Coded waveforms for optimised air-coupled ultrasonic nondestructive evaluation

Ultrasonics xxx (2014) xxx–xxx

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Coded waveforms for optimised air-coupled ultrasonic nondestructive evaluation David Hutchins a,⇑, Pietro Burrascano b, Lee Davis a, Stefano Laureti a,b, Marco Ricci b a b

School of Engineering, University of Warwick, Coventry CV4 7AL, UK Polo Scientifico Didattico di Terni, Università degli Studi di Perugia, 05100 Terni, Italy

a r t i c l e

i n f o

Article history: Received 19 June 2013 Received in revised form 28 February 2014 Accepted 13 March 2014 Available online xxxx Keywords: Ultrasonic Air-coupled Coded waveforms Cross-correlation Non-destructive evaluation

a b s t r a c t This paper investigates various types of coded waveforms that could be used for air-coupled ultrasound, using a pulse compression approach to signal processing. These are needed because of the low signal-tonoise ratios that are found in many air-coupled ultrasonic nondestructive evaluation measurements, due to the large acoustic mismatch between air and many solid materials. The various waveforms, including both swept-frequency signals and those with binary modulation, are described, and their performance in the presence of noise is compared. It is shown that the optimum choice of modulation signal depends on the bandwidth available and the type of measurement being made. Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/3.0/).

1. Introduction Air-coupled ultrasound is a non-contact technique that is of interest to nondestructive evaluation (NDE), because it can be used to test a wide range of materials, from polymers and composites to metals [1,2], It can also be used in harsh environments, where contamination is an issue, and when rapid scanning is required. The technique has been proposed for application in many areas, including the inspection of composites [3–5], the detection of contamination and changes in food quality [6] and for imaging many other materials [7]. In all cases, it is the fact that air is used as the coupling medium which makes the technique attractive. In pioneering work by Bernard Hosten and his colleagues at CNRS Bordeaux, this approach has been used for the inspection of polymer composite plates using Lamb waves [8,9]. Work was also done on the determination of the elastic constants of carbon fibre reinforced polymer (CFRP) composites [10–12]. Here, the composite sample was positioned between an ultrasonic capacitive source/receiver pair, and the angle of incidence changed by rotation of the sample relative to the ultrasonic beam axis. The ultrasonic waveforms were recorded at each angle, and the longitudinal and shear wave arrival times calculated in real time (the latter created by mode conversion within the composite plate). Changes in arrival time were then compared to theoretical ⇑ Corresponding author. E-mail address: [email protected] (D. Hutchins).

predictions, and an optimum fit between the two allowed the calculation of elastic constants to take place. Moreover in [12], the transmission coefficients at different angles were measured and used as input for an inverse procedure capable of inferring the complex viscoelastic moduli of Perspex and composite materials. The elastic properties of other materials such as wood have also been examined [13]. Note that the design of the electrostatic (or ‘‘capacitive’’) transducer is also important in determining performance [14]. Although air-coupled transducers can be scanned to form images, a higher resolution can be obtained if the transducer beam is focused. A convenient way to do this is to use external optics. This can be achieved using a Fresnel zone plate, which is aligned so as to focus on-axis at a pre-selected frequency [15]. More commonly, however, a good focus in air can be obtained across a wide bandwidth using external off-axis parabolic mirrors, which can be used for imaging thin materials and other samples [16]. It is worth mentioning that off-axis parabolic mirrors have also been exploited to increase the angular aperture of air-coupled transducers used to collect Lamb mode waves radiating from plates, allowing therefore a faster reconstruction of the dispersion curves of unknown materials [17,18]. There is, however, a problem which must be overcome if air is used to transmit ultrasonic energy into and out of a solid sample – there is a need to overcome the large acoustic impedance mismatch between air and the solid object being tested. This causes several problems. The first is that a large fraction of the incident

http://dx.doi.org/10.1016/j.ultras.2014.03.007 0041-624X/Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

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energy is reflected at the air/solid interface. This reduces the amount of energy both entering and leaving the sample. It also means that a large reflection from the incident surface results, meaning that pulse-echo operation becomes more complicated. In through-transmission, there are two basic approaches to solving the lack of signal. The first is to choose a very narrow bandwidth excitation whose central operation frequency is tuned to the through-thickness resonance of the sample, so as to maximize transmission [19]. This lack of bandwidth is a problem when it comes to conventional imaging, as discrimination of defects, for example, becomes very difficult. The second approach is to use a wider bandwidth, together with some method of retrieving very small ultrasonic signals that have passed through two separate air/sample boundaries. This second method has been investigated by several authors [20–24], and the general approach is to use a form of crosscorrelation to detect the signal. The resulting technique is known as pulse compression. The incident waveform, with a bandwidth usually defined by the transduction method used, is chosen so that this type of signal processing can be performed most effectively for a particular measurement. There are many types of signal that can be used, but they can be broadly classified within two main types: a swept-frequency ‘‘chirp’’ signal or a form of binary sequence. Fig. 1 gives an example of a chirp signal buried in noise to illustrate the pulse compression process. Cross-correlation between the output ‘‘noisy’’ signal and a reference waveform, known in the literature as a matched filter, allows this signal to be detected, with the smoothed rectified output shown. The resulting pulse compression operation has allowed the signal to be enhanced in terms of the signal to noise ratio (SNR), allowing air-coupled measurements to be made. Usually, the matched filter coincides with a replica of the sent signal. Moreover, depending on the specific application, the pulse compression procedure can use either periodic or aperiodic signals; the cross-correlation step is then realized in either cyclic or non-cyclic mode respectively. The operation shown in Fig. 1 is an example of a pulse compression operation that is not optimised. The purpose of this paper is to investigate different forms of signal modulation, and to determine the best approach for air-coupled ultrasound for a given bandwidth of transducer. To accomplish this aim, Section 2 briefly describes the basic principles behind pulse compression, and elaborates the main differences between cyclic and non-cyclic operation. Section 3 then details the various types of waveform used. Section 4 reports the results of numerical simulations that demonstrate the influence of the transducer bandwidth on the choice of the optimum waveform, whilst Section 5 illustrates experimental results that demonstrate the validity of the numerical results. Finally, conclusions are drawn in Section 6 and some further possible developments are indicated.

