I. INTRODUCTION MULTI-COMPONENT frequency modulated (FM) signals

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1 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER Application of Parameterized Time-Frequency Analysis on Multicomponent Frequency Modulated Signals Yang Yang, Zhike Peng, Xingjian Dong, Wenming Zhang, Member, IEEE, andguangmeng Abstract Parameterized time-frequency (TF) transforms, with signal-dependent kernel parameters, have been proposed to analyze multicomponent frequency modulated (FM) signals. Usually, the kernel parameters are estimated through recursive approximation of TF representation (TFR) ridge when instantaneous frequency models of the components have the same parameter settings. However, it will be inapplicable if the components have the different FM sources. In this paper, we introduce a novel method that enables the parameterized TF transform to generate the well-concentrated TFR for both the monocomponent signal and a wide class of multicomponent FM signals, whose components are modulated by either the same or the different sources. The proposed method contains two aspects: 1) estimating kernel parameters based on spectrum concentration index and 2) separating components and assembling the parameterized TFRs of the separated components. An advantage of the proposed method is that it avoids the dependence of the TFR while estimating the parameters. Moreover, it is effective at low signalto-noise rate. The validity and practical utility of the proposed method are demonstrated by both the simulated and real signals. The results show that it outperforms the traditional TF methods in providing the TFR of the improved concentration for various multicomponent FM signals. Index Terms Concentration, multicomponent signal, parameter estimation, parameterized time-frequency (TF) analysis, time-frequency representation (TFR). I. INTRODUCTION MULTI-COMPONENT frequency modulated (FM) signals arise in various applications, i.e., acoustic signal processing [1], [2], speech analysis [3], [4], fault diagnosis [5], [6], radar signal processing [7], [8], power system [9], and biomedical engineering [10]. They can be modeled as superposition of multiple monocomponent FM signals and the different components might be modulated by the different FM sources. The FM feature of the component can be characterized by instantaneous frequency (IF). Although the IF lacks a Manuscript received November 20, 2013; revised February 1, 2014; accepted March 10, Date of publication March 25, 2014; date of current version November 6, This work was supported in part by the NSFC for Distinguished Young Scholars under Grant , in part by the Excellent Young Scholars under Grant , and in part by the NSFC under Grant and Grant The Associate Editor coordinating the review process was Dr. Ruqiang Yan. The authors are with the Institute of Vibration Shock and Noise, Shanghai Jiao Tong University, Shanghai , China ( emma002@sjtu.edu.cn; z.peng@sjtu.edu.cn; donxij@sjtu.edu.cn; wenmingz@sjtu.edu.cn; gmeng@sjtu.edu.cn). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIM rigorous mathematical definition, it is still an important feature of the FM signals [11] [13]. The IF is defined to be the derivative of the phase of analytic signal, which is only tenable in the case of monocomponent signals. For a multicomponent signal, each component can be featured by its own IF. In the past few centuries, Fourier transform has been a powerful tool to analyze stationary signals [14]. However, as a global transform, it can neither reveal the local-frequency information of monocomponent FM signals nor distinguish different components for the multicomponent FM signal when the components are spectral overlapped. To give the better insight into the underlying pattern of the FM signal, time-frequency (TF) transform attracts attention of many researchers [15]. It maps a 1-D signal to a 2-D function of time and frequency so a TF representation (TFR) can be obtained to characterize the signal in the time and frequency domain simultaneously. Generally, TF transforms can be divided into two categories: 1) nonparameterized TF transform and 2) parameterized TF transform. Namely, the former is defined without signal-dependent parameters, whereas the latter has the signal-dependent parameters. In the nonparameterized category, the typical methods include short-time Fourier transform (STFT) [16], wavelet transform (WT) [6], and Wigner Ville distribution (WVD) [17]. In the case of the multicomponent FM signal, they either suffer from the poor concentration or interference of cross-term in the TFR. From a postprocessing point of view, reassignment method was proposed to improve the readability of the TFR by assigning the average of energy in a domain to the gravity center of these energy contributions [18]. Unfortunately, it may distort the IF profile for noisy signals and cannot avoid the cross-terms between the every two components of the multicomponent signal. Compared with nonparameterized TF transforms, parameterized TF transforms, with extra parameters pertaining to the FM information of the analyzed signal, are much more appealing for obtaining the well-concentrated TFR. The more concentrated the TFR is generated, the better it characterizes the real TF pattern of the signal. Otherwise, the poor concentrated TFR could result in misinterpretation of the signal in the TF plane. Chirplet transform (CT) is the oldest member of the parameterized TF transforms [19], [20]. Using a chirp rate, it is specifically designed to analyze linear FM signals. For the nonlinear FM signal, the CT cannot characterize the IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 3170 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 nonlinear IF with a single rate. An intuitive solution is to use a group of chirp rate to approximate the nonlinear IF curve, such methods have been proposed as various modified CTs [21] [25]. However, the TFR will be resulted in the distortion of IF profile. On the other hand, it is more accurate to construct nonlinear kernel to approximate the nonlinear IF. The polynomial is a potential candidate as the Weierstrass approximation theorem guarantees that any continuous function on a bounded interval can be uniformly approximated by a polynomial to any degree of accuracy [26]. Thus, the polynomial chirplet transform (PCT) uses a polynomial kernel to provide a better choice to analyze nonlinear FM signal [27]. However, if the signal has strongly oscillated IF in a large time span, the high-order polynomial approximation will suffer from Runge phenomenon. To avoid such problem, the spline chirplet transform (SCT) adopts spline kernel to deal with a class of nonlinear FM signals, whose IF cannot be approximated by the high-order polynomial function properly [28]. The SCT seems to be an ideal solution for any nonlinear FM signal, though it will be overwhelmed to determine the parameters when the strongly time-varying IF requires a great number of