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Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition

Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 480473, 13 pages doi:10.1155/2012/480473 Research Article Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition 1 2 2 2 Yufeng Lu, Alireza Kasaeifard, Erdal Oruklu, and Jafar Saniie Department of Electrical and Computer Engineering, Bradley University, Peoria, IL 61625, USA Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA Correspondence should be addressed to Erdal Oruklu, erdal@ece.iit.edu Received 7 April 2012; Accepted 31 May 2012 Academic Editor: Mario Kupnik Copyright © 2012 Yufeng Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A fractional fourier transform (FrFT) based chirplet signal decomposition (FrFT-CSD) algorithm is proposed to analyze ultrasonic signals for NDE applications. Particularly, this method is utilized to isolate dominant chirplet echoes for successive steps in signal decomposition and parameter estimation. FrFT rotates the signal with an optimal transform order. The search of optimal transform order is conducted by determining the highest kurtosis value of the signal in the transformed domain. A simulation study reveals the relationship among the kurtosis, the transform order of FrFT, and the chirp rate parameter in the simulated ultrasonic echoes. Benchmark and ultrasonic experimental data are used to evaluate the FrFT-CSD algorithm. Signal processing results show that FrFT-CSD not only reconstructs signal successfully, but also characterizes echoes and estimates echo parameters accurately. This study has a broad range of applications of importance in signal detection, estimation, and pattern recognition. 1. Introduction broadband; symmetric or skewed; nondispersive or disper- sive. In ultrasonic imaging applications, the ultrasonic signal Recently, there has been a growing attention to fractional always contains many interfering echoes due to the complex Fourier transform (FrFT), a generalized Fourier transform physical properties of the propagation path. The pattern of with an additional parameter (i.e., transform order). It the signal is greatly dependent on irregular boundaries, and was first introduced in 1980, and subsequently closed-form the size and random orientation of material microstruc- FrFT was studied [8–11] for time-frequency analysis. FrFT tures. For material characterization and flaw detection is a power signal analysis tool. Consequently, it has been applications, it becomes a challenging problem to unravel applied to different applications such as high-resolution SAR the desired information using direct measurement and imaging, sonar signal processing, blind source separation, conventional signal processing techniques. Consequently, and beamforming in medical imaging [12–15]. Short term signal processing methods capable of analyzing the nonsta- FrFT, component-optimized FrFT, and locally optimized tionary behavior of ultrasonic signals are highly desirable FrFT have also been proposed for signal decomposition [16– for signal analysis and characterization of propagation 18]. path. In practice, signal decomposition problem is essentially an optimization problem under different design criteria. Various methods such as short-time Fourier transform, The optimization can be achieved either locally or glob- Wigner-Ville distribution, discrete wavelet transform, dis- crete cosine transform, and chirplet transform have been ally, depending on the complexity of the signal, accuracy of estimation, and affordability of computational load. utilized to examine signals in joint time-frequency domain Consequently, the results of signal decomposition are not and to reveal how frequency changes with time in those signals [1–8]. Nevertheless, it is still challenging to adaptively unique due to different optimization strategies and signal models. For ultrasonic signal analysis, local responses from analyze a broad range of ultrasonic signal: narrowband or 2 Advances in Acoustics and Vibration Table 1: Parameter estimation results of two slightly overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.5 7.0 1 20 35 π/6 Estimated parameter 2.50 7.00 1.00 20.00 35.00 0.52 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 25.00 19.98 0 Table 2: Parameter estimation results of two moderately overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.7 7.0 1 20 35 π/6 Estimated parameter 2.70 7.02 1.00 20.04 33.55 0.67 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 24.87 20.38 0.01 0.5 0.5 −0.5 −0.5 −1 −1 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 2.5 3 3.5 4 4.5 5 Time (μs) Time (μs) (a) (a) −0.006 Peak at the order =−0.013 0.0013 1.5 0.04 −0.0013 −0.04 0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 012 3456789 10 Order of fractional Fourier transform (FrFT) (b) (b) Figure 1: (a) A simulated LFM signal. (b) The optimal transform Figure 2: (a) a simulated ultrasonic single echo with Θ = order tracked by Maximum amplitude of FrFT for different FrFT 2 2 [3.6 us 5 MHz 1 25 MHz 25 MHz 0]. (b) Fractional orders. Fourier transform of the signal in (a) for different transform orders. microstructure scattering and structural discontinuities are has been explored. In particular, FrFT is introduced as a more of importance for detection and material characteri- transformation tool for ultrasonic signal decomposition. zation. Chirplet covers a board range of signals representing FrFT is employed to estimate an optimal transform order, frequency-dependent scattering, attenuation and dispersion which corresponds to highest kurtosis value in the transform effects in ultrasonic testing applications. This study shows domain. The searching process of optimal transform order that FrFT has a unique property for processing chirp- is based on a segmented signal for a local optimization. type echoes. Therefore, in this paper, the application of Then, the FrFT with the optimal transform order is applied fractional Fourier transform for ultrasonic applications to the entire signal in order to isolate the dominant echo Max. amplitude Amplitude of FrFT Amplitude Amplitude Advances in Acoustics and Vibration 3 Signal s(t) containing 1 multiple echoes 0 −1 Initialization j = 1 1.5 2 2.5 3 3.5 4 Time (μs) Signal windowing: (a) s win(t) = s(t) × w (t) Fractional Fourier transform: 0 −1 FrFT (x) s win(t) 1.5 2 2.5 3 3.5 4 Search an optimal transform order, α , Time (μs) opt j = j +1 which generates a maximum Kurtosis value (b) for FrFT (x) s win(t) Fractional fourier transform: opt −1 FrFT (x) s(t) 1.5 2 2.5 3 3.5 4 Signal windowing Time (μs) opt FrFT win(x) = FrFt (x) × win (x) s(t) (c) Inverse fractional fourier transform Figure 4: (a) Simulated ultrasonic echoes (20% overlapped). (b) −α opt f (t) = FrFT (t) Θ FrFT win (x) The first signal component. (c) The second signal component Use residual (simulated signal in blue, estimated signal in red). signal for next Estimate parameters of echo estimation decomposed echo, f (t) Obtain residual signal by subtracting decomposed echo from the signal −1 Calculate energy of residual signal (E ) 1.5 2 2.5 3 3.5 4 Time (μs) No E <E r min (a) Yes Signal decomposition and −1 parameter estimation complete 1.5 2 2.5 3 3.5 4 Time (μs) Figure 3: Flowchart of FrFT-CSD algorithm. (b) −1 for parameter estimation. This echo isolation is applied 1.5 2 2.5 3 3.5 4 iteratively to ultrasonic signal until a predefined stop cri- Time (μs) terion such as signal reconstruction error or the number of iterations is satisfied. Furthermore, each decomposed (c) component is modeled using six-parameter chirplet echoes Figure 5: (a) Simulated ultrasonic echoes (50% overlapped). (b) for a quantitative analysis of ultrasonic signals. The first signal component. (c) The second signal component A bat signal is utilized as a benchmark to demonstrate (simulated signal in blue, estimated signal in red). the effectiveness of fractional Fourier transform chirplet signal decomposition (FrFT-CSD). To further evaluate the performance of FrFT-CSD, ultrasonic experimental data from different types of flaws such as flat bottom hole, side- drilled hole and disk-type cracks are evaluated using FrFT- algorithm. Section 5 performs a simulation study of FrFT- CSD. CSD and parameter estimation for complex ultrasonic The outline of the paper is as follows. Section 2 reviews signals. Sections 5 and 6 show the results of a benchmark data the properties of FrFT and the process of FrFT-based signal (i.e., bat signal); the echo estimation results of benchmark decomposition. Section 3 addresses how kurtosis, transfor- data from side-drilled hole, and disk-shape cracks; the results mation order and chirp rate are related using simulated of experimental data with high microstructure scattering data. Section 4 presents the steps involved in FrFT-CSD echoes. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude 4 Advances in Acoustics and Vibration 2. FrFT of Ultrasonic Chirp Echo For ultrasonic applications, ultrasonic chirp echo is a type of signal often encountered in ultrasonic backscattered FrFT of a signal, f (t), is given by signals accounting for narrowband, broadband, and disper- sive echoes. It can be modeled as [8]: −i ((π/4) sgn(πα/2)− (πα/4)) e 2 α (1/2)ix cot (πα/2) F (x) = e 1/2 (2π|sin(πα/2)|) (1) ∞ f (t) = β exp −α (t − τ) + i2πf (t − τ) Θ 1 c (−i (xt/ sin(πα/2))+(1/2)it cot (πα/2)) × e f (t) dt, (2) −∞ +iα (t − τ) + iθ , where α denotes transform order of FrFT and x denotes the variable in transform domain. It has been shown that if the transform order, α,changes where Θ = τf βα α θ denotes the parameter vector, c 1 2 from 0 to 4, (i.e., the rotation angle, φ,changes from 0to2π), τ is the time-of-arrival, f is the center frequency, β is the F (x) rotates the signal, f (t), and projects it onto the line amplitude, α is the bandwidth factor, α is the chirp-rate, 1 2 of angle, φ, in time-frequency domain [19]. This property and θ is the phase. contributes to FrFT-based decomposition algorithm when Hence, for the ultrasonic Gaussian chirp echo, f (t), the applied to ultrasonic signals. magnitude of F (x)given by (1) can be expressed as 1 2 α (−i(xt/ sin(πα/2)) +(1/2)it cot (πα/2)) |F (x)| = e f (t) dt 1/2 −∞ (2π|sin(πα/2)|) (3) (B −4AC)/4A = e , 1/2 2 α + (α + (1/2)cot(πα/2)) |sin(πα/2)| where the integration part can be written as (−i (xt/ sin(πα/2))+(1/2)it cot (πα/2)) e f (t) dt −∞ 2 2 2 −[t (α −α i−(1/2)icot (πα/2))+t(2α τi−2α τ−2πf i+xicsc(πα/2))+(α τ −α iτ −θ+2πf τi)] 1 2 2 1 0 1 2 0 = e dt (4) −∞ (B −4AC)/4A = e , with A = α − α i − (1/2)i cot (πα/2), B = 2α τi − 2α τ − 3. Kurtosis and FrFT Order 1 2 2 1 2 2 2πf i + xicsc(πα/2), and C = α τ − α iτ − θ +2πf τi. 0 1 2 0 Kurtosis is commonly used in statistics to evaluate the degree From (3), it can be seen that, for a linear frequency of peakedness for a distribution [20, 21]. It is defined as the modulation (LFM) signal (i.e., α = 0), if the transformation ratio of 4th-order central moment and square of 2nd-order order, α, satisfies the following equation: central moment: 1 πα πα μ (F (x)) α + cot sin = 0, 4 K(α) =   , 2 2 2 (6) μ (F (x)) (5) 2 2 1 −1 α =− tan , where μ (•) denotes 4th-order central moment and μ (•) 4 2 π 2α denotes 2nd-order central moment. A signal with high then the |F (x)| compacts to a delta function. This means kurtosis means that it has a distinct peak around the mean. that fractional Fourier transform can be used to compress the In the literatures of FrFT [18, 19, 22], kurtosis is typically duration and compact the energy of ultrasonic chirp echo used as a metric to search the optimal transform order of with an optimal transform order. Optimal transform order FrFT. Different transform order directs the degree of signal can be determined using kurtosis. The energy compaction rotation caused by FrFT, and this rotation affects the extent is a desirable property for ultrasonic signal decomposition, of energy compaction of the transformed signal. which allows using a window in FrFT domain for isolation of Figure 1(a) shows a chirp signal with the param- an echo of interest. eters, Θ = [3.6 us 5MHz 1 0 25MHz 0]. For this Advances in Acoustics and Vibration 5 1 1 0 0 −1 −1 1.5 2 2.5 3 3.5 4 1.5 2 2.5 3 3.5 4 Time (μs) Time (μs) (a) (b) −1 1.5 2 2.5 3 3.5 4 Time (μs) (c) Figure 6: (a) Simulated ultrasonic echoes (70% overlapped). (b) The first estimated echo component. (c) The second estimated echo component (simulated signal in blue, estimated signal in red). −1 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) (a) (b) Figure 7: Left column (top to bottom): decomposed bat signal components in time domain. Right column (top to down): Wigner-Ville distribution of the corresponding signals in left column. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Frequency (KHz) Frequency (KHz) Frequency (KHz) Frequency (KHz) 6 Advances in Acoustics and Vibration Table 3: Parameter estimation results of two heavily overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.7 6 1 20 55 π/6 Estimated parameter 2.70 6.11 0.97 18.87 53.79 0.72 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 25.14 20.38 0.01 0.8 0.5 0.6 0.4 −0.5 0.2 −1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 −0.2 Time (ms) (a) −0.4 −0.6 −0.8 −1 68 68.2 68.4 68.6 68.8 69 69.2 69.4 69.6 69.8 70 Time (μs) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Time (ms) Figure 10: Ultrasonic data from the front surface superimposed with the estimated chirplet (depicted in dashed red color line). (b) Figure 8: (a) Reconstructed bat signal. (b) Summed Wigner Ville distribution of the decomposed signals in (a). Transducer 25.4 mm Transducer Water 50.8 mm Water θ Disc-shaped cracks 13 mm ab Diffusion bond Crack D Crack C Titanium alloy Side-drilled hole Figure 11: Experiment setup for disc-shaped cracks in a diffusion- bonded titanium alloy. Aluminum Figure 9: Experiment setup for SDH blocks. of FrFT using different transform orders according to (3). The transform order corresponding to the maximum FrFT example, the bandwidth factor equals to zero (see (2)), among all transform orders matches the theoretical result and according to (5), the optimal transform order can be givenin(7). calculated as For ultrasonic applications, the chirp echo is band- limited. For example, Figure 2(a) shows a band-limited 2 1 −1 α =− tan =−0.013. (7) single chirp echo with the parameters Θ = [3.6us 5MHz π 2α 1 25MHz 25MHz 0]. Chirplet is a model widely used As shown in Figure 1(b), this optimal order can also be in ultrasonic NDE applications. Figure 2 illustrates the FrFT determined by direct search for the maximum amplitude of a chirplet using different transform orders. In particular, Amplitude Frequency (KHz) Amplitude Advances in Acoustics and Vibration 7 1 1 0 0 −1 −1 34 35 36 38 39 40 Time (μs) Time (μs) (a) (b) 1 1 0 0 −1 −1 39 40 41 39 40 41 Time (μs) Time (μs) (c) (d) 1 1 0 0 −1 −1 40 41 42 40 41 42 Time (μs) Time (μs) (e) (f) Figure 12: Experimental data of crack C (with normalized amplitudes) superimposed with the estimated chirplets. (a) Front surface reference signal superimposed with sum of 2 chirplets. (b) Experimental data (refracted angle 0) superimposed with sum of 2 chirplets. (c) Experimental data (refracted angle 30 at point a) superimposed with sum of 4 chirplets. (d) Experimental data (refracted angle 30 at point b) superimposed with sum of 4 chirplets. (e) Experimental data (refracted angle 45 at point a) superimposed with sum of 4 chirplets. (f) Experimental data (refracted angle 45 at point b) superimposed with sum of 4 chirplets. Table 4: Estimated parameters of chirplets (block with 1 mm SDH). Table 5: Estimated parameters of chirplets (block with 4 mm SDH). Refracted angle Refracted angle Chirplet parameters Chirplet parameters ◦ ◦ ◦ ◦ ◦ ◦ 0 30 45 0 30 45 Amplitude (m-Volt) 42.5 29 16.01 Amplitude (m-Volt) 87.75 59.34 32.61 Spherically focused Spherically focused TOA (us) 76.62 82.6 89.39 Time of arrival (us) 76.10 82.05 88.88 transducer transducer Frequency (MHz) 4.55 4.6 4.32 Frequency (MHz) 4.61 4.54 4.39 Amplitude (m-Volt) 22.71 20.43 14.53 Amplitude (m-Volt) 41.72 37.62 27.97 Planar transducer TOA (us) 76.57 82.80 89.82 Planar transducer Time of arrival (us) 76.11 82.36 89.42 Frequency (MHz) 4.48 4.67 4.81 Frequency (MHz) 4.46 4.67 4.84 the transform order from (7)(i.e., −0.013) is used for a com- parison. Our simulation shows that the optimal transform the one for the LFM echo due to the impact of bandwidth order for the band-limited echo is different compared with factor in chirp echoes. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude 8 Advances in Acoustics and Vibration 0.5 0.5 −0.5 −0.5 −1 −1 34 35 36 38 39 40 Time (μs) Time (μs) (a) (b) 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 39 40 41 39.5 40 40.5 41 41.5 Time (μs) Time (μs) (c) (d) Figure 13: Experimental data of Crack D (with normalized amplitudes) superimposed with the estimated chirplets (depicted in dashed red line). (a) Front surface reference signal superimposed with sum of 2 chirplets. (b) Experimental data (refracted angle 0) superimposed with sum of 2 chirplets. (c) Experimental data (refracted angle 30) superimposed with sum of 4 chirplets. (d) Experimental data (refracted angle 45) superimposed with sum of 4 chirplets. Table 6: Estimated parameters of chirplets (crack D). TOA (us) Center frequency (MHz) Amplitude (m-Volt) 34.583 9.42 363.3 Reference signal 34.725 10.60 54.4 38.776 10.38 4.64 Refracted angle 0 38.891 13.06 0.50 39.777 7.68 0.50 40.040 9.10 0.14 Refracted angle 30 39.674 12.57 0.18 39.861 2.18 0.03 40.677 9.85 0.17 40.956 9.85 0.07 Refracted angle 45 40.675 4.51 0.04 40.620 15.65 0.03 Amplitude Amplitude Amplitude Amplitude Advances in Acoustics and Vibration 9 (4) calculate kurtosis of FrFT (x) for different 1 Target s win(t) 0.5 orders, α: −0.5 μ FrFT (x) 4 s win(t) −1 K(α) = ; (10) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 μ FrFT (x) 2 s win(t) Time (μs) (a) (5) estimate the optimal transform order, α : opt 1 Target α = arg MAX (K(α)), opt (11) 0.5 −0.5 α corresponds to the FrFT transform order where opt −1 K(α) has the max value. In our study, a brute-force 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 search is used to estimate the optimal transform Time (μs) order. The step size of searching is set to 0.005. The computation load of calculating the kurtosis (b) and searching for the optimal order is significant. Figure 14: (a) Measured ultrasonic backscattered signal (blue) Some researchers used the maximum peak in the superimposed with the reconstructed signal consisting of 8 domi- transform domain as an alternative metric [17]. nant chirplets (red). (b) Measured ultrasonic backscattered signal For ultrasonic signal decomposition, the optimal (blue) superimposed with the reconstructed signal consisting of 23 transform order is related to the chirp rate of the chirplets (red). signal. The search range of transform order can be reduced by considering prior knowledge of ultrasonic transducer impulse response; One can conclude that the compactness in the fractional (6) apply FrFT with the estimated order α to the signal opt Fourier transform of an ultrasonic echo can be used to track opt s(t) and obtain FrFT (x) ; s(t) the optimal transform order. It is also important to point opt (7) obtain a windowed signal from FrFT (x) : s(t) out that the optimal transform order is highly sensitive to a small change in the order. Therefore, using kurtosis becomes opt FrFT (x) = FrFT (x) × win (x), (12) win j s(t) a practical approach to obtain the optimal FrFT order for ultrasonic signal analysis. (8) apply the transformation order, −α , to the signal opt FrFT win(x), then reconstruct the jth component by 4. FrFT Chirplet Signal estimating parameters of the decomposed echo: Decomposition Algorithm −α opt f (t) = FrFT (t) , (13) Θ FrFT win(x) The objective of FrFT-CSD is to decompose a highly convoluted ultrasonic signal, s(t), into a series of signal the parameter estimation process here becomes a components: single-echo estimation problem. A Gaussian-Newton algorithm used in [23–25] is adopted in FrFT-CSD; (9) obtain the residual signal by subtracting the esti- s(t) = f (t) + r(t), Θj (8) mated echo from the signal, s(t), and use the residual j=1 signal for next echo estimation; (10) calculate energy of residual signal (E )and checkcon- where f (t) denotes the jth fractional chirplet component Θj vergence: (E is predefined convergence condition) min and r(t) denotes the residue of the decomposition process. If E <E , STOP; otherwise, go to step 2. r min The steps involved in the iterative estimation of an experimental ultrasonic signal are For further clarification, the flowchart of FrFT-CSD algorithm is shown in Figure 3. It is important to mention (1) initialize the iteration index j = 1; that two windowing steps are used in FrFT-CSD algorithm. One window is used in step 2 in order to isolate a dominant (2) obtain a windowed signal s win(t) after applying a echo in time domain. It is inevitable to have an incomplete window, w (t), in time domain; echo due to windowing process. A good strategy of choosing this window is to keep as much of echo information ( ) ( ) s = s t × w t . (9) win(t) j as possible. The other window is applied in step 7. For ultrasonic chirp echoes, the energy compactness of FrFT helps to reduce the window size centered on a desired peak (3) determine the FrFT of the signal, s win(t), in the transform domain. As shown in Figure 2, a chirplet FrFT (x) ,for different orders, α; is compressed to a great extent after the transform. An s win(t) 10 Advances in Acoustics and Vibration automatic windowing process is used to detect the valleys 6. Experimental Studies of the dominant echo. In the cases of heavily overlapping For experimental studies, two aluminum blocks with differ- echoes and high noise levels (i.e., the cases of poor signal- ent size of side-drilled hole (SDH) are used [27]. One is with to-noise ratio), the performance of windowing method 1 mm diameter, another is 4 mm diameter. The experimental may be compromised. In this situation, a window with a setting is shown in Figure 9. It can be seen that the water path predetermined size can be used to isolate desirable peaks. is 50.8 mm and the depth of SDH is 25.4 mm (i.e., from the water-aluminum interface to the center of SDH). 5. Simulation and Benchmark To provide a rigorous test, two 5 MHz transducers are used to acquire ultrasonic data at normal or oblique refracted Study of FrFT-CSD angles, θ. One is planar transducer. Another is spherically To demonstrate the advantages of FrFT signal decomposition focused transducer with 172.9 mm focal length. in ultrasonic signal processing, ultrasonic chirp echoes To verify the experiment setup, the FrFT-CSD is utilized with three different overlapping scenarios are simulated, to analyze the ultrasonic data from the front surface of where chirp rate models the dispersive effect in ultrasonic the specimen. The ultrasonic data superimposed with the testing of materials. Two slightly overlapped (about 20% estimated chirplet is shown in Figure 10. overlapped) echoes is simulated using the sampling fre- It can be seen that the estimated time-of-arrival (TOA) quency of 100 MHz. The parameters of these two echoes of the front surface echo is 68.72 μs. In addition, from the are experimental setting, the TOA can be calculated as 2 × D 2 2 TOA = , (15) Θ = 2.5us7MHz120MHz 35 MHz , (14) where D denotes the water distance, and in the case of 2 2 3.0us5MHz125MHz 20 MHz 0 Θ = . incidence angle 0 this distance is 50.8 mm. The round trip of ultrasound is twice of the water distance, D.The term Figure 4 shows the simulated signal (in blue) super- v denotes the velocity of ultrasound in medium: v = imposed with estimated echoes (in red). The estimated 1.484 mm/μsfor water. parameters perfectly match the parameters of simulation From (15), the theoretical value of TOA is 68.47 μs. signal as compared in Table 1. One can conclude that The estimated TOA is in agreement (within 0.4%) with the the FrFT-CSD not only decomposes the signal efficiently, theoretical TOA. but also leads to precise parameter estimation results. A Furthermore, the parameters of chirplet are strongly moderately overlapped (about 50% overlapped) simulated related to the crack size, location, and orientation. For signal consisting of two echoes is shown in Figure 5.For example, the amplitude is a good indicator of crack size. In this simulated signal, Table 2 shows that the estimated Tables 4 and 5, the estimated amplitude from a 4 mm SDH parameters are accurate within a few percents. is roughly twice of the estimated amplitude from a 1 mm Finally, Figure 6 and Table 3 show the simulated and SDH. In NDE applications, the estimated amplitude of a estimated two heavily overlapped (about 70% overlapped) known-size crack could be used as a reference to estimate echoes. The decomposition results (Figure 6)and estimated the size of crack. As shown in (15)and (16), the estimated parameters (Table 3) confirm the robustness and effective- TOA can be used to approximate the location of crack. ness of FrFT-CSD in echo estimation for ultrasonic signal In addition, different types of cracks could have different analysis. frequency variations. From [8, 26], the response of crack An experimental bat data is commonly used as a usually shows a downshift in the frequency compared with benchmark signal in time-frequency analysis. It is a 400- the responses of grains