Novel Method for Vibration Sensor-Based Instantaneous Defect Frequency Estimation for Rolling Bearings Under Non-Stationary Conditions

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Article Novel Method for Vibration Sensor-Based Instantaneous Defect Frequency Estimation for Rolling Bearings Under Non-Stationary Conditions

Dezun Zhao 1, Len Gelman 1, Fulei Chu 2 and Andrew Ball 1,*

1 Centre for Efficiency and Performance Engineering (CEPE), School of Computing and Engineering, University of Huddersfield, Huddersfield HD1 3DH, UK; [email protected] (D.Z.); [email protected] (L.G.) 2 Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China; chufl@mail.tsinghua.edu.cn * Correspondence: [email protected]

Received: 31 July 2020; Accepted: 8 September 2020; Published: 11 September 2020

Abstract: It is proposed a novel instantaneous frequency estimation technology, multi-generalized demodulation transform, for non-stationary signals, whose true time variations of instantaneous frequencies are unknown and difficult to extract from the time-frequency representation due to essentially noisy environment. Theoretical bases of the novel instantaneous frequency estimation technology are created. The main innovations are summarized as: (a) novel instantaneous frequency estimation technology, multi-generalized demodulation transform, is proposed, (b) novel instantaneous frequency estimation results, obtained by simulation, for four types of amplitude and frequency modulated non-stationary single and multicomponent signals under strong background noise (signal to noise ratio is 5 dB), and (c) novel experimental instantaneous frequency estimation − results for defect frequency of rolling bearings for multiple defect frequency harmonics, using the proposed technology in non-stationary conditions and in conditions of different levels of noise interference, including a strong noise interference. Quantitative instantaneous frequency estimation errors are employed to evaluate performance of the proposed IF estimation technology. Simulation and experimental estimation results show high effectiveness of the proposed estimation technology.

Keywords: vibration sensor-based fault diagnosis; digital signal processing

1. Introduction Non-stationary signals occur in many fields, such as radar system, sonar, speech and mechanical systems [1]. Effective nonstationary signal processing methods are important for industrial applications. In recent years, scholars have developed some effective methods to characterize time-varying signal properties. Time-frequency analysis (TFA) [2] is an effective method to display time-varying instantaneous frequencies (IFs) and time-varying instantaneous amplitudes of nonstationary signal. The traditional TFA methods include linear TFA such as the short-time Fourier transform (STFT), the wavelet transform (WT), and the bilinear/quadratic TFA such as the Wigner Ville distribution (WVD). However, these methods are restricted by the Heisenberg uncertainty principle and cross terms. Time frequency representation (TFR) with insufficient time-frequency resolutions are not providing accurate estimations of nonstationary signal, which essentially restrict their engineering applications. For improving the time-frequency resolutions, the time-frequency reassignment methods are developed, e.g., the synchrosqueezing transform (SST), demodulated SST and synchro-extracting transform

Sensors 2020, 20, 5201; doi:10.3390/s20185201 www.mdpi.com/journal/sensors Sensors 2020, 20, 5201 2 of 30

(SET) [3]. The reassignment methods reduce energy spread via assigning the average of energy in a domain to the gravity center of these energy contributions. It is well known that synchrosqueezed time–frequency representations are concentrated on an IF. However, it is also known that [4] synchrosqueezed time–frequency representations do not provide better time–frequency resolutions for components that have high time variability, as compared to the original STFT or the WT. The synchrosqueezing just sums all the interferences, presented in the original STFT and the WT, into a more compact frequency region. However, even if the components appear to be more compact, as a result of synchrosqueezed time–frequency representations, this does not mean, that their parameters can be better evaluated. The SST is similar to the original STFT and the WT and, although being perfectly concentrated, does not necessarily provide better resolution properties than the respective original STFT and WT. Therefore, the “correct” amplitude peaks cannot be identified more easily in SST than in the original STFT and WT. This is due to a serious drawback of synchrosqueezed representations in that, in contrast to the STFT and WT, the peak amplitudes in the synchrosqueezed time–frequency representations are not proportional to the amplitudes of the corresponding spectral component; instead, they are largely determined by the parameters of the used frequency discretization. Therefore, it is known that, even if one estimated component has a smaller amplitude than another, it may still have a much higher peak in the synchrosqueezed time–frequency representations. Generally, the relationships between the peak amplitudes of synchrosqueezed representation amplitudes of the corresponding spectral component depend on the frequency discretization and these relationships are nonlinear, time-varying, and are dependent on many factors, e.g., possible amplitude and frequency modulation of the component and noise. Hence, the outcomes of IF estimation by SST will also depend on the frequency discretization, amplitude modulation (AM), frequency modulation (FM), and the presence of noise. This effect is additionally aggravated by the following factor: due to the non-smoothness of the SST, it is hard to apply peak interpolation, so, amplitude peaks remain discrete, and this amplitude discretization also affects the performance of IF evaluation. Due to the above factors, employing SST peak amplitudes for IF estimation can lead to unpredictable unstable results. Finally, if the SST is used as the postprocessing technique of the STFT, which is not able to efficiently analyze nonstationary signals, the energy concentration of SET spectra will be essentially influenced by time–frequency resolution drawbacks of the STFT, when processing data. In order to overcome the drawbacks of the SST method and obtain a better time–frequency resolution for strong FM signals, various methods have been developed in recent decades. A generalized SST was proposed [5] by demodulating the time-varying signal into a purely harmonic version. An iterative generalized SST [6] was developed to address multi-component signals with distinct FM laws. A matching SST demodulated transform [7], based on the polynomial or the Fourier series, was proposed to match time-varying IF. For all the above demodulated methods, the exact time-varying FM dependency of the signal is needed in advance. However, due to the complexity of practice diagnosis cases, it is hard to estimate the exact demodulated parameters, especially for AM-FM signals [8]. A popular method is to use the high-order information of the signal, such as the chirp rate, or frequency acceleration, to enhance the energy concentration for strong FM, which involves second-order SST [9] and higher order SST [10]. However, higher-order SST techniques are easy to be affected by noise because they are computed by the high-order derivative of the STFT and the WT. Recently, a new multi-SST (MSST) technique [11] was presented, which employs an iterative reassignment procedure to concentrate the blurry energy in a step-wise manner. MSST squeezes the time–frequency coefficients into the ridge, i.e., the curve along which the signal energy is locally maximum. However, for the ridge, MSST gives a biased estimate of an IF when dealing with nonlinear frequency modulations; therefore, it is difficult for MSST to estimate IF for nonlinear frequency modulations. Sensors 2020, 20, 5201 3 of 30

Recently, two methods were proposed to analyze the impulse signals: time-reassigned SST (TSST) [12] and transient-extracting transform (TET) [13]. TSST can achieve a concentrated time–frequency representation for strong FM, whose ridge curves are almost parallel to the frequency axis. TET uses a transient-extracting operator that gives a sharper time–frequency representation than the TSST. However, both the TSST and the TET are not suitable for a signal consisting of FM IFs in the presence of noise. The high-order SST (HSST) was proposed for IF estimation [14]. However, the computation cost is very high due to the high-order approximation of the FM function. Synchrosqueezed wavelet transform (SWT) [15] is a new method. It is called an empirical mode decomposition (EMD)-like tool. The anti-noise ability and time–frequency resolution of the SWT are improved on the basis of the WT. After the WT, the corresponding frequency of each scale is given by the derivative of the WT coefficients with respect to time. The SWT greatly improves the energy divergence and time–frequency resolution of the WT. The SWT retains the advantages of both EMD and WT. The SWT has been widely studied. Yang analyzed the statistical features of the SWT [16]. Reconstruction of the SWT was developed by Meignen [17]. The SWT was extended to the STFT [18], the wave packet transform [19], the S-transform [20], and the curvelet transform [21]. Nevertheless, the SWT and its extensions are suitable only for signals consisting of weak frequency modulated modes [22]. It is shown in [23] that in spite of the improvement in resolution, the SWT fails to identify closely spaced IFs. Their study showed that if the difference in frequency of the two closely spaced IFs is below the sampling rate, the SWT fails to separate them and demands finer sampling of the original signal. So, in summary, the use of various synchrosqueezing time–frequency representations to IF estimation has multiple essential drawbacks that outweigh its advantage. In addition, the parameterized TFA methods have been proposed to characterize time–frequency signals using signal-dependent parameters [24]. These TFA methods provide a way to qualitatively analyze the nonstationary signals. However, qualitative analysis cannot be effectively applied to applications such as IF estimation. The order analysis has been widely applied in nonstationary signal processing and engineering application [25]. In traditional order analysis, the data are measured, using special analogue hardware to sample at a rate proportional to the shaft speed. For the computed order analysis (COA), nonstationary signal and rotational speed are measured at a constant rate, and then use a software to resample the signal at constant angular increments. As a result, the time-domain nonstationary signal is transformed into an angular-domain stationary signal, and characteristic components can be quantitatively evaluated via the order spectrum, generated by the fast Fourier transform. The speed sensors require installation space, extra wiring, delicate calibration, regular maintenance, and therefore have additional cost [26], which seriously limit applications of the COA. For removing restrictions, related to usage of a speed sensors, the IF estimation based COA is developed [27], in which nonstationary instantaneous rotational frequency (RF) is extracted from TFA of nonstationary signals. However, the RF related IF is difficult to extract in noise environment. In addition, the essential disadvantage of the COA, is errors in the estimation of the shaft angular position and resampled amplitude [28], which affects its accuracy. The generalized demodulation transform (GDT) is another non-stationary signal processing method, which is proposed by Olhede [29]. The GDT can be used to transform a signal with time-varying frequency component into a signal with constant frequency component. Different to the resample in the COA, the conversion process of the GDT is based on the demodulation operator. Hence, the errors, resulting from estimation of the shaft angular position and resampled amplitude can be avoided [30]. The demodulation operator is calculated via the measured RF or estimated RF. Considering the challenges to separate the overlapped nonstationary multi-component signals, a novel GDT is proposed, based on the GDT [31]. Sensors 2020, 20, 5201 4 of 30

Cheng [32] improved the GDT, and it can be used for multi-component signal decomposition. The iterative GDT is developed to transform multi-frequency components [33]. In the iterative GDT, the true IF curve functions are required, and the multi-frequency curves are transformed into the constant frequency curves. In recent years, the GDT is widely applied in industrial applications, such as fault frequencies detection of bearings and planetary gearboxes under nonstationary conditions [34]. Cheng [35] proposed an envelope order spectrum-based GDT to detect gear fault frequencies. In the proposed method, time-varying fault frequencies are transformed into lines, and are also decomposed into some mono-component. However, the demodulation operator is difficult to obtain without a speed sensor. For example, if the signal component with IF, which is used to calculate the demodulation operator, is polluted by noise, it therefore cannot be extracted from the TFR of the original signal. Based on the above analysis, the IF estimation technique is important for machineries’ fault frequencies detection under nonstationary conditions. The tachometer-free-based methods have attracted scholars’ attention. Feng proposed [36] an iterative GDT-based energy separation method; the method can be used to process rotational machinery signals with knowledge of true IF dependency. In this method, the GDT is used to separate an interested harmonic, and the harmonic frequency can be extracted. The GDT is applied only one time to the same specific harmonic, and the number of harmonics in a signal defines the number of GDT iterations. These methods are facing the following challenge. In addition to the time-varying IFs, nonstationary signals normally exhibit time-varying instantaneous amplitudes. As a result, frequency components with relatively low amplitudes are easily contaminated by noise, and they therefore cannot be accurately extracted from the TFA by the GDT. The above GDT methods do not consider IFs of components with relatively low amplitudes that are difficult to accurately estimate. Another challenge is as follows. The GDT transforms a time-varying frequency curve into a constant frequency one, which can be quantitatively characterized by a peak in the spectrum. The transformed frequency curve has a better time–frequency concentration. However, without knowledge of a true IF curve function, the GDT does not produce IF estimations. In order to address the above-mentioned challenges, the novel IF estimation technology, named multi-generalized demodulation transform (MGDT), is proposed here. The main innovations of the paper are as follows:

1. A novel IF estimation technology is proposed for IF estimation. The proposed novel technology, MGDT, is based on the GDT and employs the novel proposition of multiple adaptive iterations of the GDT for a specific signal component to concentrate blurry TFR energy in a step-wise manner. A theoretical basis of the novel IF estimation technology is created. A novel termination criterion, that could control iterations of the MGDT method, is also proposed. By iteratively and adaptively applying multiple GDT operations to the same specific signal component, the energy of the TFR result should be concentrated in a step-wise manner. The technology does not require a priori knowledge of true IF time dependency in contrast to other advanced IF estimation technologies. This feature is important for industrial applications of the proposed technology. The main difference between the proposed MGDT and previously proposed transforms, e.g., [36], is that the MGDT is performing multiple adaptive GDT iterations for the same specific signal component and the previously proposed transforms are performing just one (initial) GDT iteration for the same specific signal component. Other differences are highlighted below. The important feature of the novel MGDT is that every new iteration of the GDT is adaptive, i.e., it is based on previous iteration results. Our extensive literature search clearly shows that nobody, in worldwide terms, has previously proposed the main important novelty of the MGDT: an adaptive iterative application of the GDT for the same signal component of a multi-component signal. 2. Novel IF estimation results, obtained by simulation, for four types of AM-FM nonstationary signals under a strong background noise (signal to noise ratio is -5dB): (i) single-component signal with nonstationary amplitude and nonstationary, nonlinear (quadratic) variation of the instantaneous frequency; (ii) multi-component signal with components that have nonstationary amplitudes and Sensors 2020, 20, 5201 5 of 30

nonstationary linear instantaneous frequencies with different chirp rates; (iii) multi-component signal with components that have nonstationary amplitudes and nonstationary, nonlinear (quadratic) instantaneous frequencies with different chirp rates and frequency accelerations; and (iv) multi-component signal with components that have nonstationary amplitudes and nonstationary, nonlinear (cubic) instantaneous frequencies with different chirp rates, frequency accelerations, and the first derivatives of frequency accelerations. 3. Novel experimental IF estimation results for defect frequency of rolling bearings for multiple defect frequency harmonics, using the proposed technology in non-stationary conditions and in conditions of six different levels of a noise interference, including of two levels of a strong noise interference.