pieces. For a special class of periodic FM signals, warblet transform with a periodic kernel was proposed to catch the periodical IF of the signal [29]. However, it is unable to deal with the signal with the nonperiodical IF due to the kernel fails to model such IF. Alternatively, the generalized wavelet transform (GWT) applies a Fourier series kernel to characterize both the periodical IF and nonperiodical IF [30]. For signals with the nonperiodical IF, the number of the Fourier series needed to be determined considering the balance between the smoothness and accuracy of the approximation. All these parameterized TF transforms are able to offer the solution to analyze the monocomponent FM signal or the multicomponent FM signal whose components have the same FM source, and the kernel parameters are obtained by iteratively fitting the ridge of the TFR. The shortcomings of the recursive fitting methods are: 1) the recursive kernel estimation requires the initialized TFR with fair concentration, or it will fail for the TFR of poor concentration and 2) since only one fixed kernel is adopted as each transform perform, it is unable to handle the multicomponent signal, whose components are modulated by different FM sources. A solution is to mask the noninterested components [31], [32] or filter the spectral-separable signal in frequency domain [33] to obtain the interested component. However, these methods will fail when the components are inseparable in the TF plane or frequency domain. Therefore, it is necessary to obtain the proper kernel parameters of the parameterized TF transform without considering the TFR, especially for the multicomponent signal, whose components are modulated by different FM sources. The kernel parameter estimation can be realized by estimating the parameters of the phase model of the signal. For example, for polynomial phase signals, maximum likelihood estimation is a simple method [35]. However, it usually requires the multidimensional search and could fall into local maximum. To reduce the searching dimension, a TF rate representation based on cubic phase function was developed, but it cannot deal with the high-order FM signal or the signal with the time-varying amplitude [35]. For the high-order polynomial phase signal, high-order ambiguity function (HAF)-based method is an order-recursive algorithm to estimate the phase parameters. However, it will be interfered by the cross-terms if the phases of different components have the same parameter [36]. Besides, the estimation error of the higher order phase will pass down to the estimated parameter of the lower order phase. To avoid the cross-term interference, product high-order ambiguity function (PHAF)-based estimation method involves multilag to improve the performance of the HAF-based method both in terms of removing the identifiability problem and its noise rejection capabilities [37]. Nevertheless, the PHAF-based method can be jeopardized by the poorly selected lags, and it still inherits the error propagation problem of the HAF-based method. Moreover, all these parameter estimation methods are effective at high signal-to-noise rate (SNR). As the parameterized TF transform can only catch the one component in one execution, there is an additional task of separating components. Empirical mode decomposition (EMD) [38], [39] is a noted approach to decompose the signal into a series of intrinsic mode function (IMF), though it cannot guarantee each IMF to be the monocomponent signal. Furthermore, the EMD suffers from the problems of mode aliasing and end effect. Matrix algebraic separation algorithm [40] was proposed to separate the two-component signal based on a slight different in their periodicity. However, it is not suitable either when the components have different phase in each period or when the signal is masked by the noise. As the demodulated component can be more easily separated, generalized demodulation was proposed [41] [43] to facilitate the component separation. Similarly, the demodulation kernel is computed by fitting the ridge extracted from a TFR, so the concentration of the TFR determines the performance of the generalized demodulation. In this paper, we introduce a novel method that enables the parameterized TF transform to generate the well-concentrated TFR for both the monocomponent signal and a wide class of multicomponent FM signals, whose components are modulated by either the same or different sources. The proposed method contains two aspects: 1) estimating kernel parameters based on spectrum concentration index (SCI) and 2) separating components and assembling the parameterized TFRs of the separated components. An advantage of the proposed method is that it avoids the dependence of the TFR while estimating the parameters. Moreover, it is effective at low SNR. The validity and practical utility of the proposed method are demonstrated by both the simulated and real signals. The results show that it outperforms the traditional TF methods in providing the TFR of the improved concentration for various multicomponent FM signals. The remainder of this paper is organized as follows. Section II introduces theoretical background of the SCI-based parameter estimation and demonstrates it by the monocomponent signal. Section III details the process of the SCI-based method in the analysis of the multicomponent signal whose components are modulated by either the same or the different sources. Section IV demonstrates the validity and practical

3 YANG et al.: APPLICATION OF PARAMETERIZED TF ANALYSIS 3171 utility of the proposed method by both the simulated and real signals. Section V draws the conclusion. II. THEORETICAL BACKGROUND A. Spectrum Concentration Index A noisy signal can be modeled as M s(t) = s k (t) + w(t) (1) k=1 where w(t) is complex white Gaussian noise of power σ 2 and mean of 0. M stands for the components number of the multicomponent FM signal. The analytic signal of the kth FM component s k (t) L 2 (R), through Hilbert transform, is defined as follows: z s,k (t) = s k (t) + jh[s k (t)]. The analytic signal considered in this paper conforms to the model as z(t) = Ae j φ(t)dt = A(t)e j[ω ct+ ϕ(t)dt]. (2) The IF of the signal is ω k (t) = φ k (t) = ω c,k + ϕ k (t), in which ω c denotes carrier frequency and ϕ(t) is modulation law. A > 0 is assumed to be time-varying amplitude of the kth component that changes much more slower than the IF. First, we define a demodulation operator as (t; P) = e j γ(t;p)dt (3) where γ(t; P) is a differentiable kernel function with the parameter set of P. Then, we propose the SCI as SCI(P) = E [ G r (ω; P) 4] (4) with G r (ω; P) = G s,k (ω; P) + G w (ω; P) k = k = k F [ z s,k (t) (t; P) ] + F[s w (t) (t; P)] A k [ e j ω c,k t+ ϕ k (t)dt ] γ(t;p) e jωt dt +F [ s w (t) (t; P) ] (5) where E(*) stands for expectation and F(*) denotes Fourier transform. Theorem 1: The SCI in (4) will reach the local maximum as long as ϕ i (t) = γ(t; P). Proof: According to (5), G s,k (ω; P) is the Fourier transform of the