2. Basic principle of the pulse-compression approach and the measurement protocols Pulse compression is a measurement technique developed for estimating the impulse response h(t) of Linear Time Invariant Systems (LTI). The basic principle at the heart of the technique can be expressed as follow: if exists a pair of signals {s(t), W(t)} such that their convolution sðtÞ  WðtÞ ¼ ^ dðtÞ ’ CdðtÞ is a good approximation of the Dirac delta function d(t), then the impulse response h(t) can be estimated by exciting the LTI system with the signal s(t) and then by convolving the system output y(t) with W(t):

yðtÞ ¼ hðtÞ  sðtÞ; sðtÞ  WðtÞ ¼ ^dðtÞ ’ C  dðtÞ; ^ ¼ yðtÞ  WðtÞ ¼ hðtÞ  sðtÞ  WðtÞ ’ C  ðhðtÞ  dðtÞÞ ¼ C  hðtÞ hðtÞ ð1Þ The better the approximation of d(t), the higher the quality of the estimation of h(t). In addition, the theory of matched filters states that, in presence of Additive White Gaussian Noise (AWGN), i.e. when y(t) = h(t)  s(t) + e(t) (where e(t) is the noise term), the choice of W(t) that maximizes the SNR uses the time-reversed replica of the input signal W(t) = s(t) as the matched filter [25]. With this choice, the approximation ^ dðtÞ of d(t) coincides with the autocorrelation function Us(s) of the input signal. In order to optimise the SNR and to ensure a good measurement resolution, Us(s) has therefore to be d-like. This requirement is fulfilled if s(t) is a wideband signal with a bandwidth that spans as much as possible the entire frequency range of response of the LTI system. A more formal description of the pulse compression theory lies beyond the scope of the present paper and the reader is referred to the literature on this topic (see for example [22]); here, it is worth stressing that a d-like feature of the signal autocorrelation can be assured both by single-shot or periodic signals, leading to two main pulse compression schemes: Acyclic Pulse Compression (APC) and Cyclic Pulse Compression (CPC), as depicted in Fig. 2. The particular features of these two approaches are highlighted below, together with a computational representation of the two procedures in terms of Discrete Fourier Transforms, to provide a general framework. Then, in Section 3, the properties of the various codes will be reviewed in the light of the differences between the two measurement methods, where specific hardware and software tools to perform the pulse compression will also be discussed. 2.1. Acyclic vs Cyclic Pulse Compression A pulse compression measurement scheme can exploit either single-shot or continuous (periodic) excitation. Depending on this choice, the impulse response of the system under test can be retrieved by using either cyclic or acyclic convolution between the system output signal and the matched filter.

Fig. 1. (a) Example of a linear chirp signal embedded in noise, and (b) the resultant pulse compression output from a cross-correlation.

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Fig. 2. Sketches of the Acyclic (left) and Cyclic (right) Pulse Compression procedures.

To gain some insight into this process, let us suppose that for the APC and CPC cases respectively there exist pairs of signals that guarantee the expression

s1 ðtÞ  s1 ðtÞ ¼ dðtÞ; s2 ðtÞ  s2 ðtÞ ¼ dðtÞ;

ð2Þ

where s1(t) is a coded waveform defined in t e [0, Ts1], s2 ðtÞ is s periodic coded waveform with period Ts2 and  dðtÞ consist of a periodic train of d(t)0 s spaced by Ts2 as illustrated in Fig. 2. In the APC case, if the output signal in a LTI system is convolved with the matched filter, we obtain exactly the impulse response h(t), while for the CPC case we obtain the convolution between h(t) and  dðtÞ. This shows the main difference between the two schemes. Indeed, if h(t) is shorter than the input signal period, the convolution between h(t) and  dðtÞ produces a periodic train 0 of non-overlapping h(t) s, whereas if h(t) is longer than the input signal period, the convolution between h(t) and  dðtÞ is no longer periodic and presents pile-up between the h(t)0 s. Therefore the CPC approach should be used only if a priori information about the maximum length of h(t) is known; in this case h(t) is easy retrieved by cutting out a single period, to prevent time-aliasing in the reconstructed h(t). Conversely, in APC, the input signal duration TS1 of s1 is not constrained by the duration of Th. This could be considered a strong limitation on the use of the cyclic scheme but actually, in air coupled ultrasonic inspection, the signal length is usually significantly longer than that of the impulse response (or at least longer than that part of h(t) that produces effects above the noise level). This is so as to achieve a sufficient SNR to overcome the large acoustic impedance mismatch between air and the solid object being tested, For this reason henceforth we consider (Ts1, Ts2) > Th. From an experimental point of view, there exist differences between the two approaches. In the APC case, the coded signal s1(t) of length TS1 excites the system that produces an output signal y1(t) = h(t)  s1(t) of duration Ty1 = Ts1 + Th. If all the h(t) should be reconstructed, the output signal has to be recorded for time T rec P T y1 and the pulse compression is implemented by filtering, i.e. convolving, the output signals with the matched filter; in the case of continuous-periodic excitation, a periodic signal s2 ðtÞ of period Ts2 > Th is switched on at t = 0 and excites the system that produces an output signal y2 ðtÞ ¼ hðtÞ  s2 ðtÞ. For a transient of duration Ts2, the output signal will be periodic with the same period Ts2 and, by performing the cyclic convolution between a single period ys.s. of the steady-state of y2 and a single period of the input signal s2 ðtÞ, h(t) is retrieved. In practice, it is sufficient to transmit two periods of the excitation signal and to record the output corresponding to the second excitation period only. Mathematically, once that the signals have been digitalized in APC, the matched filter can be applied by performing standard convolution or Fast Fourier Transform (FFT)-based convolution algorithms, such as overlap-add, overlap-save [26,27]; in cyclic mode the matched filter is applied by computing the cyclic convolution between the output signal and the matched filter. By exploiting

the Convolution Theorem for Discrete Signal, the cyclic convolution can be efficiently implemented in the frequency domain as follow:

