequency components, decays much slower and dominates in the later part of the received signal. Inspired by the spectrogram analysis in the investigation of acoustic Wood anomaly phenomenon [27,28], discriminating the time ranges of wave domination is helpful to locate the signal in the time domain for different analysis and application purposes. Please be noted that it is still not clear why shear wave is only generated at finite frequencies.
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Fig. 9. Schematic of the ultrasonic measurements. (a) The incident pulse and the waves reflected back to the upper Medium I are depicted using solid lines with arrows; the waves transmitted into the lower Medium I are depicted using dashed lines with arrows; and the waves bouncing back and forth inside Medium II are depicted using dotted lines with arrows. (b) and (c) are the schematics of the pulse-echo and through-transmission configurations, respectively. (d) is the photo of the setup for the throughtransmission configuration.
4. Application of the pulsed ultrasonic comb filtering effect in the measurement of sound velocity and thickness of thin plates In this section, we will demonstrate the capability of the comb filtering effect in measuring the longitudinal and shear velocities and the thickness of thin plate samples based on measurements in Section 3.1, and also clarify the advantages of this method over the conventional ultrasonic methods. 4.1. Principle of measurement Among the different methods for measuring ultrasonic velocity of solid plate materials [29], the most common and simple method is based on the direct measurement of the thickness (d) and the round-trip time-of-flight (TOF) in the plate (Dt), which is called TOF method:
C¼
2d Dt
ð15Þ
By applying the relationship in Eq. (11), this method can be expressed as: Fig. 10. Representation of the transmissions and reflections measured for Glass I: (a) waveform (b) amplitude spectrum.
C ¼ 2dDf
ð16Þ
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Table 3 Longitudinal velocity (CL) and shear velocity (CS) of the samples measured using TOF method as reference velocities for evaluating the accuracy of those measured based on the comb filtering effect. Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
CL (m/s) CS (m/s)
5784 3401
5645 3378
5667 3341
8383 5844a
8348 5844a
a This value is not experimentally obtained, but calculated based on the mechanical properties of the material.
The key of applying the comb filtering effect in measurement is to find the spectral interval Df. Depending on the center frequency and bandwidth (Table 2) of the pulse generated by the emitting transducer, there are two ways to obtain Df: (i) Directly obtain Df from the adjacent peaks or notches in the amplitude spectrum. This method applies to the cases in which more than one peak or notch appear as in Figs. 3, 4, 6–8 and 10. (ii) Obtain Df according to Eq. (13) or (14). Depending on the type of spectral amplitude changing (notch or peak) that the measurement has, a proper equation can be selected. For example, Eq. (13) should be chosen for analyzing the reflection spectrum (notches occurring) and Eq. (14) for the transmission spectrum (peaks occurring). To be accurate, these two methods can be applied together for cross validation. To demonstrate how to find Df from the experimental measurements, the transmission signals and their spectra for both Glass II and Silicon II are shown in Fig. 12. We can see that the transmitted pulses overlap with each other in the time domain for both samples (Fig. 12(a) and (c)), and so it is impossible to find the round-trip TOF for the samples. In Fig. 12(b) and (d) the comb filtering effect is clearly visible for both longitudinal wave and shear wave. The spectral interval Df of longitudinal wave can be
Fig. 11. Illustration of the coexistence of longitudinal and shear waves in the experimental measurements using the reflection of Silicon I measured with 10 MHz transducer: (a) waveform (b) spectrogram.
Table 4 Longitudinal velocity (CL) and shear velocity (CS) of the samples obtained based on the comb filtering effect using Eq. (16) and their percentage difference (DCL for longitudinal wave and DCS for shear wave) from the measurements using TOF method. Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
CL (m/s) DCL (%) CS (m/s) DCS (%)
5794 0.2 3465 1.9
5722 1.4 3404 0.8
5694 1.7 3399 0.6
8320 0.8 5798 0.8
8331 0.2 5798 0.8
Instead of relying on the time domain characteristics Dt, Eq. (16) offers a way for sound velocity measurement based on the frequency domain characteristics Df given a comb filtering effect occurs. For the cases in which Dt is large enough to separate the constituent pulses in the reflected or transmitted signals, Eq. (16) can give the same results as Eq. (15). However, when the constituent pulses in the reflected or transmitted waves overlap with each other, Eq. (15) fails, but Eq. (16) still works for accurate velocity measurement as long as at least one peak or notch appear in the spectral coverage of the received transducer. It is not hard to see that Eq. (16) is more applicable for the measurement of thin plates. In addition, if the sound velocity is known, the thickness of thin plates can also be measured using a variation of Eq. (16):
d¼
C 2 Df
ð17Þ
In what follows, both Eqs. (16) and (17) will be applied for demonstrating the application of the comb filtering effect in thin plate measurement.
Fig. 12. Examples of the experimentally obtained signals to demonstrate the application of the proposed methods for measuring the speed and thickness of thin plates. The transmitted wave and its spectrum of Glass II measured using 5 MHz transducer are shown in (a) and (b), respectively. The transmitted wave and its spectrum of Silicon II measured using 20 MHz transducer are shown in (c) and (d), respectively.
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obtained from the high-amplitude peaks (3 peaks in Fig. 12 (b) and 2 peaks in Fig. 12(d)), and Df values of Glass II and Silicon II are 2.01 MHz and 8.25 MHz, respectively. The spectral interval of shear wave can be obtained from the small-amplitude notches of the corresponding spectra. For examples, 2 notches exist for Glass II (Fig. 12(b)) at 2.41 MHz and 3.63 MHz and lead to a spectral interval of 1.22 MHz. Similarly, a spectral interval of 7.44 MHz can be obtained for Silicon II from the two notches at 14.89 MHz and 22.33 MHz (Fig. 12(d)). To ensure the frequency intervals obtained from directly reading (the first method proposed in this paragraph) are the ones between theoretically adjacent peaks or notches, it is always favorable to double check them through solving the simultaneous equations generated using Eq. (13) for peaks or Eq. (14) for notches (the second method proposed in this paragraph). In practical measurements, the applicability of the comb filtering based method relies on the simultaneous fulfillment of the following two conditions: (i) In the time domain, the signal for analysis must contain more than one pulse. This requirement can be readily met for the thin plate cases (the major application promoted in this paper), but for thick plate cases it has to be met by properly windowing the received signals. (ii) In the frequency domain, the spectrum of the received signal must contain at least two amplitude changing features (notch or peak). The satisfaction of this requirement depends both on the properties of the plates (thickness and sound speed) and on the spectral properties of the selected transducer (center frequency and bandwidth). As a general guide, a larger bandwidth is always desired when selecting transducers. 4.2. Measurement of sound velocity in thin plates In order to evaluate the accuracy and reliability of the sound velocity measurement based on the comb filtering effect, the longitudinal (CL) and shear (CS) velocities of the plate samples (except for the shear velocities of Silicon I and II) are experimentally measured using TOF method. Due to the small thickness, no equipment is available for this study to measure the shear velocities of the silicon samples based on TOF method. Treating silicon as ideal solid, the shear velocities are calculated using the following relationship [30]:
CS ¼
rffiffiffiffiffiffiffi c44
q
ð18Þ
where c44 is the elastic constant (79.6 GPa [31]) and q is the mass density of silicon (2331 kg/m3 [31]). All the velocities of the samples are listed in Table 3. Based on Eq. (16), the longitudinal and shear velocities of the thin plate samples are measured and listed in Table 4. As far as the focus of this work is concerned, the reflections and transmissions give almost the same results, and so only the results obtained from reflections (in pulse-echo mode) are being reported. In Table 4 are also listed the percentage difference in longitudinal (DCL) and shear (DCS) velocity measurement between the method based on comb filtering effect and the conventional TOF method. According to the values of the percentage difference it can be concluded that the velocity measurements based on the comb filtering effect has high accuracy. For measuring the sound velocity in thin plates, a big challenge that the conventional TOF based methods have is the overlapping of the received pulses (such as echoes in pulse-echo mode [25]) and the corresponding reduction of the spatial resolution [32] of
Table 5 Comparison of the center frequencies of the transducers (available for this study) that can measure the sound velocities of the sample plates using the TOF based method and the comb filtering effect based method. Samples
Glass I Glass II Glass III Silicon I Silicon II
TOF based method
Comb filtering effect based method
CL (MHz)
CS (MHz)
CL (MHz)
CS (MHz)
5 10 10 50 50
10 20 20 NA NA
5 5 5 10 20
5 5 5 10 20
Table 6 Comparison of the directly measured thickness (dD) and those measured based on the comb filtering effect (dCF) by applying Eq. (17).