inside the material. sample data digitized 2.5 μs echolocation pulse emitted by a These results indicate that the estimated parameters from large brown bat with 7 μs sampling period. To evaluate the FrFT-CSD algorithm track with reasonable accuracy the performance of FrFT-based signal decomposition algorithm, physical parameters of experimental setup. Moreover, the the bat data is utilized to demonstrate the effectiveness of FrFT-CSD algorithm provides more detailed information algorithm. describing the reflected echoes such as phase, bandwidth Through the processing of FrFT-CSD, there are four factor and chirp rate that can be used for further analysis. main chirp-type signal components identified in the bat Another experiment is set up to evaluate disk-shaped signal. The decomposed signals and their Wigner-Ville dis- cracks in a diffusion-bonded titanium alloy sample [28]. tribution (WVD) are shown in Figure 7. The reconstructed The ultrasonic data of these synthetic cracks are obtained at signal and its superimposed WVD are shown in Figure 8. normal or oblique refracted angles, θ using a 10 MHz planar The results in Figures 7 and 8 are consistent with the analysis transducer. The diameter of the transducer is 6.35 mm. The results from other techniques in time-frequency analysis water depth is 25.4 mm. The surface of diffusion bond is [26]. The FrFT-based signal decomposition algorithm not 13 mm below the front surface of water/titanium alloy inter- only reveals that the bat signal mainly contains four chirp face. Two different sizes of cracks are made with the diameter stripes in time-frequency domain, but provides a high- 0.762 mm (i.e., crack D) and the diameter 1.905 mm (i.e., resolution time-frequency representation. crack C). For crack C, the responded ultrasonic data is Advances in Acoustics and Vibration 11 Table 7: Estimated parameters of chirplets (crack C). TOA (us) Center frequency (MHz) Amplitude (m-Volt) 34.583 9.42 363.3 Reference signal 34.725 10.60 54.4 38.754 9.78 14.48 Refracted angle 0 38.863 12.93 1.86 39.784 11.02 0.58 40.029 6.06 0.19 Refracted angle 30 (point a) 40.560 7.68 0.13 40.122 10.63 0.06 40.825 9.88 0.14 41.157 9.92 0.07 Refracted angle 45 (point a) 40.795 15.64 0.04 41.658 6.87 0.05 39.757 7.78 0.49 39.536 5.32 0.11 Refracted angle 30 (point b) 39.905 4.63 0.10 39.426 11.13 0.10 40.632 9.09 0.21 40.270 9.65 0.07 Refracted angle 45 (point b) 40.468 3.40 0.16 41.100 7.97 0.07 Table 8: Estimated parameters of the 8 dominant chirplets for ultrasonic experimental data. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 2.95 3.87 1.06 20.16 13.17 2.80 Echo 2 3.47 5.53 0.63 56.86 −41.06 −2.70 Echo 3 0.33 6.57 0.54 37.45 28.66 1.76 Echo 4 1.18 7.24 0.54 27.14 30.17 3.71 Echo 5 2.08 6.66 0.53 39.13 −15.50 2.75 Echo 6 2.40 6.00 0.47 62.91 60.24 2.47 Echo 7 4.64 6.23 0.18 4.75 −0.18 −2.36 Echo 8 1.49 3.97 0.12 0.73 −0.04 −6.64 recorded from the two edges of the crack, which are marked is 38.777 μs. TOA at the angle 30 is 39.425 μs. At the angle as point a and point b. The thickness of both disk-shaped 45 ,TOA is 40.514 μs. cracks is 0.089 mm. Figure 11 shows the experiment setup for From Tables 6 and 7, it can be seen that the estimated the alloy sample [28]. TOA at angle 0 is 38.776 μs and 38.754 μs. Taking the From Figure 11, the TOA of crack at refracted angle θ is thickness of the cracks (0.089 mm) into consideration, it calculated as follows: can be asserted that the estimated TOAs at incident angle 0 are in good agreement with experimental measurements. 2 × D/ cos θ (16) TOA = TOA + , θ ref Experimental signals of crack C and crack D (with normal- ized amplitudes) superimposed with the estimated chirplets where TOA denotes the estimated TOA of reference signal (depicted in dashed line and red color) are shown in Figures ref (i.e., 34.58 μs from Tables 6 and 7). The round trip of 12 and 13. It also can be seen that the front surface reference signal and the experimental data obtained at angle 0 are well ultrasound inside titanium from the front surface to the diffusion bound is 2 × D/ cos θ,where D denotes the depth reconstructed by the FrFT-CSD algorithm (see Figures 12(a), of diffusion bond, which is 13 mm; θ denotes the refracted 12(b), 13(a) and 13(b)). Nevertheless, with the increase of angle and v denotes the velocity of ultrasound in medium: refracted angle, more chirplets needed to decompose the v = 6.2 mm/μs for titanium. Therefore, TOA at the angle 0 experimental data (see the refracted angle 30 and 45 degree θ 12 Advances in Acoustics and Vibration cases). In addition, Tables 6 and 7 show that the signal [4] G. Cardoso and J. Saniie, “Ultrasonic data compression via parameter estimation,” IEEE Transactions on Ultrasonics, energy is more evenly distributed to estimated chirplets in Ferroelectrics, and Frequency Control, vol. 52, no. 2, pp. 313– the high refracted angle cases. This spreading of signal might 325, 2005. be caused by geometrical effect of the beam profile of the [5] R. Tao, Y. L. Li, and Y. Wang, “Short-time fractional fourier planner transducer and corners/edges of disk-shaped crack. transform and its applications,” IEEE Transactions on Signal To further evaluate the performance of FrFT-based Processing, vol. 58, no. 5, pp. 2568–2580, 2010. signal decomposition algorithm, experimental ultrasonic [6] S. Zhang, M. Xing, R. Guo, L. Zhang, and Z. Bao, “Interference microstructure scattering signals are utilized to demonstrate suppression algorithm for SAR based on time frequency the effectiveness of the algorithm. The experimental signal domain,” IEEE Transaction on Geoscience and Remote Sensing, is acquired from a steel block with an embedded defect vol. 49, no. 10, pp. 3765–3779, 2011. using a 5 MHz transducer and sampling rate of 100 MHz. [7] E. Oruklu and J. Saniie, “Ultrasonic flaw detection using dis- The acquired experimental data superimposed with the crete wavelet transform for NDE applications,” in Proceedings reconstructed signal consisting of 8 dominant chirplets of IEEE Ultrasonics Symposium, pp. 1054–1057, August 2004. are shown in Figure 14(a). The estimated parameters of [8] Y. Lu, R. Demirli, G. Cardoso, and J. Saniie, “A successive dominant chirplets are listed in Table 8.Itcan be seen parameter estimation algorithm for chirplet signal decompo- that the 8 dominant chirplets not only provide a sparse sition,” IEEE Transactions on Ultrasonics, Ferroelectrics, and representation of experimental data, but successfully detect Frequency Control, vol. 53, no. 11, pp. 2121–2131, 2006. the embedded defect. [9] S. C. Pel and J. J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Transactions on Signal To improve the accuracy of signal reconstruction, FrFT- Processing, vol. 48, no. 5, pp. 1338–1353, 2000. CSD could be used iteratively to decompose the signal [10] L. B. Almeida, “Fractional fourier transform and time- further. A reconstructed signal using 23 chirplets is shown frequency representations,” IEEE Transactions on Signal Pro- in Figure 14(b). The comparison between the experimental cessing, vol. 42, no. 11, pp. 3084–3091, 1994. signal and the reconstructed signals clearly demonstrates [11] C. Candan, M. Alper Kutay, and H. M. Ozaktas, “The discrete that the FrFT-CSD is highly effective in ultrasonic signal fractional fourier transform,” IEEE Transactions on Signal decomposition. Processing, vol. 48, no. 5, pp. 1329–1337, 2000. [12] A. S. Amein and J. J. Soraghan, “The fractional Fourier trans- form and its application to High resolution SAR imaging,” 7. Conclusion in Proceedings of IEEE International Geoscience and Remote In this paper, fractional Fourier transform is studied for Sensing Symposium (IGARSS ’07), pp. 5174–5177, June 2007. ultrasonic signal processing. Simulation study reveals the [13] M. Barbu, E. J. Kaminsky, and R. E. Trahan, “Fractional link among kurtosis, the transform order, and the parameters fourier transform for sonar signal processing,” in Proceedings of MTS/IEEE OCEANS, vol. 2, pp. 1630–1635, September of each decomposed components. Benchmark and experi- mental data sets are utilized to test the FrFT-based chirplet [14] I. S. Yetik and A. Nehorai, “Beamforming using the fractional signal decomposition algorithm. Signal decomposition and fourier transform,” IEEE Transactions on Signal Processing, vol. parameter estimation results show that fractional Fourier 51, no. 6, pp. 1663–1668, 2003. transform can successfully assist signal decomposition algo- [15] S. Karako-Eilon, A. Yeredor, and D. Mendlovic, “Blind