 2 ½n ¼ IFFTfFFTfys:s: ½ng  FFTfs2 ½ngg ys:s: ½ns

ð3Þ

where IFFT is the Inverse Fast Fourier Transform [26,28]. For completeness, it should be noted that the convolution theorem for a discrete signal can be used also in the single-shot case, provided that the discrete input and output signal are padded with zeroes up to double the number of samples of the input signal, having assumed Th < Ts1. At this point of discussion, it is therefore fair to ask whether there are any advantages of the cyclic approach over the acyclic one. The first concerns the SNR attainable. By considering the optimised APC procedure in which the output is recorded for the minimum time needed to recover all the impulse response, it straightforward to see that the energy is delivered to the system for a fraction pEX = (TS1/(TS1 + Th)) of the measurement time, and that the final SNR is given by SNR(pEX) = pEX  SNRmax, where SNRmax is the limit value achievable for TS1  Th. On the other hand, in the cyclic (CPC) case, the excitation is always active during the measurement, thus saturating the SNR for a given excitation power once the condition TS2 > Th is satisfied. Moreover, in order to increase the SNR of APC for a given pEX value, averages can be executed by sending a train of single-shot excitations, but the minimum repetition time of these must be TS1 + Th; for CPC systems, averages are easily performed by considering several consecutive periods of the steady state output signal, making it more time-efficient. From a computational point of view, the input and output signals to be processed are equal in length and usually shorter (if the assumption Th < Ts1 is still valid) for the CPC than the APC case, and therefore the convolution with the matched filter is less time-consuming; moreover the cyclic convolution admits an efficient implementation through the FFT that is also at the heart of fast convolution algorithms such as overlap-add used in acyclic mode. In summary, CPC methods can only be used if a priori information about the duration of the impulse response to be measured is available; therefore its application is limited when compared to standard APC approaches. Nonetheless, when applicable, it assures the optimal SNR and computational savings for selected measurements, as will now be demonstrated. 3. The choice of waveform for pulse compression In the previous Section, it was assumed that signals exist with an ideal d-like auto-correlation. Unfortunately, such a property is not exhibited by any kind of finite-duration waveform; hence, researchers have devised strategies that use signals approximating to this condition as well as possible for pulse compression applications. Two main families of waveforms have been suggested: binary codes and frequency modulated (chirp) waveforms. In this paper, various forms of both chirp signals and binary sequences

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are investigated. In particular, the performances of linear and nonlinear chirps in terms of maximum achievable SNR and resolution are compared to two types of binary sequence, namely Maximum Length Sequences (MLS) and Golay Complementary Sequences (GCS). In this Section the main features of the various waveforms are briefly listed, highlighting the pros and the cons of each of them. For the sake of clarity, henceforth the normalised autocorrelation (both acyclic and cyclic) function of the various signals will be denoted either by ^ dðtÞ or by ^ d½n to indicate that they represent only an approximation of the ideal Dirac-delta and Unit Impulse functions respectively. 3.1. Linear and Non-linear Chirp signals A generic Chirp signal is defined as

sCHIRP ðtÞ ¼ A sinðUðtÞÞ;

ð4Þ

where U(t) is a non-linear phase function of t [29]. The key quantity of a Chirp signal is represented by the instantaneous frequency fist(t) that is related to U(t) by the expression:

fist ðtÞ ¼ 1=2p  ðdUðtÞ=dtÞ

ð5Þ

By defining a proper trajectory fist(t) in a given time interval t e [0, Ts], a chirp signal is obtained via Eq. (4) and by inverting Eq. (5). The instantaneous frequency is important because it is strictly related to the Power Spectral Density (PSD) of the resulting function. Indeed if the duration Ts of the Chirp signal is large enough, the PSD is confined within the frequency interval spanned by fist(t) and, for a given frequency value, PSD is inversely proportional to the rate of change of fist(t) at that frequency [20,30]. For instance, if fist(t) = fstart + (B/T)  t with B = fstop  fstart and t e [0, T], the usual Linear Chirp (LChirp) is retrieved, where U(t) = 2p(fstartt + (B/2T)  t2) is quadratic and the PSD is almost flat and confined within the interval f e [fstart, fstop]. In general, the frequency can be swept in either a linear or non-linear way with respect to time. Two such signals sweeping from 150 kHz to 450 kHz are shown in Fig. 3. While at first sight the waveforms do not appear to be very

different, their spectra indicate that the two signals have very different properties. This can be illustrated further by looking at the trajectory followed by fist(t) in the Time–Frequency plane, as shown in Fig. 4. In particular, the LChirp is associated with a straight line in this type of plot, whereas the trajectory of the Non-Linear Chirp (NLChirp) can be any continuous monotonic curve, defined by the type of sweep function used. A NLChirp can be defined to satisfy particular requirements, for example chirps having exponential instantaneous frequency have been used for several years to characterize non-linear systems [33,34]. Here two main aspects of NLChirps are investigated: (i) using them with knowledge of the transfer function of a probe to maximize the SNR, and (ii) generating an NLChirp by applying a window function to the PSD of an LChirp [30]. The latter case was developed to implement only in frequency domain what is commonly performed in both time and frequency domains for an LChirp by applying windowing functions [30]. It is usual to amplitude modulate an LChirp to produce spectral-windowing, in order to reduce the level of side-bands of ^ dCHIRP ðtÞ. With the general assumption W(t) = s(t), ^ dðtÞ is in fact proportional to the Inverse Fourier Transform of the PSD, according to the Wiener–Khinchin theorem. A non-windowed LChirp exhibits abrupt changes in the PSD, in proximity to the limit frequencies fstart and fstop, and this results in slowly- attenuating side-bands [31,32]. By applying a window function in time-domain to an LChirp, a smoothing of the PSD is obtained, thereby producing a faster decay of side-bands at the cost of reducing the signal energy. This is a result of losing the constant envelope feature and the effective bandwidth of the signal, causing a broadening of the main lobe of the cross-correlation. There are many different types of window that can be used, and this work has investigated three types: Gaussian, Blackman and Tukey [33,34]; here, numerical and experimental results are reported for the Tukey case only, as this is reputed to assure that an optimal trade-off exists between side-band reduction, resolution achievable and SNR. Note, however, that by applying spectral windowing through an NLChirp, the constant envelope feature is preserved, and then also the SNR, and at the same time side-bands of ^ dðtÞ are reduced with respect to an un-windowed LChirp. In this case, the effective bandwidth is also reduced. To quantify this

Fig. 3. Examples of chirp signals in the form of (a) linear and (b) non-linear frequency sweeps. Waveforms are on the left, and corresponding frequency spectra are on the right.

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Fig. 4. Trajectories in the Time–Frequency planes of linear (a) and non-linear (b) chirps.