Silicon I Silicon II
dD (mm)
DCF (mm)
Difference (%)
0.482 0.384
0.488 0.389
1.2 1.3
the emitted pulses due to the relatively short propagation path. To meet this challenge, a short ultrasound pulse must be generated by either adopting other technologies such as picosecond laser technique [25,33,34] or using high frequency transducers. Compared with TOF based methods, the method based on the comb filtering effect has lower requirements on transducers in terms of center frequency, which make it more convenient and less expensive in practical applications. Table 5 shows the comparison of the requirements on transducer center frequency for TOF based method and comb filtering effect based method in measuring the longitudinal and shear velocities of the thin plate samples using the transducers available to this study. It is readily seen that the comb filtering effect based method requires much lower center frequencies for the transducers compared with the TOF based method in order to accomplish the same measurement task. 4.3. Measurement of thickness of thin plates To show the capability and reliability of the comb filtering effect in measuring the thickness of thin plates, the two silicon samples are chosen, their thicknesses (dCF) are obtained by applying Eq. (17) and compared with the values (dD) directly measured (using a digital caliber) in Table 6. According to the percentage difference between dCF and dD in Table 6, one can see the measurements based on the comb filtering effect have relatively high accuracy. It should be noted that the longitudinal wave velocity of silicon applied to Eq. (17) is 8430 m/s, which is the tabulated value instead of the one measured in this work, because using the tabulated values is the case for most real world practice. In addition, the results shown in this section are just an example of the application of the comb filtering effect in geometric characterization. Potentially, the principle developed in this work can also be applied to characterize the geometry of the other structures such as periodic structures [32,35] or tablet-in-tablet [36]. 5. Conclusions This work presents a fundamental study of the comb filtering effect in pulsed ultrasonics intending to provide a tool for data analysis in pulsed ultrasonic measurements. Through numerical simulation, two basic types of comb filters, i.e. feedforward and feedback comb filters, are demonstrated in terms of waveform, amplitude spectrum and phase spectrum. Theoretical derivation is also performed to obtain the characteristic features of the comb
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filtering effect: (i) the formula to determine the spectral locations (frequencies) for the spectral peaks or notches, and (ii) the relationship between the relative time delay (the time domain characteristics) and the frequency interval of peaks or notches (the frequency domain characteristics). The comb filtering effect is further demonstrated and verified experimentally using the reflections and transmissions of five thin plate samples obtained in ultrasonic measurements. As examples of the practical application of the comb filtering effect in pulsed ultrasonics, the characteristic features obtained in this work are applied in the measurement of sound velocity and thickness of thin plate samples. It is proven that the method based on the comb filtering effect not only is capable of measuring sound velocity in thin plates with high accuracy, but also has the advantage of being able to simultaneously measure longitudinal and shear velocities and having lower requirement on transducers in terms of center frequency compared with the conventional time-of-flight based method. It has also been shown that the comb filter based method is also proved to be capable and reliable in thickness measurement for thin plates and hence possibly also in geometric characterization for other structures. Acknowledgement The authors wish to acknowledge the French Centre National de Recherche Scientifique (CNRS) and the Conseil Régional de Lorraine (CRL) for their financial support of this work. References [1] S.F. George, A.S. Zamanakos, Comb fitlers for pulsed radar use, Proc. Inst. Radio Eng. 42 (1954) 1159–1165. [2] R. Turner, Some thoughts on using comb filters in broadcast television transmitter and at receiver, IEEE Trans. Consum. Electron. 23 (1977) 248–257. [3] G.D. Arndt, F.M. Stuber, R.J. Panneton, Video-signal improvement using comb filtering techniques, IEEE Trans. Commun., Co 21 (1973) 331–336. [4] P.J. Brusil, T.B. Waggener, R.E. Kronauer, Using a comb filter to describe time varying biological rhythmicities, J. Appl. Physiol. 48 (1980) 557–561. [5] D.E. Veeneman, B. Mazor, A fully adaptive comb fitler for enhancing blockcoded speech, IEEE Trans. Acoust. 37 (1989) 955–957. [6] H.T. Hu, Comb filtering of noisy speech using overlap-and-add approach, Electron. Lett. 34 (1998) 16–18. [7] D. Liu, S.N. Fu, M. Tang, P. Shum, D.M. Liu, Comb filter-based fiber-optic methane sensor system with mitigation of cross gas sensitivity, J. Lightwave Technol. 30 (2012) 3103–3109. [8] H. Zou, S.Q. Lou, G.L. Yin, A wavelength-tunable fiber laser based on a twincore fiber comb filter, Opt. Laser Technol. 45 (2013) 629–633. [9] A.P. Luo, Z.C. Luo, W.C. Xu, Wavelength switchable all-fiber comb filter using a dual-pass Mach-Zehnder interferometer and its application in multiwavelength laser, Laser Phys. 20 (2010) 1814–1817. [10] G.Y. Sun, D.S. Moon, A.X. Lin, W.T. Han, Y.J. Chung, Tunable multiwavelength fiber laser using a comb filter based on erbium-ytterbium co-doped polarization maintaining fiber loop mirror, Opt. Express 16 (2008) 3652–3658. [11] J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, I. Bennion, Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters, IEEE Photonics Tech. Lett. 8 (1996) 60–62.