The objectives of the research are as follows:

1. To develop, investigate, and create the theoretical basis of the novel IF estimation technology. 2. To perform a novel validation of the proposed technology by simulation for four types of AM-FM nonstationary signals. 3. To perform comparisons of the proposed IF estimation technology with four traditional TFAs technologies by simulation. 4. To perform a novel experimental validation of the proposed technology for IF estimation for multiple harmonics of bearing defect frequency in non-stationary conditions and in conditions of diﬀerent levels of noise interference, including a strong noise interference.

The rest of the paper is structured as follows. Section2 presents the developed theoretical bases of the IF estimation technology, including brief review of the GDT, and details of the proposed technology. Section3 presents novel validation of the proposed method, using four types of the simulated signals, and comparisons of the proposed IF estimation technology with four traditional TFAs technologies by simulation. Section4 presents a novel experimental validation of the proposed technology for IF estimation for multiple harmonics of a bearing defect frequency in nonstationary conditions and in conditions of diﬀerent levels of noise interference, including a strong noise interference. Finally, the conclusions are drawn in Section5.

2. Theoretical Bases of the Proposed IF Estimation Technology In this section, we, ﬁrstly, brieﬂy review the GDT technology, and then the novel IF estimation technique, the MGDT, is presented in details.

2.1. The GDT Technology The nonstationary signals, that are AM-FM functions, can be deﬁned as:

Z t ! x(t) = A(t) cos 2π f (s)ds + ∅ (1) 0 where A(t) > 0 is the instantaneous amplitude, f (s) > 0 is the IF of the nonstationary signal. ∅ denotes the initial phase. In general, the measured signal consists of multiple frequency components, and thus can be expressed by XK Z t ! xm(t) = Ak(t) cos 2π fk(s)ds + ∅k (2) k=1 0 where K is the number of the frequency components. Sensors 2020, 20, 5201 6 of 30

The analytic signal of signal (1) should be calculated as:

xA(t) = x(t) + jH(x(t)) R t (3) A(t) exp j 2π f (s)ds + ∅ ≈ 0 where H( ) represents the Hilbert transform. · Then, the demodulation operator and the modulation operator can be expressed by Equations (4) and (5), respectively. Z t !! Φ−(t) = exp j2π fd(s)ds fct (4) − 0 − Z t !! + Φ (t) = exp j2π fd(s)ds fct (5) 0 − where fd(t) is frequency function; fc > 0 is a constant frequency. The demodulation operator is used for demodulation transform and the modulation operator is used for inverse demodulation transform. The demodulated signal can be calculated by multiplying xA(t) with Φ−(t) d ( ) = ( ) ( ) = xA t xA t Φ− t R t (6) A(t) exp j 2π fct + 2π ( f (s) fd(s))ds + ∅ 0 −

Let fd(t) = f (t), the demodulated signal will become a purely AM signal centered around the frequency fc. In the GDT technology, the frequency fc is equal to the starting frequency of frequency function f (t). ( ) d ( ) +( ) Similarly, the analytic signal xA t can be calculated by multiplying xA t with Φ t as

Z t !! d + xA(t) = xA(t)Φ (t) = A(t)exp j 2π fd(s)ds + ∅ (7) 0

2.2. Novel IF Estimation Technique The main concept of the IF estimation technique, the MGDT, is to obtain demodulation operator Φ−(t) by multiple iterations and step-wise refinement, and it is introduced, based on the mono-component analytic signal, shown in Equation (3). Before the demodulation, due to the modulation influence, the spectrum of the signal xA(t) widely spreads around its IF f (t), as shown in Figure1. Based on the GDT, with the known demodulation operator Φ−(t), the time-varying IF curve can be transformed into a constant frequency fc labelled as a blue line in Figure1. As a result, its bandwidth is narrow with high energy concentration. Hence, the spectrum of the demodulated signal is more concentrated, than the original spectrum. Based on relatively higher energy concentration of the demodulated signal, the multiple iteration and stepwise refinement strategies are proposed as follows: (a) roughly estimate the IFs of the original signal from its TFR; (b) calculate the demodulation operator, based on the estimated IF function and roughly demodulate the raw signal; as a result, the bandwidth of the IF curve becomes narrow and time–frequency concentration of the demodulated signal is enhanced and (c) obtain the high-accuracy IFs estimation by the peak searching algorithm. The implementation procedure of the MGDT is presented in Figure2. Sensors 2020, 20, x 6 of 31

𝛷 (𝑡) =exp𝑗2π 𝑓(𝑠)𝑑𝑠 − 𝑓𝑡 (5) where 𝑓(𝑡) is frequency function; 𝑓 >0 is a constant frequency. The demodulation operator is used for demodulation transform and the modulation operator is used for inverse demodulation transform. The demodulated signal can be calculated by multiplying 𝑥(𝑡) with 𝛷 (𝑡) 𝑥 (𝑡) =𝑥(𝑡)𝛷 (𝑡) = (6) 𝐴(𝑡) exp 𝑗 2𝜋𝑓𝑡+2π(𝑓(𝑠) − 𝑓(𝑠))𝑑𝑠 +∅

Let 𝑓(𝑡) =𝑓(𝑡), the demodulated signal will become a purely AM signal centered around the frequency 𝑓. In the GDT technology, the frequency 𝑓 is equal to the starting frequency of frequency function𝑓(𝑡). Similarly, the analytic signal 𝑥(𝑡) can be calculated by multiplying 𝑥 (𝑡) with 𝛷 (𝑡) as

𝑥(𝑡) =𝑥 (𝑡)𝛷 (𝑡) = 𝐴(𝑡) 𝑒𝑥𝑝 𝑗 2𝜋 𝑓(𝑠)𝑑𝑠 +∅ (7)

2.2. Novel IF Estimation Technique The main concept of the IF estimation technique, the MGDT, is to obtain demodulation operator 𝛷(𝑡) by multiple iterations and step-wise refinement, and it is introduced, based on the mono- component analytic signal, shown in Equation (3).

Before the demodulation, due to the modulation influence, the spectrum of the signal 𝑥(𝑡) widely spreads around its IF 𝑓(𝑡), as shown in Figure 1. Based on the GDT, with the known demodulation operator 𝛷(𝑡) , the time-varying IF curve can be transformed into a constant frequency 𝑓 labelled as a blue line in Figure 1. As a result, its bandwidth is narrow with high energy concentration. Hence, the spectrum of the demodulated signal is more concentrated, than the original spectrum. Based on relatively higher energy concentration of the demodulated signal, the multiple iteration and stepwise refinement strategies are proposed as follows: (a) roughly estimate the IFs of the original signal from its TFR; (b) calculate the demodulation operator, based on the estimated IF function and roughly demodulate the raw signal; as a result, the bandwidth of the IF curve becomes narrow and time-frequency concentration of the demodulated signal is enhanced and (c) obtain the high-accuracySensors 2020, 20, 5201IFs estimation by the peak searching algorithm. The implementation procedure of7 the of 30 MGDT is presented in Figure 2.

IF of raw signal: f(t) Amplitude IF of demodulated signal: fc Time (s)

Figure 1. Illustration of the GDT. Figure 1. Illustration of the GDT. SensorsThe 2020 specifics, 20, x of the implementation are described in detail as follows 7 of 31 The specifics of the implementation are described in detail as follows 1. IFs and demodulation operator extraction The rough IFs are firstly extracted from the TFR of the raw signal. The TFR is calculated, based 1. IFs and demodulation operator extraction on theThe STFT. rough The IFs STFT are firstly of the extracted signal 𝑥 from(𝑡) ∈𝐿 the(ℝ) TFR with of the the raw window signal. function The TFR 𝑔(𝑡) is calculated, ∈𝐿 (ℝ) can based be defined as 2 2 on the STFT. The STFT of the signal xA(t) L (R) with the window function g(t) L (R) can be ∈ ∈ defined as 𝑆(𝑢,) 𝜔 =Z + 𝑔(𝑡−𝑢)𝑥 (𝑡)𝑒𝑑𝑡 ∞ i2πωt (8) S(u, ω) = g(t u)x (t)e− dt (8) − A The time-frequency resolution of the TFR−∞ can be adjusted by the window width, and a wide The time–frequency resolution of the TFR can be adjusted by the window width, and a wide window results in better frequency resolution, but poor time resolution. An appropriate window window results in better frequency resolution, but poor time resolution. An appropriate window width L should be determined for obtaining an accurate result. width L should be determined for obtaining an accurate result. Then, peak searching algorithm is employed to extract the IFs from the TFR. For improving the Then, peak searching algorithm is employed to extract the IFs from the TFR. For improving the robustness against a noise, local maxima detection strategy is developed. The peak searching robustness against a noise, local maxima detection strategy is developed. The peak searching algorithm algorithm is defined as is defined as 𝐼𝐹(𝑡, 𝑓) =𝑎𝑟𝑔𝑚𝑎𝑥𝑆(𝑡, 𝑓) 𝐼𝐹(𝑡, 𝑓) =𝑎𝑟𝑔 𝑚𝑎𝑥 𝑆𝑡, 𝑓∆ ± (9)(9) ∆ ∆

where arg max represents maximize function; ∆𝑝 is the local search range, 𝐼𝐹(𝑡,𝑓) is position of where arg max represents maximize function; ∆p is the local search range, IF(ts, fs) is position of peak peak in the whole time-frequency amplitudes; 𝐼𝐹(𝑡 ,𝑓) is the output result, i = 1, 2, ⋯, I. I is in the whole time–frequency amplitudes; IF(t , f ) is the output result, i = 1, 2, , I. I is determined by determined by the window width L and the signali i length. ··· the window width L and the signal length. For fitting the discrete IFs, there are two function models. The first function model is the For fitting the discrete IFs, there are two function models. The first function model is the polynomial function model, which can be defined as polynomial function model, which can be defined as K 𝑓(𝑡) =𝑎X 𝑡 𝑘 ∈ (ℝ) (10) k fe(t) = akt k (R) (10) ∈ k=0 Based on the Weierstrass approximation theorem [37], every continuous function, defined on a closedBased interval on thecan Weierstrass be uniformly approximation approximated, theorem as closely [37], as every need, continuous by a polynomial function, function. defined This on modela closed is widely interval used can for be IF uniformly estimation approximated, of the chirp signals as closely [38]. as need, by a polynomial function. ThisThe model second is widely function used model for IF is estimation the Fourier of theseries chirp model, signals which [38]. is defined as The second function model is the Fourier series model, which is defined as 𝑓(𝑡) =𝑎 +(𝑎 𝑐𝑜𝑠(𝜔 𝑡) +𝑏 𝑠𝑖𝑛(𝜔 𝑡)) (11) K X fe(t) = a0 + (akcos(ωkt) + bksin(ωkt)) (11) The Fourier series is an expansion of ak= periodic1 function in terms of a sum of sines and cosines, and it can break up and recombine an arbitrary periodic function with whatever needed or with a The Fourier series is an expansion of a periodic function in terms of a sum of sines and cosines, practical accuracy [39]. This model also can be used for IF estimation of the chirp signals [40]. and it can break up and recombine an arbitrary periodic function with whatever needed or with In this paper, the polynomial function model is used to fit discrete IFs. Hence, the first IF curve a practical accuracy [39]. This model also can be used for IF estimation of the chirp signals [40]. function is 𝑓(𝑡) = ∑ 𝑎 𝑡 . The first demodulation and modulation operators can be calculated as

𝛷 (𝑡) =𝑒𝑥𝑝−𝑗2𝜋 𝑓(𝑠)𝑑𝑠 −𝑎 𝑡 (12)

𝛷 (𝑡) =𝑒𝑥𝑝𝑗2𝜋 𝑓(𝑠)𝑑𝑠 −𝑎 𝑡 (13)

2. Demodulation and IF accurate calculation

The first demodulation operator 𝛷 (𝑡) is used to demodulate the analytic signal 𝑥(𝑡). The first demodulated signal 𝑥 (𝑡) is calculated as 𝑥 (𝑡) =𝑥(𝑡)𝛷 (𝑡) = 𝐴(𝑡) ∙ (14) 𝑒𝑥𝑝𝑗 2𝜋𝑎 𝑡+2𝜋𝑓(𝑠) − 𝑓(𝑠)𝑑𝑠+∅

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In this paper, the polynomial function model is used to fit discrete IFs. Hence, the first IF curve e( ) = PK d1 k function is f1 t k=0 ak t . The first demodulation and modulation operators can be calculated as