Fig. 5. Comparison of autocorrelation functions of chirp signals to the non-windowed linear case (black). This is for the windowed linear Tikey (left) and windowed nonlinear Tukey (right), both in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

reduction, the following definition can be used for the effective bandwidth [35]:

Beff

pffiffiffiffiffiffi Z ¼ 12

fstop

2

ðf  fc Þ PSDðf Þdf ;

ð6Þ

fstart

pffiffiffiffiffiffi where fc is the central frequency and the factor 12 is a normalization constant used to achieve Beff = B = fstop  fstart in the case of an ideal LChirp. As an example, Fig. 5 illustrates the effect of time and the spectral windowing applied to a Lchirp signal. Time windowing provides the best reduction in side-bands both in the near and in the far region but, as expected, exhibits the largest central lobe. On the other hand, spectral windowing ensures a resolution that is almost the same as for the un-windowed Lchirp, and provides a significant reduction of the side-bands in the near region while in the far region the side-bands levels coincide. As mentioned above, a further application of the NLChirp to noisy environments is the possibility that the excitation signal has a PSD that matches the overall transfer function Hset-up of the hardware set-up, i.e. transmitter + receiver transducers and optional amplifiers [20]. In this case, the energy-bandwidth product of the output signal is optimised for a given spanned frequency range [fstart, fstop] assuring an enhancement of the SNR. Here the energy-bandwidth product is defined as

ðB  EÞoutput ¼

Z

fstop

2

ðf  fc Þ jHset-up ðf Þj2 PSDðf Þdf

ð7Þ

fstart

Note, however, that the measurement resolution is not optimised.

3.2. Binary sequences The other type of waveforms used for pulse-compression application is represented by binary sequences. Although many different types of code exist, mainly for Spread Spectrum communication protocols and error-correction schemes, here only those exhibiting the d-like autocorrelation property are considered: Golay Complementary Sequences (GCS) and Maximum Length Sequences (MLS or m-sequences). GCS consists of two binary sequences, sAGCS ½n; sBGCS ½n, defined by recursion from two seed sequences, with the peculiar property that their out-of-phase aperiodic and periodic autocorrelation coefficients sum to zero:

^dA ½n ¼ ^dB ½n 8 n–0 ; ^dA ½n þ ^dB ½n ¼ 2L  d½n; GCS GCS GCS GCS

ð8Þ

where L is the sequence length. GCS were first introduced by Marcel J.E. Golay in 1949 for infrared spectroscopy and some years later the same author gave several methods for constructing sequences of length 2N and gave examples of complementary sequences of lengths 10 and 26 [36]. The complementary property of the autocorrelation can be usefully exploited in pulse compression by taking advantage of the linear hypothesis at the heart of the method, to modify the procedures described in Section 2.1 as follows: (i) the two complementary sequences are both used as input of the LTI system under test; (ii) the matched filter is applied separately to ^A ½n ¼ ^ each output to obtain the two estimates h dAGCS ½n  h½n; GCS B ^ B ½n ¼ ^ d ½n  h½n; (iii) the impulse response is reconstructed h GCS GCS without any approximation by adding the two contributions to ob^A ½n þ h ^ B ½nÞ. In spite of the fact that it is necestain 2L  h½n ¼ ðh GCS GCS sary to use two distinct sequences, GCS are the only coded waveforms that theoretically exhibit a perfect pulse compression

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at the end of the overall procedure. This is why they are popular in ultrasonic measurements [20,37,38]. Moreover, as stated above, the complementary feature holds both for APC and CPC cases, so one can decide to choose one of the two approaches depending on the specific application. A further advantage of the GCS approach is the existence of a fast algorithm to perform correlation–convolution between a generic signal and a Golay pair with length L. Such an algorithm is known in the literature as a Fast-Golay-Correlation and it exhibits a computational complexity similar to the FFT operation but it is less demanding since it mainly involves sums and subtractions instead of multiplications [39]. This feature opened the way to digital hardware implementation of GCS pulse compression procedure for ultrasonic NDT applications [40,41].Despite the above attractive properties, there are other types of binary sequences of interest to air-coupled ultrasonic inspection. One is MLS, and here we try to highlight the main differences with respect to GCS above. MLS are periodic sequences that are generated by a Linear Feedback Shift Register (LFSR) machine and, as illustrated by Golomb [42], they exhibit the maximum period length for a given register depth. In particular for a register with N cells, the period of the MLS[n] is L = 2N  1. The configuration of the LFSR machine is derived by the theory of Galois finite fields and it assures several useful mathematical properties to those codes. For ultrasonic pulse compression applications, the most useful one is represented by the normalised cyclic auto-correlation function of these sequences that consists of a train of unit pulses with added a DC bias whose amplitude decreases inversely proportional to L: sMLS ½n  sMLS ½n ¼  d½n  ð1=LÞ. If the h[n] can be assumed to have a zero mean (where h[n] takes into account also the transfer function of the transducers), and the considerations about the h[n] duration are satisfied, then it is straightforward to see that by adopting the cyclic operation a perfect estimate of h[n] can be retrieved. Fortunately, this hypothesis is true in almost all cases, so that MLS can be fully exploited in CPC application. Indeed, after the work of Golomb [43], MLS-based impulse response schemes were successfully introduced into acoustic measurements. Nevertheless, even in the most general case, by adopting the cyclic protocol, h[n] can be perfectly retrieved by applying specific processing [44] or by slightly modifying the amplitude of MLS[n] to attain a ‘‘Perfect Sequence’’ [45]. MLS are therefore equivalent to GCS if CPC is possible, and in fact may be preferred since only one sequence is necessary. Moreover as for GCS, there exists a fast algorithm to perform cyclic correlation with MLS sequences derived by Hadamard Transform: the so-called Fast-m-Transform [46]. Note that, if APC has to be used, MLS methods are not suitable, since consistent self-noise appears in the aperiodic autocorrelation function and therefore GCS are highly preferable. A further advantage of MLS with respect GCS, provided by the decimation property, is the possibility in CPC to under-sample the output signal of any factor 2k attaining after 2k periods exactly the original signal, thus allowing the use of Analog to Digital Converters with sampling rate equal or lower than the signal bandwidth itself [47]. Note also that MLS can be used in multi-input/multi-output systems by exploiting their pseudo-orthogonality [48,49]. It should also be noted that Legendre sequences [43] have very similar properties to MLS. However, they cannot be generated recursively, and a fast algorithm for their correlation does not exist; hence, the MLS approach has historically been preferred. Moreover it has been shown recently how both GCS and MLS pulse compression schemes are extremely robust against quantization noise introduced by ADC’s, and they both allow the measurement of impulse responses with dynamics well below the quantization step [24,50]. For both GCS and MLS, it is important to stress that the autocorrelation properties aforementioned hold for numerical sequences but, when changing sequences of numbers into practical voltage

