A new signal processing time-frequency transform for guided

waves characterization

A. Marzani*, E. Viola

DISTART, University of Bologna Viale del Risorgimento 2, 40136 Bologna, Italy

L. De Marchi, N. Speciale

DEIS, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

ABSTRACT

Guided waves (GWs) have characteristic dispersive time-frequency representations (TFRs). Unfortunately, any TFRis subjected to the time-frequency uncertainty principle. This limits the capability of the TFR to distinguish multiple,closely spaced guided modes over a wide frequency range. To this aim we implemented a new time-frequency signalprocessing tool, that we call Warped Frequency Transform (WFT), that in force of a more flexible tiling of thetime-frequency domain presents enhanced guided modes extraction capabilities. Such tiling can be chosen to matchthe spectro-temporal structure of the different propagating modes by selecting an appropriate warping map whichgenerates non-linearly frequency modulated atoms. The proposed transformation is fast, invertible, and covariant togroup velocity-delay shifts. In particular, in this work we propose design and calculation strategies for non-smoothmaps tailored to Lamb Waves propagating in a single aluminium plate and to Flexural waves propagating in ahollow steel pipe. Time-transient guided wave propagation events are obtained artificially by means of dedicatedFinite Element Simulations. The results show that the WFT limits interference patterns which appears with othersTFRs and produces a sparse representation of the guided wave pattern that can be suitable for identification andcharacterization purposes.

Keywords: Time-Frequency Representations, Warped Frequency Transforms, Group-delay Covariance, Lamb Waves

1. INTRODUCTION

Stress Guided Waves (GWs) are mechanical waves that can propagate along a structural component, such a plate, arod, a rail, that is generally termed as waveguide. When a certain waveguide is excited one or more GWs can propagateat a given frequency. Each wave propagates with defined wavelength, phase velocity, energy velocity, attenuation andshape over the waveguide cross-section. These waves are characterized, in general, by a dispersive behaviour forwhich the above mentioned wave features modify varying the frequency of propagation. Multimodal and dispersionproperties of GWs generates non-stationary time transient signals when the waveguide is excited. The generated pulse,in fact, change its shape depending on the traveled distance because within the pulse exist several different frequencycomponents each characterized by a different propagation speed. It follows that the extraction of the energy contentof different propagating waves from a recorded time-waveform is not an easy task. This, in general, is attempted bymeans of time-frequency representations (TFRs) such us the Short Time Fourier Transform (STFT), as well as theWavelet Transform (WT) (see e.g. Refs.1,2).

Unfortunately, any TFR is subjected to the time-frequency uncertainty principle that limits the capability ofdistinguishing multiple, closely spaced guided modes. To this aim we implemented a new Warped Frequency Transform(WFT) which, in force of a more flexible tiling of the time-frequency domain, presents enhanced GWs extractioncapabilities. Such tiling is designed to match the spectro-temporal structure of a propagating guided wave. This tilingis obtained by selecting an appropriate warping function that reshape the signal frequency axis in agreement with thegroup velocity curve of the selected guided mode. This can be done once the group velocity dispersion curves of theconsidered waveguide can be predicted at priori for the frequency range of interest.

In this manuscript, an application on Lamb waves propagating in an isotropic aluminum plate is proposed toshow the potential of the proposed TFR. Lamb waves dispersion curves are predicted by using a semi-analyticalformulation while time-transient events are produced numerically via a Finite Element simulation. Next the extractednon stationary signals are post-processed by means of the WFT tuned on both the lowest fundamental S0 and A0

modes. Comparison of the WFT results with those produced by the STFT and WT reveal the enhanced capabilitiesof the WFT to extract the energy content of a particular guided mode, making the proposed transform suitable forguided waves applications.