Z t !! e d1 Φe1−(t) = exp j2π f1(s)ds a0 t (12) − 0 −

Z t !! e+( ) = e( ) d1 Φ1 t exp j2π f1 s ds a0 t (13) 0 −

2. Demodulation and IF accurate calculation

e ( ) ( ) The ﬁrst demodulation operator Φ1− t is used to demodulate the analytic signal xA t . The ﬁrst d1( ) demodulated signal xA t is calculated as !!! Z t d1 d1 e xA (t) = xA(t)Φe1−(t) = A(t) exp j 2πa0 t + 2π f (s) f1(s) ds + ∅ (14) · 0 −

The ﬁrst IF curve function ef1(t) is not identical to the actual IF curve function f (t) of the analytic d1( ) signal. Hence, the spectrum of the demodulated signal xA t is not centered around the carrier d1 frequency a0 , but has narrower frequency band than the original spectrum. e ( ) e ( ) The second demodulation operator Φ2− t is calculated for further demodulation, where Φ2− t is e( ) PK d2 k ( ) obtained by the second IF function f2 t k=0 ak t , which is extracted from the TFR S2 u, ω of the d1( ) demodulated signal xA t . With the iteration procedure, the actual IF function fm(t) is calculated as

XK d1 dm k dm fm(t) = (a +, , a )t + a (15) k ··· k 0 k=1

Furthermore, the actual demodulation and modulation operators can be calculated as

Z t !! dm Φm− (t) = exp j2π fm(s)ds a0 t (16) − 0 −

Z t !! + dm Φm (t) = exp j2π fm(s)ds a0 t (17) 0 −

Hence, the MGDT result of the analytic signal xA(t) can be shown as

dm dm xA (t) = xA(t)Φm− (t) = A(t)exp j 2πa0 t + ∅ (18)

3. Termination criterion and quantitative evaluation

The iteration can stop if an obvious peak can be captured in the spectrum of the demodulated dm signal and the biggest peak frequency PFm is approximately equal to the starting frequency a of the 0 dm dm IF function ffm(t). Namely, PFm a /a δ. δ is the termination threshold. In this paper, δ = 0.01. − 0 0 ≤ dm( ) The spectrum of the MGDT signal xA t is obtained, using the Fourier transform. The quantitative indicators of the IF estimation error and transform error are deﬁned to evaluate performance of the MGDT. The IF estimation error FE is deﬁned as    fm(t) f (t)  FE = max −  (19)  f (t)  Sensors 2020, 20, 5201 9 of 30

The transform error TE is deﬁned as

PFm PF TE = | − | (20) PF where PF is peak frequency in the spectrum of the demodulated signal with theoretical demodulation dm( ) operator; PFm denotes the biggest peak frequency in the spectrum of the demodulated signal xA t . The sensitivity of the MGDT is determined by both the initialization parameters in Figure2 and noise level. The noise level is the most important parameter. If the signal to noise ratio (SNR) is relatively high, the IFs can be calculated under few iterations. On the contrary, more iterations should Sensorsbe considered 2020, 20, x under a low SNR. 9 of 31

Figure 2. ImplementationImplementation procedure procedure of IF estimation technique. 3. Validation of the Proposed IF Estimation Technology via Simulations 3. Validation of the Proposed IF Estimation Technology via Simulations 3.1. Single-Component Non-Stationary Signal Analysis with Nonlinear (Quadratic) FM 3.1. Single-Component Non-Stationary Signal Analysis with Nonlinear (Quadratic) FM For evaluating the performance of the MGDT, a simulated signal, consisting of one frequency componentFor evaluating and the Gaussianthe performance white noise of then( tMGDT,) is constructed a simulated as signal, consisting of one frequency component and the Gaussian white noise n(t) is constructed as 1 ( ) =1 2t 4 3 + 2 + + ( ) 𝑥(x𝑡)t= 𝑒e − 𝑠𝑖𝑛2𝜋sin 2π(−36𝑡36t +227 227𝑡t +260 260𝑡t )+𝑛n t (𝑡) (21)(21) 5 5 − The IF function function of of the the simulated simulated signal signal is is 𝑓f((𝑡t))== −108𝑡108t2 + 454𝑡454t + 260 + 260;; thethe samplingsampling frequency − is 2000 Hz andand thethe timetime durationduration isis 22s. s. It It is is note noted,d, that that this this function function is is not used in the proposed technology; this function isis justjust usedused forfor IFIF errorerror estimation.estimation. TheThe SNRSNR isis −55dB, dB, which which is is calculated calculated as − 𝑝 ! 𝐴 p A 2 𝑆𝑁𝑅= = 10𝑙𝑜𝑔 x =10𝑙𝑜𝑔= x (22) SNR 10log10 𝑝 10log10 𝐴 (22) pn An

where 𝑝 and 𝐴 denote power and standard deviation of the noiseless signal, respectively; 𝑝 where px and Ax denote power and standard deviation of the noiseless signal, respectively; pn and Ax and 𝐴 are power and standard deviation of the noise, respectively. are powerFigure and 3 shows standard the deviationsimulated ofsignal the noise, with respectively.a noise, its IF curve and its instantaneous amplitude curve. The instantaneous amplitude of the simulated signal is time-varying, as shown in Figure 3b, i.e. the instantaneous amplitude is relatively low in the initial part of the signal and is relatively high in the final part of the signal. The analytic signal obtained by using the Hilbert transform, is presented in Figure 4a. TFR of the analytic signal, generated by the STFT, and its local zooms are shown in Figure 4b–d, respectively. From the local zooms it can be seen, that the time-frequency ridge in the initial part is much blurry, than that in the final part. The main reason is that the noise pollutes the weak part of the signal.

Sensors 2020, 20, 5201 10 of 30

Figure3 shows the simulated signal with a noise, its IF curve and its instantaneous amplitude curve. The instantaneous amplitude of the simulated signal is time-varying, as shown in Figure3b, i.e., the instantaneous amplitude is relatively low in the initial part of the signal and is relatively high in the ﬁnal part of the signal. The analytic signal obtained by using the Hilbert transform, is presented in Figure4a. TFR of the analytic signal, generated by the STFT, and its local zooms are shown in Figure4b–d, respectively. From the local zooms it can be seen, that the time-frequency ridge in the initial part is much blurry, than that in the ﬁnal part. The main reason is that the noise pollutes Sensors 2020, 20, x 10 of 31 the weakSensors part 2020, of 20, thex signal. 10 of 31 (a) (b) (a) (b)

Figure 3. Characteristics of the simulated signal: (a) instantaneous frequencies (IF); (b) instantaneous FigureamplitudeFigure 3. Characteristics 3. Characteristics (IA); and (c of) simulated theof the simulated simulated signal signal:with signal: a noise. (a) instantaneous frequencies frequencies (IF); (IF); (b) (instantaneousb) instantaneous amplitudeamplitude (IA); (IA); and and (c) simulated(c) simulated signal signal with with a a noise. noise.

FigureFigure 4. TFR: 4. TFR: (a) (a envelope) envelope signal; signal; ((bb)) TFR;TFR; ( (cc)) local local zoom zoom 1; 1;and and (d) ( dlocal) local zoom zoom 2. 2. Figure 4. TFR: (a) envelope signal; (b) TFR; (c) local zoom 1; and (d) local zoom 2. TheThe IF estimation IF estimation technology, technology, thethe MGDT, is is employ employeded to the to envelope the envelope signal signalfor calculating for calculating the the IF.IF. FigureFigureThe IF 5 showsestimationshows the the firsttechnology, ﬁrst iteration iteration the results. MGDT, results. Here, is employ the Here, lengthed the to of the the length envelope Gaussian of thesignal window Gaussian for iscalculating L = 128, window 50% the is IF. Figure 5 shows the first iteration results. Here, the length of the Gaussian window is L = 128, 50% L = 128,overlap, 50% the overlap, local search and range the localis ∆𝑝 search = 186Hz range. The extracted is ∆p = IFs,186 fittedHz. IF The curve extracted (𝑓(𝑡) =2.96𝑡 IFs, ﬁtted+ IF 170.64𝑡overlap, the + 422.73local) andsearch the rangeactual isIF ∆𝑝curve = are 186Hz presented. The extracted in Figure IFs, 5a. fitted It can IF be curve found, (𝑓 that(𝑡) =2.96𝑡due to the+ inaccurate170.64𝑡 +extraction 422.73) and of the IFs actual in the IFinitial curve part are of presented the signal, in the Figure fitted 5a. IF It curve can be has found, a clear that error, due which to the inaccurate extraction of IFs in the initial part of the signal, the fitted IF curve has a clear error, which is 62.6%, calculated by Equation (20). Based on Equation (12), the demodulation operator 𝛷 (𝑡) = is 62.6%, calculated by Equation (20). Based on Equation (12), the demodulation operator 𝛷 (𝑡) = 𝑒𝑥𝑝 −𝑗2𝜋 (2.96𝑠 + 170.64𝑠 + 422.73)𝑑𝑠 − 422.73𝑡 is calculated. The first demodulated signal 𝑒𝑥𝑝 −𝑗2𝜋 (2.96𝑠 + 170.64𝑠 + 422.73)𝑑𝑠 − 422.73𝑡 is calculated. The first demodulated signal is shown in Figure 5b, and its TFR is presented in Figure 5c. The spectrum of the demodulated signal, is shown in Figure 5b, and its TFR is presented in Figure 5c. The spectrum of the demodulated signal,

Sensors 2020, 20, 5201 11 of 30

2 curve (ef1(t) = 2.96t + 170.64t + 422.73) and the actual IF curve are presented in Figure5a. It can be found, that due to the inaccurate extraction of IFs in the initial part of the signal, the ﬁtted IF curve has a clear error, which is 62.6%, calculated by Equation (20). Based on Equation (12), R t 2 Sensorsthe demodulation 2020, 20, x operator Φe (t) = exp j2π 2.96s + 170.64s + 422.73 ds 422.73t is calculated.11 of 31 1− − 0 − The ﬁrst demodulated signal is shown in Figure5b, and its TFR is presented in Figure5c. The spectrum generatedof the demodulated by the Fourier signal, transform, generated is byshown the Fourier in Figure transform, 5d, from is which shown it incan Figure be found,5d, from that which there itis nocan clear be found, no-smeared that there peak. is noFigure clear 5 no-smeared also shows peak.the traditional Figure5 also GDT shows results. the It traditional can be found, GDT that results. the traditionalIt can be found GDT thatis not the performing traditional well GDT for is this not performingcase, i.e. some well instantaneous for this case, amplitudes i.e., some instantaneous are very low andamplitudes are polluted are very by a low noise. and are polluted by a noise.

Figure 5. The first demodulation results by the traditional GDT: (a) extracted IFs; (b) first demodulated Figure 5. The first demodulation results by the traditional GDT: (a) extracted IFs; (b) first signal; (c) TFR of first demodulated signal; and (d) the spectrum. demodulated signal; (c) TFR of first demodulated signal; and (d) the spectrum. Let continue to the second iteration. The parameters of window function L and peak Let continue to the second iteration. The parameters of window function L and peak search range search range ∆p are the same, as the parameters in the first iteration procedure. The new IFs ∆p are the same, as the parameters in the first iteration procedure. The new IFs are extracted from the are extracted from the TFR of the first demodulated signal and they are plotted in Figure 6a. TFR of the first demodulated signal and they2 are plotted in Figure 6a. The fitted IF curve (𝑓(𝑡) = The fitted IF curve (ef2(t) = 109.1t + 279.1t + 262.4) is calculated using Equation (10), −109.1𝑡 + 279.1𝑡 + 262.4) is calculated− using Equation (10), and it is presented as blue line in Figure and it is presented as blue line in Figure6a. It could be seen a close coincidence between 6a. It could be seen a close coincidence between the extracted IFs and the fitted IF curve. Then, the the extracted IFs and the fitted IF curve. Then, the new demodulation operator is calculated new demodulationR operatort is calculated 𝛷(𝑡) =𝑒𝑥𝑝−𝑗2𝜋 (−109.1𝑠 + 279.1𝑠 + 262.4)𝑑𝑠 − Φe (t) = exp j2π 109.1s2 + 279.1s + 262.4 ds 262.4t . With the new demodulation operator, 2− − 0 − − 262.4𝑡the second. With demodulated the new demodulation signal is obtained, operator, as shown the second in Figure demodulated6b. Its TFR signal and spectrum is obtained, are shownas shown in inFigure Figure6c,d, 6b. respectively. Its TFR and Therespectrum is a constantare shown time–frequency in Figure 6c,d, ridge respectively. in the TFR. There A clearis a constant non-smeared time- frequencypeak, whose ridge frequency in the TFR. is 262.2 A clear Hz, cannon-smeared be seen in pe theak, spectrum. whose frequency Based on is the 262.2 termination Hz, can be criterion, seen in i.e., 262.2 262.4 /262.4 0.01, the iteration process is stopped. the spectrum.| − Based| on ≤the termination criterion, i.e. |262.2 − 262.4|/262.4 ≤ 0.01, the iteration The actual IF function f (t) = 106.1t2 + 449.7t + 262.4 can be calculated based on process is stopped. 2 − Equation (15). Figure7a shows that the calculated IF curve (blue line) coincides with the actual IF curve (red line), and the error is 0.85%. Then, the actual demodulation operator R t Φ (t) = exp j2π 106.1s2 + 449.7s ds 262.4t is calculated based on Equation (16). With the 2− − 0 − − d ( ) actual demodulation operator, the demodulated signal xA t is calculated, as presented in Figure7b. Its TFR and the spectrum are presented in Figure7c,d, respectively. As shown in the TFR and

Sensors 2020, 20, x 12 of 31

Sensors 2020, 20, 5201 12 of 30 the spectrum, the time-varying time–frequency ridge is transformed into a line, which can be Sensors 2020, 20, x 12 of 31 quantitatively characterized.