kth component shifted by the time-varying frequency of ϕ k (t) γ(t; P). The shifted component remains the same energy with the original signal based on Parseval s theorem as Gs,k (ω; P) F 2 [ dω = zs,k (t) (t; P) ] 2 dω zs,k = (t) (t; P) 2 dt = z s,k (t) 2 dt = A k 2 = I k. (6) Discretely, G s,k (ω; P) can be given in the form as G s,k (ω; P) = B k δ(ω ν k ) (7) where ν k = {ω} uk [ω a,ω b ] stands for the sampled frequencies of the component between ω a and ω b, and ω a ω c,k + ϕ k (t) γ(t; P) ω b. u k N denotes the number of the elements in ν k and N is the number of the signal samples. In order to explain the SCI in different scenarios, B k is assumed to be the constant and it satisfies the Parseval s theorem as I k B k = = A k. (8) u k uk Thus, the expectation of the first term in the right-hand side of (5) can be obtained as E [ G s,k (ω; P) ] = B k. (9) k The expectation of the second term in the right-hand side of (5) equals the zero, so (4) becomes SCI(P) = E [ G r (ω; P) 4] = ( ) 2 Ik (10) u k k where I k /u k is denoted as the mean energy of the signal in the frequency dimension. From (10), it can be seen that the expectation of G r (ω; P) only depends on I k /u k instead of the noise. Thus, the SCI is determined by the mean energy in frequency domain with respect to the kth demodulated component. According to the Parseval s theorem, the mean energy will be smaller when the band of the signal becomes broader. Thus, the SCI indicates the spectrum concentration degree of the multicomponent FM signal with respect to the interested component. Next, the discussion about the maximum of the SCI in three cases is provided. Case 1: k = 1 ( ) I 2 SCI(P) = I 2, u 1. (11) u Only when γ(t; P) = ϕ(t), andu =1, then SCI = I 2. It means that the energy of the monocomponent signal is concentrated around the carrier frequency after the demodulation. Thus, only when the demodulation operator has the best matched kernel, i.e., γ(t; P) = ϕ(t), does the maximum of the SCI appear. Case 2: k > 1, and ϕ k (t) = ϕ(t), fork = 1,...,M M ( ) 2 Ik SCI(P) = Ik 2, u k 1. (12) k=1 u k Only when γ(t; P) = ϕ(t), i.e., u k = 1, for k = 1,...,M, then SCI = k I 2 k. When the multicomponent signal has the single FM law, the different components will have the same FM law. The well demodulation is able to make the energy of all components concentrate around their carrier frequencies, respectively. It implies that only when the kernel of the demodulation operator matches the FM law, does the SCI reach the maximum. k

4 3172 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 Case 3: k > 1, and ϕ k (t) = ϕ l (t), fork, l = 1,...,M M ( ) 2 Ik SCI(P) = Ii 2 + ( ) 2 Ik, u k 1. (13) u k u k k=1 k =i When γ(t; P) = ϕ i (t), for i (1,...,M), u i = 1, then SCI = Ii 2 + k =i (I k/u k ) 2. When the multicomponent signal has multiple FM sources, each component will have a unique FM law. It suggests that the SCI will obtain the local maximum as long as the kernel of the demodulation operator matches the IF of the each component. Ideally, if the other components cover much broader bandwidth after the demodulation with respect to the interested component, i.e., u k 1, k = i, the SCI will become SCI = Ii 2. (14) In such situation, the SCI is only determined by the spectral concentration of the ith component after the demodulation. It should be mentioning that in Case 3, we assume that the IFs of the components are quite different from the each other. Meanwhile, the mean energy of any component in the frequency domain after the demodulation is assumed to be less than the total energy of the interested component. In the above three cases, the optimization of the SCI is able to obtain the parameters with respect to the each component. However, when the sum of the mean energy of other components after the demodulation is more than the total energy of the interested component, searching the maximum of SCI will not guarantee the accurate parameters estimation for the interested component. This is because that the local maximum of the SCI for the weaker component can be suppressed by the stronger component. Especially, for the signal with the components whose IF rates are slightly different from the each other, the optimal SCI might result in demodulating both the unwanted components and the interested component partially. In this case, it is recommended to adopt the second case to deal with the signal, i.e., considering it as the multicomponent signal whose components are modulated by the same source. In addition, since the SCI is consisted of the kernel function of the demodulation operator, and the kernel function can be formulated by the polynomial, spline, Fourier series, and so forth. As long as the formulation of the kernel is able to approximate the IF of the signal, the SCI-based method will be effective to determine the parameters of such kernel function for the considered signal. Therefore, the parameterized TF transforms mentioned in Section I, i.e., PCT, SCT, and GWT, are potential candidates. While using different kernel functions to construct the SCI, the corresponding parameterized TF transform will be adopted to generate the TFR for the analyzed signal. Hereinafter, the polynomial kernel and the PCT are applied to demonstrate the proposed method. B. PCT and Parameter Estimation for Monocomponent Signal In real practices, it constantly confronted the situations where the IF of the signal is unknown or difficult to determine in advance. An alternative solution is to select a phase model for the analyzed signal from a group of the determined mathematical models. The selected model usually is required to approximate the characteristics of the IF with the reasonable accuracy. In this paper, we apply the polynomial function to construct the phase of the signal as j z s,k (t) = A k (t)e ( 2π f c t+2π n+1 k=2 ) c k 1 k t k (15) where {c 1,...,c n } are parameters of the polynomial phase. According to Weierstrass approximation theorem, it is reasonable to use the polynomial phase signal to characterize a large number of the FM signals. For the signal in (15), the PCT was proposed to generate the well concentrated TFR. The PCT is given by [27] PCT s (m,ω,α 1,...,α n ; σ) = z s (t) α R 1,...,α n (t) α Q 1,...,α n (t, m)g σ (t m)e jωt dt (16) with and R α 1,...,α n (t) = e j Q α 1,...,α (t, m) = e j n+1 1 k α k 1(t) k k=2 (17) n+1 α k 1 m (k 1) t k=2 (18) where α R 1,...,α n (t) and α Q 1,...,α n (t, m) are rotation operator and shifting operator, respectively. {α 1,...,α n } are the transform parameters of the PCT. g(t) L 2 (R) denotes a nonnegative, symmetric, and normalized real window, usually taken as the Gaussian function expressed as g σ (t) = 1 e 1 2 (t/σ)2. (19) 2πσ When {α 1,...,α n } = {c 1,...,c n }, the PCT will offer the minimum resolution and the best concentration for the analyzed signal. Conventionally, these parameters are attained by iteratively fitting the extracted ridge from the TFR with the polynomial function. However, it is difficult to determine the proper initial TFR without prior knowledge. Even though the iterative fitting is able to reduce the random errors caused by the noise, the poor concentration of the initial TFR or the significant interference of the noise could destroy the estimation process. Therefore, it is important to obtain the proper parameters for the PCT without considering the TFR, especially at low SNR. Accordingly, we apply the polynomial kernel into the SCI as n+1 γ(t; α 1,...,α n ) = α k 1 t (k 1). (20) k=2 According to the first case of the SCI, the optimization is a single-objective problem, particularly { α 1,..., α n }=arg max SCI(α 1,...,α n ). (21) ω It worth mentioning that choosing the order of the approximating polynomial is an open issue. In this paper, we determine the order of the polynomial function by trial. Usually,