*Corresponding author: Alessandro Marzanie-mail: alessandro.marzani@unibo.it, Telephone: +39 051 2093504

2. WARPED FREQUENCY TRANSFORM

2.1 The Discrete Operator

For a given discrete-time signal, the WFT introduces a deformation of the frequency axis f with a proper warpingfunction w(f). In order to guarantee invertibility, w(f) must be chosen so that it maps the f axis on itself, which is:

w(f) > 0 = a.e. ⇒ ∃w−1, w−1(w(f)) = f (1)

where w(f) represents the first derivative of the map w(f). The warping function w(f) is defined in the interval[−1/2, 1/2] and it can be extended as w(f + k) = k + w(f), with k ∈ Z. In addition, in order to guarantee that areal signal is transformed into a real signal, w(f) must be an odd function. The frequency warping operator W canbe defined as

Ww = F−1Fw (2)

where F−1 is the inverse Discrete-Time Fourier Transform and Fw is a modified Discrete-Time Fourier Transformthat, for a given discrete signal s(n), is defined as:

[Fws] (f) =√

w(f)∑

n∈Z

s(n)e−j2πnw(f) (3)

In Eq. (3) the factor√

w(f) has been introduced to render the operator unitary, i.e. to preserve orthogonality.However, the numerical implementation of Eq. (3) involves unfeasible continuous operations along the frequency axis.So we introduced a sampling operation on M discrete frequencies fk = k/M, k ∈ {−M/2, . . . ,M/2} to get a modifiedDiscrete Fourier Transform:

[FwDs] (fk) =√

w(fk)∑

n∈Z

s(n)e−j2πnw(fk) (4)

The final discrete warped frequency operator is defined as:

WwD = F−1M FwD (5)

where F−1M is the inverse of the Discrete Fourier Transform of size M×M . Through this decomposition of the frequency

warped operator we can achieve a fast computation. In fact WD can be efficiently factorized with the NonuniformFast Fourier Transform algorithm3,4 .

2.2 Group Delay Shifts Covariance

The group delay is defined as the derivative of signal’s phase response and it is a measure of time delay introduced ineach sinusoidal component. TFRs are classified as Group Delay Shifts Covariant (GDSC) when the TFR of a signals which undergoes a change τ(f) in the group delay (s → sch) corresponds to the TFR of the original signal shiftedby τ(f), i.e. when:

TFR[sch](t, f) = TFR[s](t − τ(f), f) (6)

It has been shown5 that a GDSC representation can be obtained by warping a signal and then analyzing it with aTFR of the Cohen class, such as the Short Time Frequency Transform (STFT), if:

Kdw−1(f)

df= τ(f) (7)

where K is an arbitrary constant. For a stress guided wave signal s detected at a traveled distance D from the source,whose response in the frequency domain is:

S(f) = e−j2π

∫

Dcg(f)

df· P (f) (8)

being P (f) is the Fourier Transform of the incipient pulse and cg(f) the mode group velocity dispersive relation, thegroup delay is simply τ(f) = D/cg(f). Therefore, for an assumed value of K, we can design a GDSC TFR appropriatefor a given dispersive guided wave with a group velocity cg(f) by setting the opportune warping function. Startingfrom Eq. (7) the inverse of the warping function can be calculated according to the following relation:

w−1(f) =1

K

∫ f

0

1

cg(f)df (9)

As w−1(f) is the functional inverse of the warping function we have to apply, we can compute the discrete warpingoperator by this formula:

WwD = W †

w−1D, (10)

i.e. by computing the operator of the inverse map (Ww−1D) and then exploiting the property of warping which allowsto invert the operator with a simple adjoint transposition denoted by †.

2.3 Warpogram and reassigned warpogram

Once the warping operator tuned on the considered mode is defined, the GDSC time frequency representation, thatwe will call from now on warpogram, is obtained with the following interpolation:

TFRGDSC [s](t, f) = STFT [WwDs](tKcg(f), w−1(f)) (11)

Generally speaking, first the signal is warped in frequency (WwDs), then a Short–Time Fourier Transform isperformed, and finally the warped axes are reparametrized to provide the correct time-frequency alignment. It isworth to notice that the compensation of the dispersive effects acts independently from the distance D, which, infacts, does not appear in (9), i.e. the formula used for map design. In order to further increase the readability of thenew TFR, the reassignment method6 could be applied to the STFT of the warped signal. This method moves eachvalues of the spectrogram computed at a given point (t, f) to another point (t, f) which is the center of gravity of thesignal energy distribution around (t, f), thus enhancing the resolution of the representation.

