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Analysis of Fractional Fourier Transform for Ultrasonic NDE Applications Yufeng Lu*, Erdal Oruklu**, and Jafar Saniie**

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Analysis of Fractional Fourier Transform for Ultrasonic NDE Applications Yufeng Lu*, Erdal Oruklu**, and Jafar Saniie** 10.1109/ULTSYM.2011.0123 Analysis of Fractional Fourier Transform for Ultrasonic NDE Applications Yufeng Lu*, Erdal Oruklu**, and Jafar Saniie** ** Department of Electrical and Computer Engineering *Department of Electrical and Computer Engineering Illinois Institute of Technology Bradley University Chicago, Illinois 60616 Peoria, Illinois 61625 Abstract—In this study, a Fractional Fourier transform-based It has been utilized in different applications such as high signal decomposition (FrFT-SD) algorithm is utilized to resolution SAR imaging, sonar signal processing, blind analyze ultrasonic signals for NDE applications. FrFT, as a deconvolution, and beamforming in medical imaging [7-10]. transform tool, enables signal decomposition by rotating the Short term FrFT, component optimized FrFT, and locally signal with an optimal transform order. The search of optimal optimized FrFT have also been proposed for signal transform order is conducted by searching the highest kurtosis decomposition [11-13]. There are many different ways to value of the signal in the transformed domain. Simulation decompose a signal. Essentially it is an optimization problem study reveals the relationship among the kurtosis, the under different criteria. The optimization is conducted either transform order of FrFT, and the chirp rate parameter. globally or locally. For ultrasonic NDE applications, local Furthermore, parameter estimation is applied to the responses from scattering microstructure and discontinuities decomposed components (i.e., chirplets) for echo characterization. Simulations and experimental results show are more interesting for flaw detection and material that FrFT-SD not only reconstructs signal successfully, but also characterization. estimates parameters accurately. Therefore, FrFT-SD could In our previous work [14], FrFT is introduced as a be an effective tool for analysis of ultrasonic signals. transformation tool for ultrasonic signal decomposition. FrFT is employed to estimate an optimal transform order, Keywords – Fractional Fourier transform (FrFT), signal which is associated with highest kurtosis value in fractional decomposition, kurtosis, ultrasonic NDE transform domain. The searching process of optimal transform order is based on a segmented signal for a local I. INTRODUCTION optimization. Once the optimal transform order is obtained, Ultrasonic signals contain important information pertain the FrFT with the transform order is applied to the entire to the physical properties of the propagation path. These signal. A signal component is acquired by applying a backscattered echoes often interfere with each other due to window in the fractional domain and inverse FrFT. limited resolution of the ultrasonic transducer, closed Ultrasonic signal is iteratively decomposed into a group of locations, boundaries, orientations, and sizes of random signal components until a predefined stop criterion such as reflectors and may also be corrupted by measurement noise signal reconstruction error or the number of iterations is and/or undesired scattering echoes. Consequently, it becomes satisfied. a challenging problem and conventional signal analysis In this investigation, analysis of FrFT for ultrasonic techniques fail to unscramble the desired signal information nondestructive evaluation applications is studied. FrFT- for material characterization. Therefore, signal processing based signal decomposition (FrFT-SD) leads to an efficient methods capable of characterizing the non-stationary signal decomposition for simulated ultrasonic signals. behavior of ultrasonic signal for NDE applications are highly Furthermore, each decomposed component is modeled using desired. six-parameter chirplet echoes. As a result, parameter Time-frequency analysis has been widely used to estimation is performed for a quantitative analysis of examine signals in joint time-frequency domain and probe ultrasonic signals. The chirplet covers a board range of how frequency changes with time in those signals. The signals representing frequency-dependent scattering, advantage and disadvantage of time-frequency analysis attenuation and dispersion effects in ultrasonic applications. methods such as short-time Fourier transform, Wigner-Ville Moreover, the relationship among kurtosis, transformation distribution, wavelet transform, and chirplet transform are order of FrFT and chirp rate is addressed and studied through well-known [1-6]. It remains a challenging problem to find a simulations. method which is applicable to a broad range of ultrasonic The outline of the paper is as follows. Section II reviews echoes, narrowband or broadband; symmetric or skewed; FrFT-based signal decomposition (FrFT-SD). Section III nondispersive or dispersive. addresses how kurtosis, transformation order and chirp rate Recently, there has been a growing attention in Fractional are related through simulations. Section IV discusses Fourier transform. As a generalized Fourier transform with performance evaluation of FrFT-SD and parameter an additional parameter (i.e., transform order), FrFT has estimation for simulated data. Section V shows the results of potentials in all areas where Fourier transform can be applied. experimental study. Section VI concludes the paper. 