5 YANG et al.: APPLICATION OF PARAMETERIZED TF ANALYSIS 3173 Fig. 2. TFR obtained by the PCT with SCI-based parameter estimation. (a) PCT. (b) IF estimation and approximation. Fig. 1. TFR obtained by iterative PCT and ridge approximation. (a) Initial PCT (STFT). (b) IF estimation and approximation. (c) Second PCT. (d) IF estimation and approximation. (e) Third PCT. (f) IF estimation and approximation. a relatively large order can be adopted for the polynomial function at the beginning. If the IF of the signal can be well fitted with a polynomial function of the relative low order, then the estimated high-order polynomial coefficients would be close to zero. The nonlinear optimization problem in (21) can be realized by various techniques, such as genetic algorithm [44], neural network [45], and particle swarm optimization (PSO) [46]. As fewer parameters to be adjusted in the PSO, it is applied for the SCI optimization. Next, the estimated parameters based on the SCI optimization can be used by the PCT. Traditionally, the kernel parameters can be estimated through the recursive approximation of the TFR ridge in the case of either the monocomponent signal or the multicomponent signal whose the components are modulated by the same source. To illustrate the differences between the parameters estimation based on the SCI optimization and the TFR ridge approximation, a polynomial phase signal with exponential decay amplitude is taken as an example, particularly z s (t) = 2e ( 0.08t) e j2π(30t+0.5t 2 0.4t 3 /3) (22) whose IF law is f (t) = 30 + t 0.4t 2. The amplitude decays exponentially with the rate of The signal is masked by the noise and the SNR is 10 db. The parameters to be estimated are {α 1,α 2 }={1, 0.4}. Fig. 1 shows the process of the iteratively fitting the extracted ridge from the TFR. The initial PCT is essentially the STFT as its parameters are set to zeroes. Hereinafter, without special note, the window length is set to 512. After three iterations, the parameters are estimated as {ˆα 1, ˆα 2 }= { , }, which are significantly different with the real values. From Fig. 1, it can be seen that the TFR becomes to spread more and the extracted ridge is far away from the real IF after the iterative process. It implies that this method fails when the initial TFR is of poor concentration and the SNR is low. On the other hand, Fig. 2 shows that the TFR generated by PCT with the parameters estimated based on the SCI optimization. The estimated results are {ˆα 1, ˆα 2 }={0.9966, }, which is much more accurate than the iterative fitting method. In the PSO, the size of the population is set to 30, and the search region is set to [ 10 10] and [ 1 1]forα 1,α 2, respectively. The global maximum is found at the 87 generation, and it totally takes s. It can be seen that the resulting TFR reveals the real TF feature of the signal and the extracted ridge is closely near to the real IF trajectory. III. PARAMETERIZED TF ANALYSIS OF MULTICOMPONENT FM SIGNAL According to the model of the multicomponent FM signal given by (1), it becomes the superposition of multiple polynomial phase signals when the polynomial phase is applied as z r (t) = k j A k (t)e [ 2π f c,k +2π n 1 i=0 ] c i,k i+1 (t)i+1 + w(t) (23) in which the IF of the kth component is f k (t) = f c,k + n 1 i=0 c i,k(t) i. A. SCI-Based Parameter Estimation for FM Components With the Same FM Law In some applications, the components of the multicomponent FM signal are encoded in the distinctive carrier frequency by the same FM source. For example, the changes in the running condition, such as the speed and load, could cause the modulation in the vibration responses of machinery.

6 3174 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 The vibration of the rotary machine during the speed varying process is a typical FM process, because normally the rotating frequency and its multiples dominate the vibration signal, and these harmonics follow the rotating frequency during the speed varying process. Thus, the vibration responses during such transitional processes contain the multicomponent characteristic and the components have the same FM law. For such signal, the proposed method falls into the second case of the SCI, where the optimization is essentially a single-objective problem as in (21). B. SCI-Based Parameter Estimation for FM Components With Different FM Laws In this section, we focus on the situations when the signal is resulted from the different FM sources. For example, in speech processing, the speech signals are modeled as a superposition of the time-varying resonances and the each FM component is modeled as a single resonance [47]. For high-frequency radar in the multiple targets tracking application, the clutter is modeled to be the multicomponent signal with different FM laws [48]. In rotating machine, there are multiple vibration excitations in the machinery. The transfer path of a vibration signal from the each excitation source to a vibration transducer usually consists of multiple components, and each component has its own dynamic characteristics and operational condition [43]. In these situations, the multicomponent signal is composed of the FM components with the different FM laws. In this case, the third case of the SCI is considered, and its optimization is a typical multiobjective problem, particularly { P 1,..., P k }=arg max ω SCI (k)(p) (24) where P k denotes the estimated parameters for the kth component. Equation (24) can be realized by the PSO designed particularly for multiobjective problem [49], [50]. Since the parameterized transform cannot use multiple kernels at one time, we develop a practical method for the application of parameterized TF analysis on the multicomponent FM signals. It mainly contains two related tasks: 1) estimating parameters based on the SCI and 2) separating the component and assembling the parameterized TFRs of the separated components. In the first task, the parameters of the FM law with respect to one component are estimated by the SCI optimization. The complete demodulation results in the best concentration in the spectrum at the carrier frequency of the component. Meanwhile, the other components will be severely