3. NUMERICAL APPLICATION

3.1 Simulation of propagative Lamb waves

A transient Finite Element analysis with the commercial software STRAUS7, is carried out to simulate Lamb wavespropagating in an isotropic plate. The effectiveness of conventional finite element packages for modeling elastic wavespropagating in structural components has been shown in the past (see e.g.7 and8). The three-dimensional wavepropagation problem is reduced to a bi-dimensional one by assuming a plane strain condition. An isotropic aluminum

-1.E-04

0.E+00

1.E-04

0 20 40 60 80 100 120 140 160 180 200

time [ s]

u(t)

[mm

]

(b)

P(t)

2.54 mm

u(t)

1220 mm

150 mm 45°

(a)

Acr6FA4.tmp 1Acr6FA4.tmp 1 02/12/2008 1.24.1202/12/2008 1.24.12

Figure 1. (a) Sketch of the plate used in the finite element simulation. (b) time-waveform u(t) extracted for further TFRspost-processing.

plate (Young modulus E = 69 GPa, Poisson’s coeff. ν = 0.33, density ρ = 2700 kg/m3) of length 1220 mm andthickness 2.54 mm has been considered. Lamb waves were excited by imposing an inclined concentrated force P (t),acting on the left hand edge of the plate, as shown in Fig. 1a. The force has been shaped in time as a triangularwindow with total duration equal to 0.7 µs in order to excite consistent Lamb Waves up to 1.5 MHz. To satisfythe requirements for simulation accuracy7 , the integration time step was set equal to 0.02 µs, and the plate wasdiscretized by using four nodes plane elements of dimension 0.25×0.254 mm. In Fig. 1b the out-of-plane displacementu(t), occurring at a point on the top side of the plate located at 150 mm away from the left edge, i.e. away fromthe excitation point, is represented. This time-waveform is next post-processed to test the suitability of the WFT incomparison with the STFT and the WT.

3.2 Time-frequency post processing: STFT and WT

The STFT (spectrogram) and the WT (scalogram) post-processing of the time-waveform u(t) recorded at 150 mmfrom the left edge of the plate are shown in Fig. 2(a) and (b), respectively. As it can be seen from these figures, thefinite time-frequency resolution of both TFRs produces interference patterns and limits the capability of distinguishingclosely spaced Lamb modes. This is particular evident for both TFRs at about 750 kHz where the fundamental A0

and S0 modes cross each other originating a unique energy interference pattern. In addition, it can be seen that thescalogram presents lower capability to distinguish the energy content of different waves at very low frequency.

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Figure 2. Spectrogram (left) and scalogram (right) of the simulated signal u(t).

3.3 Time-frequency post processing: WFT and reassigned WFT

In order to apply to proposed procedure, first the group velocity dispersion curves for the considered waveguide mustbe predicted.

0 0.5 1 1.5 2−2

−1

0

1

2

3

4

5

6

7

Frequency [MHz]

c g [Km

/s]

S0

A0

A1 S

1

S2 A

2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Norm. frequency

w(f)

S0

A0

Figure 3. (left) Lamb waves group velocity dispersion curves for an aluminum plate (thickness h = 2.54 mm, Young modulusE = 69 GPa, Poisson’s coeff. ν = 0.33, density ρ = 2700 kg/m3). (right) warping functions w(f) obtained by means of Eq. (9)considering the group velocity curve for the S0 mode (continuous line) and for the A0 mode (dashed line)

