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Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition

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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 Yufeng Lu, 1 Alireza Kasaeifard,2 Erdal Oruklu,2 and Jafar Saniie2 1 Department of Electrical and Computer Engineering, Bradley University, Peoria, IL 61625, USA 2 Department of Electrical and Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA Correspondence should be addressed to Erdal Oruklu, [email protected] 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 Various methods such as short-time Fourier transform, an optimization problem under different design criteria. Wigner-Ville distribution, discrete wavelet transform, dis- The optimization can be achieved either locally or glob- crete cosine transform, and chirplet transform have been ally, depending on the complexity of the signal, accuracy utilized to examine signals in joint time-frequency domain of estimation, and affordability of computational load. and to reveal how frequency changes with time in those Consequently, the results of signal decomposition are not signals [1–8]. Nevertheless, it is still challenging to adaptively unique due to different optimization strategies and signal analyze a broad range of ultrasonic signal: narrowband or models. For ultrasonic signal analysis, local responses from 2 Advances in Acoustics and Vibration Table 1: Parameter estimation results of two slightly overlapped ultrasonic echoes. 2 2 τ (us) fc (MHz) βα1 (MHz) α2 (MHz) θ (Rad) 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) fc (MHz) βα1 (MHz) α2 (MHz) θ (Rad) 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 1 1 0.5 0.5 0 0 − . Amplitude 0 5 −0.5 Amplitude −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) 10 2 Peak at the order =−0.013 −0.006 8 0.0013 1.5 −0.0013 0.04 6 −0.04 4 1 of FrFT 2 Amplitude 0.5 Max. amplitude Max. 0 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0 012345678910 Order of fractional Fourier transform (FrFT) x (b) (b) Figure 1: (a) A simulated LFM signal. (b) The optimal transform Θ = order tracked by Maximum amplitude of FrFT for different FrFT Figure 2: (a) a simulated ultrasonic single echo with . 2 2 orders. [3 6 us 5 MHz 1 25 MHz 25 MHz 0]. (b) Fractional 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 Advances in Acoustics and Vibration 3 Signal s(t) containing 1 multiple echoes 0 −1 Amplitude Initialization j = 1 1.5 2 2.5 3 3.5 4 Time (μs) Signal windowing: (a) s win(t) = s(t) × wj (t) 1 Fractional Fourier transform: 0 α − FrFT (x)s win(t) 1 Amplitude 1.5 2 2.5 3 3.5 4 Search an optimal transform order, αopt, Time (μs) j = j +1 which generates a maximum Kurtosis value α (b) for FrFT (x)s win(t) 1 Fractional fourier transform: 0 αopt − FrFT (x)s(t) 1 Amplitude 1.5 2 2.5 3 3.5 4 Signal windowing Time (μs) αopt FrFT win(x) = FrFt (x)s(t) × winj (x) (c) Inverse fractional fourier transform Figure 4: (a) Simulated ultrasonic echoes (20% overlapped). (b) f t = −αopt t Θj ( ) FrFT ( )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, fΘj (t) Obtain residual signal by subtracting decomposed echo from the signal 1 0 − Calculate energy of residual signal (Er ) 1 Amplitude 1.5 2 2.5 3 3.5 4 μ No Time ( s) Er <Emin (a) Yes 1 Signal decomposition and 0 − parameter estimation complete 1 Amplitude 1.5 2 2.5 3 3.5 4 Time (μs) Figure 3: Flowchart of FrFT-CSD algorithm. (b) 1 0 for parameter estimation. This echo isolation is applied −1 iteratively to ultrasonic signal until a predefined stop cri- Amplitude 1.5 2 2.5 3 3.5 4 terion such as signal reconstruction error or the number Time (μs) of iterations is satisfied. Furthermore, each decomposed (c) component is modeled using six-parameter chirplet echoes for a quantitative analysis of ultrasonic signals. Figure 5: (a) Simulated ultrasonic echoes (50% overlapped). (b) A bat signal is utilized as a benchmark to demonstrate The first signal component. (c) The second signal component the effectiveness of fractional Fourier transform chirplet (simulated signal in blue, estimated signal in red). 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. 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 f t FrFT of a signal, ( ), is given by signals accounting for narrowband, broadband, and disper- sive echoes. It can be modeled as [8]: e−i ((π/4) sgn(πα/2)− (πα/4)) Fα x = e(1/2)ix2cot (πα/2) ( ) 1/2 (2π|sin(πα/2)|) (1) 2 ∞ fΘ(t) = β exp −α1(t − τ) + i2πfc(t − τ) e(−i (xt/ sin(πα/2))+(1/2)it2cot (πα/2)) f t dt × ( ) , (2) −∞ 2 +iα2 (t − τ) + iθ , where α denotes transform order of FrFT and x denotes the variable in transform domain.
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