source rithm by identifying the dominant echo in successive esti- separation based on the fractional Fourier transform,” in mation iteration. Parameter estimation is further performed Proceedings of the 4th International Symposium on Independent based on the echo isolation. The FrFT-CSD algorithm could Component Analysis and Blind Signal Separation, pp. 615–620, have a broad range of applications in signal analysis including target detection and pattern recognition. [16] A. T. Catherall and D. P. Williams, “High resolution spec- trograms using a component optimized short-term fractional Fourier transform,” Signal Processing, vol. 90, no. 5, pp. 1591– Acknowledgments 1596, 2010. [17] M. Bennett, S. McLaughlin, T. Anderson, and N. McDicken, The authors wish to thank Curtis Condon, Ken White, and “Filtering of chirped ultrasound echo signals with the frac- Al Feng of the Beckman Institute of the University of Illinois tional fourier transform,” in Proceedings of IEEE Ultrasonics for the bat data and for permission to use it in the study. Symposium, pp. 2036–2040, August 2004. [18] L. Stankovic, ´ T. Alieva, and M. J. Bastiaans, “Time-frequency signal analysis based on the windowed fractional Fourier References transform,” Signal Processing, vol. 83, no. 11, pp. 2459–2468, [1] S. Mallat, A Wavelet tour of Signal Processing: The Sparse Way, 2003. Academic Press, 2008. [19] Y. Lu, A. Kasaeifard, E. Oruklu, and J. Saniie, “Performance [2] I. Daubechies, “The wavelet transform, time-frequency local- evaluation of fractional Fourier transform(FrFT) for time- ization and signal analysis,” IEEE Transactions on Information frequency analysis of ultrasonic signals in NDE applications,” Theory, vol. 36, no. 5, pp. 961–1005, 1990. in Proceedings of IEEE International Ultrasonics Symposium (IUS ’10), pp. 2028–2031, October 2010. [3] S. Mann and S. Haykin, “The chirplet transform: physical [20] F. Millioz and N. Martin, “Circularity of the STFT and considerations,” IEEE Transactions on Signal Processing, vol. 43, no. 11, pp. 2745–2761, 1995. spectral kurtosis for time-frequency segmentation in Gaussian Advances in Acoustics and Vibration 13 environment,” IEEE Transactions on Signal Processing, vol. 59, no. 2, pp. 515–524, 2011. [21] R. Merletti,A.Gulisashvili, andL.R.LoConte,“Estimation of shape characteristics of surface muscle signal spectra from time domain data,” IEEE Transactions on Biomedical Engineering, vol. 42, no. 8, pp. 769–776, 1995. [22] Y. Lu, E. Oruklu, and J. Saniie, “Analysis of Fractional Fouriter transform for ultrasonic NDE applications,” in Proceedings of IEEE Ultrasonic Symposium, Orlando, Fla, USA, October 2011. [23] R. Demirli and J. Saniie, “Model-based estimation of ultra- sonic echoes part I: analysis and algorithms,” IEEE Transac- tions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 3, pp. 787–802, 2001. [24] R. Demirli and J. Saniie, “Model-based estimation of ultra- sonic echoes part II: nondestructive evaluation applications,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 3, pp. 803–811, 2001. [25] R. Demirli and J. Saniie, “Model based time-frequency estima- tion of ultrasonic echoes for NDE applications,” in Proceedings of IEEE Ultasonics Symposium, pp. 785–788, October 2000. [26] Y. Lu, E. Oruklu, and J. Saniie, “Ultrasonic chirplet signal decomposition for defect evaluation and pattern recognition,” in Proceedings of IEEE International Ultrasonics Symposium (IUS ’09), ita, September 2009. [27] Ultrasonic Benchmark Data, World Federation of NDE, 2004, http://www.wfndec.org/. [28] Ultrasonic Benchmark Data, World Federation of NDE, 2005, http://www.wfndec.org/. 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Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition

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Copyright © 2012 Yufeng Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 480473, 13 pages doi:10.1155/2012/480473 Research Article Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition 1 2 2 2 Yufeng Lu, Alireza Kasaeifard, Erdal Oruklu, and Jafar Saniie Department of Electrical and Computer Engineering, Bradley University, Peoria, IL 61625, USA Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA Correspondence should be addressed to Erdal Oruklu, erdal@ece.iit.edu Received 7 April 2012; Accepted 31 May 2012 Academic Editor: Mario Kupnik Copyright © 2012 Yufeng Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A fractional fourier transform (FrFT) based chirplet signal decomposition (FrFT-CSD) algorithm is proposed to analyze ultrasonic signals for NDE applications. Particularly, this method is utilized to isolate dominant chirplet echoes for successive steps in signal decomposition and parameter estimation. FrFT rotates the signal with an optimal transform order. The search of optimal transform order is conducted by determining the highest kurtosis value of the signal in the transformed domain. A simulation study reveals the relationship among the kurtosis, the transform order of FrFT, and the chirp rate parameter in the simulated ultrasonic echoes. Benchmark and ultrasonic experimental data are used to evaluate the FrFT-CSD algorithm. Signal processing results show that FrFT-CSD not only reconstructs signal successfully, but also characterizes echoes and estimates echo parameters accurately. This study has a broad range of applications of importance in signal detection, estimation, and pattern recognition. 1. Introduction broadband; symmetric or skewed; nondispersive or disper- sive. In ultrasonic imaging applications, the ultrasonic signal Recently, there has been a growing attention to fractional always contains many interfering echoes due to the complex Fourier transform (FrFT), a generalized Fourier transform physical properties of the propagation path. The pattern of with an additional parameter (i.e., transform order). It the signal is greatly dependent on irregular boundaries, and was first introduced in 1980, and subsequently closed-form the size and random orientation of material microstruc- FrFT was studied [8–11] for time-frequency analysis. FrFT tures. For material characterization and flaw detection is a power signal analysis tool. Consequently, it has been applications, it becomes a challenging problem to unravel applied to different applications such as high-resolution SAR the desired information using direct measurement and imaging, sonar signal processing, blind source separation, conventional signal processing techniques. Consequently, and beamforming in medical imaging [12–15]. Short term signal processing methods capable of analyzing the nonsta- FrFT, component-optimized FrFT, and locally optimized tionary behavior of ultrasonic signals are highly desirable FrFT have also been proposed for signal decomposition [16– for signal analysis and characterization of propagation 18]. path. In practice, signal decomposition problem is essentially an optimization problem under different design criteria. Various methods such as short-time Fourier transform, The optimization can be achieved either locally or glob- Wigner-Ville distribution, discrete wavelet transform, dis- crete cosine transform, and chirplet transform have been ally, depending on the complexity of the signal, accuracy of estimation, and affordability of computational load. utilized to examine signals in joint time-frequency domain Consequently, the results of signal decomposition are not and to reveal how frequency changes with time in those signals [1–8]. Nevertheless, it is still challenging to adaptively unique due to different optimization strategies and signal models. For ultrasonic signal analysis, local responses from analyze a broad range of ultrasonic signal: narrowband or 2 Advances in Acoustics and Vibration Table 1: Parameter estimation results of two slightly overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.5 7.0 1 20 35 π/6 Estimated parameter 2.50 7.00 1.00 20.00 35.00 0.52 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 25.00 19.98 0 Table 2: Parameter estimation results of two moderately overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.7 7.0 1 20 35 π/6 Estimated parameter 2.70 7.02 1.00 20.04 33.55 0.67 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 24.87 20.38 0.01 0.5 0.5 −0.5 −0.5 −1 −1 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 2.5 3 3.5 4 4.5 5 Time (μs) Time (μs) (a) (a) −0.006 Peak at the order =−0.013 0.0013 1.5 0.04 −0.0013 −0.04 0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 012 