978-1-4577-1252-4/11/$26.00 ©2011 IEEE 512 2011 IEEE International Ultrasonics Symposium Proceedings Authorized licensed use limited to: Illinois Institute of Technology. Downloaded on October 02,2020 at 06:07:00 UTC from IEEE Xplore. Restrictions apply. II. FRFT-BASED SIGNAL DECOMPOSITION 4. Calculate kurtosis of α for different FrFT (x)s _ win (t ) The FrFT-SD for ultrasonic signal is reviewed as follows. order,α . FrFT of a signal, f ()t , is given by M ()FrFT α ( x) K ()α = 4 s _ win (t ) (5) α 2 []M ()FrFT ( x) π πα xt 1 πα 2 s _ win (t ) − − −i + it2 cot i( ) 1 πα πα 2 2 4 4 i x2 cot ∞ sin (1) α e 2 2 2 FrFT (x) = e e f (t)dt th 1 −∞ where M ()• denotes 4 order central moment and πα 2 4 π nd 2 sin M ()• denotes 2 order central moment. 2 2 5. Find the optimal transformation order, α . where α : denotes transform order of FrFT est α = ()α est arg α MAX K ( ) x : denotes the variable in transform domain. α corresponds to the FrFT transform order where est The transform order,α , is related to the rotation angle , φ , K ()α has the max value. πφ 6. Apply FrFT with the estimated orderα to the signal α = est with . α s ( t ) and obtain est . 2 FrFT (x) s(t ) α For ultrasonic applications, ultrasonic chirp echo is a type 7. Get a windowed signal from est . FrFT (x) s(t ) of signal often encountered in ultrasonic backscattered signal α = est ∗ accounting for narrowband, broadband, and dispersive FrFT _ win(x) FrFT (x) s(t) wini (x) echoes. It can be modeled as [3-5]: 8. Get a decomposed component, FCi (t) , by applying the transformation order, − α , to the signal , est ()= β (− α (− τ )2 ) f Θ t exp 1 t (2) FrFT _ win(x) . + π ()− τ + θ + α ()− τ 2 i 2 f c t i i 2 t −α = est FC i (t) FrFT (t) FrFT _ win ( x ) Θ= τ β α α θ where [ fc 1 2 ] denotes the parameter τ 9. Obtain the residual signal by subtracting the vector, is the time-of-arrival, f c is the center frequency, estimated echo, FCi (t) and update the signal, s(t) . α α β is the amplitude, 1 is the bandwidth factor, 2 is the 10. Calculate energy of residual signal( E ) and check chirp-rate, and θ is the phase. r convergence: ( Emin is predefined convergence The objective of FrFT-SD is to decompose a highly condition) convoluted ultrasonic signal, s(t) , into a series of signal If E < E , STOP; otherwise, go to step 2. () r min components, FC j t . N III. KURTOSIS, TRANSFORM ORDER AND CHIRP RATE = ()+ (3) s(t) FC j t r (t) j =1 Kurtosis is commonly used in statistics to evaluate the degree of peakedness for a distribution. It is defined as the () th where FC j t denotes the jth signal component and r (t) ratio of 4 central moment and square of variance [see denotes the residue of the decomposition process. Equation 5 in section II]. A signal with high Kurtosis means that it has a distinct peak around the mean. In the literatures The steps involved in the iterative estimation of an of FrFT [11-14], Kurtosis is mostly used as a metric to experimental ultrasonic signal are outlined as follows: search the optimal transform order of FrFT. Different 1. Initialize the iteration index = . transform order directs the degree of signal rotation caused i 1 by FrFT. 2. Get a windowed signal after applying a s _ win(t) For ultrasonic signals that can be modeled by chirplets, window, , in time domain. wi (t) the chirp rate parameter can be interpreted as an indicator of rotation in time-frequency representation. To show the link = ∗ s _ win(t) s(t) wi (t) (4) between these two types of rotations, a simulation study is performed. A single chirp echo is simulated with the 3. Obtain FrFT of the signal, s _ win(t) , for different Θ= 2 α parameters [3.6us 5MHz 1 0 25MHz 2 ] where transformation order, α . α the chirp rate, 2 , varies from 1 to 100. α denotes FrFT of the signal FrFT (x) s _ win (t ) FrFT-SD is applied to the chirp echo. Kurtosis and the s _ win(t) with a various of orders α . corresponding optimal transform order are recorded. The connections among kurtosis, optimal transform order of 513 2011 IEEE International Ultrasonics Symposium Proceedings Authorized licensed use limited to: Illinois Institute of Technology. Downloaded on October 02,2020 at 06:07:00 UTC from IEEE Xplore. Restrictions apply. FrFT and chirp rate of chirp echoes are shown in Figures 1, Θ =[3.6us 8MHz 1 pi/6 25MHz2 25MHz2 ] 2 and 3. It can be seen that the optimal transform order is a 1 Θ = 2 2 monotonic function of chirp rate; meanwhile kurtosis is a 2 [3.9us 5MHz 1 0 25MHz 30MHz ] monotonic function of transform order.
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