1612 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 3, MARCH 2012 Spline-Kernelled Chirplet Transform for the Analysis of Signals With Time-Varying Frequency and Its Application Y. Yang, Z. K. Peng, G. Meng, and W. M. Zhang
Abstract—The conventional time–frequency analysis (TFA) Among a number of TFA methods, short-time Fourier trans- methods, including continuous wavelet transform, short-time form (STFT) [17]–[19], wavelet transform (WT) [10], [20], and Fourier transform, and Wigner–Ville distribution, have played Wigner–Ville distribution (WVD) [3], [5], [5] have been widely important roles in analyzing nonstationary signals. However, they often show less capability in dealing with nonstationary signals used; they are nonparameterized methods [21]. For the STFT with time-varying frequency due to the bad energy concentration method, the signal is assumed to be piecewise stationary at the in the time–frequency plane. On the other hand, by introducing an scale of the window width as the STFT relies on traditional extra transform kernel that matches the instantaneous frequency Fourier transform, and therefore, the STFT cannot produce of the signal, parameterized TFA methods show powerful ability an accurate estimation for time-varying IFs. Essentially, the in characterizing time–frequency patterns of nonstationary signals with time-varying frequency. In this paper, a novel time–frequency WT is a kind of STFT method with adjustable window size, transform, called spline-kernelled chirplet transform (SCT), is using large window for low-frequency components and small proposed. By introducing a frequency-rotate operator and a window for high-frequency components. Thus, it is no wonder frequency-shift operator constructed with spline kernel func- that the WT cannot achieve an accurate estimation for time- tion, the SCT is particularly powerful for the strongly nonlinear varying IFs as well [22]. The WVD can present excellent TFR frequency-modulated signals. In addition, an effective algorithm is developed to estimate the parameters of transform kernel in for signals in terms of energy concentration, but its bilinear the SCT. The capabilities of the SCT and parameter estimation structure creates the redundant cross terms that do not indicate algorithm are validated by their applications for numerical signals the true time–frequency structure of the signal, which leads to and a set of vibration signal collected from a rotor test rig. inaccurate estimation of IFs. Index Terms—Chirplet transform (CT), instantaneous In order to capture the accurate time–frequency pattern frequency (IF), spline-kernelled chirplet transform (SCT), of chirplike signal, chirplet transform (CT), a parameterized time–frequency representation (TFR). time–frequency transform, has been developed, and now, it has been often used in the analysis of linear-frequency-modulated I. INTRODUCTION (LFM) signals [23], [24]. Compared with STFT, CT possesses an extra chirp kernel, which is characterized by a chirping NSTANTANEOUS FREQUENCY (IF) [1], [2], as an in- rate parameter. With the transform kernel, the time–frequency I formative time–frequency characteristic of a signal, plays atoms of the CT can be shifted and sheared to suit the sig- an important role in nonstationary signal analysis. Estimation nal. Nonetheless, the CT fails to characterize the underly- of the IF is desirable in the fields of rotary machine [3]– ing time–frequency pattern of nonlinear-frequency-modulated [8], power system [9], [10], electronic system [11], speech (NLFM) signals as it is impossible for the chirp kernel to match [12], wind turbine [13], transportation system [14], and others, the nonlinear IF of NLFM signal. To overcome the shortcoming where the IFs are often used to characterize important physical of the CT, Chassande-Mottin and Pai [25] developed a gen- parameters of the signals. To estimate the IF, time–frequency eral chirplet chain (CC) method to analyze gravitational wave analysis (TFA) methods [15], [16] are often applied, and the es- signals with nonlinearly time-varying IFs. By maximizing an timation accuracy usually depends on the energy concentration optimal statistics, the method is used to find the best CC whose of the time–frequency representation (TFR) produced by TFA IF is essentially a piecewise linear approximation of the IF to be methods. detected. Mihovilovic and Bracewell [26] proposed an adaptive CT to improve the TFR so as to estimate IF