[12] P.M. Narins, E.F. Evans, G.F. Pick, J.P. Wilson, Comb-filtered noise generation for use in auditory neurophysicological and psychophysical experiments, IEEE Trans. Biomed. Eng. 26 (1979) 43–47. [13] B. Santhanam, P. Maragos, Harmonic analysis and restoration of separation methods for periodic signal mixtures: algebraic separation versus comb filtering, Signal Process. 69 (1998) 81–91. [14] M. Niedzwiecki, M. Meller, Generalized adaptive comb filters/smoothers and their application to the identification of quasi-periodically varying systems and signals, Automatica 49 (2013) 1601–1613. [15] M. Niedzwiecki, P. Kaczmarek, Generalized adaptive notch and comb filters for identification of quasi-periodically varying systems, IEEE Trans. Signal Process. 53 (2005) 4599–4609. [16] H. Prance, R.J. Prance, P.B. Stiffell, Hardware comb filter enhances dynamic range and noise performance of sensors in noisy environments, Rev. Sci. Instrum. 78 (2007). [17] A.G. Dempster, Use of comb filters in GPS L1 receivers, GPS Solut. 12 (2008) 179–185. [18] J.J. Guo, Y.H. Yang, G.D. Peng, Analysis of polarization-independent tunable optical comb filter by cascading MZI and phase modulating Sagnac loop, Opt. Commun. 284 (2011) 5144–5147. [19] A. Beydoun, V.T. Nguyen, L. Naviner, P. Loumeau, Optimal digital reconstruction and calibration for multichannel time interleaved Sigma Delta ADC based on comb-filters, Int. J. Electron. Commun. 67 (2013) 329–339. [20] R. Wu, C.M. Long, D.E. Leaird, A.M. Weiner, Directly generated Gaussianshaped optical frequency comb for microwave photonic filtering and picosecond pulse generation, IEEE Photonics Tech. Lett. 24 (2012). [21] T. Halmrast, Orchestral timbre: comb-filter coloration from reflections, J. Sound Vibrat. 232 (2000) 53–69. [22] A. Galembo, A. Askenfelt, Signal representation and estimation of spectral parameters by inharmonic comb filters with application to the piano, IEEE Trans. Speech Audio Process. 7 (1999) 197–203. [23] W. Jin, X. Liu, M.S. Scordilis, L. Han, Speech enhancement using harmonic emphasis and adaptive comb filtering, IEEE Trans. Audio Speech Lang. 18 (2010) 356–368. [24] H.T. Hu, C. Yu, C.H. Lin, Usefulness of the comb filtering output for voiced/ unvoiced classification and pitch detection, in: Proceedings of the 2009 International Conference on Signal Processing Systems, 2009, pp. 135–139. [25] N.W. Pu, W.I. Kuo, Laser picosecond ultrasonic measurement of longitudinal sound velocity in nanometer thin films, Review of Progress in Quantitative Nondestructive Evaluation 27a and 27b (975) (2008) 286–293. [26] J.O. Smith, Physical Audio Signal Processing, W3K Publishing, 2010. [27] J.F. Liu, N.F. Declercq, Investigation of the origin of acoustic Wood anomaly, J. Acoust. Soc. Am. 138 (2015) 1168–1179. [28] J. Liu, N.F. Declerq, Time-frequency analysis of wood anomalies in acoustics, Proc. Meet. Acoust. 19 (2013) 055075. [29] A.S.f.T.a. Materials, Standard Practice for Measuring Ultrasonic Velocity in Materials, in, ASTM International 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428–2959, United States, 2015. [30] M.J.P. Musgrave, Crystal Acoustics, The Acoustical Society of America, 2003. [31] H.J. Mcskimin, W.L. Bond, E. Buehler, G.K. Teal, Measurement of the elastic constants of silicon single crystals and their thermal coefficients, Phys. Rev. 83 (1951), 1080–1080. [32] J. Liu, N.F. Declercq, Ultrasonic geometrical characterization of periodically corrugated surfaces, Ultrasonics 53 (2013) 853–861. [33] C. Thomsen, H.T. Grahn, H.J. Maris, J. Tauc, Surface generation and detection of phonons by picosecond light pulses, Phys. Rev. B 34 (1986) 4129–4138. [34] O.B. Wright, Thickness and sound-velocity measurement in thin transparent films with laser picosecond acoustics, J. Appl. Phys. 71 (1992) 1617–1629. [35] A. Shaw, J. Liu, S.W. Yoon, N.F. Declercq, Characterization of the geometry of microscale periodic structures using acoustic microscopy, Ultrasonics 70 (2016) 258–265. [36] J.F. Liu, C. Cetinkaya, Mechanical and geometric property characterization of dry-coated tablets with contact ultrasonic techniques, Int. J. Pharmaceut. 392 (2010) 148–155.
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Pulsed ultrasonic comb filtering effect and its applications in the measurement of sound velocity and thickness of thin plates Jingfei Liu ⇑, Nico F. Declercq Georgia Institute of Technology, Laboratory for Ultrasonic Nondestructive Evaluation ‘‘LUNE”, Georgia Tech-CNRS UMI2958, Georgia Tech Lorraine, 2, rue Marconi, 57070 Metz, France
a r t i c l e
i n f o
Article history: Received 28 July 2016 Received in revised form 28 November 2016 Accepted 3 December 2016 Available online 8 December 2016 Keywords: Comb filter Pulsed ultrasonics Spectral peaks and notches Sound velocity Thin plates Comb filtering effect
a b s t r a c t An analytical and experimental study of the pulsed ultrasonic comb filtering effect is presented in this work intending to provide a fundamental tool for data analysis and phenomenon understanding in pulsed ultrasonics. The basic types of comb filter, feedforward and feedback filters, are numerically simulated and demonstrated. The characteristic features of comb filters, which include the formula for determining the locations of the spectral peaks or notches and the relationship between its temporal characteristics (relative time delay between constituent pulses) and its spectral characteristics (frequency interval between peaks or notches), are theoretically derived. To demonstrate the applicability of the comb filtering effect, it is applied to measuring the sound velocities and thickness of a thin plate sample. It is proven that the comb filtering effect based method not only is capable of accurate measurements, but also has advantages over the conventional time-of-flight based method in thin plate measurements. Furthermore, the principles developed in this study have potential applications in any pulsed ultrasonic cases where the output signal shows comb filter features. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction A comb filter refers to a filter whose frequency spectrum consists of a number of equally spaced elements resembling the tines of a comb [1]. As a physical tool, it has been used to improve the signal-to-noise ratio in pulsed radar systems [1], video signal transmission systems [2,3] as well as in biological signal detection [4] and speech transmission [5,6]. It has also been applied to designing gas sensors [7] and multi-wavelength fiber lasers [8–11]. In the auditory physiological and psycho-physical studies, it is even applied for generating noises [12]. As an algorithm of signal processing, its application for separating mixed signals [13], tracking slow changes in system frequencies [14,15], improving sensor performance [16] and GPS code tracking performance [17] have been investigated. The most recent research is focused on its application in optics [18], electronics [19] and photonics [20]. Within the scope of acoustics the comb filtering effect has been mostly investigated in room acoustics such as its role in the sound coloration of orchestra and opera halls [21], in musical acoustics such as its application in musical sound representation [22] and in speech acoustics such as speech enhancement [23] and pitch detection [24]. In audio engineering, the comb filtering effect, ⇑ Corresponding author. E-mail address: [email protected] (J. Liu). http://dx.doi.org/10.1016/j.ultras.2016.12.003 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.
which normally results from the interference of sound with its delayed duplicates, is either deliberately avoided to eliminate undesired colored sound, or used to create flanging effects. All these acoustic investigations are performed in the low frequency regime, and to the authors’ knowledge no investigation has been done in the high frequency regime such as in ultrasonics. In ultrasonic applications, the comb filtering effect occurs in various cases such as the reflection and transmission of the pulses normally incident unto thin plates (which is studied in this work). If this effect can be well understood, it will definitely be helpful for extracting desired information from ultrasonic measurements or properly interpreting experimental observations. The current work serves as a fundamental study of the comb filtering effect in pulsed ultrasonics. The theoretical investigation of the pulsed ultrasonic comb filtering effect starts with a numerical model of a pulse with a Gaussian spectrum. Two basic forms of comb filter, namely feedforward and feedback, are first theoretically described using a block diagram and the difference equation; and are then numerically simulated based on the defined pulse. The important features of the comb filtering effect in pulsed ultrasonics, namely the locations of spectral amplitude enhancing and reducing effects and the relationship between the temporal and spectral characteristic features, are analytically derived. Then, a special case of the feedback comb filter, in which all the pulses overlap in the time domain, is introduced to demonstrate the comb
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filtering effect in extreme situations. The pulsed ultrasonic comb filtering effect is further demonstrated experimentally based on a series of pulse-echo and through-transmission measurements on thin plate samples. In addition to the theoretical analysis and experimental demonstration, and in order to show its practical usefulness in ultrasonics, the characteristic relationship of the comb filtering effect is applied to measuring the sound velocity and thickness of thin plate samples as the solution of a common problem in pulsed ultrasonics. For sound velocity measurement, the method based on the comb filtering effect not only is capable of simultaneously obtaining both longitudinal and shear velocities, but is even capable of doing so in cases where the conventional time-of-flight (TOF) based ultrasonic method fails in the case of very thin plates where an overlap of ultrasonic echoes occurs in addition to the inherent reduction in temporal resolution [25]. Furthermore, the same principle derived from the comb filtering effect appears even applicable to acquire thickness measurement of thin plates with high accuracy. The rest of this paper is outlined as follows. The comb filtering effect in pulsed ultrasound is theoretically described in Section 2 and experimentally demonstrated in Section 3. Section 4 shows the application of this effect in measuring the sound velocity and thickness of thin plate samples as a case study of potential application of the study presented in this work. Section 5 provides the concluding remarks. 2. Pulsed ultrasonic comb filtering effect
amplitude A0 n and the ‘phase’ of the phase spectrum refers to the angle of the complex amplitude obtained in the fast Fourier transform (FFT) calculation. An example of a Gaussian amplitude spectrum and the resultant pulse waveform as well as its phase spectrum is shown in Fig. 1. If more than one pulse is defined in one wave, their phase difference is realized by choosing a proper pulse phase u0 for each pulse in addition to a relative time delay t0. 2.2. Feedforward comb filter The block diagram and difference equation of the feedforward comb filter are shown in Fig. 2 and Eq. (3), respectively. It is easy to see that the output of this filter is a linear combination of the direct and delayed input signals. From the perspective of pulsed ultrasonics, this filter is the physical model of the reality in which a pulse is superposed onto a delayed and attenuated version of itself. The reason of delay can be either extra distance or different materials that the input pulse propagates through, or both. And the source of the attenuation can possibly be absorption, diffraction or spherical spreading loss.