3456789 10 Order of fractional Fourier transform (FrFT) (b) (b) Figure 1: (a) A simulated LFM signal. (b) The optimal transform Figure 2: (a) a simulated ultrasonic single echo with Θ = order tracked by Maximum amplitude of FrFT for different FrFT 2 2 [3.6 us 5 MHz 1 25 MHz 25 MHz 0]. (b) Fractional orders. Fourier transform of the signal in (a) for different transform orders. microstructure scattering and structural discontinuities are has been explored. In particular, FrFT is introduced as a more of importance for detection and material characteri- transformation tool for ultrasonic signal decomposition. zation. Chirplet covers a board range of signals representing FrFT is employed to estimate an optimal transform order, frequency-dependent scattering, attenuation and dispersion which corresponds to highest kurtosis value in the transform effects in ultrasonic testing applications. This study shows domain. The searching process of optimal transform order that FrFT has a unique property for processing chirp- is based on a segmented signal for a local optimization. type echoes. Therefore, in this paper, the application of Then, the FrFT with the optimal transform order is applied fractional Fourier transform for ultrasonic applications to the entire signal in order to isolate the dominant echo Max. amplitude Amplitude of FrFT Amplitude Amplitude Advances in Acoustics and Vibration 3 Signal s(t) containing 1 multiple echoes 0 −1 Initialization j = 1 1.5 2 2.5 3 3.5 4 Time (μs) Signal windowing: (a) s win(t) = s(t) × w (t) Fractional Fourier transform: 0 −1 FrFT (x) s win(t) 1.5 2 2.5 3 3.5 4 Search an optimal transform order, α , Time (μs) opt j = j +1 which generates a maximum Kurtosis value (b) for FrFT (x) s win(t) Fractional fourier transform: opt −1 FrFT (x) s(t) 1.5 2 2.5 3 3.5 4 Signal windowing Time (μs) opt FrFT win(x) = FrFt (x) × win (x) s(t) (c) Inverse fractional fourier transform Figure 4: (a) Simulated ultrasonic echoes (20% overlapped). (b) −α opt f (t) = FrFT (t) Θ FrFT win (x) The first signal component. (c) The second signal component Use residual (simulated signal in blue, estimated signal in red). signal for next Estimate parameters of echo estimation decomposed echo, f (t) Obtain residual signal by subtracting decomposed echo from the signal −1 Calculate energy of residual signal (E ) 1.5 2 2.5 3 3.5 4 Time (μs) No E <E r min (a) Yes Signal decomposition and −1 parameter estimation complete 1.5 2 2.5 3 3.5 4 Time (μs) Figure 3: Flowchart of FrFT-CSD algorithm. (b) −1 for parameter estimation. This echo isolation is applied 1.5 2 2.5 3 3.5 4 iteratively to ultrasonic signal until a predefined stop cri- Time (μs) terion such as signal reconstruction error or the number of iterations is satisfied. Furthermore, each decomposed (c) component is modeled using six-parameter chirplet echoes Figure 5: (a) Simulated ultrasonic echoes (50% overlapped). (b) for a quantitative analysis of ultrasonic signals. The first signal component. (c) The second signal component A bat signal is utilized as a benchmark to demonstrate (simulated signal in blue, estimated signal in red). the effectiveness of fractional Fourier transform chirplet signal decomposition (FrFT-CSD). To further evaluate the performance of FrFT-CSD, ultrasonic experimental data from different types of flaws such as flat bottom hole, side- drilled hole and disk-type cracks are evaluated using FrFT- algorithm. Section 5 performs a simulation study of FrFT- CSD. CSD and parameter estimation for complex ultrasonic The outline of the paper is as follows. Section 2 reviews signals. Sections 5 and 6 show the results of a benchmark data the properties of FrFT and the process of FrFT-based signal (i.e., bat signal); the echo estimation results of benchmark decomposition. Section 3 addresses how kurtosis, transfor- data from side-drilled hole, and disk-shape cracks; the results mation order and chirp rate are related using simulated of experimental data with high microstructure scattering data. Section 4 presents the steps involved in FrFT-CSD echoes. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude 4 Advances in Acoustics and Vibration 2. FrFT of Ultrasonic Chirp Echo For ultrasonic applications, ultrasonic chirp echo is a type of signal often encountered in ultrasonic backscattered FrFT of a signal, f (t), is given by signals accounting for narrowband, broadband, and disper- sive echoes. It can be modeled as [8]: −i ((π/4) sgn(πα/2)− (πα/4)) e 2 α (1/2)ix cot (πα/2) F (x) = e 1/2 (2π|sin(πα/2)|) (1) ∞ f (t) = β exp −α (t − τ) + i2πf (t − τ) Θ 1 c (−i (xt/ sin(πα/2))+(1/2)it cot (πα/2)) × e f (t) dt, (2) −∞ +iα (t − τ) + iθ , where α denotes transform order of FrFT and x denotes the variable in transform domain. It has been shown that if the transform order, α,changes where Θ = τf βα α θ denotes the parameter vector, c 1 2 from 0 to 4, (i.e., the rotation angle, φ,changes from 0to2π), τ is the time-of-arrival, f is the center frequency, β is the F (x) rotates the signal, f (t), and projects it onto the line amplitude, α is the bandwidth factor, α is the chirp-rate, 1 2 of angle, φ, in time-frequency domain [19]. This property and θ is the phase. contributes to FrFT-based decomposition algorithm when Hence, for the ultrasonic Gaussian chirp echo, f (t), the applied to ultrasonic signals. magnitude of F (x)given by (1) can be expressed as 1 2 α (−i(xt/ sin(πα/2)) +(1/2)it cot (πα/2)) |F (x)| = e f (t) dt 1/2 −∞ (2π|sin(πα/2)|) (3) (B −4AC)/4A = e , 1/2 2 α + (α + (1/2)cot(πα/2)) |sin(πα/2)| where the integration part can be written as (−i (xt/ sin(πα/2))+(1/2)it cot (πα/2)) e f (t) dt −∞ 2 2 2 −[t (α −α i−(1/2)icot (πα/2))+t(2α τi−2α τ−2πf i+xicsc(πα/2))+(α τ −α iτ −θ+2πf τi)] 1 2 2 1 0 1 2 0 = e dt (4) −∞ (B −4AC)/4A = e , with A = α − α i − (1/2)i cot (πα/2), B = 2α τi − 2α τ − 3. Kurtosis and FrFT Order 1 2 2 1 2 2 2πf i + xicsc(πα/2), and C = α τ − α iτ − θ +2πf τi. 0 1 2 0 Kurtosis is commonly used in statistics to evaluate the degree From (3), it can be seen that, for a linear frequency of peakedness for a distribution [20, 21]. It is defined as the modulation (LFM) signal (i.e., α = 0), if the transformation ratio of 4th-order central moment and square of 2nd-order order, α, satisfies the following equation: central moment: 1 πα πα μ (F (x)) α + cot sin = 0, 4 K(α) =   , 2 2 2 (6) μ (F (x)) (5) 2 2 1 −1 α =− tan , where μ (•) denotes 4th-order central moment and μ (•) 4 2 π 2α denotes 2nd-order central moment. A signal with high then the |F (x)| compacts to a delta function. This means kurtosis means that it has a distinct peak around the mean. that fractional Fourier transform can be used to compress the In the literatures of FrFT [18, 19, 22], kurtosis is typically duration and compact the energy of ultrasonic chirp echo used as a metric to search the optimal transform order of with an optimal transform order. Optimal transform order FrFT. Different transform order directs the degree of signal can be determined using kurtosis. The energy compaction rotation caused by FrFT, and this rotation affects the extent is a desirable property for ultrasonic signal decomposition, of energy compaction of the transformed signal. which allows using a window in FrFT domain for isolation of Figure 1(a) shows a chirp signal with the param- an echo of interest. eters, Θ = [3.6 us 5MHz 1 0 25MHz 0]. For this Advances in Acoustics and Vibration 5 1 1 0 0 −1 −1 1.5 2 2.5 3 3.5 4 1.5 2 2.5 3 3.5 4 Time (μs) Time (μs) (a) (b) −1 1.5 2 2.5 3 3.5 4 Time (μs) (c) Figure 6: (a) Simulated ultrasonic echoes (70% overlapped). (b) The first estimated echo component. (c) The second estimated echo component (simulated signal in blue, estimated signal in red). −1 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) −1 0 0.5 1 1.5 2 0.5 1 1.5 2 Time (ms) Time (ms) (a) (b) Figure 7: Left column (top to bottom): decomposed bat signal components in time domain. Right column (top to down): Wigner-Ville distribution of the corresponding signals in left column. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Frequency (KHz) Frequency (KHz) Frequency (KHz) Frequency (KHz) 6 Advances in Acoustics and Vibration Table 3: Parameter estimation results of two heavily overlapped ultrasonic echoes. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 Actual parameter 2.7 6 1 20 55 π/6 Estimated parameter 2.70 6.11 0.97 18.87 53.79 0.72 Echo 2 Actual parameter 3.0 5 1 25 20 0 Estimated parameter 3.00 5.00 1.00 25.14 20.38 0.01 0.8 0.5 0.6 0.4 −0.5 0.2 −1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 −0.2 Time (ms) (a) −0.4 −0.6 −0.8 −1 68 68.2 68.4 68.6 68.8 69 69.2 69.4 69.6 69.8 70 Time (μs) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Time (ms) Figure 10: Ultrasonic data from the front surface superimposed with the estimated chirplet (depicted in dashed red color line). (b) Figure 8: (a) Reconstructed bat signal. (b) Summed Wigner Ville distribution of the decomposed signals in (a). Transducer 25.4 mm Transducer Water 50.8 mm Water θ Disc-shaped cracks 13 mm ab Diffusion bond Crack D Crack C Titanium alloy Side-drilled hole Figure 11: Experiment setup for disc-shaped cracks in a diffusion- bonded titanium alloy. Aluminum Figure 9: Experiment setup for SDH blocks. of FrFT using different transform orders according to (3). The transform order corresponding to the maximum FrFT example, the bandwidth factor equals to zero (see (2)), among all transform orders matches the theoretical result and according to (5), the optimal transform order can be givenin(7). calculated as For ultrasonic applications, the chirp echo is band- limited. For example, Figure 2(a) shows a band-limited 2 1 −1 α =− tan =−0.013. (7) single chirp echo with the parameters Θ = [3.6us 5MHz π 2α 1 25MHz 25MHz 0]. Chirplet is a model widely used As shown in Figure 1(b), this optimal order can also be in ultrasonic NDE applications. Figure 2 illustrates the FrFT determined by direct search for the maximum amplitude of a chirplet using different transform orders. In particular, Amplitude Frequency (KHz) Amplitude Advances in Acoustics and Vibration 7 1 1 0 0 −1 −1 34 35 36 38 39 40 Time (μs) Time (μs) (a) (b) 1 1 0 0 −1 −1 39 40 41 39 40 41 Time (μs) Time (μs) (c) (d) 1 1 0 0 −1 −1 40 41 42 40 41 42 Time (μs) Time (μs) (e) (f) Figure 12: Experimental data of crack C (with normalized amplitudes) superimposed with the estimated chirplets. (a) Front surface reference signal superimposed with sum of 2 chirplets. (b) Experimental data (refracted angle 0) superimposed with sum of 2 chirplets. (c) Experimental data (refracted angle 30 at point a) superimposed with sum of 4 chirplets. (d) Experimental data (refracted angle 30 at point b) superimposed with sum of 4 chirplets. (e) Experimental data (refracted angle 45 at point a) superimposed with sum of 4 chirplets. (f) Experimental data (refracted angle 45 at point b) superimposed with sum of 4 chirplets. Table 4: Estimated parameters of chirplets (block with 1 mm SDH). Table 5: Estimated parameters of chirplets (block with 4 mm SDH). Refracted angle Refracted angle Chirplet parameters Chirplet parameters ◦ ◦ ◦ ◦ ◦ ◦ 0 30 45 0 30 45 Amplitude (m-Volt) 42.5 29 16.01 Amplitude (m-Volt) 87.75 59.34 32.61 Spherically focused Spherically focused TOA (us) 76.62 82.6 89.39 Time of arrival (us) 76.10 82.05 88.88 transducer transducer Frequency (MHz) 4.55 4.6 4.32 Frequency (MHz) 4.61 4.54 4.39 Amplitude (m-Volt) 22.71 20.43 14.53 Amplitude (m-Volt) 41.72 37.62 27.97 Planar transducer TOA (us) 76.57 82.80 89.82 Planar transducer Time of arrival (us) 76.11 82.36 89.42 Frequency (MHz) 4.48 4.67 4.81 Frequency (MHz) 4.46 4.67 4.84 the transform order from (7)(i.e., −0.013) is used for a com- parison. Our simulation shows that the optimal transform the one for the LFM echo due to the impact of bandwidth order for the band-limited echo is different compared with factor in chirp echoes. Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude 8 Advances in Acoustics and Vibration 0.5 0.5 −0.5 −0.5 −1 −1 34 35 36 38 39 40 Time (μs) Time (μs) (a) (b) 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 39 40 41 39.5 40 40.5 41 41.5 Time (μs) Time (μs) (c) (d) Figure 13: Experimental data of Crack D (with normalized amplitudes) superimposed with the estimated chirplets (depicted in dashed red line). (a) Front surface reference signal superimposed with sum of 2 chirplets. (b) Experimental data (refracted angle 0) superimposed with sum of 2 chirplets. (c) Experimental data (refracted angle 30) superimposed with sum of 4 chirplets. (d) Experimental data (refracted angle 45) superimposed with sum of 4 chirplets. Table 6: Estimated parameters of chirplets (crack D). TOA (us) Center frequency (MHz) Amplitude (m-Volt) 34.583 9.42 363.3 Reference signal 34.725 10.60 54.4 38.776 10.38 4.64 Refracted angle 0 38.891 13.06 0.50 39.777 7.68 0.50 40.040 9.10 0.14 Refracted angle 30 39.674 12.57 0.18 39.861 2.18 0.03 40.677 9.85 0.17 40.956 9.85 0.07 Refracted angle 45 40.675 4.51 0.04 40.620 15.65 0.03 Amplitude Amplitude Amplitude Amplitude Advances in Acoustics and Vibration 9 (4) calculate kurtosis of FrFT (x) for different 1 Target s win(t) 0.5 orders, α: −0.5 μ FrFT (x) 4 s win(t) −1 K(α) = ; (10) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 μ FrFT (x) 2 s win(t) Time (μs) (a) (5) estimate the optimal transform order, α : opt 1 Target α = arg MAX (K(α)), opt (11) 0.5 −0.5 α corresponds to the FrFT transform order where opt −1 K(α) has the max value. In our study, a brute-force 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 search is used to estimate the optimal transform Time (μs) order. The step size of searching is set to 0.005. The computation load of calculating the kurtosis (b) and searching for the optimal order is significant. Figure 14: (a) Measured ultrasonic backscattered signal (blue) Some researchers used the maximum peak in the superimposed with the reconstructed signal consisting of 8 domi- transform domain as an alternative metric [17]. nant chirplets (red). (b) Measured ultrasonic backscattered signal For ultrasonic signal decomposition, the optimal (blue) superimposed with the reconstructed signal consisting of 23 transform order is related to the chirp rate of the chirplets (red). signal. The search range of transform order can be reduced by considering prior knowledge of ultrasonic transducer impulse response; One can conclude that the compactness in the fractional (6) apply FrFT with the estimated order α to the signal opt Fourier transform of an ultrasonic echo can be used to track opt s(t) and obtain FrFT (x) ; s(t) the optimal transform order. It is also important to point opt (7) obtain a windowed signal from FrFT (x) : s(t) out that the optimal transform order is highly sensitive to a small change in the order. Therefore, using kurtosis becomes opt FrFT (x) = FrFT (x) × win (x), (12) win j s(t) a practical approach to obtain the optimal FrFT order for ultrasonic signal analysis. (8) apply the transformation order, −α , to the signal opt FrFT win(x), then reconstruct the jth component by 4. FrFT Chirplet Signal estimating parameters of the decomposed echo: Decomposition Algorithm −α opt f (t) = FrFT (t) , (13) Θ FrFT win(x) The objective of FrFT-CSD is to decompose a highly convoluted ultrasonic signal, s(t), into a series of signal the parameter estimation process here becomes a components: single-echo estimation problem. A Gaussian-Newton algorithm used in [23–25] is adopted in FrFT-CSD; (9) obtain the residual signal by subtracting the esti- s(t) = f (t) + r(t), Θj (8) mated echo from the signal, s(t), and use the residual j=1 signal for next echo estimation; (10) calculate energy of residual signal (E )and checkcon- where f (t) denotes the jth fractional chirplet component Θj vergence: (E is predefined convergence condition) min and r(t) denotes the residue of the decomposition process. If E <E , STOP; otherwise, go to step 2. r min The steps involved in the iterative estimation of an experimental ultrasonic signal are For further clarification, the flowchart of FrFT-CSD algorithm is shown in Figure 3. It is important to mention (1) initialize the iteration index j = 1; that two windowing steps are used in FrFT-CSD algorithm. One window is used in step 2 in order to isolate a dominant (2) obtain a windowed signal s win(t) after applying a echo in time domain. It is inevitable to have an incomplete window, w (t), in time domain; echo due to windowing process. A good strategy of choosing this window is to keep as much of echo information ( ) ( ) s = s t × w t . (9) win(t) j as possible. The other window is applied in step 7. For ultrasonic chirp echoes, the energy compactness of FrFT helps to reduce the window size centered on a desired peak (3) determine the FrFT of the signal, s win(t), in the transform domain. As shown in Figure 2, a chirplet FrFT (x) ,for different orders, α; is compressed to a great extent after the transform. An s win(t) 10 Advances in Acoustics and Vibration automatic windowing process is used to detect the valleys 6. Experimental Studies of the dominant echo. In the cases of heavily overlapping For experimental studies, two aluminum blocks with differ- echoes and high noise levels (i.e., the cases of poor signal- ent size of side-drilled hole (SDH) are used [27]. One is with to-noise ratio), the performance of windowing method 1 mm diameter, another is 4 mm diameter. The experimental may be compromised. In this situation, a window with a setting is shown in Figure 9. It can be seen that the water path predetermined size can be used to isolate desirable peaks. is 50.8 mm and the depth of SDH is 25.4 mm (i.e., from the water-aluminum interface to the center of SDH). 