for NLFM signals. Manuscript received March 18, 2011; revised June 27, 2011; accepted With a frequency drift rate, this method is able to adjust the July 8, 2011. Date of publication August 4, 2011; date of current version October 25, 2011. This work was supported in part by the National Natural tilt angle of time–frequency atom to match the NLFM signals Science Foundation of China under Grants 10902068 and 10732060 and in part in the time–frequency plane. The local frequency drift rate can by Shanghai Pujiang Program under Grant 10PJ1406000. be estimated by the slope of the line which approximates the The authors are with the State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China (e-mail: energy ridge locally in the TFR. Cui and Wong [27] proposed [email protected]; [email protected]; [email protected]; another adaptive CT based on matching pursuit method and [email protected]). applied it to characterize the time-dependent behavior of the Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. visual evoked potential from its initial transient portion to the Digital Object Identifier 10.1109/TIE.2011.2163376 steady-state portion. In their study, the signal is assumed to
0278-0046/$26.00 © 2011 IEEE YANG et al.: SCT FOR THE ANALYSIS OF SIGNALS WITH TIME-VARYING FREQUENCY AND ITS APPLICATION 1613 be a sum of a series of weighted chirplets and residue, and According to definition (1), the CT of a signal is equivalent the method iteratively projects the residue of the signal into to the STFT of its analytical signal multiplied by the complex a predefined chirplet dictionary. The best-matched chirplets window ψ(t, α). The CT can also be rewritten as can be determined by maximizing a similarity degree. In the ∞ essence, the aims of these “adaptive” CTs are to find the best-fit straight lines with arbitrary slopes to approximate the signal’s CT(τ,ω,α; σ)=A z¯(t)wσ(t − τ)exp(−jωt)dt (3) energy ridge locally in the time–frequency plane. However, it −∞ is obvious that the broken line is neither very smooth nor very accurate approximation for highly nonlinear curve. Moreover, with the boundary effect of decomposed chirplets in the TFR can R S mislead the identification and extraction of the IF. z¯(t)=z(t)Φ (t, α)Φ (t, τ, α) In this paper, a novel time–frequency transform, named as R 2 Φ (t, α) = exp(−jαt /2) spline-kernelled CT (SCT), is proposed based on the conven- tional CT. By implementing a frequency-rotate operator and ΦS(t, τ, α) = exp(jατt) a frequency-shift operator constructed with spline kernel, it is A τ,α −jτ2α/ able to characterize the underlying time–frequency pattern of ( ) = exp( 2) the signal with nonlinearly time-varying IF as it can be well approximated by a spline function. Moreover, the parameters where A(τ,α) is a complex value with modulus |A(τ,α)| =1. of the spline kernel of the SCT can be estimated through the In the time–frequency analysis, it is the modulus of the TFR, spline approximation. Therefore, the SCT is able to produce |CT(τ,ω,α; σ)|, that is usually of interests and meaningful, so a TFR with an excellent energy concentration for signal with the CT can be simplified as nonlinearly time-varying IF so as to facilitate the accurate ∞ IF estimation. The rest of this paper is organized as follows. CT(τ,ω,α; σ)= z¯(t)wσ(t − τ)exp(−jωt)dt Section II briefly introduces the conventional CT, the spline (4) function, and the underlying principle of the SCT. In Section III, −∞ the parameter estimation algorithm for the SCT is presented. R S A signal contaminated by the noise is used to demonstrate the where Φ (t, α) and Φ (t, τ, α) are named as the frequency- effectiveness of the proposed algorithm. Section IV evaluates rotate operator and the frequency-shift operator, respectively. the proposed SCT algorithm on several numerical examples and To elaborate the functionality of these two operators, an LFM a set of signals collected from a rotor test rig. The conclusions signal is considered, i.e., are drawn in Section V. 2 s(t) = sin(2πω0t + πλt ). (5) II. SCT The IF of the signal is Ω(t)=ω0 + λt, where λ is the slope In this section, the detail of the SCT is described after a brief of the IF trajectory and ω0 is the initial frequency. In the introduction of the CT and the spline function. CT of s(t), ΦR(t, α) rotates the analytical signal z(t) by an angle arctan(−α) in the time–frequency plane. If the chirping A. CT rate α equals λ, then the frequency of the analytical signal is ω S t, τ, α