yðnÞ ¼ b0 xðnÞ þ bM xðn MÞ
ð3Þ
To demonstrate the effect of this feedforward comb filter in pulsed ultrasonics, a wave containing a pulse defined in Section 2.1 and its delayed and attenuated version is assumed as shown in Fig. 3(a). Compared with the first pulse, the second pulse is delayed by 1 ls and attenuated to 80% of the amplitude. Because the second pulse is the delayed and attenuated version of the first pulse,
There are two basic types of comb filters: feedforward filter and feedback filter [26]. In this section, we will first define an ultrasonic pulse commonly appearing in pulsed ultrasonic technology, and then numerically demonstrate these two types of comb filters in both the time domain and the frequency domain. Together with the relationship for determining the frequencies of the peaks/ notches, the relationship between the time domain characteristics (relative time delay between pulses) of the output of these filters and their frequency domain characteristics (distance between the spectral peaks/notches) will be analytically derived. A special case, in which the output pulses of these filters overlap in the time domain, will also be demonstrated. 2.1. Defining a pulse In most cases of ultrasonic testing the spectra of ultrasonic pulses are not exactly Gaussian per se, but they are to a good approximation Gaussian-like. For the convenience of theoretical analysis a Gaussian spectrum is assumed for all the ultrasonic pulses in this study. The amplitude of the Gaussian spectrum is defined as
Aðf Þ ¼ A0 e
ðf f 0 Þ2 2D2
ð1Þ
in which A0 is the maximum amplitude, f0 the center frequency, and D the width deviation of the spectrum. An ultrasonic pulse corresponding to this spectrum is defined by the following Fourier series, ! ! K K X X 1 1 0 i2pf n t ið2pf n ðtt 0 Þþu0 Þ xðtÞ ¼ Re An e An e ¼ Re ð2Þ K K n¼1 n¼1 in which x(t) is the pulse waveform, K the total number of component harmonics, An the complex amplitude, fn the frequency, u0 the pulse phase, i.e. a phase offset of all the harmonic components representing the given pulse, and t0 is the time delay of the pulse. Unless otherwise clearly stated, in what follows, the ‘amplitude’ of an amplitude spectrum means the absolute value of a complex
Fig. 1. (a) A Gaussian amplitude spectrum with unity maximum amplitude, center frequency of 10 MHz and width deviation of 2 MHz, (b) the resultant ultrasonic pulse defined on this spectrum and (c) the phase spectrum of the defined pulse. Zero phase offset and zero time delay are chosen for all the component harmonics in this pulse.
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201
Fig. 2. Block diagram of the feedforward comb filter.
Fig. 4. Illustration of the feedforward comb filter using a time domain signal containing a pulse and its delayed (by 1 ls) and attenuated (80% in amplitude) variation with a phase offset of p: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
Fig. 3. Illustration of the feedforward comb filter using a time domain signal containing a pulse and its delayed (by 1 ls) and attenuated (80% in amplitude) version: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
there is no phase shift between them. The amplitude spectrum of this wave, which is compared with that of its first pulse, is shown in Fig. 3(b). This figure clearly shows the comb shape of this filter: compared with its first pulse, the spectral amplitude of this twopulse wave is enhanced at some frequencies (such as 6, 7, 8, 9, 10, 11, 12, 13 and 14 MHz) and reduced at other frequencies (such as 6.5, 7.5, 8.5, 9.5, 10.5, 11.5, 12.5, and 13.5 MHz). The amplitude enhancement and reduction result from the constructive and destructive interference of the two pulses of the wave, respectively. It is interesting to observe from Fig. 3(c) that phase shifts occur at the frequencies of the notches in Fig. 3(b), while no phase shifts can be observed at the frequencies of the peaks. This observation is a confirmation of the occurrence of constructive and destructive interferences: amplitude is enhanced or reduced at the frequencies with a phase difference smaller or larger than p/2, respectively. Besides delay and attenuation, the second pulse of this twopulse model of the feedforward comb filter could also have phase difference from the first pulse. As a special case, a phase difference of p is assumed for the second pulse in addition to the 1 ls delay and 80% amplitude attenuation shown in Fig. 4. The purpose of showing this case is that such a model is often encountered in
reality such as the second pulse being the reflection of the first pulse from a rigid interference (causing a phase difference of p). Similar to Fig. 3, a comb shape of amplitude spectrum can be observed in Fig. 4(b) and the phase shifts occurring at the notch frequencies can also be observed in Fig. 4(c). Different from Fig. 3, the spectral locations for amplitude enhancement (peaks) and reduction (notches) are shifted by 0.5 MHz as shown in Fig. 4(b), which is the result of the phase offset (p) between the two pulses in the waveform (Fig. 4(a)). 2.3. Feedback comb filter The block diagram and difference equation of the feedback comb filter are shown in Fig. 5 and Eq. (4), respectively. The output of this filter is a linear combination of the input signal and a series of exponentially decaying versions of the input signal, which are uniformly spaced in time domain. This filter can work as the
Fig. 5. Block diagram of the feedback comb filter.