signals, some aspects should be considered. In particular, at any ‘‘1’’ or ‘‘1’’ of the codes, a voltage level exists with a finite duration dT related to the update rate of generation Rgen of the sequences by dT = (Rgen)1. At the same time, the correlation step needed for pulse compression depends on the sampling rate RADC of the ADC process. To correctly apply pulse compression, RADC must be a multiple of Rgen, i.e. RADC = m Rgen with m e N, and some pre-processing has to be performed when m > 1, so as to exploit the fast correlation algorithms. Usually this is the case since while Rgen determines the actual PSD of excitation, a greater RADC is needed to faithfully reconstruct output signals. Indeed, the generated signals have power spectra close to that of a boxcar pulse of width dT, so that it is better to choose Rgen in order to concentrate the energy in the actual bandpass of the system. In the following, it will be shown that a good criterion is 2fc < Rgen < 4fc where fc is the central frequency of the transducers. At the same time, as a general rule RADC should be at least 5–10 times greater than the upper frequency of interest. Of course, it is possible to choose Rgen = RADC to consider the input signal as the ideal numeric sequences, but at the cost of exciting a bandwidth larger than the effective one. This could waste a lot of energy, due to the band-pass behaviour of a typical ultrasonic experimental system. In contrast to this, chirpbased pulse compression schemes allow Rgen and RADC to be chosen independently, providing that the matched filter is sampled at RADC. From an experimental point of view, in most of the cases the transducers have a bandwidth B that is a fraction of the central frequency fc. The spectra of binary sequences thus do not fit very well with the frequency range of interest, causing a decrease of the effective energy delivered to the system. For this reason, in order to increase the energy transfer ratio by better matching the system bandpass, it has been proposed in literature to modulate the GCS or MLS sequences by replacing ‘‘1’’ or ‘‘1’’ with short sequences of binary values to perform some spectral shaping [51]. The easiest solution here considered it to replace each ‘‘1’’ with the combination ‘‘1, 1’’ and each ‘‘1’’ with the opposite sub-code ‘‘1, 1’’. This technique is largely adopted in acoustics where it has been named Inverse Repeated Sequences (IRS) [52]. Examples of waveforms and spectra of GCS and MLS sequences and their respective IRS’s are shown in Figs. 6 and 7, where it is evident how the IRS signal spectra match the transfer function of a typical ultrasonic transducer more effectively. In the following Sections, all the various signals introduced will be compared for use in air-coupled ultrasound. Their performance will be determined for various levels of noise first numerically by considering measurement systems with different bandwidth of operation and then experimentally, so that their suitability in NDE can be determined. 4. Numerical investigation of modulation scheme performance The optimal choice of the coded waveforms for air-coupled ultrasonic NDE strongly depends on the transfer function of the measuring system utilized [52]. For instance, on the basis of that illustrated in Section 3, intuitively one can expect that for narrowband transducers it is better to use a chirp, tailored to the system response, rather than that of a flat excitation provided by binary sequences. In this Section, in order to quantify the effectiveness of one excitation with respect to the others, the results of numerical simulations executed considering the various types of excitation are reported. In particular, the maximum SNR attainable for a fixed excitation duration and peak power is investigated by varying the transducer bandwidth and the noise level. In order to make the analyses as general as possible, the simulations assumed normalised frequencies. In particular, the bandwidth B of the overall Tx–Rx system is given as a percentage of

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Fig. 6. Examples of waveforms useful for ultrasonic air-coupled signals derived from Standard and Inverse Repeated Golay and MLS sequences. The ‘‘standard’’ waveforms were optimised for a central frequency of 300 kHz, see Fig. 7, and the waveforms were generated to all have the same duration.

Fig. 7. Power Spectral Density curves for the waveforms of Fig. 6. The red line in the Golay cases represents the sum of the two PSD that exhibits the complementary property as the autocorrelation. It can be seen that, unless for the DC component, the PSD of MLSs is half the sum of the PSD of the two Golay sequences, as expected, since the overall Golay duration is double.

Fig. 8. Examples of impulse responses and Power Spectral Densities of FIR filters used to simulate the effect of measurement set-up (Tx–Rx transducers + amplifier) at different values of bandpass width.

the central frequency fC of the system varying from 10% to 200%, which represents the limit case of an almost-ideal broadband system in the range [0,B]. The sampling rate considered was RADC = 20fC. The system impulse response is simulated by a bandpass

FIR filter and some examples of the corresponding impulse response and of the magnitude of the transfer function are reported in Fig. 8. The following combinations of waveforms and pulse compression schemes are considered, which represent the most promising

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cases, both for easiness of the procedure, maximum available SNR, computational resources, etc.: 1. Acyclic Pulse Compression & Tukey-windowed Linear Chirp (LChirpAPC). 2. Cyclic Pulse Compression & Tukey-windowed Non-Linear Chirp (NLChirpCPC). 3. Acyclic Pulse Compression & Golay (GCS–APC). 4. Cyclic Pulse Compression & MLS (MLS–CPC); 5. Acyclic Pulse Compression & IRS-Golay (IRSGCS–APC). 6. Cyclic Pulse Compression & IRS-MLS (IRSMLS–CPC). For this comparison, each waveform has appoximately the same duration (GCS and CHIRPs defined with k29 samples while MLS with k(29  1) samples with k = 5) and the same peak amplitude. This assures that binary sequences have a doubled power and an additional 3 dB of SNR with respect to chirps. Moreover, Golaybased codes lead to another factor of 3 dB in the theoretical SNR, since they use of a pair of sequences. The binary sequences were optimised to cover the bandwidth [0–2fC], the IRS-binary sequence were defined to have the maximum of the PSD at fC, Chirp signals instead were defined both with a fixed bandwidth B = 2fC and with a varying B following the one of the Tx–Rx transducers system. This latter case is denoted as a ‘‘Matching Chirp’’ (MChirp). Four main quantities were calculated in presence of noise to consider different aspects related to detection, resolution, etc.: (a) the width of the main lobe (‘‘MLW’’); (b) the SNR in the main lobe (‘‘ML-SNR’’); (c) the near side-lobe level (‘‘NSL’’) and (d) the far side-lobe level (‘‘FSL’’). Each quantity is computed within a distinct region of the measured signal; to gain deeper insight, consider the three regions labelled in Fig. 9 by the capital letters A, B, C: A indicates the main lobe, that is the region outside of which the envelope of the expected signal is 20 dB below the maximum; B indicates near side-lobes, i.e. two regions at the sides of the main lobe, each with the same width as the main lobe; C indicates far side-lobes, that is the outer regions beyond near side-lobes. MLW provides a measure of the resolution of the pulse compression procedure, and can be defined as: ex