CT is basically the inner product of the signal to be analyzed equivalent to 0 all the time, and then, Φ ( ) relocates the τ and a family of analysis windows. For a frequency-modulated frequency component of the signal at time with an increment 2 ατ z t τ signal s(t) ∈ L (R), its CT at τ is defined as [28] of . It is noticed that the frequency of ( ) at time after CT would be equal Ω(τ) when α = λ. Therefore, the CT can +∞ be essentially decomposed into a series of operations: First, the α 2 CT(τ,ω,α; σ)= z(t)exp −j (t − τ) wσ(t − τ) −α 2 signal is rotated by a degree arctan( ) in the time–frequency −∞ plane. Second, it is shifted by a frequency increment of ατ in × exp(−jωt)dt (1) the time–frequency plane. Next, it is processed by the STFT with the window wσ. τ α ∈ R where and stand for time and chirping rate, respec- It is worth noticing that the frequency resolution of the CT z t s t tively. ( ) is the analytical signal of ( ), calculated by a depends on the chirping rate α and the length of the Gaussian H z t s t jH s t Hilbert transform [29] , i.e., ( )= ( )+ [ ( )]. Since window. Taking the signal denoted in (5) as an example, the the Hilbert transform is only very effective for the high sam- time–frequency atom adopted by the CT is of bandwidth of pling rate [30], the sampling rate should be set appropriately σ|λ − α| +1/σ and duration of σ to divide the time–frequency w t in order to obtain the correct analytical signal. σ( ) is a plane. When α = λ, the time–frequency atom has a minimum nonnegative, symmetric, and normalized real window, which is bandwidth of 1/σ and a duration of σ, which is the same as often taken as the Gaussian window function defined as that of the Gaussian window. Obviously, the CT degrades to 2 STFT when α =0. Given the proper chirping rate, the CT 1 1 t wσ(t)=√ exp − . (2) can produce a TFR with an optimal energy concentration, and 2πσ 2 σ |CT(τ,ω,α; σ)| achieves global maximum at (ω,α)=(ω0,λ). 1614 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 3, MARCH 2012
B. Spline Function It seems reasonable for the adaptive CT to calculate the approximation when the straight line can accurately fit the IF in a properly determined interval. However, the obvious disadvantage of the broken line approximation lies in its lack of smoothness as well as its excessive computational load for in- terval determination. According to Weierstrass approximation theorem [31], polynomial is a better choice to provide smoother and more efficient approximation to continuous function than the broken lines. However, polynomial approximation is not necessarily good. The Gram matrix would be ill-conditioned Fig. 1. Illustration of SCT principal. (Solid line) IF of signal. (Dashed line) in solving the least square approximation with a polynomial of IF trajectory after rotation. (Dotted line) IF trajectory after rotation and shift. high order, which is worse as the order of polynomial increase. C. SCT Moreover, if a function is to be approximated on a larger interval, the order of the approximating polynomial has to be In order to characterize the time–frequency patterns of sig- chosen unacceptably large. Thus, it will be difficult to compute nals with nonlinearly time-varying frequency, the SCT is pro- the least square polynomial when the Gram matrix is highly ill- posed. By replacing the kernel of the frequency-rotate operator conditioned, which leads to inaccurate approximation for the and frequency-shift operator in the conventional CT with a data to be matched. spline kernel, the proposed SCT extends the capability of the Spline is known to be effective in approximating high dy- conventional CT and can be applied to a wider class of signals, namic shapes on a larger interval. It can be defined as a i.e., the NLFM signals whose IF trajectory can be approximated piecewise polynomial function with boundary constraints at by a spline function as in (9). The SCT of a signal is defined as jointed breaks. The broken line is the spline of order 2. On an +∞ interval [a, b] with breaks a = t1 < ··· the SCT is determined by both ΔIFs(τ; σ) and the bandwidth for i =1:N − 1, can be measured. As long as D(xo) > Δ, of the Gaussian function 1/σ, i.e., the frequency resolution at in which Δ is the threshold, the intermittent point of (xo,yo) i i τ equals ΔIFs(τ; σ)+1/σ.Ifqk is well selected to equal pk, is determined as an interior break. In this case, the constraint for all i and k, ΔIFs(τ; σ) will become zero all