distorted and their energies will be spread all over the spectrum. As illustrated in Fig. 3, there is a two-component signal, whose real TF signature is shown in Fig. 3(a) and the bandwidth of components are overlapped in the frequency domain [Fig. 3(b)]. After the demodulation with respect to the first component, the spectrum shows the higher peak and the narrower bandwidth at the carrier frequency of this component, whereas the second component has the lower peak and broader bandwidth [Fig. 3(c)]. Therefore, the first component can be easily identified. By taking the advantage of such feature in the spectrum after the demodulation, the second task can be Fig. 3. Demodulation with respect to one component. (a) Real TF signature. (b) Predemodulated spectrum. (c) Spectrum after demodulation. developed based on the procedure of demodulation filtering recovering. The component separation in the second task is mainly designed to avoid the aforementioned two scenarios that might cause the malfunction of the optimization: 1) the energy of the components is of great difference and 2) the IFs of the components are of slight difference. To sum up, the proposed method includes the following steps: 1) estimating the parameters with respect to a component by optimizing the SCI; 2) demodulating the component with the estimated parameters; 3) filtering the signal with a bandpass filter that covers the carrier frequency of the interested component and occupies as small bandwidth as possible; 4) recovering the filtered component with the estimated parameters; 5) performing the PCT on the recovered component with the estimated parameters; 6) conducting the above loop until only one component is left; 7) combining all the TFRs obtained by the PCT. Especially, in the step 3), the carrier frequency can be estimated simply by the peak of the demodulated spectrum or the techniques developed in [51] and [52]. To illustrate this procedure, a two-component signal is considered as z s (t) = z 1 (t) + z 2 (t) = e j2π n 1 i=0 c 1,i i+1 t i+1 + e n 1 j2π i=0 c 2,i i+1 t i+1 (25) where {c 1,1,...,c 1,n } and {c 2,1,...,c 2,n } are set to c 1,0 = c 2,0 = 0, c 1,1 = 8, c 1,2 = 8, c 2,1 = 2, c 2,2 = 4, c 1,i = c 2,i = 0,(i > 2), particularly z s (t) = e j (16πt+8πt2) + e j (4πt+4πt2 ) (26) whose component IFs are f 1 (t) = 8 + 8t and f 2 (t) = 2 + 4t, respectively. In this case, the SCI optimization turns to be the

7 YANG et al.: APPLICATION OF PARAMETERIZED TF ANALYSIS 3175 to be acquired with prior knowledge or via nonparameterized TF transform, such as the STFT. Even if the proposed method seems to be incapable of analyzing the signal with the intersected components, the unwanted components will be spread while generating the TFR concentrated at the interested component with the properly estimated parameters. As a result, the proposed method still manages to provide the improved concentration for such signals. Fig. 4. TFRs in several steps of the proposed method. (a) STFT of the original signal. (b) Demodulation of the second component. (c) Filtered component. (d) PCT of the recovered component. (e) PCT of the first component. (f) Assembled PCT. two-objective optimization problem as { } α1,1,..., α 1,n ; α 2,1,..., α 2,n =arg max SCI (2) (α 1,...,α n ). ω (27) Fig. 4 shows main steps of the proposed method. All results are displayed via the TFRs for the sake of the consistency. It is shown in Fig. 4(a) that the TFR generated by the STFT is too smeared to characterize both the IF of the two components. With the estimated parameter, the second component is demodulated and it is well concentrated at the carrier frequency along the time axis in the TFR as shown in Fig. 4(b). The demodulated component is then filtered to be a stationary signal, and Fig. 4(c) shows its TFR generated by the STFT. Next, the filtered component is recovered by modulating with the estimated parameters. For the recovered signal, the TFR generated by the PCT obtains better concentration [Fig. 4(d)]. Fig. 4(e) shows the PCT of the reminder of the signal after filtering. With the estimated parameters corresponding to the first component, the resulting TFR attains the better concentration for the second component. Finally, the assembled PCT is well concentrated at both the IFs of the two components as shown in Fig. 4(f). It should be mentioned that the number of the components is assumed IV. COMPARISON AND VALIDATION In this section, the effectiveness of the proposed method is verified on both the simulated and real signals. First, a two-component polynomial phase signal with the exponential decay amplitude is given by z s (t) = 2e ( 0.08t) e j2π(30t 5t t 3 /3) +2e ( 0.02t) e j2π(25t 5t t 3 /3) (28) whose IF laws of the two component are f 1 (t) = 30 10t + 0.4t 2 and f 2 (t) = 25 10t + 0.4t 2, respectively. The sampling frequency is 100 Hz and the sampled points are The corresponding amplitudes decay exponentially with the rate of 0.08 and 0.02, respectively. The signal is masked by the noise and the SNR is 10 db. The parameters to be estimated are {α 1,α 2 }={ 10, 0.4}. The parameters estimated by the SCI optimization are {ˆα 1, ˆα 2 } = { , }. Inthe PSO, the size of the population is set to 20, and the search region is set to [ 20 0] and [ 1 1]forα 1,α 2, respectively. For the comparison purpose, several traditional TF analysis methods are considered, i.e., the STFT, an adaptive STFT (ASTFT), the WT, the WVD, and the reassigned pseudo- WVD. In particular, the ASTFT applies an adaptation rule of the maximum correlation criterion [53] and the PCT uses the parameters estimated by the SCI optimization. Fig. 5 shows the TFRs obtained by the different methods. In Fig. 5(a), the severely smeared TFR generated by the STFT can neither reveal the TF signatures of the each component nor distinguish the two components. This is because it uses the constant window so as to obtain the constant TF resolution. The ASTFT adjusts the window length based on the rate that the IF varies to achieve the adaptive resolution. However, in Fig. 5(b), the concentration of the TFR is still not satisfying. Moreover, it is even worse when the signal attenuates significantly and the noise prevents to select the proper window length. Comparatively, the WT uses the window length inversely proportional to the frequency. However, it has the worse concentration at the higher frequency where the IF varies faster than at the lower frequency. As a representative of the bilinear TF transform, the WVD has the best concentration. Nevertheless, the cross-terms destroy the readability of the resulting TFR, especially with the presence of noise. To overcome this shortcoming, the reassigned pseudo-wvd improves readability using reassignment technique. However, it cannot avoid the interference of the cross-term completely. On the contrary, the TFR generated by the PCT with the proper parameters shows the better concentration, so the TF pattern of the signal is accurately characterized and the two components are well identified.