Figure 6. Second demodulation results: (a) extracted IFs; (b) second demodulated signal; (c) TFR of second demodulated signal; and (d) the spectrum.

The actual IF function 𝑓(𝑡) = −106.1𝑡 + 449.7𝑡 + 262.4 can be calculated based on Equation (15). Figure 7a shows that the calculated IF curve (blue line) coincides with the actual IF curve (red line), and the error is 0.85%. Then, the actual demodulation operator 𝛷 (𝑡) = 𝑒𝑥𝑝 −𝑗2𝜋 (−106.1𝑠 + 449.7𝑠)𝑑𝑠 − 262.4𝑡 is calculated based on Equation (16). With the actual demodulation operator, the demodulated signal 𝑥 (𝑡) is calculated, as presented in Figure 7b. Its TFR and the spectrum are presented in Figure 7c,d, respectively. As shown in the TFR and the spectrum, the time-varying time-frequency ridge is transformed into a line, which can be Figure 6. Second demodulation results: (a) extracted IFs; (b) second demodulated signal; (c) TFR of quantitatively characterized. secondFigure demodulated6. Second demodulation signal; and results: (d) the spectrum.(a) extracted IFs; (b) second demodulated signal; (c) TFR of second demodulated signal; and (d) the spectrum.

Figure 7. MGDT results: (a) calculated IF curve; (b) demodulated signal; (c) TFR; and (d) the spectrum. Figure 7. MGDT results: (a) calculated IF curve; (b) demodulated signal; (c) TFR; and (d) the spectrum. In addition, the frequency component can also be separated from noise, using the inverse +( ) MGDT algorithm for avoiding noise interference. The actual modulation operator Φ2 t can be d ( ) determined, based on Equation (17). The TFR of the demodulated signal xA t with one constant time–frequency ridge has well-localized zones of concentration, as shown in Figure7c. Thus, the band-pass ﬁlter is employed to the demodulated signal with the central frequency 262.2 Hz. +( ) ( ) With the actual modulation operator Φ2 t , the modulation signal xA f t of the ﬁltered component

Figure 7. MGDT results: (a) calculated IF curve; (b) demodulated signal; (c) TFR; and (d) the spectrum.

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In addition, the frequency component can also be separated from noise, using the inverse MGDT algorithm for avoiding noise interference. The actual modulation operator 𝛷 (𝑡) can be determined, based on Equation (17). The TFR of the demodulated signal 𝑥 (𝑡) with one constant time-frequency ridge Sensorshas well-localized2020, 20, 5201 zones of concentration, as shown in Figure 7c. Thus, the band-pass filter is employed13 of 30 to the demodulated signal with the central frequency 262.2 Hz. With the actual modulation operator 𝛷 (𝑡) , the modulation signal 𝑥(𝑡) of the filtered component 𝑥 (𝑡) is calculated as 𝑥(𝑡) = d f d f + x (t) is calculated as x (t) = x (t)Φ (t), which is presented in Figure8a. Figure8b is the TFR of 𝑥A (𝑡)𝛷 (𝑡), which is presentedA f inA Figure2 8a. Figure 8b is the TFR of the modulation signal. Its local zoom theis shown modulation in Figure signal. 8c, from Its local which zoom it can is be shown found, in that Figure there8c, are from no whichinterference it can components. be found, that there are no interference components.

Figure 8. Separated frequency component: (a) separated signal; (b) TFR; and (c) local zoom. Figure 8. Separated frequency component: (a) separated signal; (b) TFR; and (c) local zoom. For comparison of the MGDT with the traditional TFA methods, the GDT, WT, the SST and the For comparison of the MGDT with the traditional TFA methods, the GDT, WT, the SST and the SET are employed. SET are employed. There are many different families of wavelet functions for various engineering applications. As the There are many different families of wavelet functions for various engineering applications. As proposed MGDT is employing complex analytical signals, it requires a choice of complex wavelet the proposed MGDT is employing complex analytical signal, it requires a choice of complex wavelet types for comparison purposes [41]. In the present work, the complex Morlet mother wavelet function type for comparison purpose [41]. In the present work, the complex Morlet mother wavelet function is selected due to the following main reasons: (i) the Morlet function employs the Gaussian window is selected due to the following main reasons: (i) the Morlet function is employing the Gaussian and, therefore, provides a highly efficient balance between time and frequency resolutions [42]; (ii) the window and, therefore, is providing highly efficient balance between time and frequency resolutions Morlet wavelet is not a sharp edge function; therefore, it minimizes the spectral leakage [43]; (iii) the [42] (ii) the Morlet wavelet is not sharp edge function; therefore, it minimizes the spectral leakage Morlet wavelet allows the selection of a highly efficient balance between time and frequency resolutions [43]; (iii) the Morlet wavelet allows selection of highly efficient balance between time-frequency by two parameters: the center frequency and the bandwidth parameter; (iv) the wavelet transform with resolutions by two parameters: the center frequency and the bandwidth parameter (iv) the wavelet the Morlet function is computationally efficient due to its link to the fast Fourier transform. For the transform with the Morlet function is computationally efficient due to its link to the fast Fourier WT, the wavelet name is “cmor4-2”, the central frequency is 2 Hz, and the bandwidth parameter is 4s2. transform. For the WT, the wavelet name is “cmor4-2”, the central frequency is 2 Hz and bandwidth For the SST, the window length is 100. For the SET, the window length is 120. The TFRs, calculated parameter is 4s2. by the traditional methods, the WT, the SST, and the SET are shown in Figure9a–c, respectively. For the SST, the window length is 100. For the SET the window length is 120. The TFRs, calculated The IFs are extracted, using the peak search algorithm from the TFRs, and the extracted IFs are by the traditional methods, the WT, the SST, and the SET are shown in Figure 9a–c, respectively. presented in Figure9d. In addition, the actual IF and IF, estimated by the novel MGDT, are also The IFs are extracted, using the peak search algorithm from the TFRs, and the extracted IFs are presented in Figure9d. All peak search parameters are the same, as applied to the proposed MGDT. presented in Figure 9d. In addition, the actual IF and IF, estimated by the novel MGDT, are also From Figure9d, it can be found that the extracted IFs by the GDT, the WT, the SST, and the SET have presented in Figure 9d. All peak search parameters are the same, as applied to the proposed MGDT. relatively large errors in the initial duration of the signal. Their total estimation errors are 55.9%, 29.6%, From Figure 9d, it can be found, that the extracted IFs by GDT, the WT, the SST and the SET have 60.2%, and 113.8%, respectively. The IF estimation error of the GDT, 55.9%, is obtained by use of the relatively large errors in the initial duration of the signal. Their total estimation errors are 55.9%, traditional GDT just once for the raw signal. The IF estimation errors for different methods are listed 29.6%, 60.2%, and 113.8%, respectively. The IF estimation error of GDT, 55.9%, is obtained by use of in Table1. It could be seen that the proposed method has a better ability to calculate IFs than the traditional TFA methods, for the simulated signal that features a strong nonlinear FM, a time-variable signal amplitude, and a very noisy environment, the signal to noise ratio is 5dB. It could be also seen − from Figure9d that the main contributions to the estimation errors of the WT, the SST, and the SET are Sensors 2020, 20, x 14 of 31

the traditional GDT just one time for the raw signal. The IF estimation errors for different methods are listed in Table 1. It could be seen, that the proposed method has better ability to calculate IFs, than the traditional TFA methods, for the simulated signal, that is featuring of a strong nonlinear FM, a time variable signal amplitude and a very noisy environment, the signal to noise ratio is −5dB. It Sensorscould2020 be, also20, 5201 seen from Figure 9d, that the main contributions to estimation errors of the WT, the14 of SST 30 and the SET are in a very noisy area of a signal, characterized by low signal amplitude: i.e. in the time slot of (0–0.6) s. in a very noisy area of a signal, characterized by a low signal amplitude, i.e., in the time slot of (0–0.6) s.

Figure 9. Comparison results: TFRs obtained by (a) wavelet transform (WT), (b) synchrosqueezing Figure 9. Comparison results: TFRs obtained by (a) wavelet transform (WT), (b) synchrosqueezing transform (SST) and (c) synchro-extracting transform (SET); and (d) extracted IFs from TFRs, the actual transform (SST) and (c) synchro-extracting transform (SET); and (d) extracted IFs from TFRs, the IF and IF, extracted by the MGDT technology. actual IF and IF, extracted by the MGDT technology. Table 1. IF estimation errors for different methods. Table 1. IF estimation errors for different methods. Method IFs Estimation Error MGDTMethod IFs Estimation 0.85% Error GDTMGDT 0.85% 55.9% WTGDT 55.9% 29.6% SSTWT 29.6% 60.2% SETSST 60.2% 113.8% SET 113.8% 3.2. Multi-Component Nonstationary Signal with Linear FM 3.2. ForMulti-Component further validation Non-Stationa of the IFry Signal estimation with Linear technology, FM another multi-component signal with componentsFor further that have validation time-varying of the amplitudesIF estimation and technology, linear FMs another with diff erentmulti-component chirp rates is signal simulated with oncomponents, the background that have of a noisetime varying as amplitudes and linear FMs with different chirp rates is simulated on the background of a noise as x(1t) = 1 e2t 4 sin 2π 45.4t2 + 260t + 0.91 1 e2t 4 sin 2π 11.35t2 + 65t 5 − × × 5− × 𝑥(𝑡) = 𝑒 ×sin2𝜋(45.4𝑡1 +2t 260𝑡4 )+0.9× 2𝑒 ×sin2𝜋 (11.35𝑡 + 65𝑡) (23) 5 +0.8 5 e − sin 2π 2.27t5 + 13t + n(t) 1 × × (23) +0.8 × 𝑒 ×sin2𝜋(2.27𝑡 + 13𝑡)+𝑛(𝑡) The IF functions of the three5 components of the simulated signal are fa(t) = 90.8t + 260, The IF functions of the three components of the simulated signal are 𝑓 (𝑡) = 90.8𝑡 + 260, fb(t) = 22.7t + 65, and fc(t) = 4.54t + 13, respectively. It is noted that these functions are not used in the𝑓(𝑡) proposed = 22.7𝑡 technology; +65, and these𝑓(𝑡) functions = 4.54𝑡 are + 13,just respectively. used for IF It error is noted, estimations. that these The functions sampling are frequency not used isin 2000 the Hz,proposed signal timetechnology; duration these is 2s, functions and the SNR are isjust5 used dB. for IF error estimations. The sampling − frequencyFigure 10is 2000 shows Hz, the signal multi-component time duration signal is 2s and and the the SNR TFR is of − the5dB. envelope signal. From the TFR, it can be found that the time–frequency ridge in the initial part of the signal is much more blurry than that in the final part of the signal.

Sensors 2020, 20, x 15 of 31 Sensors 2020, 20, x 15 of 31

Figure 10 shows the multi-component signal and the TFR of envelope signal. From the TFR, it Figure 10 shows the multi-component signal and the TFR of envelope signal. From the TFR, it can be found, that the time-frequency ridge in the initial part of the signal is much blurry, than that can be found, that the time-frequency ridge in the initial part of the signal is much blurry, than that Sensorsin the2020 final, 20, part 5201 of the signal. 15 of 30 in the final part of the signal.

FigureFigure 10. 10.(a )(a Multi-component) Multi-component signal; signal; and and (b ()b its) it TFR.s TFR. Figure 10. (a) Multi-component signal; and (b) its TFR.

TheThe IF IF estimation estimation technology, technology, the the MGDT, MGDT, is is employed employed to to the the envelope envelope signal signal for for calculating calculating the the The IF estimation technology, the MGDT, is employed to the envelope signal for calculating the IF with the highest amplitude. The extracted IF function fa(t) = 89.1t + 256.2 is obtained. Figure 11 IF with highest amplitude. The extracted IF function, 𝑓(𝑡) = 89.1𝑡 + 256.2 is obtained. Figure 11 IF with highest amplitude. The extracted IF function, 𝑓(𝑡) = 89.1𝑡 + 256.2 is obtained. Figure 11 showsshows, that that the the extracted extracted IF curveIF curve (blue (blue line) line) coincides is coincide with with the theoretical the theoretical IF curve IF curve (red line),(red line), and the and shows, that the extracted IF curve (blue line) is coincide with the theoretical IF curve (red line), and estimationthe estimation error iserror 1.46%. is 1.46%. the estimation error is 1.46%. Based on the estimated IF fa(t), the first component can be removed from the raw signal. Then, Based on the estimated IF 𝑓(𝑡), the first component can be removed from the raw signal. Then Based on the estimated IF 𝑓(𝑡), the first component can be removed from the raw signal. Then thethe MGDT MGDT is is applied applied to to the the signal signal without without the the first first component, component, and and the the IF IF of of the the second second component component isthe extracted MGDT is as appliedf (t) = to22.2 thet signal+ 66.8 without. The estimation the first component, error is 2.77%. and the Then, IF of by the removing second component the second is extracted asb 𝑓(𝑡) = 22.2𝑡 + 66.8. The estimation error is 2.77%. Then by removing the second is extracted as 𝑓(𝑡) = 22.2𝑡 + 66.8. The estimation error is 2.77%. Then by removing the second component, the IF of the third component can be extracted as fc(t) = 4.65t + 13.58, whose estimation component, the IF of the third component can be extracted as 𝑓(𝑡) = 4.65𝑡 + 13.58 , whose component, the IF of the third component can be extracted as 𝑓(𝑡) = 4.65𝑡 + 13.58 , whose errorestimation is 4.46%. error Hence, is 4.46%. the proposed Hence, IFthe estimation proposed technologyIF estimation could technology be also used could for be nonstationary also used for estimation error is 4.46%. Hence, the proposed IF estimation technology could be also used for multi-componentnonstationary multi-component signals with a time-varying signals with amplitude time varying and amplitude linear FM. and linear FM. nonstationary multi-component signals with time varying amplitude and linear FM. ItIt should should be be noted noted, that that the the amplitudes amplitudes of of components components of of the the signal signal are are di different,fferent, as as shown shown by by It should be noted, that the amplitudes of components of the signal are different, as shown by EquationEquation (23), (23), and and the IF IF of of the the component component with with a ahigher higher amplitude amplitude is isestimated estimated more more accurately. accurately. Equation (23), and IF of the component with a higher amplitude is estimated more accurately.