ex

MLW ¼ minfD s: n t: h ½n < n1 ; n > n1 þ D < 0:1 Maxðh ½nÞ; 8ng ð9Þ Here, the smaller the MLW, the better the temporal-spatial resolution of the measurement, so that this parameter is of particular interest in Time-of-Flight measurement or in range detection. MLSNR expresses the capability of detecting a signal in presence of noise and it is defined as the ratio between the energy of the expected impulse response after pulse compression without noise hex[n] = hFIR[n]  s[n]  W[n] to the Energy of the noise inside the main lobe:

P

2 ex nMainLobe ðh ½nÞ 2 ex m nMainLobe ðh ½n  h ½nÞ

ML-SNR ¼ P

;

ð10Þ

where hFIR[n] is the FIR impulse response used to simulate the system, and hm[n] = (hFIR[n]  s[n] + e[n])  W[n] is the measured impulse response in presence of noise e[n]. NSL is defined as the ratio (in dB) between the maximum absolute amplitude of hex[n] and the mean value of the envelope of hm[n] in the Near Sidelobes region, given by ex

NSL ¼

Maxðh ½nÞ m MeanðEnvðh ½nÞÞnNearSidelobes

ð11Þ

Also in a similar way to NSL, FSL is defined as the ratio (in dB) between the maximum absolute amplitude of hex[n] and the maximum value of the envelope of hm[n] in the far side-lobes region ex

FSL ¼

Maxðh ½nÞ m MaxðEnvðh ½nÞÞnFarSidelobes

ð12Þ

NSL and FSL quantify the level of noise and at the same time express the limit value that a secondary signal should not be detected. These quantities are of utmost importance when complex impulse responses have to be measured, as in the case of multi – path reflection, multi-layered structures, etc. Fig. 10 shows surface plots of the ML-SNR versus the noise power and the bandwidth of the Tx–Rx system. The results are quite similar for all the cases and in general higher SNR are attained with broadband systems, as expected for this choice of excitations. Moreover with respect to ‘‘standard’’ Linear Chirp, GCS and MLS, some improvements are attained by using a non-linear chirp and the IRS approach. The IRS sequences, both GCS and MLS, attain the highest value of ML-SNR. If we let the bandwidth of the chirp follow that of the experimental system, then an improvement of the SNR of the Chirp is achieved, as expected from our previous discussion. Fig. 11 shows the surface plot of the ML-SNR for the MChirp compared to that of the IRSGCS–APC and in particular the subplot on the right maps the Noise Power – System BW region for which IRSGCS–APC exhibits a SNR higher than MChirp. The area where IRSGCS–APC is better than MChirp is coloured white, otherwise is black. For BW P 90%fC (approximately), the IRSGCS (and also the IRSMLS) perform better than Matched Chirp; for BW < 90%fC a higher SNR is attained by using a Chirp (Linear or Non-linear) ‘‘matched’’ with the system band-pass. This result is reasonable, since almost half of the input energy is filtered out by the transducers for BW = fC, erasing therefore the theoretical advantage of binary sequences At the same time, the matched chirp has the worst resolution among the various schemes, corresponding to a large MLW, as shown by Fig. 12 that reports the trends of the MLW for all the excitation types considered. While all the various codes with fixed bandwidth have practically the same resolution, the matched chirp achieves a better SNR, with a significant decrease of the resolution that is accentuated for BW < 90%fC. Figs. 13 and 14 report a comparison analogous to that illustrated in Fig. 11 for the NSL and the FSL respectively.

Fig. 9. Definition of MainLobe (A), Near Sidelobes (B) and Far Sidelobes (C) regions.

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Fig. 10. ML-SNR versus the noise power and the system bandpass width for various pulse compression schemes.

Fig. 11. Comparison of the ML-SNR achievable for IRSGCS–ACP and MChirp schemes. In the plot on the right, in white is evidenced the Noise-BW region in which the IRS Golay attains higher SNR than matched chirp.

Fig. 12. Main lobe width (MLW) versus the system band-pass width for various pulse compression schemes.

In summary, the numerical simulations show that for sufficiently-broadband systems, the binary sequences are preferable both in terms of resolution, SNR and side-lobe levels. Conversely, a chirp signal matching the bandwidth of the experimental system outperforms binary sequences in terms of the SNR

of the main lobe detection when the bandwidth of the measuring system is below the central frequency value, i.e. when B < fC, even at the cost of deteriorating the resolution. Indeed, the band-pass behaviour of chirp signals counterbalances the higher energy carried within binary sequences by fitting better to the overall transfer function of the experimental system. This effect becomes even more relevant if narrow-bandwidth systems are used. In this case, by using a standard binary sequence, only a small portion of the input energy can be transferred to the measured system. In contrast to this, the energy spectrum of a chirp signal can be tailored to the transducer bandwidth. Hence, the input energy in this case is always almost effectively delivered to the system, so that chirp signals are a good choice for transducers with a narrow bandwidth. In any case, since it is desirable to use binary signals both for hardware and computational reduced costs, sub-modulation and spectrum design techniques can be applied also to this case, in order to further improve the SNR. It is worth stressing that, among the binary sequences, the IRS approach achieves the best results, so it is could be useful to extend this approach in various applications of air-coupled ultrasonic NDE, and at the same time optimizing the performances by introducing more complex modulation of the sequences [51]. By using

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Fig. 13. Comparison of the NSL achievable for IRSGCS–ACP and MChirp schemes. In the plot on the right, in white is evidenced the Noise-BW region in which the IRS Golay attains higher NSL than MatchedChirp.