8 3176 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 Fig. 6. Signal given by (29). (a) Waveform. (b) Real TF signature. Fig. 5. TFRs of the multicomponent FM signal with single FM law. (a) STFT. (b) ASTFT. (c) WT. (d) WVD. (e) Reassigned pseudo-wvd. (f) PCT. Second, a three-component FM signal is considered as z s (t) = z 1 (t) + z 2 (t) + z 3 (t) = 1.3e 0.3t e j (200πt 24πt2 +2πt 3 ) +0.5e 0.3t e j (20πt+18πt2) + 5e j (10πt+4πt2 ) (29) whose component IFs are f 1 (t) = t + 3t 2, f 2 (t) = t and f 3 (t) = 5 + 4t, respectively. Two components have the time-varying amplitude and the amplitude of the third component is constant. The signal is masked by the noise with the SNR of 10 db. The sampling frequency is 260 Hz and the sampled points are Its waveform and the real TF signature are shown in Fig. 6. In the PSO, the size of the population is set to 30, and the search region is set to [ 30 30] and[05]forα 12,α 22,α 23 and α 13,α 23,α 33, respectively. The parameters estimated by the SCI optimization are listed in Table I. It implies that the SCI optimization-based parameter estimation works well at low SNR and is effective for the signal with time-varying amplitude. For the comparison purpose, Fig. 7 gives the TFRs generated by the different methods. The STFT shows the poor concentration due to the constant resolution as long as the window is fixed, so it fails for the components having the time-varying IFs [Fig. 7(a)]. In the ASTFT, the stronger component dominates the selection of the window length, and the improvement in terms of the concentration is also not Fig. 7. TFR of the signal in (29). (a) STFT. (b) ASTFT. (c) WT. (d) WVD. (e) Reassigned pseudo-wvd. (f) Assembled PCT. satisfied [Fig. 7(b)]. The WT, adjusting the window length inversely toward the center frequency of the window, shows the extremely poor concentration at high frequency due to the worse frequency resolution [Fig. 7(c)]. The reassigned pseudo-wvd sustains the concentration of WVD, yet it cannot suppress the cross-terms between the each components. On the other hand, the assembled PCT shows the best concentration so the IFs of all components are characterized accurately. The third example is the real bat echolocation signal [54]. The sampling frequency is Hz and the sampled points are 400. The size of the analysis window is set to 256. Similarly, the STFT, ASTFT, WT, WVD, pseudo-wvd, and reassigned pseudo-wvd are also considered for the sake

9 YANG et al.: APPLICATION OF PARAMETERIZED TF ANALYSIS 3177 TABLE I PARAMETER ESTIMATION BASED ON SCI OPTIMIZATION FOR THE SIGNAL IN (29) TABLE II PARAMETER ESTIMATION BASED ON SCI OPTIMIZATION FOR BAT ECHOLOCATION SIGNAL Fig. 8. TFR of bat echolocation signal. (a) STFT. (b) ASTFT. (c) WT. (d) WVD. (e) Reassigned pseudo-wvd. (f) Assembled PCT. of comparison. The TFRs obtained by these methods are shown in Fig. 8, respectively. In the PSO, the size of the population is set to 30, and the search region is set to [ 10 0], [0 1], and [ 1 1] for α i1,α i2,α i3, i = 1, 2, 3, respectively. It can be seen that the STFT applies the constant TF resolution that is not suitable to characterize the timevarying IFs of the components. The ASTFT selects the smaller window at the slow change of the IF, but it does not improve the concentration significantly. The WT uses the scaled TF resolution, which leads to the worse frequency resolution at the higher frequency. The two WVD-related methods show the best concentration at the autoterm, but both of them are interfered by the cross-terms. In Fig. 7(f), the assembled PCT shows the enhanced concentration and reveals the real IFs of the three FM components. Table II lists the parameters estimated by the SCI optimization. In many practical applications, it is usually necessary to closely investigate some components of the multicomponent signal to acquire their TF information. The last example is the vibration signal of hydraulic turbine during cast-down stage. The sampling frequency is 16 Hz and the sampled points are The size of the analysis window is set to 256. Fig. 9 shows the TFRs generated by the STFT, ASTFT, WT, WVD, reassigned pseudo-wvd, and PCT. In the PSO, the size of the population is set to 30, and the search regions are set to [ 1 1] for all parameters to be estimated. As the fundamental component and its three harmonics cover the most vibration energy of the hydraulic turbine, the proposed method is used to obtain these components. Table III provides the estimated kernel parameters based on the SCI optimization with respect to the each component. In Fig. 9(a), the STFT cannot obtain the TFR concentration for the fundamental component as it varies with the time 10 s. In Fig. 9(b), although the ASTFT improves the TFR concentrated for the fundamental component, it fails to adequately characterize the harmonics. The WT shares the same problem with the ASTFT, and its TFR shows worse frequency resolution at high frequency [as shown in Fig. 9(c)]. Due to the signal having the nonlinearity and multiple components, the WVD introduces plenty of cross-terms that damages the readability of the TFR, as shown in Fig. 9(d). In Fig. 9(e), it can be seen that the reassigned pseudo-wvd suppresses the cross-terms caused by the nonlinearity of the signal, though the cross-terms between the components are still remained. Comparatively, the assembled PCT is able to obtain the TFR of well concentration for the interested components, i.e., fundamental frequency component, 1 harmonics, 2 harmonics, 3 harmonics, and 4 harmonics. The TF features of these separated components

10 3178 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 TABLE III PARAMETER ESTIMATION FOR VIBRATION SIGNAL OF HYDRAULIC TURBINE separated components, it is able to improve the concentration of the TFR for both the monocomponent signal and a wide class of multicomponent FM signals whose components are modulated by either the same or the different sources. The tests on both the simulated and the real signals show that the proposed method works well for the above signals, especially at the low SNR. In addition, compared with the STFT, ASTFT, WT, WVD, and reassigned WVD, the proposed method outperforms in obtaining the TFR with the better concentration. Fig. 9. TFR of hydraulic turbine vibration signal. (a) STFT. (b) Adaptive STFT. (c) WT. (d) WVD. (e) Reassigned pseudo WVD. (f) Assembled PCT. can be further used for the instantaneous speed estimation or amplitude extraction of the harmonics in the applications of fault diagnosis, and so forth. V. CONCLUSION In this paper, we proposed a novel method for applying the parameterized TF transform, i.e., PCT, to analyze multicomponent FM signals. By executing the two related tasks: 1) estimating kernel parameters based on SCI and 2) separating components and assembling the parameterized TFRs of the REFERENCES [1] B. Barkat and B. Boashash, A high-resolution quadratic time-frequency distribution for multicomponent signals analysis, IEEE Trans. Signal Process., vol. 49, no. 10, pp , Oct [2] E. R. Thompson, B. D. Simpson, and N. Iyer, Multicomponent signal detection: Tones in noise and amplitude modulation detection, J. Acoust. Soc. Amer., vol. 133, no. 5, p. 3285, [3] Y. Pantazis, O. Rosec, and Y. Stylianou, Adaptive AM FM signal decomposition with application to speech analysis, IEEE Trans. Audio, Speech, Lang. Process., vol. 19, no. 2, pp , Feb [4] A. Aïssa-El-Bey, K. Abed-Meraim, and Y. Grenier, Blind separation of underdetermined convolutive mixtures using their time frequency representation, IEEE Trans. Audio, Speech, Lang. Process., vol. 15, no. 5, pp , Jul [5] Y. Amirat, V. Choqueuse, and M. Benbouzid, EEMD-based wind turbine bearing failure detection using the generator stator current homopolar component, Mech. Syst. Signal Process. vol. 41, no. 1, pp , [6] E. C. Lau and H. W. Ngan, Detection of motor bearing outer raceway defect by wavelet packet transformed motor current signature analysis, IEEE Trans. Instrum. Meas., vol. 59, no. 10, pp , Oct [7] M. Yeary et al., Multichannel receiver design, instrumentation, and first results at the national weather radar testbed, IEEE Trans. Instrum. Meas., vol. 61, no. 7, pp , Jul [8] C. Gu, C. Li, J. Lin, J. Long, H. Jiangtao, and R. Lixin, Instrumentbased noncontact Doppler radar vital sign detection system using heterodyne digital quadrature demodulation architecture, IEEE Trans. Instrum. Meas., vol. 59, no. 6, pp , Jun [9] R. Naidoo and P. Pillay, A new method of voltage sag and swell detection, IEEE Trans. Power Del., vol. 22, no. 2, pp , Apr [10] S. S. Mahmoud, Z. M. Hussain, I. Cosic, and Q. Fang, Time-frequency analysis of normal and abnormal biological signals, Biomed. Signal Process. Control, vol. 1, no. 1, pp , [11] L. Rankine, M. Mesbah, and B. Boashash, IF estimation for multicomponent signals using image processing techniques in the time-frequency domain, Signal Process., vol. 87, no. 6, pp , [12] I. Orovic, M. Orlandic, S. Stankovic, and Z. Uskokovic, A virtual instrument for time-frequency analysis of signals with highly nonstationary instantaneous frequency, IEEE Trans. Instrum. Meas., vol. 60, no. 3, pp , Mar [13] P. F. Pai, Online tracking of instantaneous frequency and amplitude of dynamical system response, Mech. Syst. Signal Process., vol. 24, no. 4, pp , 2010.