5. Simulation and Benchmark To provide a rigorous test, two 5 MHz transducers are used to acquire ultrasonic data at normal or oblique refracted Study of FrFT-CSD angles, θ. One is planar transducer. Another is spherically To demonstrate the advantages of FrFT signal decomposition focused transducer with 172.9 mm focal length. in ultrasonic signal processing, ultrasonic chirp echoes To verify the experiment setup, the FrFT-CSD is utilized with three different overlapping scenarios are simulated, to analyze the ultrasonic data from the front surface of where chirp rate models the dispersive effect in ultrasonic the specimen. The ultrasonic data superimposed with the testing of materials. Two slightly overlapped (about 20% estimated chirplet is shown in Figure 10. overlapped) echoes is simulated using the sampling fre- It can be seen that the estimated time-of-arrival (TOA) quency of 100 MHz. The parameters of these two echoes of the front surface echo is 68.72 μs. In addition, from the are experimental setting, the TOA can be calculated as 2 × D 2 2 TOA = , (15) Θ = 2.5us7MHz120MHz 35 MHz , (14) where D denotes the water distance, and in the case of 2 2 3.0us5MHz125MHz 20 MHz 0 Θ = . incidence angle 0 this distance is 50.8 mm. The round trip of ultrasound is twice of the water distance, D.The term Figure 4 shows the simulated signal (in blue) super- v denotes the velocity of ultrasound in medium: v = imposed with estimated echoes (in red). The estimated 1.484 mm/μsfor water. parameters perfectly match the parameters of simulation From (15), the theoretical value of TOA is 68.47 μs. signal as compared in Table 1. One can conclude that The estimated TOA is in agreement (within 0.4%) with the the FrFT-CSD not only decomposes the signal efficiently, theoretical TOA. but also leads to precise parameter estimation results. A Furthermore, the parameters of chirplet are strongly moderately overlapped (about 50% overlapped) simulated related to the crack size, location, and orientation. For signal consisting of two echoes is shown in Figure 5.For example, the amplitude is a good indicator of crack size. In this simulated signal, Table 2 shows that the estimated Tables 4 and 5, the estimated amplitude from a 4 mm SDH parameters are accurate within a few percents. is roughly twice of the estimated amplitude from a 1 mm Finally, Figure 6 and Table 3 show the simulated and SDH. In NDE applications, the estimated amplitude of a estimated two heavily overlapped (about 70% overlapped) known-size crack could be used as a reference to estimate echoes. The decomposition results (Figure 6)and estimated the size of crack. As shown in (15)and (16), the estimated parameters (Table 3) confirm the robustness and effective- TOA can be used to approximate the location of crack. ness of FrFT-CSD in echo estimation for ultrasonic signal In addition, different types of cracks could have different analysis. frequency variations. From [8, 26], the response of crack An experimental bat data is commonly used as a usually shows a downshift in the frequency compared with benchmark signal in time-frequency analysis. It is a 400- the responses of grains inside the material. sample data digitized 2.5 μs echolocation pulse emitted by a These results indicate that the estimated parameters from large brown bat with 7 μs sampling period. To evaluate the FrFT-CSD algorithm track with reasonable accuracy the performance of FrFT-based signal decomposition algorithm, physical parameters of experimental setup. Moreover, the the bat data is utilized to demonstrate the effectiveness of FrFT-CSD algorithm provides more detailed information algorithm. describing the reflected echoes such as phase, bandwidth Through the processing of FrFT-CSD, there are four factor and chirp rate that can be used for further analysis. main chirp-type signal components identified in the bat Another experiment is set up to evaluate disk-shaped signal. The decomposed signals and their Wigner-Ville dis- cracks in a diffusion-bonded titanium alloy sample [28]. tribution (WVD) are shown in Figure 7. The reconstructed The ultrasonic data of these synthetic cracks are obtained at signal and its superimposed WVD are shown in Figure 8. normal or oblique refracted angles, θ using a 10 MHz planar The results in Figures 7 and 8 are consistent with the analysis transducer. The diameter of the transducer is 6.35 mm. The results from other techniques in time-frequency analysis water depth is 25.4 mm. The surface of diffusion bond is [26]. The FrFT-based signal decomposition algorithm not 13 mm below the front surface of water/titanium alloy inter- only reveals that the bat signal mainly contains four chirp face. Two different sizes of cracks are made with the diameter stripes in time-frequency domain, but provides a high- 0.762 mm (i.e., crack D) and the diameter 1.905 mm (i.e., resolution time-frequency representation. crack C). For crack C, the responded ultrasonic data is Advances in Acoustics and Vibration 11 Table 7: Estimated parameters of chirplets (crack C). TOA (us) Center frequency (MHz) Amplitude (m-Volt) 34.583 9.42 363.3 Reference signal 34.725 10.60 54.4 38.754 9.78 14.48 Refracted angle 0 38.863 12.93 1.86 39.784 11.02 0.58 40.029 6.06 0.19 Refracted angle 30 (point a) 40.560 7.68 0.13 40.122 10.63 0.06 40.825 9.88 0.14 41.157 9.92 0.07 Refracted angle 45 (point a) 40.795 15.64 0.04 41.658 6.87 0.05 39.757 7.78 0.49 39.536 5.32 0.11 Refracted angle 30 (point b) 39.905 4.63 0.10 39.426 11.13 0.10 40.632 9.09 0.21 40.270 9.65 0.07 Refracted angle 45 (point b) 40.468 3.40 0.16 41.100 7.97 0.07 Table 8: Estimated parameters of the 8 dominant chirplets for ultrasonic experimental data. 2 2 τ (us) f (MHz) βα (MHz) α (MHz) θ (Rad) c 1 2 Echo 1 2.95 3.87 1.06 20.16 13.17 2.80 Echo 2 3.47 5.53 0.63 56.86 −41.06 −2.70 Echo 3 0.33 6.57 0.54 37.45 28.66 1.76 Echo 4 1.18 7.24 0.54 27.14 30.17 3.71 Echo 5 2.08 6.66 0.53 39.13 −15.50 2.75 Echo 6 2.40 6.00 0.47 62.91 60.24 2.47 Echo 7 4.64 6.23 0.18 4.75 −0.18 −2.36 Echo 8 1.49 3.97 0.12 0.73 −0.04 −6.64 recorded from the two edges of the crack, which are marked is 38.777 μs. TOA at the angle 30 is 39.425 μs. At the angle as point a and point b. The thickness of both disk-shaped 45 ,TOA is 40.514 μs. cracks is 0.089 mm. Figure 11 shows the experiment setup for From Tables 6 and 7, it can be seen that the estimated the alloy sample [28]. TOA at angle 0 is 38.776 μs and 38.754 μs. Taking the From Figure 11, the TOA of crack at refracted angle θ is thickness of the cracks (0.089 mm) into consideration, it calculated as follows: can be asserted that the estimated TOAs at incident angle 0 are in good agreement with experimental measurements. 2 × D/ cos θ (16) TOA = TOA + , θ ref Experimental signals of crack C and crack D (with normal- ized amplitudes) superimposed with the estimated chirplets where TOA denotes the estimated TOA of reference signal (depicted in dashed line and red color) are shown in Figures ref (i.e., 34.58 μs from Tables 6 and 7). The round trip of 12 and 13. It also can be seen that the front surface reference signal and the experimental data obtained at angle 0 are well ultrasound inside titanium from the front surface to the diffusion bound is 2 × D/ cos θ,where D denotes the depth reconstructed by the FrFT-CSD algorithm (see Figures 12(a), of diffusion bond, which is 13 mm; θ denotes the refracted 12(b), 13(a) and 13(b)). Nevertheless, with the increase of angle and v denotes the velocity of ultrasound in medium: refracted angle, more chirplets needed to decompose the v = 6.2 mm/μs for titanium. Therefore, TOA at the angle 0 experimental data (see the refracted angle 30 and 45 degree θ 12 Advances in Acoustics and Vibration cases). In addition, Tables 6 and 7 show that the signal [4] G. Cardoso and J. 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