CalculatedCalculated IF IF

Harmonic fa Harmonic fa

IFIF curve curve

Frequency(Hz) Harmonic fb Frequency(Hz) Harmonic fb

Harmonic fc Harmonic fc

FigureFigure 11. 11.Calculated Calculated IF IF curves. curves. Figure 11. Calculated IF curves. 3.3. Multi-Component Nonstationary Signal with Nonlinear (Quadratic) FM 3.3. Multi-Component Non-Stationary Signal with Nonlinear(Quadratic) FM 3.3. Multi-Component Non-Stationary Signal with Nonlinear(Quadratic) FM For further validation of the proposed IF estimation technology, another multi-component signal For further validation of the proposed IF estimation technology, another multi-component signal withFor components further validation that have of time-varying the proposed amplitudesIF estimation and technology, nonlinear another (quadratic) multi-component FMs with diff signalerent with components, that have time varying amplitudes and nonlinear (quadratic) FMs with different chirpwith components, rates and diff thaterent have frequency time varying accelerations amplitu isdes simulated and nonlinear on the background (quadratic) ofFMs a noise with as different chirp rates and different frequency accelerations is simulated on the background of a noise as chirp rates and different frequency accelerations is simulated on the background of a noise as 1 1 2t 4 3 2 𝑥(𝑡) =1 𝑒x(t) =𝑠𝑖𝑛2𝜋e −(−7.2𝑡sin 2π +7.2 45.4𝑡t ++45.4 260𝑡t +)260t 𝑥(𝑡) = 𝑒5 𝑠𝑖𝑛2𝜋5 (−7.2𝑡 + 45.4𝑡− + 260𝑡) (24) 5 1 2t 4 3 2 (24) 1 +0.9 5 e − sin 2π 14.4t + 90.8t + 520t + n(t)# (24) +0.9 ×1 𝑒 𝑠𝑖𝑛2𝜋× (−14.4𝑡−+ 90.8𝑡 + 520𝑡)+𝑛(𝑡) +0.9 × 𝑒5 𝑠𝑖𝑛2𝜋(−14.4𝑡 + 90.8𝑡 + 520𝑡)+𝑛(𝑡) 5 2 The IF functions of the two components of the simulated signal are fa(t) = 21.6t + 90.8t + 260 − and f (t) = 43.2t2 + 181.6t + 520, respectively. It is noted that these functions are not used in the b − Sensors 2020, 20, x 16 of 31 Sensors 2020, 20, x 16 of 31 The IF functions of the two components of the simulated signal are 𝑓 (𝑡) = −21.6𝑡 + 90.8𝑡 + Sensors 2020, 20, 5201 16 of 30 260andThe 𝑓 IF(𝑡 )functions= −43.2𝑡 of +the 181.6𝑡 two components + 520, respectively. of the simulated It is noted, signal that these are 𝑓functions(𝑡) = −21.6𝑡 are not+ used 90.8𝑡 in + the260 proposedand 𝑓(𝑡) =technology; −43.2𝑡 + these 181.6𝑡 functions + 520, respectively. are just used It is noted,for IF thaterror these estimations. functions Theare notsampling used in proposedfrequencythe proposed technology; is 2000 technology; Hz and these the functionsthese time durationfunctions are just is usedare 2s. Thejust for IFSNRused error is for − estimations.5dB. IF error estimations. The sampling The frequency sampling is 2000frequencyFigure Hz and is 12 the2000 shows time Hz durationtheand multi-component the time is 2s. duration The SNR signal is is 2s. 5 Theand dB. SNR the TFR is −5dB. of envelope signal. From the TFR, it − can beFigureFigure found 12 12 that shows shows the thetime-frequency the multi-component multi-component ridge signalin signal the initial and and the thepart TFR TFR of ofthe of the signalenvelope envelope is much signal. signal. blurry From From than the the thatTFR, TFR, in it itthecan can final be be found foundpart of that that the the thesignal. time-frequency time–frequency ridge ridge in in the the initial initial part part of of the the signal signal is is much much blurry more blurrythan that than in thatthe infinal the part final of part the ofsignal. the signal.

Figure 12. (a) Multi-component signal; and (b) its TFR. Figure 12. (a) Multi-component signal; and (b) its TFR. Figure 12. (a) Multi-component signal; and (b) its TFR. The IFIF estimationestimation technology,technology, MGDT, MGDT, is is employed employed to to the the signal signal for for calculating calculating the the IF withIF with the highestThe amplitude. IF estimation The estimatedtechnology, IF MGDT,function is of employed the first componentto the signal is for𝑓 (𝑡calculating) = −24.1𝑡2 the+94.6𝑡+ IF with highest amplitude. The estimated IF function of the first component is fa(t) = 24.1t + 94.6t + 262.2. (−) Figure262.2highest. Figure13 amplitude. shows 13 thatshows theThe that estimatedestimated the estimated IF IF curve function IF (blue curve of line) the(blu coincides firste line) component coincides with the is theoreticalwith 𝑓 the𝑡 =theoretical IF −24.1𝑡 curve (red+94.6𝑡+ IF curve line), and(red262.2 theline),. Figure estimation and 13the shows estimation error that is 1.2%. the error estimated is 1.2%. IF curve (blue line) coincides with the theoretical IF curve (redBased line), andon the the estimated estimation IF error 𝑓(𝑡 )is, the1.2%. first component can be removed from the raw signal. Then Based on the estimated IF fa(t), the first component can be removed from the raw signal. Then, 𝑓 (𝑡) the MGDTBased is on applied the estimated to thethe signalsignal IF withoutwithout, the first thethe component firstfirst component,component, can be andand removed thethe IFIF offromof thethe the secondsecond raw signal. componentcomponent Then isthe estimated MGDT isas applied 𝑓(𝑡) = to −47.2𝑡 the signal2 + 186.6𝑡 without+ the 526.8 first. The component, estimation anderror the is 1.4%.IF of theHence, second it is component validated, is estimated as fb(t) = 47.2t + 186.6t + 526.8. The estimation error is 1.4%. Hence, it is validated 𝑓 (𝑡) =− −47.2𝑡 + 186.6𝑡 + 526.8 thatis estimated the the proposed proposed as IF IF estimation estimation technology technology could could be. The bealso estimation also used used for multi-component forerror multi-component is 1.4%. Hence, signals it signals is withvalidated, withtime avariablethat time-variable the signalproposed amplitude signal IF estimation amplitude and nonlinear andtechnology nonlinear (quadratic could (quadratic) be) frequency also used frequency modulationsfor multi-component modulations with different with signals diff erentchirp with chirprates time ratesandvariable different and signal different frequency amplitude frequency accelerations. and accelerations. nonlinear (quadratic) frequency modulations with different chirp rates and different frequency accelerations.

Calculated IF Calculated IF Harmonic fb Harmonic f IF curve b IF curve Frequency(Hz)

Frequency(Hz) Harmonic fa Harmonic fa

Figure 13. Estimated IF curves. Figure 13. Estimated IF curves. 3.4. Multi-Component Non-Stationary Nonstationary Signal Signal with with Nonlinear Nonlinear (Quadratic) (Quadratic) FM FM 3.4. Multi-Component Non-Stationary Signal with Nonlinear (Quadratic) FM For furtherfurther validationvalidation of of the the proposed proposed IF estimationIF estimation technology, technology, another another multi-component multi-component signal withsignal componentsFor with further components, thatvalidation have that time-varying haveof the time proposed varying amplitudes IF amplitudes estimation and nonlinear and techno nonlinear (cubic)logy, (cubic)another FMs with FMs multi-componentdi withﬀerent different chirp signal with components, that have time varying amplitudes and nonlinear (cubic) FMs with different

Sensors 2020, 20, 5201 17 of 30

Sensors 2020, 20, x 17 of 31 Sensors 2020, 20, x 17 of 31 rates, different frequency accelerations, and different first derivatives of frequency acceleration is chirp rates, different frequency accelerations and different first derivatives of frequency acceleration simulatedchirp rates, on thedifferent background frequency of aaccelerations noise as and different first derivatives of frequency acceleration is simulated on the background of a noise as is simulated on the background of a noise as 1 1 2t 4 4 3 2 𝑥(𝑡)x=(t1) =𝑒 e 𝑠𝑖𝑛2𝜋− sin (2𝑡π −7.2𝑡t 7.2t+45.4𝑡+ 45.4t++ 260𝑡260)t 𝑥(𝑡) = 5𝑒5 𝑠𝑖𝑛2𝜋 (𝑡 −7.2𝑡− +45.4𝑡 + 260𝑡) (25) 51 2t 4 4 3 2 (25) +0.91 5 e − sin 2π 2t 14.4t + 90.8t + 520t + n(t) +0.9 ×1 𝑒× 𝑠𝑖𝑛2𝜋(2𝑡 −14.4𝑡− +90.8𝑡 + 520𝑡)+𝑛(𝑡) (25) +0.9 × 5𝑒 𝑠𝑖𝑛2𝜋(2𝑡 −14.4𝑡 +90.8𝑡 + 520𝑡)+𝑛(𝑡) 5 3 2 TheThe IF IF functions functions ofof the the simulated simulated signal signal are are𝑓(𝑡)fa=4𝑡(t) =− 21.6𝑡4t +21.6 90.8𝑡t + +90.8 260andt + 260𝑓(𝑡)and= The IF functions of the simulated signal are 𝑓(𝑡) =4𝑡 − 21.6𝑡 −+ 90.8𝑡 + 260and 𝑓(𝑡) = f (t8𝑡) =−43.2𝑡8t3 +43.2 181.6𝑡t2 + 181.6 + 520. t It+ is520 noted,. It that is noted these that functions these ar functionse not used are in notthe proposed used in the technology; proposed b 8𝑡−43.2𝑡 + 181.6𝑡 + 520. It is noted, that these functions are not used in the proposed technology; these functions− are just used for IF error estimations. The sampling frequency is 2000 Hz and the time technology;these functions these are functions just used are for just IF error used estimation for IF errors. The estimations. sampling Thefrequency sampling is 2000 frequency Hz and isthe 2000 time Hz duration is 2s. The SNR is −5dB. andduration the time is duration2s. The SNR is 2 is s. − The5dB. SNR is 5 dB. Figure 14 shows the multi-component− signal and the TFR of envelope signal. From the TFR, it FigureFigure 14 14 shows shows the the multi-component multi-component signalsignal andand thethe TFRTFR ofof theenvelope envelope signal. signal. From From the TFR, the TFR,it can be found that the time-frequency ridge in the initial part of the signal is much blurry than that in it cancan bebe foundfound that the the time-frequency time–frequency ridge ridge in in the the initial initial part part of ofthe the signal signal is much is much blurry more than blurry that in than the final part of the signal. thatthe in final the finalpart of part the of signal. the signal.