over the time at (xo,yo) needs to be loosened, and thus the spline kernel is span, and the frequency resolutions of the SCT at any moments broken into separate segments. can reach the minimum 1/σ. An iterative procedure based on the TFR is proposed to esti- mate the parameters from the unknown signal with nonlinearly III. PARAMETER ESTIMATION METHOD time-varying frequency. At first, the TFR is obtained by using i the SCT with initialized parameter matrix q(i, k)=qk =0,for It is crucial to develop a parameter estimation method for the all i and k, which makes SCT degrade to STFT. Since the SCT to facilitate the construction of the spline kernel function energy of a signal is mainly distributed around the IF of the that can approximate the IF of the signal in real applications. signal in the time–frequency plane, the IF can be estimated Only with the appropriately determined parameters can SCT through the time–frequency peak detection. The position of the attain a satisfactory TFR with an excellent concentration and energy peak in the TFR is denoted as the estimated IF of the achieve the IF estimation with significant accuracy. The spline considered signal approximation is introduced to estimate the parameters of the transform kernel of the SCT. According to [33], any spline func- Ω( t) = arg max (|SCT(t, ω, Q; σ)|) . (15) tion of a given degree and smoothness can be represented as a ω weighted sum of basic splines (b-splines). As another way to t represent a spline, B-form has become a standard representation Second, a spline function is selected to approximate Ω( ) by other than the ppform for its convenience of constructing and solving (14). Once the best fitted spline is calculated, a proper shaping a function, which is defined as frequency-rotate operator and frequency-shift operator of the m SCT can be constructed for the SCT. Next, using the estimated S(x)= cjBj,n(x) for a ≤ x ≤ b (11) parameters, the SCT generates an improved TFR with a better j=1 energy concentration. This procedure keeps iterating until no more evident modification is observed in the estimated IF, i.e., where m is the number of the b-splines and cj is the control B j n point. j,n denotes the th b-spline of order for a knot Ωiter+1(t) − Ωiter(t) sequence, i.e., v1 ≤ v2 ...≤ vm+n. It should be noted that the ζiter = mean dt < δ (16) knots are different from the aforementioned breaks in that they Ωiter(t) are repeated. Simply, apart from the interior breaks, the knot sequence should contain the two endpoints, a and b,ofthe where δ is a predetermined threshold and the basic interval exactly n times. Thus, breaks can be the interior subscript denotes the index of the iteration. The breaks of knot sequence, which may be repeated depending detail of the algorithm is described as follows. B on the required constraints. In particular, j,n is nonnegative Initialization: v ,...,v piecewise polynomial with breaks j j+n and is zero Initialized parameter matrix, Q, number of spline pieces, outside the interval (vj,...,vj+n), which can be computed as minor integer δ, window size, maximum iteration Lmax, 1 if vj ≤x Fig. 2. SCT-based TFR and estimated IF in the first iteration. Fig. 3. SCT-based TFR and estimated IF in the second iteration. Fig. 4. SCT-based TFR and estimated IF in the third iteration. In order to demonstrate the proposed parameter estimation the STFT. The energy of the signal is scattered along both the method, an NLFM signal with piecewise IF denoted as follows time and frequency axes, and it is unable to achieve an accurate is considered: estimation of the IF. A spline that is modified at the breaking point is used to approximate the estimated IF curve, as shown 30 − 11t +1.2t2, 0 ≤ t<5s in Fig. 2. With the estimated parameters, the SCT improves the Ω(t)= 100 13 2 (17) − 3 +26.5t − 6 t , 5s ≤ t<10s. concentration of the TFR, and the extracted IF becomes closer to the true IF of the signal as shown in Figs. 3–6. The best TFR A Gaussian noise with a standard deviation of 0.6 and a with a superior energy concentration and the precise estimation mean of zero is artificially added to the signal. The SNR is of the IF are shown in Fig. 7. The criterion of the termination thus 1.5468 dB. The window size is chosen as 512. δ is 0.015, condition is listed in Table I. In addition, in order to quantify the and the sampling frequency is set to be 100 Hz. It takes six accuracy of the estimated IF compared to the true IF, the relative iterations before