Here, the Semi-analytical Finite Element (SAFE) formulation developed by Bartoli et al.9 , that does not suffer ofthe mode crossing problem in the computation of the group velocity curves, is used. The number of SAFE elementshas been set in agreement with the accuracy crterion described in Ref.10 . In Fig. 3 are represented the group velocitydispersion curves for the Lamb waves existing in the 0 − 2 MHz frequency range for the plate considered in section3.1. It can be seen that up to six waves can exist, namely the fundamental A0 and S0 waves, and four higher orderwaves. Below 600 kHz only the two fundamental waves can propagate. These cg(f) curves can therefore be used todesign the warping function w(f) according to Eq. (9) as shown in Fig. 3(right).In Fig. 4 is represented the tiling of the time-frequency plane for the STFT (left), with constant time-frequency atoms,and that for the WFT produced by using the warping function w(f) tuned on the S0 Lamb Wave. The constant Kwas selected so that w(0.5) = 0.5. Overlapped as a thicker lines in Fig. 4 are the group delay curve for the S0 mode ata distance D = 100 mm (continuous line) and at a distance D = 200 mm (dashed line). For the WFT time-frequencyplane, it is worth noting how the atoms shapes are suited to the group delay of the S0 mode, and also how the atomsinclination varies as time increases, leading to the WFT the desired group covariant property. Both things consideredthe WFT support is suitabile for the analysis of dispersive systems. In particular, the tiling for the WFT plane resultsfrom warping the regular tiling of the STFT, by using w(f). In fact, Eq. (11) can be rewritten as:

TFRGDSC [s]

(

t

K

(

dw

df

)−1

, w(f)

)

= STFT [WwDs](t, f)). (12)

The application of Eq.(11) gives the warpogram tuned on the S0 mode as represented in Fig. 5(left). It can be

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 106

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 106

Time [s]

Freq

uenc

y [H

z]

Figure 4. Tiling of the time-frequency plane produced by the STFT (left) and by the GDSC TFR tuned on the S0 group velocitycurve (right). Over imposed as a thicker continuous (dashed) line is the group delay of the S0 mode for the plate described inFig. 3 for a distance D = 100 mm (D = 200 mm).

seen from this figure that the S0 mode is well captured all over the excited frequency range, while the A0 mode, asdesired, is barely visible. The benefit of the proposed WFT can be clearly seen by comparing the sparsity of the S0

mode in the warpogram of Fig. 5(left) with those of the STFT and WT represented in Fig. 2. The further resolutionimprovement produced by reassigning the warpogram is visible in Fig. 5(right). Similarly, shaping the map on thegroup velocity dispersion curve branch of the A0 mode and applying the proposed GDSC TFR, yields the warpogram

of Fig. 6b. It can be seen that also the A0 mode is well defined in this plot, even if in this case the presence of theS0 mode is still consistent. This is true in particular in the frequency range where both the A0 and S0 modes presentlow dispersive behavior, for which the maps designed via Eq.(9) present similar aspect.

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Time[s]

Freq

uenc

y[H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Figure 5. (left) Warpogram of the simulated signal u(t) calibrated on the S0 mode. (right) reassinged warpogram.

3.4 Dispersive data extraction

For characterization purposes, extraction of the dispersion curves from the TFR of the processed time-waveform iscrucial. The simplest way to do it consists in extracting the spectrum maximum at several frequencies. However, thissimple way to proceed produces consistent results in a broad frequency range only when a sparse representation ofthe existing waves within the signal is achieved. It can be seen from Fig. 7, where the theoretical SAFE dispersioncurves are overlapped with the energy peaks extracted from the reassigned spectrogram of the processed signal u(t),and with those obtained from the S0 warpogram, the benefit of the proposed WFT.

4. CONCLUSIONS

In this work we presented a new time-frequency representation (TFR) that can efficiently extract the dispersive energycontent of multiple guided waves within a non-stationary recorded signal. The new tool based its strength on thetime-frequency map used to process the signal. This map, in fact, can be tuned to match the spectro-temporalstructure of a dispersive guided wave by non-linearly frequency modulated atoms. In particular, in our approach themap is shaped on the group velocity curve of a particular mode. This leads to the transformation a group delay shifts

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Time [s]

Freq

uenc

y [H

z]

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0

2.5

5

7.5

10

12.5

15x 105

Figure 6. (left) Warpogram of the simulated signal u(t) calibrated on the A0 mode. (right) reassigned warpogram.