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physical model of a wave bouncing back and forth within two parallel surfaces while being received at one side of the parallel surfaces.
yðnÞ ¼ b0 xðnÞ aM xðn MÞ
ð4Þ
In pulsed ultrasonics, the effect of this feedback comb filter can be demonstrated with a time domain signal shown in Fig. 6(a), which contains six pulses that are in phase, equally spaced in time but exponentially decayed in amplitude by a factor of 0.6. Similar to the amplitude spectrum of the feedforward comb filter (Fig. 4(b)), the spectral amplitude of the feedback filter shown in Fig. 6(b) are also enhanced and reduced, but the effect of enhancement is much more visible. This is because all the six pulses are in phase, and each one starting from the second one enforces the effect of the amplitude enhancement at a specific set of frequencies (such as 6, 7, 8, 9, 10, 11, 12, 13 and 14 MHz), while in between these frequencies the amplitude is generally reduced. Fig. 6(c) also shows a series of phase shifts that are corresponding to local notches (less visible) in Fig. 6(b). To clearly demonstrate the amplitude reducing effect of the feedback filter, the signal chosen is shown in Fig. 7(a), which is similar to wave in Fig. 6(a) except that the phase offset between the first pulse and the other five pulses is p (while these five pulses are in phase). Due to the phase offset of p, the spectral amplitudes at the same set of frequencies, at which the amplitude enhancement is observed in Fig. 6(b), are reduced, while at other frequencies the spectral amplitudes are somehow enhanced as shown in Fig. 7(b). Again, phase shifts occur in Fig. 7(c) at the frequencies where the amplitude reducing effect occurs in Fig. 7(b). Fig. 7. Illustration of the feedback comb filter using a time domain signal containing a pulse and five of its variations. Each of the variations has a 1 ls delay and 60% amplitude compared with its previous one. All these five variations are in phase and they are all p out of phase with the first pulse. (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
2.4. Analytical derivation of important features of pulsed comb filtering effect After demonstrating the two types of comb filters in Sections 2.2 and 2.3, an important question arises: how to determine the frequencies at which spectral amplitude enhancement or reduction occurs. In this section, we will analytically investigate this question. Taking a signal containing two pulses (feedforward filter) as an example, the spectral minima or maxima are related to the relative time delay (Dt0) and the initial phase shift (Du0) of the two pulses. Assuming that the harmonic components of a pulse take the following general form in terms of its particle displacement (u):
u ¼ A expði2pf ðt t 0 Þ þ iu0 Þ
ð5Þ
with A as the amplitude, f the frequency, t0 the time delay and u0 the initial phase. The effective phase shift (Du) caused by Dt0 and Du0 that possibly leads to spectral amplitude reduction (notch) is
Du ¼ 2pf m Dt0 Du0 ¼ ð2m þ 1Þp
ð6Þ
where m is an integer, and fm is the notch frequency corresponding to the value m. This expression can be rewritten as
fm ¼ Fig. 6. Illustration of the feedback comb filter using a time domain signal containing a pulse and five of its duplicates, each has a 1 ls delay and 60% amplitude compared with its previous one: (a) waveform, (b) amplitude spectrum and (c) phase spectrum.
2m þ 1 Du0 þ 2 Dt 0 2p
ð7Þ
For the case shown in Fig. 3, the time interval between two pulses (Dt0) is 1 ls and the phase offset (Du0) is 0, thus Eq. (7) can be simplified as
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f m ¼ m þ 0:5
ð8Þ
which explains why the spectral notches appear at the frequencies of 6.5, 7.5, 8.5, 9.5, 10.5, 11.5, 12.5, and 13.5 MHz. In the same manner, Eq. (7) can also explain the case shown in Fig. 4. For spectral amplitude enhancement (peaks), the effective phase shift (Du0) should take the form:
Du ¼ 2pf n Dt 0 Du0 ¼ 2np
ð9Þ
with n being an integer. And the frequency for amplitude enhancement fn can be expressed as
fn ¼
n Du 0 þ Dt 0 2p
ð10Þ
This equation explains the amplitude enhancement shown in Figs. 3 and 4 if considering their respective values for Dt0 and Du0. Although Eqs. (7) and (10) are derived from the feedforward comb filter (a two-pulse model), it also applies to the feedback comb filter (multiple-pulse model) if we treat all the pulses except for the first one as one when considering the initial phase shift (Du0). One thing that should be noted is that only one type of spectral amplitude changing effect, either amplitude enhancement or reduction, can be distinctly identified in the feedback comb filter. This is the consequence of the accumulation of one type of spectral amplitude changing effect caused by the multiple in-phase pulses (except for the first one) in the output signal of the filter. In the preceding investigations (Sections 2.2 and 2.3), an interesting relationship is observed between the time domain and frequency domain characteristic features of the signals: the time delay Dt0 between adjacent pulses and the frequency interval (Df) between adjacent spectral peaks or notches. For all the cases shown in Figs. 3, 4, 6 and 7, Df is 1 MHz, and it is exactly the reciprocal of Dt0 (1 ls). In other words, the time domain characteristics (Dt0) of the filter output and their frequency domain characteristics (Df) satisfy the following relationship:
Df ¼
1 Dt 0
203
constituent pulses are close enough or even overlap with each other to appear as a single wave. In this section, we take the feedback comb filter as an example to show the cases with pulses overlapping in the output signal in order to demonstrate a more complete picture of the comb filtering effect. In the examples of the feedback comb filter shown in Figs. 6 and 7, the time delay between pulses is 1 ls. To show the case in which the pulses are overlapping, this time delay is reduced to 0.25 ls, and the six pulses in Figs. 6(a) and 7(a) become a single wave as shown in Fig. 8(a) and (c), respectively. According to Eq. (11), as the time delay Dt0 decreases, the frequency interval Df becomes larger. As a consequence, in Fig. 8(b) and (d) only two, instead of 9 as in Figs. 6(b) and 7(b), spectral peaks and notches respectively appear in the spectra of the corresponding waves. Although both waveform and amplitude spectrum in this case have distinct change compared with those with output pulses clearly separated, the characteristic features of the comb filter, i.e. the relationships shown in Eq. (11), Eqs. (7) and (10) or Eqs. (13) and (14), hold.
ð11Þ
This relationship actually can be derived from both Eqs. (7) and (10) as shown in Eqs. (12a) and (12b), respectively. This means that this relationship is the intrinsic properties of the pulsed comb filters.
Df ¼ f mþ1 f m ¼ Df ¼ f nþ1 f n ¼
1 Dt 0
1 Dt 0
ð12aÞ ð12bÞ
Considering the relationship in Eq. (11), the frequency for the occurrence of spectral amplitude enhancing and reducing effects can also be determined using Eqs. (13) and (14), which are the variation of Eqs. (7) and (10), respectively.
1 Du 0 fm ¼ m þ Df þ 2 2p f n ¼ nDf þ
Du0 2p
ð13Þ
ð14Þ
2.5. A special case of temporally overlapping pulses In order to clearly demonstrate the constituent pulses of the output of the comb filters, the time delay (Dt0) between adjacent pulses in the waves shown in Sections 2.2 and 2.3 is chosen to be large enough for the pulses being distinctly separated. However, this is not always the case in reality because in many cases the
Fig. 8. Illustration of the cases of the feedforward comb filter in which the pulses are overlapping with each other. (a) and (c) are the variations (with a 0.25 ls time delay between pulses) of Figs. 6(a) and 7(a), respectively. (b) and (d) are the corresponding variations of Figs. 6(b) and 7(b), respectively.
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3. Experimental demonstration of the pulsed ultrasonic comb filtering effect To experimentally demonstrate the comb filtering effect in pulsed ultrasonics, a series of measurements are performed on thin plate samples using immersion transducers. In addition, these experimental results will also be used in the next section for demonstrating the application of the comb filtering effect in measuring the sound velocity and thickness of thin plates.