FigureFigure 14. 14.(a ()a Multi-component) Multi-component signal;signal; andand (b) it itss TFR. TFR. Figure 14. (a) Multi-component signal; and (b) its TFR. TheThe IF estimationIF estimation technology, technology, the the MGDT, MGDT, is employedis employed to to the the signal signal for for estimating estimating the the IFs. IFs Figure. Figure 15 The IF estimation technology, the MGDT, is employed to the signal for estimating the IFs. Figure shows15 shows that the that estimated the estimated IF curves IF curves (blue (blue lines) lines) coincide coincide with with the the theoretical theoretical IF curvesIF curves (red (red lines) lines) for 15 shows that the estimated IF curves (blue lines) coincide with the theoretical IF curves (red lines) bothfor components. both components. The estimation The estimation error for errors the first for componentthe first component is 1.2% and is 1.2% for the and second for the component second for both components. The estimation errors for the first component is 1.2% and for the second it iscomponent 1.6%. is 1.6%. component is 1.6%. Hence,Hence, it it is is validated validated, thatthat the the proposed proposed IF IFesti estimationmation technology technology could could be also be used also for used multi- for Hence, it is validated, that the proposed IF estimation technology could be also used for multi- multi-componentcomponent signals signals with time with variable a time-variable signal ampl signalitude amplitudeand nonlinear and (cubic) nonlinear frequency (cubic) modulations frequency component signals with time variable signal amplitude and nonlinear (cubic) frequency modulations modulationswith different with dichirpfferent rates, chirp different rates, di fffrequencyerent frequency accelerations accelerations, and different and diff erentfirst derivatives first derivatives of with different chirp rates, different frequency accelerations and different first derivatives of frequency acceleration. offrequency acceleration. acceleration. Calculated IF Calculated IF

Harmonic fb Harmonic fb IF curve IF curve

Harmonic fa Harmonic fa

Figure 15. Estimated IF curves and theoretical IF curves. FigureFigure 15. 15.Estimated Estimated IF curves and and theoretical theoretical IF IF curves. curves. The proposed method could be used for effective online estimation of time variable linear and TheThe proposed proposed method method could could be be used used for for the effective eﬀective online online estimation estimation of oftime time-variable variable linear linear and and nonlinear IFs. The method is designed to estimate IFs from vibration data automatically. Upon nonlinearnonlinear IFs. IFs. The The method method is designed is designed to estimate to estima IFste from IFs vibrationfrom vibration data automatically. data automatically. Upon receiving, Upon

Sensors 2020, 20, 5201 18 of 30 automatically, an IF estimation, related to the ﬁrst iteration, a termination will be automatically used to determine if the method should continue to iterate or stop further iterations. If an estimation error is lower than a deﬁned termination threshold, the method will stop working. Otherwise, the method will continue to iterate until an estimation error that is lower than a termination threshold is obtained. The method is computationally fast and provides near real-time estimation results, e.g., the calculation time is 1.7 s for the IF estimation of 2-second vibration data, the frequency range of the STFT is 1000 Hz, the frequency resolution is 7.8 Hz, the overlapping for the STFT is 50%, the sampling frequency is 2000 Hz, and there are two iterations that are employed to achieve the required estimation accuracy.

4. Validation of the Proposed IF Estimation Technology via Experimental Trials In this section, experimental validation of the proposed IF estimation technology for a nonstationary multi-component signal from the rolling element bearing is performed. The rolling bearing is one of the most widely used mechanical parts in rotating machinery. Its faults are important reasons of shutdowns of machines. For reducing economic losses and avoiding disastrous accidents, novel IF estimations for rolling bearing has become an important topic [44–47]. The captured nonstationary bearing vibration signal from a faulty bearing is considered for experimental MGDT validation. The main purpose of the validation is to estimate the non-stationary instantaneous bearing defect frequencies for the fundamental harmonic and the higher harmonics, using the proposed IF estimation technology for different levels of noise interference at a very early stage of damage development. Figure 16 shows the structural sketch of the experimental set-up. The AC motor (Marathon, Beloit, WI, USA )(3 phase, Hz: 60/50, RPM: 3450/2850, voltage: 208–230/460) is used to drive the steel shaft, which is supported by two bearings (Rexnord, Milwaukee, WI, USA), including a faulty one. The employed motor has a star connection. In delta connection, each winding will get phase to phase voltage. Alternatively, in star connection, there will be two windings in series across two phases. Hence, motor windings get a higher voltage in delta connection. In star connection, the stator phase voltage is about 58 % of the line voltage. The torque of the motor is directly proportional to the square of the applied voltage. In star connection, the torque of the motor is about 33 % of the maximum rated torque of the motor. The torque delivering capacity of the motor in the delta connection is more than the torque-delivering capacity in the star connection, i.e., torque (star connection) = 33 % of torque (delta connection). Motor vibration amplitudes are propositional to the motor torque. Therefore, it is easier to estimate IFs of narrowband motor components in the delta connection, compared with the star connection. Two flywheels (SpectraQuest, Richmond, VA, USA) are installed on the shaft to apply load to the system. The RF is measured by a built-in tachometer (SpectraQuest, Richmond, VA, USA) (one pulse per revolution analogue transistor-transistor logic output for DAQ purposes) and the RF can be adjusted by the AC converter. An accelerometer (PCB, Depew, NY, USA) (PCB 352A60) is mounted on the top surface of the faulty bearing housing by a tapped hole to capture the vibration signal. This miniature accelerometer achieves an extremely high frequency response from 5 Hz to 60 kHz. The sensitivity of the accelerometer is 10mV/g; electrical filter corner frequency is 45 kHz; electrical filter roll-off is 10 dB/decade; resonant frequency is 95 kHz; measurement range is up to 500 g; broadband resolution is 0.002 g rms; ≥ nonlinearity is 1%; transverse sensitivity is 5%; overload limit is 5000 g pk; temperature response is ≤ ≤ 54 to +121 °C; base strain sensitivity is 0.05 g/µε; excitation voltage is 18 to 30 VDC; constant current − ≤ excitation is 2 to 20 mA; output impedance is 100Ohm; output bias voltage is from 8 to 12 VDC; ≤ discharge time constant is from 0.02 to 0.06 s; spectral noise (10 Hz) is 160µg/ √Hz; spectral noise (100 Hz) is 40 µg/ √Hz; spectral noise (1 kHz) is 15 µg/ √Hz; spectral noise (10 kHz) is 10 µg/ √Hz; height is 21.6 mm; weight is 6 gm; sensing element is ceramic; size-hex is 3/8in; sensing geometry is shear; housing material is stainless steel; sealing is welded hermetic; electrical connector is 5-44 coaxial; Sensors 2020, 20, 5201 19 of 30 electrical connection position is the top; mounting is integral stud; and mounting thread is 10-32 male. TheSensors accelerometer 2020, 20, x is shown in Figure 16. 19 of 31 The signal collecting system is mainly consisted of a DAQ card (Model: NI-PCI6259, number of channelsThe issignal 16 di collectingfferential orsystem 32 single-ended, is mainly consiste ADC resolutiond of a DAQ is card 16 bits, (Model: single NI-PCI6259, channel maximum number isof 1.25channels MS/s, is and 16 multichanneldifferential or maximal 32 single sample ended, rate AD isC 1resolution MS/s) and is the16 NI-DAQmxbits, single channel software. maximum is 1.25MS/s,The bearing multichannel type is maximal ER-10K, sample number rate of ballsis 1 MS/s) is 8, balland diameterthe NI-DAQmx is 8.94 software. mm, pitch diameter is 33.5 mm,The contactbearing angle type isis 0,ER-10K, and its FCCsnumber of theof balls outer is race, 8, ball inner diameter race, and is ball8.94mm, are 3.05, pitch 4.95, diameter and 1.99, is respectively.33.5mm, contact A defect angle in is the0, and form its of FCCs a 0.1 of mm outer diameter race, inner hole israce, considered and ball inare the 3.05, outer 4.95, raceway and 1.99, of therespectively. bearing; the A defect relative in damage the form size, of 0.1 related mm diameter to the outer hole race is considered circumference, in the is 0.2%.outer raceway The bearing of the is shownbearing; in the Figure relative 16. damage size, related to the outer race circumference, is 0.2%. The bearing is shown in Figure 16.

Figure 16. Experimental set-up, a sensor and a bearing. Figure 16. Experimental set-up, a sensor and a bearing. First, the raw bearing vibration signal is collected. The collected vibration signal from the First, raw bearing vibration signal is collected. The collected vibration signal from the bearing bearing with outer race fault is presented in Figure 17a; the estimated from speed sensor RF function is with outer race fault is presented in Figure 17a; the estimated from speed sensor RF function is 𝑓(𝑡) = f (t) = 2.9t + 19.9 Hz; the RF increases from 19.9 to 31.5 Hz over 4 s. The sampling frequency is 48,000 Hz 2.9𝑡 + 19.9 Hz; the RF increases from 19.9 Hz to 31.5 Hz during 4s. The sampling frequency is 48,000 and the time duration is 4 s. It should be noted that speed sensor data are not used for the proposed Hz and the time duration is 4s. It should be noted, that speed sensor data are not used for the estimation technology, as the technology does not require a priori knowledge of a true time-varying IF proposed estimation technology, as the technology is not required a priori knowledge of a true time dependency. These data are used only for IF error estimations for the proposed technology. varying IF dependency. These data are used only for IF error estimations for the proposed For selection of the sampling frequency, the following frequencies should be taken into account: technology. the fundamental outer race bearing defect frequency that varies from 60.7 to 96.1 Hz and a frequency For selection of the sampling frequency, the following frequencies should be taken into account: band that is related to transient impulses, generated by a bearing outer race local fault. Before the the fundamental outer race bearing defect frequency, that is varying from 60.7 Hz to 96.1 Hz and a ﬁnal selection of the sampling frequency, a preliminary assessment of a frequency band that is related frequency band, that is related to transient impulses, generated by bearing outer race local fault. to transient impulses, generated by a bearing local outer race fault, was performed by spectral Before the final selection of the sampling frequency, a preliminary assessment of a frequency band, kurtosis [48–50], and it was found that this band is (9–12) kHz. So, based on the estimated frequency that is related to transient impulses, generated by bearing local outer race fault, was performed by band, it is clear that the bearing outer race defect frequency is not critical in the selection of the sampling the spectral kurtosis [48–50], and it was found, that this band is (9–12) kHz. So, based on the estimated frequency, and the estimated frequency band is critical in sampling frequency selection. In order to frequency band, it is clear, that the bearing outer race defect frequency is not critical in selection of avoid aliasing, the sampling frequency is selected as four times the maximum frequency of a frequency the sampling frequency, and the estimated frequency band is critical in sampling frequency selection. band that is related to transient impulses, generated by bearing local faults, i.e., 48 kHz. In order to avoid aliasing, the sampling frequency is selected as four times of the maximum frequency Firstly, the proposed technology is validated for IF estimation of the ﬁrst harmonic of the outer of a frequency band, that is related to transient impulses, generated by bearing local faults, i.e. 48 race defect frequency. The envelope signal, shown in Figure 17b, is estimated by the Hilbert transform. kHz. Figure 17c,d show the TFR and the spectrum of the envelope signal, respectively. From the TFR, it can Firstly, the proposed technology is validated for IF estimation of the first harmonic of outer race be found that due to the noise, the weak IFCFs in the initial duration of the signal are polluted by defect frequency. The envelope signal, shown in Figure 17b, is estimated by the Hilbert transform. noise. In addition, because of the time-varying operation condition, the IFCF curve is time-varying Figure 17c, d shows the TFR and the spectrum of the envelope signal, respectively. From the TFR, it and cannot be quantitatively characterized by the classical Fourier spectrum, as shown in Figure 17d. can be found, that due to the noise, the weak IFCFs in the initial duration of the signal are polluted by noise. In addition, because of the time-varying operation condition, the IFCF curve is time-varying and cannot be quantitatively characterized by the classical Fourier spectrum, Figure 17d.

Sensors 2020, 20, 5201 20 of 30 Sensors 2020, 20, x 20 of 31

Figure 17. Raw signal: (a) signal; (b) envelope signal; (c) TFR; and (d) the spectrum. Figure 17. Raw signal: (a) signal; (b) envelope signal; (c) TFR; and (d) the spectrum. The proposed IF estimation technology is employed to the envelope signal for obtaining the IFCF. The proposed IF estimation technology is employed to the envelope signal for obtaining the IFCF. Here, the length of the Gaussian window is L = 12000, 50% overlap, the local search range ∆p = 100 Hz. Here, the length of the Gaussian window is L = 12000, 50% overlap, the local search range ∆𝑝 = 100Hz. The main criterion for window length selection depends on an instantaneous frequency change, The main criterion for window length selection depends on an instantaneous frequency change, characterized by the chirp rate in the case of linear frequency variation and by the chirp rate and the characterized by the chirp rate in the case of the linear frequency variation and by the chirp rate and frequency acceleration in the case of quadratic frequency variation, and so on [51–54]. the frequency acceleration in the case of quadratic frequency variation, and so on [51–54]. The employed criterion of window length selection is clearly explained below on the basis of the The employed criterion of window length selection is clearly explained below on the basis of the two main cases of combinations of the instantaneous frequency change fj(t) of the components j(t) two main cases of combinations of instantaneous frequency change 𝑓 (𝑡) of components 𝑗(𝑡) and and frequency resolution, defined by a window length. frequency resolution, defined by a window length. Case 1. The instantaneous frequency change fchj of IF of component fj(t) during the duration Case 1. The instantaneous frequency change 𝑓 of IF of component 𝑓(𝑡) during duration of of the selected window is within the frequency limits ( fp f /2; fp + f /2), defined by the frequency the selected window is within frequency limits (𝑓 −𝑓/2− ; 𝑓 +𝑓/2), defined by the frequency resolution f at a particular central frequency fp of a frequency bin of the STFT. The considered case resolutioncould be viewed f at a particular as a quasi-stationary central frequency case within𝑓 of a frequency a window bin length of the and STFT. noadditional The considered errors case are couldprovided be viewed that are as related a quasi-stationary to STFT estimation case within for a nonstationary a window length signal, and because it is provided the time-variable no additional IF is errors,not moving related from to STFT one frequency estimation bin for to non-stationary another during signal, the selected because window time variable length. IF is not moving from one frequency bin to another during the selected window length. Case 2. The instantaneous frequency change fchj of IF of component fj(t) during the duration of Case 2. The instantaneous frequency change 𝑓 of IF of component 𝑓 (𝑡) during duration of the selected window is more than the frequency limits ( fp f /2; fp + f /2), defined by the frequency the selected window is more, than the frequency limits (𝑓 −𝑓/2− ; 𝑓 +𝑓/2), defined by the frequency resolution f at the particular central frequency fp of a frequency bin. resolutionThis case f at isthe definitely particular not central a quasi-stationary frequency 𝑓 case of anda frequency additional bin. errors are provided that are related to STFTThis estimation case is definitely for a nonstationary not a quasi-stationary signal, as the case time-variable and it is provided IF is moving additional from oneerrors, frequency related bin to STFTto another estimation during for the non-stationary selected window signal, length, as time i.e., variable the IF will IF be,is moving at least, from in two one di fffrequencyerent frequency bin to anotherbins (or during even more, the selected depending window on the length: speed i.e. of IF the will frequency be, at least, change) in two fordifferent the selected frequency duration bins (or of eventhe window. more, depending on speed of frequency change) for the selected duration of the window. Based on the considered two cases,cases, thethe clearclear criterioncriterion forfor windowwindow selectionselection, that that we we employed employed, is is as follows: the instantaneous frequency change 𝑓 of IF of component 𝑓 (𝑡) during duration of as follows: the instantaneous frequency change fchj of IF of component fj(t) during the duration of the 𝑓 −𝑓/2 𝑓 +𝑓/2 theselected selected window window should should be within be within the frequency the frequency limit ( fplimitf / 2( ;fp + f /;2 ), defined), bydefined the frequency by the − frequencyresolution withresolution a “safety with factor” a “safety within factor” the threshold within threshold range (1.5–3) range for (1.5–3) a “safety for “safety factor”, factor”, where a where safety 𝑆=𝑓/𝑓 afactor safety is factor defined is defined as S = f as/ f ch. . The threshold threshold range range (1.5–3) (1.5–3) is is defined defined as as follows: follows: it itis isa danger a danger to toselect select a threshold a threshold around around 1, as 1, anas anIF IFstill still could could move move from from one one frequency frequency bin bin to to another another during during the the selected selected window window length. length. If If the selected threshold rangerange is(1.5–3), (1.5–3), then, then it the will IF be will IF notnon-movement move from one from frequency one frequency bin to anotherbin to another during during the selected window length. It is possible to have a safety factor more than 3; however, a