Fig. 14. Comparison of the FSL achievable for IRSGCS–ACP and MChirp schemes. In the plot on the right, in white is evidenced the Noise-BW region in which the IRS Golay attains higher FSL than MatchedChirp.

such techniques the binary sequences approach could out-perform chirps, even with narrow-band transducers. In general, MLS and GCS attained very close results, so that when CPC can be used, there could be advantages in terms of measurement and computational efficiency in using MLS instead of GCS. 5. Experimental investigation of modulation scheme performance The initial data was collected using a pair of air-coupled transducers from The Ultran Group (Boston, USA), centred at fc = 200 kHz and aligned on-axis in air. The transmitter was directly driven by an Arbitrary Function Generator PXI NI-PXI 5412 from National Instruments Inc., providing a voltage signal of ±6 V on a load of 50 X, while the receiving transducer was directly digitalized by an ADC NI-PXI 5105 having 12bit of resolution and 60 MS/s of maximum sampling rate. Different levels of SNR were

reproduced by placing sheets of various materials: wood, plastic, steel, etc. between the transducers. The transducer pair presents a quite irregular transfer function, characterised by two main lobes, as depicted in Fig. 15, where high-rate ‘‘Standard’’ GCS– APC and MLS–CPC were used to characterize the system transfer function. It can be seen that, after proper normalization, GCS and MLS gave very similar results, as expected from simulations. In the following experimental comparisons, all the binary sequences types (GCS–APC, MLS–CPC, IRSGCS–APC, IRSMLS–CPC) were used while for the Chirp waveforms a LChirp-APC and a NLChirp-CPC with fc = 200 kHz and BW = 120,000 = 60%fc were considered. Moreover, due to the peculiar transfer function of the system, also a Non-linear chirp designed to have the same PSD of the system itself was employed in CPC [20], labelled henceforth as NLChirpT-CPC. Fig. 16 compares the results in air with the aforementioned schemes; also Figs. 17–19 compares the results when a wood

Fig. 15. Experimental characterisation of the Tx–Rx system by using broadband GCS and MLS. On the left the impulse response h(t) is reported, on the right the PSD.

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Fig. 16. Comparison of performaces in air for different excitaion schemes.

Fig. 17. Comparison of performaces through wood for different excitaion schemes.

Fig. 18. Comparison of performaces through plasitc for different excitaion schemes.

sample, a plastic sample and a steel plate are placed between the probes respectively. The plots show the envelopes of the reconstructed impulse response, and the scale is normalised so that 1Volt of amplitude corresponds to 0 dB. In all these Figures, the upper subplot refers to the chirp waveforms (Black: LChirp-APC, Red: NLChirp-CPC and Blue: NLChirp-CPC) while the lower subplot refers to binary sequences (Black: GCS–APC; Red: MLS–CPC; Blue:IRSGCS–APC and Green:IRSMLS–CPC). Note that the Golay-

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Fig. 19. Comparison of performaces through steel for different excitaion schemes.

based schemes are expected to have 3db over the MLS’s and Chirp’s schemes due to the use of two sequences that actually double the energy delivered to the system. Finally, to better compare Chirp and Binary waveforms, in Fig. 20 a direct comparison of LChirpAPC with IRSGCS–APC and IRSMLS–CPC is reported for the wood and plastic samples respectively. In these cases, the envelopes were normalised to their relative maxima, so as to better appreciate the differences in resolution and side-lobe levels. In agreement with the numerical simulations, the bandwidth of the transducers makes the chirp the optimal choice in terms of side-lobe levels. On the other hand, IRS sequences maintain a better resolution with respect to the chirp case in the presence of strong attenuation, as demonstrated by the width of the main lobe. In particular, the rise-time of the signal in the IRS case is significantly shorter than for the chirp, making such strategies preferable if time-of-flight is to be measured with high precision. To further investigate the differences between the various excitations, data were also recorded using a pair of capacitive transducers, which have been described elsewhere [1,2]. They use a thin metallised Mylar membrane attached to a micromachined backplate, so as to operate in air with a high efficiency. The experimental impulse response of the devices used in the current experiments is shown in Fig. 21. It can be seen that the experimental amplitude response for a pair of transducers peaks at approximately 250 kHz, which was assumed as fC in this case. The response also led to the choice of a frequency sweep range of 100– 400 kHz for the chirp signals. The voltage drive signals were generated in LabView™ software, and sent to the transmitter via a National Instruments PXI system. Note that a DC bias was applied to the transmitter via a decoupling circuit and a charge amplifier was used to provide amplification and a DC bias to the receiver, before being recorded via the PXI system. Moreover, although the central frequency is near to that of the previous results, the impulse response for the capacitive transducers is significantly shorter, due to their broadband nature with respect to the ULTRAN probes described earlier. The initial experiments again recorded signals that were transmitted by each of a set of sequences across a simple air gap. This was done to determine the initial properties of the recorded signals in ideal conditions. The ultrasonic waveforms will not be the same as those provided by the driving voltage sequences, due to the amplitude and phase response of the transducers, the effect of diffraction by the finite apertures of the transducer themselves, and frequency-dependent ultrasonic attenuation in air. Moreover in this case two further steps of amplification were introduced, that affect the overall system transfer function. The experimental results can

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Fig. 20. Left: comparison of performaces through wood for LChirp (black) and IRSGCS (blue). Right: comparison of performaces through plastic for LChirp (black) and IRSMLS (green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 21. Experimental characterisation of the capacitive transducers Tx–Rx system by using pulse excitation. On the left the impulse response h(t) is reported, on the right the PSD.

be illustrated by looking at two particular examples – a LChirp-ACP and a MLS–CPC. The chirp had a bandwidth defined by the chosen start and finish frequencies (100–400 kHz), whereas the MLS sequence had a generation rate of a single bit is Rgen = 107/17 600 kHz, in compliance with the empirical rule is 2fc < Rgen < 4fc previously introduced, see Section 3.2. These signals were used to drive the ultrasonic air-coupled transmitter, and the signals detected by the receiver after propagation through air fc recorded. The received experimental spectra in each case are shown in Fig. 22. Note that, as stated above, the system exhibited a wider proportional bandwidth with respect to the experiments with ULTRAN transducers, and for the MLS case non-vanishing components extends for all the region f e [0, 2fC]. The pulse-compression output is therefore expected to be very concentrated in a small time interval, as shown in Fig. 23. It was found that the received MLS signal amplitude was approximately twice that of the LChirp-APC, and that the width of the main cross-correlation peak was narrower. The broadband feature of the transducer makes them very suitable for performing thickness measurement by resonance, which represents an important application of air-coupled ultrasonic systems in the NDE of solid samples.