0 0.25 0.5 0.75 1.0 1.11.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Frequency [MHz]

c g [km

/s]

S0 predicted via SAFE

A0 predicted via SAFE

S0 extract from reassigned spectrogram

S0 extract from warpogram

Acr6FB0.tmp 1Acr6FB0.tmp 1 02/12/2008 1.32.4002/12/2008 1.32.40Figure 7. S0 mode energy peaks extracted from the reassigned spectrogram and from the warpogram tuned on the S0 mode.

covariant property that makes the Warped Frequency Transform (WFT) suitable to process high dispersive signals.In essence, the proposed WFT allows to a sparser time-frequency representation of the considered guided wave. Anapplication to propagating Lamb waves in a single layer isotropic aluminum plate was presented to show the potentialof the proposed procedure. Time-transient events were obtained by a dedicated Finite Element simulations. Theresults showed that: i) the WFT produces a sparser representation of a particular guided wave pattern if compared toScalogram and Spectrogram; ii) the energy peaks extraction is a simple task and can be performed to obtain reliablemode representation; iii) the quality of the energy peaks extraction is independent on the distance waves source -waves receiver, thanks to the group velocity covariant property. These properties make the WFT a suitable tool foridentification and characterization procedures based on non-stationary signals.

ACKNOWLEDGMENTS

Funding for this project was provided by the Italian Ministry for University and Scientific and Technological ResearchMIUR (40%). This topic is one of the research thrusts of the Centre of Study and Research for the Identification ofMaterials and Structures (CIMEST) of the University of Bologna (Italy).

REFERENCES

[1] M. Niethammer, L. Jacobs, J. Qu, and J. Jarrzynski, “Time-frequency representations of lamb waves,” J. Acoust.

Soc. Am. 109(5), pp. 1841–1847, 2001.

[2] H. Kuttig, M. Niethammer, S. Hurlebaus, and L. J. Jacobs, “Model-based analysis of dispersion curves,” J.

Acoust. Soc. Am. 119, pp. 2122–2130, 2006.

[3] A. Ware, “Fast approximate fourier transforms for irregularly spaced data,” SIAM Rev. 40(4), pp. 838–856, 1998.

[4] S. Caporale, L. De Marchi, and N. Speciale, “Analytical computation of fast frequency warping,” in IEEE ICASSP

International Conference on Acoustics, Speech and Signal Processing, pp. 3793–3796, 2008.

[5] A. Papandreou-Suppappola, R. Murray, B. Iem, and G. F. Boudreaux-Bartels, “Group delay shift covariantquadratic time-frequency representations,” IEEE Trans. Signal Process. 49(11), pp. 2549–2564, 2001.

[6] F. Auger and P. Flandrin, “Improving the readability of time-frequency and time-scalerepresentations by thereassignment method,” IEEE Trans. Signal Process. 43(5), pp. 1068–1089, 1995.

[7] F. Moser, L. Jacobs, and J. Qu, “Modeling elastic wave propagation in waveguides with the finite elementmethod,” NDT&E International 32, pp. 225–234, 1999.

[8] I. Bartoli, F. Lanza di Scalea, M. Fateh, and E. Viola, “Modeling guided wave propagation with application tothe long-range defect detection in railroad tracks,” NDT&E International 38, pp. 325–334, 2005.

[9] I. Bartoli, A. Marzani, F. Lanza di Scalea, and E. Viola, “Modeling wave propagation in damped waveguides ofarbitrary cross-section,” Journal of Sound and Vibration 295, pp. 685–707, 2006.

[10] A. Marzani, “Time-transient response for ultrasonic guided waves propagating in damped cylinders,” International

Journal of Solids and Structures 45(25-26), pp. 6347 – 6368, 2008.