11 YANG et al.: APPLICATION OF PARAMETERIZED TF ANALYSIS 3179 [14] M. A. Platas-Garza and J. A. de la O Serna, Dynamic harmonic analysis through Taylor Fourier transform, IEEE Trans. Instrum. Meas., vol. 60, no. 3, pp , Mar [15] D. Marelli, M. Aramaki, R. Kronland-Martinet, and C. Verron, Timefrequency synthesis of noisy sounds with narrow spectral components, IEEE Trans. Audio, Speech, Lang. Process., vol. 18, no. 8, pp , Nov [16] W. T. Kuang and A. S. Morris, Using short-time Fourier transform and wavelet packet filter banks for improved frequency measurement in a Doppler robot tracking system, IEEE Trans. Instrum. Meas., vol. 51, no. 3, pp , Jun [17] L. V. White and B. Boualem, On estimating the instantaneous frequency of a Gaussian random signal by use of the Wigner-Ville distribution, IEEE Trans. Audio, Speech, Lang. Process., vol. 36, no. 3, pp , Mar [18] F. Auger and P. Flandrin, Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans. Signal Process., vol. 43, no. 5, pp , May [19] F. Peng, D. Yu, and J. Luo, Sparse signal decomposition method based on multi-scale chirplet and its application to the fault diagnosis of gearboxes, Mech. Syst. Signal Process., vol. 25, no. 2, pp , [20] L. Angrisani, P. Daponte, and M. D Apuzzo, A measurement method based on time-frequency representations for testing GSM equipment, IEEE Trans. Instrum. Meas., vol. 49, no. 5, pp , Oct [21] L. Angrisani and M. D Arco, A measurement method based on a modified version of the chirplet transform for instantaneous frequency estimation, IEEE Trans. Instrum. Meas., vol. 51, no. 4, pp , Aug [22] J. Li and H. Ling, Application of adaptive chirplet representation for ISAR feature extraction from targets with rotating parts, IEE Proc.- Radar, Sonar Navigat., vol. 150, no. 4, pp , Aug [23] Z. Zeng, F. Wu, L. Huang, F. Liu, and J. Sun, The adaptive chirplet transform and its application in GPR target detection, Appl. Geophys., vol. 6, no. 2, pp , [24] Q. Yin, S. Qian, and A. Feng, A fast refinement for adaptive Gaussian chirplet decomposition, IEEE Trans. Signal Process., vol. 50, no. 6, pp , Jun [25] Y. Lu, R. Demirli, G. Cardoso, and J. Saniie, A successive parameter estimation algorithm for chirplet signal decomposition, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 53, no. 11, pp , Nov [26] Weierstrass s Theorem [Online]. Available: com/weierstrassstheorem.html [27] Z. K. Peng, G. Meng, F. L. Chu, Z. Q. Lang, W. M. Zhang, and Y. Yang, Polynomial chirplet transform with application to instantaneous frequency estimation, IEEE Trans. Instrum. Meas., vol. 60, no. 9, pp , Sep [28] Y. Yang, Z. K. Peng, G. Meng, and W. M. Zhang, Spline-kernelled chirplet transform for the analysis of signals with time-varying frequency and its application, IEEE Trans. Ind. Electron., vol. 59, no. 3, pp , Mar [29] L. Angrisani, M. D Arco, R. Schiano Lo Moriello, and M. Vadursi, On the use of the Warblet transform for instantaneous frequency estimation, IEEE Trans. Instrum. Meas., vol. 54, no. 4, pp , Aug [30] Y. Yang, Z. K. Peng, G. Meng, and W. M. Zhang, Characterize highly oscillating frequency modulation using generalized Warblet transform, Mech. Syst. Signal Process., vol. 26, pp , Jan [31] V. G. Reju, S. N. Koh, and Y. Soon, Underdetermined convolutive blind source separation via time-frequency masking, IEEE Trans. Audio, Speech, Lang. Process., vol. 18, no. 1, pp , Jan [32] D. Grimaldi, S. Rapuano, and L. De Vito, An automatic digital modulation classifier for measurement on telecommunication networks, IEEE Trans. Instrum. Meas., vol. 56, no. 5, pp , Oct [33] E. Van der Ouderaa, J. Schoukens, and J. Renneboog, Peak factor minimization using a time-frequency domain swapping algorithm, IEEE Trans. Instrum. Meas., vol. 37, no. 1, pp , Mar [34] S. Gazor and R. R. Far, Adaptive maximum windowed likelihood multicomponent AM-FM signal decomposition, IEEE Trans. Audio, Speech, Lang. Process., vol. 14, no. 2, pp , Mar [35] P. O Shea, A fast algorithm for estimating the parameters of a quadratic FM signal, IEEE Trans. Signal Process., vol. 52, no. 2, pp , Feb [36] B. Porat and B. Friedlander, Accuracy analysis of estimation algorithms for parameters of multiple polynomial-phase signals, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., May 1995, pp [37] S. Barbarossa, A. Scaglione, and G. B. Giannakis, Product highorder ambiguity function for multicomponent polynomial-phase signal modeling, IEEE Trans. Signal Process., vol. 46, no. 3, pp , Mar [38] R. Yan and R. X. Gao, Hilbert Huang transform-based vibration signal analysis for machine health monitoring, IEEE Trans. Instrum. Meas., vol. 55, no. 6, pp , Dec [39] A. Y. Goharrizi and N. Sepehri, Internal leakage detection in hydraulic actuators using empirical mode decomposition and Hilbert spectrum, IEEE Trans. Instrum. Meas., vol. 61, no. 2, pp , Feb [40] B. Santhanam and P. Maragos, Multicomponent AM-FM demodulation via periodicity-based algebraic separation and energy-based demodulation, IEEE Trans. Commun., vol. 48, no. 3, pp , Mar [41] S. Olhede and A. T. Walden, A generalized demodulation approach to time-frequency projections for multicomponent signals, Proc. Roy. Soc. A, Math., Phys. Eng. Sci., vol. 461, no. 2059, pp , [42] Z. Feng, F. Chu, and M. J. Zuo, Time-frequency analysis of timevarying modulated signals based on improved energy separation by iterative generalized demodulation, J. Sound Vibrat., vol. 330, no. 6, pp , [43] J. Cheng, Y. Yang, and D. Yu, The