3.1. Experiments Five thin plates with different thickness are chosen for ultrasonic measurements. Among the five samples, three are glass plates (Glass I, II and III) and the other two are silicon wafers (Silicon I and II) with [100] as the thickness direction. The thicknesses of the samples are measured using a Mitutoyo digital caliper (Model No. CD-15DC) and their average values (A) and the ratios of standard deviation to average (SD/A) obtained from 10 measurements for each sample are listed in Table 1. Four pairs of immersion transducers are used in the ultrasonic measurements and their basic information is listed in Table 2. These transducers are fabricated by either Valpey Fisher (VP) or NDT Systems (NDTS). For each transducer, the center frequency (f0) is provided by the manufacturer, while the temporal pulse length (Ds) and the spectral pulse bandwidth (B) are measured using an echo reflected from a flat steel surface in a pulse-echo test. The thresholds for measuring Ds and B are 5% of the maximum amplitude of the echo waveform and 10% of the maximum value of echo spectrum, respectively. The schematic of the ultrasound signal propagation is shown in Fig. 9(a), in which an ultrasound pulse is perpendicularly incident onto Medium II (thin plates) and the reflected waves (solid lines with arrows) and the transmitted waves (dashed lines with arrows) are received in the pulse-echo mode (Fig. 9(b)) and the through-transmission mode (Fig. 9(c)), respectively. All the measurements are performed in water, and so Medium I in Fig. 9 is water. In addition, a photo (Fig. 9(d)) is also provided for the through-transmission configuration in order to clearly display the experimental setup. Experimental measurements are performed in water using a customer-designed scanner (Inspection Technology Europe BV, Netherlands). This scanner has an angular resolution of 0.01°, which facilitates the accurate positioning of the transducer and accordingly the incident beams. The experimental data are acquired by the data acquisition software ‘Winspect’ using a
Table 1 Thickness of the samples: average value (A) and ratio of standard deviation to average (SD/A). Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
A (mm) SD/A (%)
2.84 0.11
1.42 0.57
0.97 0.59
0.48 0.87
0.38 1.12
Table 2 Basic information of the immersion transducers used in the ultrasonic measurements: manufacturer, center frequency (f0) and temporal pulse length (Ds) and spectral pulse bandwidth (B). Manufacturer
VF
NDTS
NDTS
VF
f0 (MHz) Ds (ls) B (MHz)
5 0.71 8.01
10 0.36 12.78
20 0.26 44.77
50 0.08 58.16
sampling frequency of 100 MHz. The spectra of the received signals are obtained using the Fast Fourier Transform (FFT) functions in ‘Matlab’.
3.2. Demonstration of the comb filtering effect It is not hard to see that the physical model of the ultrasound measurements in Fig. 9 is the feedback comb filter. To demonstrate the comb filtering effect in these measurements, the transmission and reflection measurements of Glass I obtained using 10 MHz transducers are shown in Fig. 10. From the comparison of the amplitude spectra of the reflection and transmission in Fig. 10(b) we can see that transmitted and reflected signals give the same results concerning the locations for spectral enhancements (peaks) and reductions (notches). An interesting observation can be obtained from Fig. 10(b) that there are two small spectral anomalies appearing at 10.36 MHz and 13.40 MHz. This is actually the comb filtering effect of the shear waves generated in the plate samples due to the coexistence of longitudinal and shear waves in the experimental measurements. The fact that the anomalies at 10.36 MHz and 13.40 MHz are caused by shear wave can be confirmed by comparing the measured frequency interval (3.04 MHz) with the frequency interval Df, which can be obtained by applying the relationship in Eqs. (11) and (15) using the measured thickness (Table 1) and shear velocity (Table 3) of Glass I. Theoretically, for the normal incidence of plane waves onto a plate with two parallel surfaces such as the samples in this work, the coexistence of longitudinal and shear waves in the reflection and transmission is impossible. But in this work, longitudinal and shear waves are observed as coexistent, especially for thinner plate samples. Fig. 11 is an example of the coexistence of the longitudinal and shear waves. In the reflection waveform of Silicon I shown in Fig. 11(a), the longitudinal and shear waves are mainly contained in the main bang (before 150 ls) and tail (after 150 ls), respectively. The reason that the wave in the tail of the reflection is identified as shear wave is that the speed of the wave in the tail obtained using the method proposed in this study (Section 4) matches the shear wave speed of the tested material with very high accuracy (Table 4). As to the mechanism of the shear wave generation in the current experimental configuration, a mode conversion effect must take place because all the measurements are performed in water (in which only longitudinal waves is supported), although the exact mechanism is not clear to us. Since the incident wave is a bounded beam, the incident wavefront is not a perfect plane, which leads to the fact that a portion of energy in the normally incident beam will hit the interfaces at slightly small angles and this portion of obliquely incident energy will probably lead to the generation of shear waves (first mode conversion). When the shear wave in the solid plates encounters the interfaces again, it will be converted to longitudinal wave again (the second mode conversion) and received as either the reflected wave (in pulse-echo mode) or the transmitted wave (in throughtransmission mode). Moreover, it is also observed from the spectrogram such as Fig. 11(b) that the longitudinal wave has large frequency coverage (as the emitting transducer), decays faster and dominates in the earlier part of the received signal; while the shear wave normally contains several discrete frequency components, decays much slower and dominates in the later part of the received signal. Inspired by the spectrogram analysis in the investigation of acoustic Wood anomaly phenomenon [27,28], discriminating the time ranges of wave domination is helpful to locate the signal in the time domain for different analysis and application purposes. Please be noted that it is still not clear why shear wave is only generated at finite frequencies.
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Fig. 9. Schematic of the ultrasonic measurements. (a) The incident pulse and the waves reflected back to the upper Medium I are depicted using solid lines with arrows; the waves transmitted into the lower Medium I are depicted using dashed lines with arrows; and the waves bouncing back and forth inside Medium II are depicted using dotted lines with arrows. (b) and (c) are the schematics of the pulse-echo and through-transmission configurations, respectively. (d) is the photo of the setup for the throughtransmission configuration.
4. Application of the pulsed ultrasonic comb filtering effect in the measurement of sound velocity and thickness of thin plates In this section, we will demonstrate the capability of the comb filtering effect in measuring the longitudinal and shear velocities and the thickness of thin plate samples based on measurements in Section 3.1, and also clarify the advantages of this method over the conventional ultrasonic methods. 4.1. Principle of measurement Among the different methods for measuring ultrasonic velocity of solid plate materials [29], the most common and simple method is based on the direct measurement of the thickness (d) and the round-trip time-of-flight (TOF) in the plate (Dt), which is called TOF method:
C¼
2d Dt
ð15Þ
By applying the relationship in Eq. (11), this method can be expressed as: Fig. 10. Representation of the transmissions and reflections measured for Glass I: (a) waveform (b) amplitude spectrum.
C ¼ 2dDf
ð16Þ
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Table 3 Longitudinal velocity (CL) and shear velocity (CS) of the samples measured using TOF method as reference velocities for evaluating the accuracy of those measured based on the comb filtering effect. Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
CL (m/s) CS (m/s)
5784 3401
5645 3378
5667 3341
8383 5844a
8348 5844a
a This value is not experimentally obtained, but calculated based on the mechanical properties of the material.
The key of applying the comb filtering effect in measurement is to find the spectral interval Df. Depending on the center frequency and bandwidth (Table 2) of the pulse generated by the emitting transducer, there are two ways to obtain Df: (i) Directly obtain Df from the adjacent peaks or notches in the amplitude spectrum. This method applies to the cases in which more than one peak or notch appear as in Figs. 3, 4, 6–8 and 10. (ii) Obtain Df according to Eq. (13) or (14). Depending on the type of spectral amplitude changing (notch or peak) that the measurement has, a proper equation can be selected. For example, Eq. (13) should be chosen for analyzing the reflection spectrum (notches occurring) and Eq. (14) for the transmission spectrum (peaks occurring). To be accurate, these two methods can be applied together for cross validation. To demonstrate how to find Df from the experimental measurements, the transmission signals and their spectra for both Glass II and Silicon II are shown in Fig. 12. We can see that the transmitted pulses overlap with each other in the time domain for both samples (Fig. 12(a) and (c)), and so it is impossible to find the round-trip TOF for the samples. In Fig. 12(b) and (d) the comb filtering effect is clearly visible for both longitudinal wave and shear wave. The spectral interval Df of longitudinal wave can be
Fig. 11. Illustration of the coexistence of longitudinal and shear waves in the experimental measurements using the reflection of Silicon I measured with 10 MHz transducer: (a) waveform (b) spectrogram.