reaching the termination condition. The TFRs error is used to measure the difference between the estimated IF generated by the SCT and the estimated IFs in each iteration and the true IF. The relative error is defined as are shown in Figs. 2–7, respectively. The blue line denotes ⎛ ⎞ the estimated IF, and the red line denotes the approximated Ωiter(t) − Ω(t) error = mean ⎝ dt⎠ . (18) polynomial. The TFR generated by the SCT with initialized |Ω(t)| parameter q =0as illustrated in Fig. 2 is actually the result of YANG et al.: SCT FOR THE ANALYSIS OF SIGNALS WITH TIME-VARYING FREQUENCY AND ITS APPLICATION 1617 Fig. 5. SCT-based TFR and estimated IF in the fourth iteration. Fig. 6. SCT-based TFR and estimated IF in the fifth iteration. Fig. 7. SCT-based TFR and estimated IF in the sixth iteration. TABLE I examples are given in the form of S(t) = sin[2πϕ(t)], whose VALUE OF THE TERMINATION CONDITION IFs are given by ∂ϕ(t) 10(t−5) Ω(t)= = +10, (0≤t≤10s) (19) ∂t 1+(t−5)4 TABLE II ∂ϕ(t) ERRORS OF THE ESTIMATED IF t . t t2/ −t3/ , ≤t≤ s . Ω( )= ∂t =10+2 5 + 3 40 (0 18 ) (20) Both of them are artificially added with additive Gaussian noise with a standard deviation of 0.6 and a mean of zero. The Relative errors are listed in Table II, which clearly shows sampling frequency is set to be 100 Hz, and the window length that the estimated IF is closer to the true IF in the proposed is 512. The TFR obtained by the proposed SCT is compared parameter estimation procedure. with the TFRs obtained by conventional CT, continuous WT (CWT), and WVD. The parameters of the SCT are estimated using the proposed method presented in Section III. IV. NUMERICAL STUDIES The SNR of the first signal is 1.6482 dB. The TFRs generated In this section, three examples are used to demonstrate by the CT, CWT, WVD, and SCT are shown in Fig. 8. The the effectiveness of the proposed SCT method. The first two chirping rate of the CT is set to be 3.8 Hz/s. The TFR generated 1618 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 3, MARCH 2012 Fig. 8. TFR of the signal with IF in (20) by (a) CT, (b) CWT, (c) WVD, and (d) SCT. TABLE III the reciprocal relationship between the central frequency of the PARAMETERS OF THE KERNEL OF THE SCT [FIG. 8(d)] wavelet function and its window length. As shown in Fig. 9(c), the existence of spurious trajectories in the TFR obtained from the cross terms of the WVD strongly interferes the recognition of the true IF. However, in Fig. 9(d), the SCT provides the TFR with an excellent concentration, based on which the IF can be estimated precisely. The estimated coefficients of the SCT are by the CT, shown in Fig. 8(a), can barely reveal the inherent listed in Table IV. time–frequency pattern of the signal. To be specific, the TFR All of the tests are implemented by MATLAB version 7.11.0 only shows the clear IF trajectory over 0–10 s due to a good (R2010b) on a PC with AMD Athlon 64 X2 dual-core processor approximation by a linear function with a changing rate of 4000 + 2.10 GHz and 1-GB RAM. The calculation times of the 3.8 Hz/s. However, the TFR after 10 s is too blur to reveal the IF CT, the CWT, the WVD, and the one iteration of SCT with trajectory. In Fig. 8(b), it can be seen that the TFR generated by proper parameters for these two examples are listed in Table V. the CWT scatters the energy around the IF at the high-frequency It can be seen that the calculation time of the SCT in one region due to its coarse frequency resolution, and it is inaccurate iteration is similar to that of the WVD in both cases, which to estimate the IF. As shown in Fig. 8(c), it is also difficult is much less than that of CWT but slight more than that of CT. for the TFR generated by the WVD to differentiate the true IF In addition, a set of vibration signals collected by ac- trajectory from the spurious frequency contents introduced by celerometers on a rotor test rig during speed-up and shut-down the cross terms. On the other hand, as shown in Fig. 8(d), it processes is considered. The speed-up and shut-down processes is evident that the SCT outperforms the CT, the WT, and the of rotary machine usually contain valuable information related WVD as it clearly reveals the true time–frequency pattern of to the machine health condition, which plays an important role the signal. The estimated coefficients of the SCT are listed in in machine condition monitoring. In this part, the SCT is used Table III. to implement the TFA for a set of the vibration