envelope order spectrum based on generalized demodulation time-frequency analysis and its application to gear fault diagnosis, Mech. Syst. Signal Process., vol. 24, no. 2, pp , [44] F. M. Janeiro and P. M. Ramos, Impedance measurements using genetic algorithms and multiharmonic signals, IEEE Trans. Instrum. Meas., vol. 58, no. 2, pp , Feb [45] J. Krabicka, G. Lu, and Y. Yan, Profiling and characterization of flame radicals by combining spectroscopic imaging and neural network techniques, IEEE Trans. Instrum. Meas., vol. 60, no. 5, pp , May [46] H. A. Nguyen, H. Guo, and K. S. Low, Real-time estimation of sensor node s position using particle swarm optimization with log-barrier constraint, IEEE Trans. Instrum. Meas., vol. 60, no. 11, pp , Nov [47] A. Potamianos and P. Maragos, Speech formant frequency and bandwidth tracking using multiband energy demodulation, J. Acoust. Soc. Amer., vol. 99, no. 6, pp , [48] M. W. Y. Poon, R. H. Khan, and S. Le-Ngoc, A singular value decomposition (SVD) based method for suppressing ocean clutter in high frequency radar, IEEE Trans. Signal Process., vol. 41, no. 3, pp , Mar [49] J.F.Schutte,J.A.Reinbolt,B.J.Fregly,R.T.Haftka,andA.D.George, Parallel global optimization with the particle swarm algorithm, Int. J. Numer. Methods Eng., vol. 61, no. 13, pp , [50] J. Riget and J. S. Vesterstrøm, A diversity-guided particle swarm optimizer-the ARPSO, Dept. Comput. Sci., Univ. Aarhus, Aarhus, Denmark, Tech. Rep. 2, [51] T. K. Moon, The expectation-maximization algorithm, IEEE Signal Process. Mag., vol. 13, no. 6, pp , Nov [52] S. Umesh and D. W. Tufts, Estimation of parameters of exponentially damped sinusoids using fast maximum likelihood estimation with application to NMR spectroscopy data, IEEE Trans. Signal Process., vol. 44, no. 9, pp , Sep [53] H. K. Kwok and D. L. Jones, Improved instantaneous frequency estimation using an adaptive short-time Fourier transform, IEEE Trans. Signal Process., vol. 48, no. 10, pp , Oct [54] Bat Echolocation Chirp [Online]. Available: software/bat-echolocation-chirp Yang Yang received the B.S. and M.S. degrees in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2006 and 2009, respectively, where she is currently pursuing the Ph.D. degree with the Mechanical Engineering Department. She studied at the Intelligent Maintenance System Center, University of Cincinnati, Cincinnati, OH, USA, from 2007 to She is currently with the State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University. Her current research interests include signal processing, system identification, machine health diagnosis, and prognostics.

12 3180 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 12, DECEMBER 2014 Zhike Peng received the B.Sc. and Ph.D. degrees from Tsinghua University, Beijing, China, in 1998 and 2002, respectively. He was a Research Associate with the City University of Hong Kong, Hong Kong, from 2003 to 2004, and a Research Officer with the Cranfield University, Cranfield, U.K. After that, he was with the University of Sheffield, Sheffield, U.K., for four years. He is currently a Distinguished Professor with the State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, China. His current research interests include nonlinear vibration, signal processing and condition monitoring, and fault diagnosis for machines and structures. Wenming Zhang (M 11) received the B.S. degree in mechanical engineering and the M.S. degree in mechanical design and theories from Southern Yangtze University, Wuxi, China, and the Ph.D. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000, 2003, and 2006, respectively. He is currently a Professor with the State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University. He has been involved in the dynamics and control for micro/nanoelectromechanical systems (MEMS/NEMS). His current research interests include nonlinear dynamics and chaos control, nonlinear vibration and control, coupled parametrically excited microresonators, and the reliability analysis and assessment for MEMS/NEMS applications. Xingjian Dong received the B.S. degree in aircraft design engineering and the M.S. degree in solid mechanics from Northwestern Polytechnical University, Xi an, China, and the Ph.D. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 1999, 2002, and 2006, respectively. He is currently a Lecturer with the Shanghai Jiao Tong University. His current research interests include vibration analysis, smart structures, and fatigue analysis of structures. Guang Meng received the Ph.D. degree from Northwestern Polytechnical University, Xi an, China, in He was a Professor and the Director of the Vibration Engineering Institute at the Northwestern Polytechnical University, in From 1989 to 1993, he was a Research Assistant with Texas A&M University, College Station, TX, USA, an Alexander von Humboldt Fellow with Technical University Berlin, Berlin, Germany, and a Research Fellow with New South Wales University, Sydney, Australia. From 2000 to 2008, he was with Shanghai Jiao Tong University, Shanghai, China, as the Cheung Kong Chair Professor, the Associate Dean, and the Dean of the School of Mechanical Engineering. He is currently a Professor and the Director of the State Key Laboratory of Mechanical System and Vibration at the Shanghai Jiao Tong University. His current research interests include dynamics and vibration control of mechanical systems, nonlinear vibration, and microelectromechanical systems.