Sensors 2020, 20, 5201 21 of 30 the selected window length. It is possible to have a safety factor of more than 3; however, a drawback of such a big safety factor is that the frequency resolution for this case becomes poorer and this will potentially compromise an IF estimation. It is discussed here how this criterion was employed for the considered experimental trials. As the main effort is for estimation, firstly, the IF of the first harmonic of the bearing outer race defect frequency is considered a time variation of the IF of this harmonic, which is 3.05 f (t), where × f (t) = 2.9t + 19.9 f (t) is the time variation of the fundamental rotation frequency, and the coefficient of 3.05 is the FCC of the bearing outer race. For the selected window length of 12000 (i.e., time duration of this window length is 0.25 s), the instantaneous frequency change fch1 of the IF of the first harmonic of the bearing outer race defect frequency is fchj= (0.25s x 2.9 Hz/s x 3.05) = 2.21 Hz and the frequency resolution related to this window length is f = (1/0.25 Hz) = 4 Hz. So, for this selection of the window, the instantaneous frequency change (i.e., 2.21 Hz) during the duration of the window is within the frequency resolution (i.e., 4 Hz) and, therefore, it is possible to avoid additional errors related to STFT estimation for the first harmonic of the bearing outer race defect frequency. This avoidance is because the time-variable IF is not moving from one frequency bin to another during window length. The “safety factor” for this estimation is S = (4 Hz/2.21 Hz) = 1.81 and it is inside the specified threshold range (1.5–3). It is possible to select a window length, e.g., of 6000 (i.e., the time duration of this window length is 0.125s). In this case, the instantaneous frequency change fch1 of the IF of the first harmonic of the bearing outer race defect frequency is fchj= (0.125s x 2.9 Hz/s x 3.05) = 1.11 Hz and the frequency resolution related to this window length is f = (1/0.125 Hz) = 8 Hz. So, in this case, the instantaneous frequency change during the duration of the window (i.e., 1.11 Hz) is also within the frequency resolution (8 Hz) and it is also possible to avoid additional errors related to STFT estimation for the first harmonic of the bearing outer race defect frequency. However, the “safety factor” for this estimation is too big at S = (8 Hz/1.11 Hz) = 7.21. A drawback of this big safety is that the frequency resolution for this case (8 Hz) is poorer, compared with window length of 12000 that, potentially, will compromise an IF estimation. If a longer window length, e.g., 18000, is selected, then the instantaneous frequency change fchj of the IF of the first harmonic of the bearing outer race defect frequency is (0.3755s x 2.9 Hz/s x 3.05) = 3.32 Hz, and the frequency resolution related to this window length is f = (1/0.375 Hz) = 2.67 Hz. In this case, the instantaneous frequency change during the duration of the window (i.e. 3.32 Hz) is not within the frequency resolution (2.67 Hz) and it is not possible to avoid additional errors related to STFT estimation for the first harmonic of the bearing outer race defect frequency. This non-avoidance is because the time-variable IF is moving from one frequency bin to another during the window length. So, the selection of a longer window length of 18000 is not acceptable. The same criterion was used for the selection of the window length for simulation trials. For the simulation of two harmonics with nonlinear frequency modulation, the selected window length is 128. For the selected window length of 128 (i.e., the time duration of this window length is 0.064 s), the instantaneous frequency change fch1 of the IF of the first harmonic is 5.72 Hz and the frequency resolution related to this window length is f = (1/0.064 Hz) = 15.625 Hz. So, for this selection of the window, the instantaneous frequency change during the duration of the window (i.e., 5.72 Hz) is within the frequency resolution (i.e., 15.625 Hz) and it is possible to avoid additional errors related to STFT estimation for the first harmonic. The “safety factor” for this estimation S = (15.625 Hz /5.72 Hz) = 2.73 which is inside the specified threshold range (1.5–3). So, in order to implement the described criterion for the proposed method, what need to be known, approximately, are the maximum chirp rates for signals with linear variation of the IF and the maximum chirp rates and frequency accelerations for quadratic variation of the IF, etc. Figure 18 shows the first iteration demodulation results. As shown in Figure 18a, the first extracted IFCF has a relatively big error; the error between the first fitted IFCF curve (ef (t) = 7.6t2 28.6t + 99) 1 − and the actual IFCF curve (3.05 f (t)) is 63.1%. A clear peak, without smearing, cannot be captured in × Sensors 2020, 20, 5201 22 of 30 the spectrum, as shown in Figure 18d. The second iteration demodulation results are shown in Figure 19. From Figure 19a, it can be found that the extracted IFCF is coincident with the fitted IFCF curve (ef (t) = 7.6t2 + 37.1t + 58.8). There is a clear non-smeared peak in the spectrum, presented in 2 − Figure 19d,Sensors and 2020 the, 20 frequency, x of that peak is very close to the starting frequency of ef2(22t )of. 31 Based on ef1(t), ef2 (t) and Equation (16), the estimated IFCF curve function is f2(t) = 8.5t + 58.8, (𝑓(𝑡) = −7.6𝑡 + 37.1𝑡 + 58.8). There is a clear non-smeared peak in the spectrum, presented in Figure ( ) and the IF19d, estimation and the frequency error betweenof that peak the is very estimated close to the IF starting function frequencyf2 t ofand 𝑓(𝑡) the. obtained one from the speed sensor functionBased on 3.05𝑓(𝑡), 𝑓f((𝑡t)) andis 4.4%. Equation Applying (16), the the estima terminationted IFCF curve threshold function of is 0.01, 𝑓(𝑡) the=8.5𝑡+ third iteration × is not needed.58.8, and the IF estimation error between the estimated IF function 𝑓(𝑡) and the obtained from the speed sensor function 3.05 × 𝑓(𝑡) is 4.4%. Applying the termination threshold of 0.01, the third Based on Equation (17), the demodulation operator is estimated as Φ−(t) = iteration is not needed. 2 R t exp j2π 8.5Basedsds 58.8on tEquation. With (17), the the obtained demodu demodulationlation operator operator,is estimated the demodulatedas 𝛷 (𝑡) = signal 0 − 𝑒𝑥𝑝 −𝑗2𝜋− 8.5𝑠𝑑𝑠 −58.8𝑡. With the obtained demodulation operator, the demodulated signal is is estimated and is shown in Figure 20b. Figure 20c,d are the TFR and the spectrum of the demodulated signal. Fromestimated the TFR, and ais constantshown in Figure time–frequency 20b. Figure 20 curvec,d are can TFR be and found, the spectrum and its of value the demodulated is 58.8 Hz, which is signal. From the TFR, it can be found a constant time-frequency curve; its value 58.8 Hz, which is a a peak frequencypeak frequency in the in spectrum, the spectrum, as Figure shown 20d. in Figure 20d.

Figure 18.FigureFirst 18. demodulation First demodulation results results (the (the traditional traditional GDT): GDT): (a) (extracteda) extracted IFs; (b IFs;) demodulated (b) demodulated signal; signal; Sensors 2020, 20, x 23 of 31 (c) TFR; and(c) TFR; (d) spectrum. and (d) spectrum.

Figure 19. SecondFigure 19. iteration Second iteration demodulation demodulation results: results: ((aa) extracted extracted IFs; IFs; (b) demodulated (b) demodulated signal; (c signal;) TFR; (c) TFR; and (d) spectrum.and (d) spectrum.

Figure 20. Multiple demodulation results: (a) calculated instantaneous fault characteristic frequency (IFCF) curve; (b) demodulation signal; (c) TFR; and (d) the spectrum.

Then, the proposed technology is further validated for estimation of IFs of the second and the third harmonics of outer race defect frequency. Figure 21a,b is the TFR and the spectrum of the demodulated signal, related to the second harmonic. Figure 21c,d is the TFR and the spectrum of the demodulated signal, related to the third harmonic. A constant time-frequency ridge and a peak can be detected in the TFR and the spectrum of demodulated signal, related to the second harmonic (Figure 21a,b). The peak frequency is 117.7 Hz, which is very close to the value 121.4 Hz, estimated by the speed sensor. Similarly, a constant time-

Sensors 2020, 20, x 23 of 31

Sensors 2020, 20, 5201Figure 19. Second iteration demodulation results: (a) extracted IFs; (b) demodulated signal; (c) TFR; 23 of 30 and (d) spectrum.

Figure 20. FigureMultiple 20. Multiple demodulation demodulation results: results: (a ()a) calculated calculated instantaneous instantaneous fault faultcharacteristic characteristic frequency frequency (IFCF) curve;(IFCF) (b )curve; demodulation (b) demodulation signal; signal; (c) ( TFR;c) TFR; and and ((d)) the the spectrum. spectrum.

Then, the proposed technology is further validated for estimation of IFs of the second and the Then,third the harmonics proposed of technologyouter race defect is furtherfrequency. validated Figure 21a,b for is the estimation TFR and the of spectrum IFs of the of the second and third harmonicsdemodulated of the signal, outer related race to defect the second frequency. harmonic. FigureFigure 21c,d 21a,b is the is theTFR TFRand the and spectrum the spectrum of the of the demodulateddemodulated signal, relatedsignal, related to the to secondthe third harmonic.harmonic. Figure 21c,d is the TFR and the spectrum of the demodulated signal,A constant related time-frequency to the third ridge harmonic. and a peak can be detected in the TFR and the spectrum of demodulated signal, related to the second harmonic (Figure 21a,b). The peak frequency is 117.7 Hz, A constantwhich is time-frequency very close to the value ridge 121.4 and Hz, aestimate peakd can by the be speed detected sensor. in Simi thelarly, TFR a constant and the time- spectrum of demodulated signal, related to the second harmonic (Figure 21a,b). The peak frequency is 117.7 Hz, which is very close to the value 121.4 Hz, estimated by the speed sensor. Similarly, a constant time-frequencySensors 2020 ridge, 20, x and a peak can be detected in the TFR and the spectrum of the24 of demodulated 31 signal, relatedfrequency to theridge 3 and third a peak harmonic can be detected (Figure in 21thec,d). TFR and The the peak spectrum frequency of the demodulated is 176.7 Hz, signal, which is very close to therelated value to the 182.1 3 third Hz, harmonic estimated (Figure by 21c,d). the speed The peak sensor. frequency is 176.7 Hz, which is very close to the value 182.1 Hz, estimated by the speed sensor.

Figure 21.FigureIFCF 21. harmonic-relatedIFCF harmonic-related demodulation demodulation results; results; (a) 2nd (a) 2ndharmonic-related harmonic-related TFR; (b) 2nd TFR; (b) 2nd harmonic-related spectrum; (c) 3rd harmonic-related TFR; and (d) 3rd harmonic-related spectrum. harmonic-related spectrum; (c) 3rd harmonic-related TFR; and (d) 3rd harmonic-related spectrum. The IF estimation technology is further experimentally validated at different added noise levels. The The IFGaussian estimation white noise technology is added isto furtherthe experi experimentallymental vibration signal validated and the atSNRs diﬀ areerent −4dB added (strong noise levels. The Gaussian white noise is added to the experimental vibration signal and the SNRs are 4 background interference), 2dB (strong background interference), 8dB, 30dB, 40dB, and 50dB, respectively. − (strong backgroundThe IF estimation interference), technology 2 (strong is applied background to these interference),signals. All technology 8, 30, 40, parameters and 50 dB.are remaining the same. The obtained IFCF estimation errors under different noise levels are listed in Table 2. It can be found, that the results have no change under noise levels from 50dB to 8dB compared with the experimental signal without additional noise. The estimation error slightly increases to 5.8% for noise level 2dB. The estimation error further increases to 7.6%, for noise level - 4dB. Hence, it is experimentally confirmed a high immunity level against noise, provided by the proposed technology in non-stationary conditions of very early stage of local bearing damage development, the relative damage size is 0.2%. It should be noted, that the experimental raw signal has an initial noise level.