signal collected The SNR of the second signal is 2.1159 dB. The TFRs during a speed-up and shut-down process. The test rig is shown generated by the CT, CWT, WVD, and the SCT are shown in Fig. 10. in Fig. 9. The chirping rate of the CT is set to be 9.5 Hz/s. Fig. 11 shows the collected vibration signal. The sampling As shown in Fig. 9(a), it is difficult for the TFR generated frequency is 100 Hz, and the window size is set to be 512. by the CT to identify the underlying time–frequency pattern The spline kernel of order 4 with 18 pieces is adopted in the of the signal since it is inadequate for the CT to analyze the SCT. The termination condition in (16) is applied, and the NLFM signals. The TFR generated by the CWT, as shown in threshold is set to be δ =0.1%. Three iterations have been Fig. 9(b), shows poor time or frequency resolution because of conducted before reaching the termination condition. Figs. 12 YANG et al.: SCT FOR THE ANALYSIS OF SIGNALS WITH TIME-VARYING FREQUENCY AND ITS APPLICATION 1619 Fig. 9. TFR of signal with IF in (19) by (a) CT, (b) CWT, (c) WVD, and (d) SCT. TABLE IV PARAMETERS OF THE KERNEL OF THE SCT [FIG. 9(d)] Fig. 10. Rotor test rig. TABLE V COMPARISON OF THE TIME COST Fig. 11. Set of vibration signal. and 13 show the TFRs and the estimated instantaneous speed obtained by the STFT and the SCT, respectively. It is clear that the TFR shown in Fig. 13 has much better energy concentration 1620 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 3, MARCH 2012 rig as an example, the energy at the frequency components other than the fundamental frequency component in the TFR generated by the SCT is more scattered than that of the TFR generated by STFT. The authors are currently working on extending the SCT to the signals containing multiple-frequency components. V. C ONCLUSION The conventional time–frequency transform methods can- not characterize the accurate time–frequency patterns for the Fig. 12. TFR generated by (a) the STFT and (b) the extracted ridge and the signals with time-varying frequency due to the poor energy estimated instantaneous rotating speed. concentration. A novel time–frequency transform has been proposed to characterize the underlying time–frequency feature of the signal with nonlinearly time-varying IF trajectory. With the effective algorithm developed to estimate the parameters of transform kernel of the SCT, its potential and effectiveness are validated through analyzing several numerical simulation signals and a set of experimental vibration signals. The compar- ison results show that the SCT outperforms the STFT, the CWT, the conventional CT, and the WVD in providing the TFR of better energy concentration and achieving more accurate IF es- timation for the signals with nonlinearly time-varying IF. Future work will focus on extending the SCT from monocomponent Fig. 13. TFR generated by (a) the SCT and (b) the extracted ridge and the signal analysis to multicomponent signal analysis. estimated instantaneous rotating speed. TABLE VI REFERENCES PARAMETERS OF TRANSFORM KERNEL OF THE SCT (FIG. 13) [1] B. 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Her research interests include sig- [16] L. Cohen, Time–Frequency Analysis. Upper Saddle River: Prentice-Hall, nal processing, machine health diagnosis, and prognostics. 1995, pt. 2, pp. 30–52. [17] S. H. Nawab and T. F. Quatieri, Short-Time Fourier Transform. Upper Saddle River, NJ: Prentice-Hall, 1987. [18] H. K. Kwok and D. L. Jones, “Improved instantaneous frequency estima- Z. K. Peng received the B.Sc. and Ph.D. degrees tion using an adaptive short-time Fourier transform,” IEEE Trans. Signal from Tsinghua University, Beijing, China, in 1998 Process., vol. 48, no. 10, pp. 2964–2972, Oct. 2000. and 2002, respectively. [19] P. J. Kootsookos, B. C. Lovell, and B. Boashash, “A unified approach From 2003 to 2004, he was with the City Uni- to the STFT, TFDs, and instantaneous frequency,” IEEE Trans. Signal versity of Hong Kong, Kowloon, Hong Kong, as a Process., vol. 40, no. 8, pp. 1971–1982, Aug. 2002. Research Associate, and then, he was with Cranfield [20] A. Bouzida, O. Touhami, R. Ibtiouen, M. Fadel, A. Rezzoug, and University, Cranfield, U.K., as a Research Officer. A. Beloucherani, “Fault diagnosis in industrial induction machines After that, he was with the University of Sheffield, through discrete wavelet transform,” IEEE Trans. Ind. Electron., vol. 58, Sheffield, U.K., for four years. He is currently