To illustrate such application, samples of Plexiglas with different thickness (0.1 mm, 1 mm and 5 mm) were placed normal to the ultrasonic beam. This material has a longitudinal velocity of 2700 ms1, corresponding to a wavelength at the central frequency of 10.8 mm. Since resonance appears when the thickness D of the sample contains an integer number of half-wavelengths for some frequency, and since the effective bandwidth extend from 100 kHz to 400 kHz, resonance should appear when

D

[nk1 nk2  ; ; 2 2 n

ð13Þ

2700½m=s 2700½m=s where k1 ¼ 400;000½cycles=s ’ 6:75½mm=cycles; k2 ¼ 100;000½cycles=s ’ 27½mm=cycles. The only experimental object that should exhibit resonance is therefore the last one with D = 5 mm, and this was confirmed by the results reported in Fig. 24, where the impulse responses retrieved for all the Plexiglas samples and by using both LChirpAPC and MLS–CPC were depicted. The two responses are effectively similar for the two thinner samples. This is because their thickness is small enough that the frequency of the sample’s fundamental resonance (at 1.3 MHz for a 1 mm thickness) is above that of

Fig. 22. Experimental spectra measured with the capacitive transducers Tx–Rx system by using (left) LChirp-APC and (right) MLS–CPC.

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Fig. 23. Experimental impulse response envelopes reconstructed with the capacitive transducers Tx–Rx system by using (left) LChirp-APC and (right) MLS–CPC.

Fig. 24. Impulse response for three different thicknesses of Plexiglas: (a) 0.1 mm, (b) 1 mm and (c) 5 mm and fo LChirpAPC (left) and MLS–CPC (right).

Fig. 25. Frequency spectra arising from the impulse responses for the 5 mm thick Plexiglas plate: (left) LChirp-APC (right) MLS–CPC solid black with superimposed LChirpAPC dashedred. All vertical scales are normalised to the maximum values of the spectra.

the highest frequencies transmitted. At a thickness of 5 mm, however, where the solid plate through-thickness resonance would be expected at 267 kHz, the impulse response is that of a decaying resonance. This is exactly what would be expected from such a system.

It is interesting to look at the frequency spectra corresponding to the impulse responses for the 5 mm thick Plexiglas sample, and these are shown in Fig. 25 for the two signals. Both have produced a good spectral response, indicating that the frequency of resonance could be estimated from the peak value.

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Indeed in the right subplot of Fig. 25, the spectra of both MLS and the linear chirp are normalised and plotted together: it can be seen that the narrow peak corresponding to the resonance frequency coincides in the two cases and give thickness values of 5.2 mm, assuming a longitudinal velocity of 2700 ms1. However, the larger received amplitude of the impulse response for the MLS sequences (Fig. 24) can make these codes preferable for thickness measurements were the sidelobes level is less important. 6. Conclusions This paper has revised through numerical and experimental data most of the types of signal used for air-coupled ultrasound and their relative measurement procedures. The results have demonstrated that it is extremely important that the type of excitation signal used is chosen so as to match the bandwidth available in the experimental system. With respect to the enhancement of the SNR, binary and Chirp signals achieve very similar results for wide bandwidth measurements even if binary codes assure high SNR values. For a narrower available bandwidth, a chirp signal would be preferred; alternatively, some spectral shaping must be introduced into binary codes, as shown by the Inverted Repeated Sequences approach. For time-of-flight and thickness measurement, the binary codes have been seen to be the best choice. They give the highest received amplitude, and the shortest impulse response when used in a through-transmission measurement. For defect detection purpose, where very low side-bands level are needed, linear windowed chirps appear to be the best solution for narrowband systems, at the cost of reducing the resultant spatial resolution. Acknowledgements DAH acknowledges financial support via EPSRC United Kingdom. PB, SL and MR acknowledge PRIN 2009 project ‘‘Diagnostica non ditruttiva ad ultrasuoni tramite sequenze pseudo-ortogonali per imaging e classificazione automatica di prodotti industriali’’ and ‘‘Fondazione CARIT’’ Italia for financial support. References [1] D.A. Hutchins, A. Neild, Air-borne ultrasound transducers, in: K. Nakamura (Ed.), Ultrasonic Transducers, Woodhead publishers, 2012. [2] F. Lionetto, A. Tarzia, A. Maffezzoli, Air-coupled ultrasound: a novel technique for monitoring the curing of thermosetting matrices, IEEE Trans. Ultrason. Ferroelectrics Freq. Control 54 (2007) 1437. [3] M. Castaings, P. Cawley, R. Farlow, G. Hayward, Single sided inspection of composite materials using air coupled ultrasound, J. Nondestruct. Eval. 17 (1998) 37. [4] R. Kazys, A. Demcenko, E. Zukauskas, L. Mazeika, Air-coupled ultrasonic investigation of multi-layered composite materials, Ultrasonics 44 (1) (2006) 819. [5] M. Masmoudi, M. Castaings, Three-dimensional hybrid model for predicting air-coupled generation of guided waves in composite material plates, Ultrasonics 52 (2012) 81. [6] P. Pallav, D.A. Hutchins, T.H. Gan, Air-coupled ultrasonic evaluation of food materials, Ultrasonics 49 (2009) 244. [7] S. Delrue, K. Van Den Abeele, E. Blomme, J. Deveugele, P. Lust, O.B. Matar, Twodimensional simulation of the single-sided air-coupled ultrasonic pitch–catch technique for non-destructive testing, Ultrasonics 50 (2010) 188. [8] B. Hosten, C. Biateau, Finite element simulation of the generation and detection by air-coupled transducers of guided waves in viscoelastic and anisotropic materials, Acoust. Soc. Am. 123 (2008) 1963. [9] M. Castaings, B. Hosten, Lamb and SH waves generated and detected by aircoupled ultrasonic transducers in composite material plates, NDT&E Int. 34 (2001) 249. [10] B. Hosten, D.A. Hutchins, D.W. Schindel, Air-Coupled Ultrasonic Bulk Waves to Measure Elastic Constants in Composite Materials, Rev. Prog. Quant. NDE (1996) 1075. [11] B. Hosten, D.A. Hutchins, D.W. Schindel, Measurement of elastic constants in composite materials using air-coupled ultrasonic bulk waves, J. Acoust. Soc. Am. 99 (1996) 2116.

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Please cite this article in press as: D. Hutchins et al., Coded waveforms for optimised air-coupled ultrasonic nondestructive evaluation, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.03.007