Table 4 Longitudinal velocity (CL) and shear velocity (CS) of the samples obtained based on the comb filtering effect using Eq. (16) and their percentage difference (DCL for longitudinal wave and DCS for shear wave) from the measurements using TOF method. Sample
Glass I
Glass II
Glass III
Silicon I
Silicon II
CL (m/s) DCL (%) CS (m/s) DCS (%)
5794 0.2 3465 1.9
5722 1.4 3404 0.8
5694 1.7 3399 0.6
8320 0.8 5798 0.8
8331 0.2 5798 0.8
Instead of relying on the time domain characteristics Dt, Eq. (16) offers a way for sound velocity measurement based on the frequency domain characteristics Df given a comb filtering effect occurs. For the cases in which Dt is large enough to separate the constituent pulses in the reflected or transmitted signals, Eq. (16) can give the same results as Eq. (15). However, when the constituent pulses in the reflected or transmitted waves overlap with each other, Eq. (15) fails, but Eq. (16) still works for accurate velocity measurement as long as at least one peak or notch appear in the spectral coverage of the received transducer. It is not hard to see that Eq. (16) is more applicable for the measurement of thin plates. In addition, if the sound velocity is known, the thickness of thin plates can also be measured using a variation of Eq. (16):
d¼
C 2 Df
ð17Þ
In what follows, both Eqs. (16) and (17) will be applied for demonstrating the application of the comb filtering effect in thin plate measurement.
Fig. 12. Examples of the experimentally obtained signals to demonstrate the application of the proposed methods for measuring the speed and thickness of thin plates. The transmitted wave and its spectrum of Glass II measured using 5 MHz transducer are shown in (a) and (b), respectively. The transmitted wave and its spectrum of Silicon II measured using 20 MHz transducer are shown in (c) and (d), respectively.
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obtained from the high-amplitude peaks (3 peaks in Fig. 12 (b) and 2 peaks in Fig. 12(d)), and Df values of Glass II and Silicon II are 2.01 MHz and 8.25 MHz, respectively. The spectral interval of shear wave can be obtained from the small-amplitude notches of the corresponding spectra. For examples, 2 notches exist for Glass II (Fig. 12(b)) at 2.41 MHz and 3.63 MHz and lead to a spectral interval of 1.22 MHz. Similarly, a spectral interval of 7.44 MHz can be obtained for Silicon II from the two notches at 14.89 MHz and 22.33 MHz (Fig. 12(d)). To ensure the frequency intervals obtained from directly reading (the first method proposed in this paragraph) are the ones between theoretically adjacent peaks or notches, it is always favorable to double check them through solving the simultaneous equations generated using Eq. (13) for peaks or Eq. (14) for notches (the second method proposed in this paragraph). In practical measurements, the applicability of the comb filtering based method relies on the simultaneous fulfillment of the following two conditions: (i) In the time domain, the signal for analysis must contain more than one pulse. This requirement can be readily met for the thin plate cases (the major application promoted in this paper), but for thick plate cases it has to be met by properly windowing the received signals. (ii) In the frequency domain, the spectrum of the received signal must contain at least two amplitude changing features (notch or peak). The satisfaction of this requirement depends both on the properties of the plates (thickness and sound speed) and on the spectral properties of the selected transducer (center frequency and bandwidth). As a general guide, a larger bandwidth is always desired when selecting transducers. 4.2. Measurement of sound velocity in thin plates In order to evaluate the accuracy and reliability of the sound velocity measurement based on the comb filtering effect, the longitudinal (CL) and shear (CS) velocities of the plate samples (except for the shear velocities of Silicon I and II) are experimentally measured using TOF method. Due to the small thickness, no equipment is available for this study to measure the shear velocities of the silicon samples based on TOF method. Treating silicon as ideal solid, the shear velocities are calculated using the following relationship [30]:
CS ¼
rffiffiffiffiffiffiffi c44
q
ð18Þ
where c44 is the elastic constant (79.6 GPa [31]) and q is the mass density of silicon (2331 kg/m3 [31]). All the velocities of the samples are listed in Table 3. Based on Eq. (16), the longitudinal and shear velocities of the thin plate samples are measured and listed in Table 4. As far as the focus of this work is concerned, the reflections and transmissions give almost the same results, and so only the results obtained from reflections (in pulse-echo mode) are being reported. In Table 4 are also listed the percentage difference in longitudinal (DCL) and shear (DCS) velocity measurement between the method based on comb filtering effect and the conventional TOF method. According to the values of the percentage difference it can be concluded that the velocity measurements based on the comb filtering effect has high accuracy. For measuring the sound velocity in thin plates, a big challenge that the conventional TOF based methods have is the overlapping of the received pulses (such as echoes in pulse-echo mode [25]) and the corresponding reduction of the spatial resolution [32] of
Table 5 Comparison of the center frequencies of the transducers (available for this study) that can measure the sound velocities of the sample plates using the TOF based method and the comb filtering effect based method. Samples
Glass I Glass II Glass III Silicon I Silicon II
TOF based method
Comb filtering effect based method
CL (MHz)
CS (MHz)
CL (MHz)
CS (MHz)
5 10 10 50 50
10 20 20 NA NA
5 5 5 10 20
5 5 5 10 20
Table 6 Comparison of the directly measured thickness (dD) and those measured based on the comb filtering effect (dCF) by applying Eq. (17).
Silicon I Silicon II
dD (mm)
DCF (mm)
Difference (%)
0.482 0.384
0.488 0.389
1.2 1.3
the emitted pulses due to the relatively short propagation path. To meet this challenge, a short ultrasound pulse must be generated by either adopting other technologies such as picosecond laser technique [25,33,34] or using high frequency transducers. Compared with TOF based methods, the method based on the comb filtering effect has lower requirements on transducers in terms of center frequency, which make it more convenient and less expensive in practical applications. Table 5 shows the comparison of the requirements on transducer center frequency for TOF based method and comb filtering effect based method in measuring the longitudinal and shear velocities of the thin plate samples using the transducers available to this study. It is readily seen that the comb filtering effect based method requires much lower center frequencies for the transducers compared with the TOF based method in order to accomplish the same measurement task. 4.3. Measurement of thickness of thin plates To show the capability and reliability of the comb filtering effect in measuring the thickness of thin plates, the two silicon samples are chosen, their thicknesses (dCF) are obtained by applying Eq. (17) and compared with the values (dD) directly measured (using a digital caliber) in Table 6. According to the percentage difference between dCF and dD in Table 6, one can see the measurements based on the comb filtering effect have relatively high accuracy. It should be noted that the longitudinal wave velocity of silicon applied to Eq. (17) is 8430 m/s, which is the tabulated value instead of the one measured in this work, because using the tabulated values is the case for most real world practice. In addition, the results shown in this section are just an example of the application of the comb filtering effect in geometric characterization. Potentially, the principle developed in this work can also be applied to characterize the geometry of the other structures such as periodic structures [32,35] or tablet-in-tablet [36]. 5. Conclusions This work presents a fundamental study of the comb filtering effect in pulsed ultrasonics intending to provide a tool for data analysis in pulsed ultrasonic measurements. Through numerical simulation, two basic types of comb filters, i.e. feedforward and feedback comb filters, are demonstrated in terms of waveform, amplitude spectrum and phase spectrum. Theoretical derivation is also performed to obtain the characteristic features of the comb