Table 2. IFCF estimation errors for different noise levels.

SNR IFCF Estimation Error No additional noise 4.4% 50dB 4.4% 40dB 4.4% 30dB 4.4% 8dB 4.4% 2dB 5.8% −4dB 7.6%

As a result of experimental investigation at different SNR of the interference, it is obtained that satisfactory estimation error of 5.8% is achieved at additional noise level of 2dB SNR. As experimental vibration data are already contaminated by a noise, it is believed that 5.8% error is achieved, at least,

Sensors 2020, 20, 5201 24 of 30

The IF estimation technology is applied to these signals. All technology parameters remain the same. The obtained IFCF estimation errors under diﬀerent noise levels are listed in Table2. It can be found that the results have no change under noise levels from 50 to 8 dB compared with the experimental signal without additional noise. The estimation error slightly increases to 5.8% for noise level 2 dB. The estimation error further increases to 7.6%, for noise level 4 dB. Hence, it is − experimentally conﬁrmed that a high immunity level against noise is provided by the proposed technology in nonstationary conditions of the very early stage of local bearing damage development, and the relative damage size is 0.2%.

Table 2. IFCF estimation errors for diﬀerent noise levels.

SNR IFCF Estimation Error No additional noise 4.4% 50 dB 4.4% 40 dB 4.4% 30 dB 4.4% 8 dB 4.4% 2 dB 5.8% 4 dB 7.6% −

It should be noted that the experimental raw signal has an initial noise level. As a result of the experimental investigation at different SNRs of the interference, it is obtained that a satisfactory estimation error of 5.8% is achieved at the additional noise level of 2 dB SNR. As experimental vibration data are already contaminated by noise, it is believed that a 5.8% error is achieved, at least, in the range of (0–2 dB) SNR. Thus, the (0–2) dB SNR range is a successful immunity level against interference achieved by the proposed method via experiments. It is also shown that the 5 dB SNR is a successful immunity level against noise achieved by the − proposed method via simulation, and the simulation error range is (1.2%–4.46%). In the current experimental setup, the investigated faulty bearing does not slip. In the majority of bearing operation conditions under nominal loads, bearing slippage is insignificant and it is not needed to make any changes to the proposed method. However, at high speeds and light loads or at no loads, bearing slippage may be essential. A minimum load must be applied to a bearing in order to avoid slippage. The value of this minimum load increases with speed. If bearing slippage appears, the fault characteristic periods change and, therefore, the bearing defect frequencies are affected by the slippage. Bearing slippage needs to be taken into account for IF estimation of bearing defect frequencies. For bearing slippage conditions, a novel signal segmentation approach by slicing the bearing vibrations into small realizations (periods), considering simultaneously the relatively small number, e.g., 5–10, of these realizations (i.e., groups of realizations) for IF estimation/fault diagnosis purposes, and then tracking IF estimations/fault diagnosis results via multiple groups has been proposed, developed, investigated, and comprehensively experimentally validated [55–60]. This approach takes into account that bearing slippage is not a very fast developing process and, therefore, allows to consider simultaneously 5–10 realizations. It is possible to apply this novel approach for the proposed method for bearing slippage conditions by considering the STFT of a small number of realizations (i.e., groups), applying the method for multiple STFTs of groups, starting with groups characterized by relatively high SNR regions, and tracking the obtained estimation results via multiple groups in order to obtain the overall IF estimation for the whole nonstationary bearing vibration signal. The proposed method is developed, firstly, as a generic IF estimation method and, secondly, with a specific view of application for IF estimation related to the bearing defect frequencies, for the purpose of bearing fault diagnosis. It is well known that all bearing defect frequencies, i.e., outer race defect frequencies, inner race defect frequencies, rolling element defect frequencies, and cage Sensors 2020, 20, 5201 25 of 30 defect frequencies, are directly proportional to the fundamental rotational frequency and constants of proportionalities which are known as fault characteristic coefficients (FCCs). Therefore, taking into account these direct proportionalities, it is known that for both stationary and nonstationary conditions of bearing operations and for a simultaneous appearance of multiple bearing local faults (e.g., outer race faults, inner race faults, etc.), the bearing defect frequencies of all harmonics are not intersecting. Even if a rotating shaft with two different rolling element bearings is considered, the bearing defect frequencies of all harmonics are also not intersecting. Thus, the current version of the proposed IF estimation method is designed for non-intersecting IFs. Other known GDT approaches are also designed for non-intersecting IFs. Taking into account the nature of the proposed method, the method does not have any limitations on the number of non-intersecting harmonics. It is shown above that the proposed method can successfully estimate multiple IFs for the following cases in a very noisy environment, where the SNR is 5 dB: − Multi-component signal with three harmonic components with time-variable amplitudes and • linear FMs. Multi-component signal with two harmonic components with time-variable amplitudes and • nonlinear (quadratic) FMs. Multi-component signal with two harmonic components with time-variable amplitudes and • nonlinear (cubic) FMs.

Therefore, the proposed technology has the potential ability to diagnose multiple bearing faults by IF estimations of bearing defect frequencies, related to local faults of outer race, inner race, cage, and roller elements. As the proposed technology has good immunity against interference, it allows to diagnose early stages of local bearing defects. In the performed experiment, the IF of the outer race defect is successfully estimated at the very early stage of outer race damage development, and the relative damage size is 0.2%. As the proposed method does not require a priori knowledge about “true IF time dependency”, it could be applied to two scenarios: (i) where bearing defect frequencies are known; and (ii) where bearing defect frequencies are unknown. The last case is very important for industrial applications. It is possible to apply the proposed method for IF estimations also for induction motors, including inverter-fed induction motors. The proposed method could be used for IF estimations for harmonics related to issues other than bearing faults, such as mechanical faults in induction motors in stationary and nonstationary conditions. In most cases, such an application will be related to IF estimations for variable rotation speed harmonics. In nonstationary conditions of motor operations, these harmonics, normally, have time-variable amplitudes and time-variable nonlinear frequency modulations (pure linear frequency modulations do not often appear in the exploitation of induction motors). Therefore, the three important advantages of the proposed method: the ability of effective IF estimation in conditions of time-variable harmonic amplitude, the ability of effective IF estimation in conditions of time-variable nonlinear frequency modulations, and the ability of the estimation of IF of multiple harmonics, could be exploited for IF estimations for harmonics related to mechanical faults in induction motors. The effects of inverter harmonics on motor vibration signatures are studied in detail in the literature. It is shown theoretically and experimentally that the vibration signatures, caused by the inverter, are related to the multiple harmonics of the grid supply frequency. It is a possibility that those harmonics will affect the reliability of IF estimations for induction motors as strong inverter-induced harmonics could dominate in vibration spectra and, therefore, will mask weaker harmonics of bearing defect frequencies. However, it is possible to eliminate the influence of inverter-induced harmonics from motor raw vibrations in the same way as gearbox mesh harmonics are removed from gearbox raw vibrations in order to obtain gearbox vibration residual signals, e.g., [48]. In these motor vibration residual signals, masking effects of inverter-induced harmonics will be essentially reduced and, thus, effective IF estimations of motor narrowband harmonics could be performed by the proposed technology. Sensors 2020, 20, 5201 26 of 30

5. Conclusions The novel IF estimation technology, the multi-generalized demodulation transform (MGDT), is developed and investigated for nonstationary signals, whose nonstationary instantaneous frequencies are unknown and difficult to extract from time–frequency representations in an essentially noisy environment. The theoretical basis of the novel IF estimation technology is created. The main novel feature of the IF estimation technology is the proposition of multiple adaptive iterations of the GDT for the same specific signal component to concentrate blurry TFR energy in a step-wise manner. The important feature of the novel MGDT is that every new iteration of the GDT is adaptive, i.e., it is based on previous iteration results. Our extensive literature search clearly shows that nobody, in worldwide terms, has previously proposed the main important novelty of the MGDT: an adaptive iterative application of the GDT for the same signal component of a multi-component signal. The proposed IF estimation technology allows to remove the GDT’s restriction of knowledge (a priori or in-advance estimation from a speed sensor) of a true instantaneous frequency that is important for industrial applications of the proposed technology. With the proposed adaptive iterations of the generalized demodulation procedure for the same specific signal component, the energy concentration on the time–frequency ridge is enhanced and a specific signal component with the instantaneous frequency curve, that is polluted by noise, can be accurately estimated. Novel validation of the proposed technology is successfully performed by the simulation for four types of amplitude- and frequency-modulated nonstationary signals under a strong background noise (signal to noise ratio is 5 dB): (i) single-component signal with a nonstationary amplitude and − nonstationary, nonlinear (quadratic) time variation of the instantaneous frequency; (ii) multi-component signal with components that have nonstationary amplitudes and nonstationary, linear instantaneous frequencies with different chirp rates; (iii) multi-component signal with components that have nonstationary amplitudes and nonstationary, nonlinear (quadratic) instantaneous frequencies with different chirp rates and frequency accelerations; and (iv) multi-component signal with components that have nonstationary amplitudes and nonstationary, nonlinear (cubic) instantaneous frequencies with different chirp rates, frequency accelerations, and the first derivatives of frequency accelerations. The achieved estimation errors for the simulated nonstationary signal in the presence of a strong interference ( 5 dB signal to noise ratio) are in the following ranges: (i) (1.46–4.46)% − for a multi-component signal with linear frequency modulations and nonstationary amplitudes of components, (ii) (1.2–1.4)% for a multi-component signal with nonlinear (quadratic) frequency modulation and nonstationary variation of amplitudes of components, (iii) (1.2–1.6)% for a multi-component signal with components that have nonstationary amplitudes and nonstationary, nonlinear (cubic) time variations of the instantaneous frequency with different chirp rates, frequency accelerations, and the first derivatives of frequency accelerations, and (iv) 0.85% for a single-component signal with a nonstationary amplitude and nonstationary, nonlinear (quadratic) time variation of the instantaneous frequency. It is validated by experimental trails that the novel IF estimation technology successfully estimates nonstationary bearing defect frequencies for the first, second, and third harmonics of bearing defect frequencies (without a priori knowledge of true time-varying instantaneous frequency dependency) under time-varying rotation speeds and time-varying amplitudes of these harmonics in the presence of interference and at the very early stage of outer race damage development: the relative local damage size is 0.2%. It is shown that the estimation error for the instantaneous defect characteristic frequency estimation is 4.4%. The instantaneous bearing defect characteristic frequencies of the first, second, and third harmonics, that are polluted by noise in the time–frequency representation, are accurately estimated by the proposed technology. The estimation errors for nonstationary bearing defect frequencies of the first, second, and third harmonics are experimentally evaluated for six difference levels of added Gaussian noise interference, characterized by a signal to noise ratio (SNR) from 50 to 4 dB, including for two levels of a strong − noise interference. It is shown that the estimation errors are unchanged (i.e., staying at a level of 4.4%) Sensors 2020, 20, 5201 27 of 30 under noise levels from 50 to 8 dB, compared with the case without additional noise. The estimation error slightly increases to 5.8% for noise level 2 dB (a strong noise interference) and it further increases to 7.6% for noise level 4 dB (a strong noise interference). These results experimentally confirm − a high immunity level against noise delivered by the proposed technology. Based on the experimental validation, for an acceptable IF estimation accuracy, the SNR for the proposed technology should not be lower than (0–2 dB). The IF estimation technology, the MGDT, is a promising novel concept that could be widely employed for sensor-based vibration instantaneous frequency estimation for nonstationary signals of electromechanical systems in nonstationary operations, in a noisy environment, and without knowledge (a priori or by in-advance estimation from a speed sensor) of true instantaneous frequency variations. The proposed IF estimation technology could be employed for rotating components/machineries such as rolling element bearings, gearboxes, compressors, and induction motors, and could be also effectively employed for other technologies, e.g., low-frequency acoustic non-destructive technologies, e.g., [61,62], motor current signature analysis for induction motors, e.g., [63], ultrasound fault detection technologies, etc.

Author Contributions: Conceptualization, D.Z. and A.B. Methodology, D.Z., F.C. and L.G. Software, D.Z. Validation, D.Z. and F.C. Investigation, D.Z. and A.B. Writing—original draft preparation, D.Z. Writing—review and editing, L.G. Visualization, D.Z. Supervision L.G. All authors have read and agreed to the published version of the manuscript. Funding: This study is supported by the National Natural Science Foundation of China (Grant No. 51905292) and China Postdoctoral Science Foundation (2019M660615 and 2020T130348). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

GDT Generalized demodulation transform MGDT Multi-generalized demodulation transform IF Instantaneous frequency TFR time–frequency representation TFA time–frequency analysis STFT Short-time Fourier transform WT Wavelet transform SST Synchrosqueezing transform TSST Time-reassigned SST MSST Multi-SST HSST High-order SST TET Transient-extracting transform SET Synchro-extracting transform COA Computed order analysis RF Rotational frequency SNR Signal to noise ratio IFCF Instantaneous fault characteristic frequency FCC Fault characteristic coeﬃcient AM-FM Amplitude-modulated and frequency-modulated SWT Synchrosqueezed wavelet transform EMD Empirical mode decomposition

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