a no. 9, pp. 4385–4395, Sep. 2011. Research Professor with the State Key Laboratory [21] F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic of Mechanical System and Vibration, Shanghai Jiao time–frequency signal representations,” IEEE Signal Process. Mag., Tong University, Shanghai, China. His main expertise relates to the subject vol. 9, no. 2, pp. 21–67, Apr. 1992. areas of nonlinear vibration, signal processing and condition monitoring, and [22] G. Kaiser, A Friendly Guide to Wavelets (Modern Birkhäuser Classics). fault diagnosis for machines and structures. Cambridge, MA: Birkhäuser Boston, 2011, pt. 1, pp. 60–77. [23] S. Mann and S. Haykin, “The chirplet transform: Physical considera- tions,” IEEE Trans. Signal Process., vol. 43, no. 11, pp. 2745–2761, Nov. 1995. [24] D. Mihovilovicˇ and R. N. Bracewell, “Whistler analysis in the G. Meng received the Ph.D. degree from Northwest- time–frequency plane using chirplets,” J. Geophys. Res., vol. 97, no. A11, ern Polytechnical University, Xi’an, China, in 1988. pp. 17 199–17 204, Nov. 1992. In 1993, he was a Professor and the Director of [25] E. Chassande-Mottin and A. Pai, “Best chirplet chain: Near-optimal Vibration Engineering Institute with Northwestern detection of gravitational wave chirps,” Phys.Rev.D, vol. 73, no. 4, Polytechnical University. From 1989 to 1993, he was pp. 42 003–42 026, Feb. 2006. also a Research Assistant with Texas A&M Univer- [26] D. Mihovilovic and R. N. Bracewell, “Adaptive chirplet representation sity, College Station, an Alexander von Humboldt of signals on time–frequency plane,” Electron. Lett., vol. 27, no. 13, Fellow with Technical University Berlin, Berlin, pp. 1159–1161, Jun. 1991. Germany, and a Research Fellow with New South [27] J. Cui and W. Wong, “The adaptive chirplet transform and visual Wales University, Sydney, Australia. From 2000 to evoked potentials,” IEEE Trans. Biomed. Eng., vol. 53, no. 7, pp. 1378– 2008, he was with Shanghai Jiao Tong University, 1384, Jul. 2006. Shanghai, China, as the Cheung Kong Chair Professor, the Associate Dean, [28] S. Mann and S. Haykin, ““Chirplets” and “warblets”: Novel time– and the Dean of the School of Mechanical Engineering. He is currently a frequency methods,” Electron. Lett., vol. 28, no. 2, pp. 114–116, Professor and the Director of the State Key Laboratory of Mechanical System Jan. 1992. and Vibration, Shanghai Jiao Tong University. His research interests include [29] B. Boashash, Time Frequency Signal Analysis and Processing: A Compre- dynamics and vibration control of mechanical systems, nonlinear vibration, and hensive Reference. Amsterdam, The Netherlands: Elsevier, 2003, pt. 4, microelectromechanical systems. pp. 229–368. [30] G. L. Xu, X. T. Wang, and X. G. Xu, “Time-varying frequency-shifting signal assisted empirical mode decomposition method for AM–FM signals,” Mech. Syst. Signal Process., vol. 23, no. 8, pp. 2458–2469, W. M. Zhang was born in May 1978. He received Nov. 2009. the B.S. degree in mechanical engineering and the [31] [Online]. Available: http://mathworld.wolfram.com/ M.S. degree in mechanical design and theories from WeierstrassApproximationTheorem.html Southern Yangtze University, Wuxi, China, in 2000 [32] C. De Boor, A Practical Guide to Splines. New York: Springer-Verlag, and 2003, respectively, and the Ph.D. degree in 2001, pt. 7, pp. 17–20. mechanical engineering from Shanghai Jiao Tong [33] J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and University, Shanghai, China, in 2006. Their Applications. New York: Academic, 1967, pt. 1, pp. 1–8. He is currently an Associate Professor with [34] M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. the State Key Laboratory of Mechanical System I. Theory,” IEEE Trans. Signal Process., vol. 41, no. 2, pp. 821–833, and Vibration, School of Mechanical Engineering, Feb. 1993. Shanghai Jiao Tong University. He has deep expe- [35] M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. riences in the dynamics and control for micro-/nanoelectromechanical sys- II. Efficiency design and applications,” IEEE Trans. Signal Process., tems (MEMS/NEMS). His researches involve nonlinear dynamics and chaos vol. 41, no. 2, pp. 834–848, Feb. 1993. control, nonlinear vibration and control, coupled parametrically excited mi- [36] D. Lasser and J. Hoschek, Fundamentals of Computer Aided Geometric croresonators, and the reliability analysis and assessment for MEMS/NEMS Design. New York: Elsevier, 2002. applications.