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filtering effect: (i) the formula to determine the spectral locations (frequencies) for the spectral peaks or notches, and (ii) the relationship between the relative time delay (the time domain characteristics) and the frequency interval of peaks or notches (the frequency domain characteristics). The comb filtering effect is further demonstrated and verified experimentally using the reflections and transmissions of five thin plate samples obtained in ultrasonic measurements. As examples of the practical application of the comb filtering effect in pulsed ultrasonics, the characteristic features obtained in this work are applied in the measurement of sound velocity and thickness of thin plate samples. It is proven that the method based on the comb filtering effect not only is capable of measuring sound velocity in thin plates with high accuracy, but also has the advantage of being able to simultaneously measure longitudinal and shear velocities and having lower requirement on transducers in terms of center frequency compared with the conventional time-of-flight based method. It has also been shown that the comb filter based method is also proved to be capable and reliable in thickness measurement for thin plates and hence possibly also in geometric characterization for other structures. Acknowledgement The authors wish to acknowledge the French Centre National de Recherche Scientifique (CNRS) and the Conseil Régional de Lorraine (CRL) for their financial support of this work. References [1] S.F. George, A.S. Zamanakos, Comb fitlers for pulsed radar use, Proc. Inst. Radio Eng. 42 (1954) 1159–1165. [2] R. Turner, Some thoughts on using comb filters in broadcast television transmitter and at receiver, IEEE Trans. Consum. Electron. 23 (1977) 248–257. [3] G.D. Arndt, F.M. Stuber, R.J. Panneton, Video-signal improvement using comb filtering techniques, IEEE Trans. Commun., Co 21 (1973) 331–336. [4] P.J. Brusil, T.B. Waggener, R.E. Kronauer, Using a comb filter to describe time varying biological rhythmicities, J. Appl. Physiol. 48 (1980) 557–561. [5] D.E. Veeneman, B. Mazor, A fully adaptive comb fitler for enhancing blockcoded speech, IEEE Trans. Acoust. 37 (1989) 955–957. [6] H.T. Hu, Comb filtering of noisy speech using overlap-and-add approach, Electron. Lett. 34 (1998) 16–18. [7] D. Liu, S.N. Fu, M. Tang, P. Shum, D.M. Liu, Comb filter-based fiber-optic methane sensor system with mitigation of cross gas sensitivity, J. Lightwave Technol. 30 (2012) 3103–3109. [8] H. Zou, S.Q. Lou, G.L. Yin, A wavelength-tunable fiber laser based on a twincore fiber comb filter, Opt. Laser Technol. 45 (2013) 629–633. [9] A.P. Luo, Z.C. Luo, W.C. Xu, Wavelength switchable all-fiber comb filter using a dual-pass Mach-Zehnder interferometer and its application in multiwavelength laser, Laser Phys. 20 (2010) 1814–1817. [10] G.Y. Sun, D.S. Moon, A.X. Lin, W.T. Han, Y.J. Chung, Tunable multiwavelength fiber laser using a comb filter based on erbium-ytterbium co-doped polarization maintaining fiber loop mirror, Opt. Express 16 (2008) 3652–3658. [11] J. Chow, G. Town, B. Eggleton, M. Ibsen, K. Sugden, I. Bennion, Multiwavelength generation in an erbium-doped fiber laser using in-fiber comb filters, IEEE Photonics Tech. Lett. 8 (1996) 60–62.
[12] P.M. Narins, E.F. Evans, G.F. Pick, J.P. Wilson, Comb-filtered noise generation for use in auditory neurophysicological and psychophysical experiments, IEEE Trans. Biomed. Eng. 26 (1979) 43–47. [13] B. Santhanam, P. Maragos, Harmonic analysis and restoration of separation methods for periodic signal mixtures: algebraic separation versus comb filtering, Signal Process. 69 (1998) 81–91. [14] M. Niedzwiecki, M. Meller, Generalized adaptive comb filters/smoothers and their application to the identification of quasi-periodically varying systems and signals, Automatica 49 (2013) 1601–1613. [15] M. Niedzwiecki, P. Kaczmarek, Generalized adaptive notch and comb filters for identification of quasi-periodically varying systems, IEEE Trans. Signal Process. 53 (2005) 4599–4609. [16] H. Prance, R.J. Prance, P.B. Stiffell, Hardware comb filter enhances dynamic range and noise performance of sensors in noisy environments, Rev. Sci. Instrum. 78 (2007). [17] A.G. Dempster, Use of comb filters in GPS L1 receivers, GPS Solut. 12 (2008) 179–185. [18] J.J. Guo, Y.H. Yang, G.D. Peng, Analysis of polarization-independent tunable optical comb filter by cascading MZI and phase modulating Sagnac loop, Opt. Commun. 284 (2011) 5144–5147. [19] A. Beydoun, V.T. Nguyen, L. Naviner, P. Loumeau, Optimal digital reconstruction and calibration for multichannel time interleaved Sigma Delta ADC based on comb-filters, Int. J. Electron. Commun. 67 (2013) 329–339. [20] R. Wu, C.M. Long, D.E. Leaird, A.M. Weiner, Directly generated Gaussianshaped optical frequency comb for microwave photonic filtering and picosecond pulse generation, IEEE Photonics Tech. Lett. 24 (2012). [21] T. Halmrast, Orchestral timbre: comb-filter coloration from reflections, J. Sound Vibrat. 232 (2000) 53–69. [22] A. Galembo, A. Askenfelt, Signal representation and estimation of spectral parameters by inharmonic comb filters with application to the piano, IEEE Trans. Speech Audio Process. 7 (1999) 197–203. [23] W. Jin, X. Liu, M.S. Scordilis, L. Han, Speech enhancement using harmonic emphasis and adaptive comb filtering, IEEE Trans. Audio Speech Lang. 18 (2010) 356–368. [24] H.T. Hu, C. Yu, C.H. Lin, Usefulness of the comb filtering output for voiced/ unvoiced classification and pitch detection, in: Proceedings of the 2009 International Conference on Signal Processing Systems, 2009, pp. 135–139. [25] N.W. Pu, W.I. Kuo, Laser picosecond ultrasonic measurement of longitudinal sound velocity in nanometer thin films, Review of Progress in Quantitative Nondestructive Evaluation 27a and 27b (975) (2008) 286–293. [26] J.O. Smith, Physical Audio Signal Processing, W3K Publishing, 2010. [27] J.F. Liu, N.F. Declercq, Investigation of the origin of acoustic Wood anomaly, J. Acoust. Soc. Am. 138 (2015) 1168–1179. [28] J. Liu, N.F. Declerq, Time-frequency analysis of wood anomalies in acoustics, Proc. Meet. Acoust. 19 (2013) 055075. [29] A.S.f.T.a. Materials, Standard Practice for Measuring Ultrasonic Velocity in Materials, in, ASTM International 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428–2959, United States, 2015. [30] M.J.P. Musgrave, Crystal Acoustics, The Acoustical Society of America, 2003. [31] H.J. Mcskimin, W.L. Bond, E. Buehler, G.K. Teal, Measurement of the elastic constants of silicon single crystals and their thermal coefficients, Phys. Rev. 83 (1951), 1080–1080. [32] J. Liu, N.F. Declercq, Ultrasonic geometrical characterization of periodically corrugated surfaces, Ultrasonics 53 (2013) 853–861. [33] C. Thomsen, H.T. Grahn, H.J. Maris, J. Tauc, Surface generation and detection of phonons by picosecond light pulses, Phys. Rev. B 34 (1986) 4129–4138. [34] O.B. Wright, Thickness and sound-velocity measurement in thin transparent films with laser picosecond acoustics, J. Appl. Phys. 71 (1992) 1617–1629. [35] A. Shaw, J. Liu, S.W. Yoon, N.F. Declercq, Characterization of the geometry of microscale periodic structures using acoustic microscopy, Ultrasonics 70 (2016) 258–265. [36] J.F. Liu, C. Cetinkaya, Mechanical and geometric property characterization of dry-coated tablets with contact ultrasonic techniques, Int. J. Pharmaceut. 392 (2010) 148–155.